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Fundamentals and exchange rate forecastability withsimple machine learning methods

Christophe Amat, Tomasz Michalski, Gilles Stoltz

To cite this version:Christophe Amat, Tomasz Michalski, Gilles Stoltz. Fundamentals and exchange rate forecastabilitywith simple machine learning methods. 2018. �halshs-01003914v6�

Fundamentals and exchange rate forecastability with simplemachine learning methodsI

Christophe Amata, Tomasz Michalskib,∗, Gilles Stoltzc

aEcole Polytechnique, Palaiseau, FrancebHEC Paris – GREGHEC, Jouy-en-Josas, France

cHEC Paris – CNRS, Jouy-en-Josas, France

Abstract

Using methods from machine learning we show that fundamentals from simple exchangerate models (PPP or UIRP) or Taylor-rule based models lead to improved exchange rateforecasts for major currencies over the floating period era 1973–2014 at a 1-month forecasthorizon which beat the no-change forecast. Fundamentals thus contain useful informationand exchange rates are forecastable even for short horizons. Such conclusions cannot beobtained when using rolling or recursive OLS regressions as used in the literature. Themethods we use – sequential ridge regression and the exponentially weighted averagestrategy, both with discount factors – do not estimate an underlying model but combinethe fundamentals to directly output forecasts.

JEL Classification: C53, F31, F37

Keywords: exchange rates, forecasting, machine learning, purchasing power parity,uncovered interest rate parity, Taylor-rule exchange rate models

1. Introduction

We show that fundamentals from “classic” exchange rate models and Taylor-rule ex-change rate models are useful in forecasting short-term changes in exchange rates formajor currencies. Using prediction techniques borrowed from machine learning thatcome with performance guarantees – sequential ridge regression with discount factorsand the exponentially weighted average strategy with discount factors – we are broadlyable to improve forecasting forward the 1-month exchange rate for the floating rate pe-riod 1973–2014 in terms of the usual root mean square error (RMSE) criterion, and

IWe would like to thank an anonymous referee, Philippe Bacchetta, Charles Engel, Refet Gurkaynak,Robert Kollmann, Jose Lopez, Evren Ors, Cavit Pakel, Martin Puhl, Romain Ranciere, Helene Rey,Barbara Rossi, Micha l Rubaszek, Kenneth D. West and the seminar participants at Bilkent University,HEC Paris, the National Bank of Poland, OECD, UN, EEA 2015 conference, Royal Economic Society2015, 8th Financial Risks International Forum 2015 (Institut Louis Bachelier) for helpful comments anddiscussions. All remaining errors are ours.

∗Corresponding authorEmail addresses: christophe.amat@polytechnique.edu (Christophe Amat), michalski@hec.fr

(Tomasz Michalski), stoltz@hec.fr (Gilles Stoltz)

Preprint submitted to Elsevier May 28, 2018

obtain lower Theil ratios than the no-change (random walk) prediction. Fundamentalsfrom purchasing power parity models (PPP), uncovered interest rate parity (UIRP), andTaylor-rule exchange rate models – which traditional exchange rate forecasting literature(based on OLS methods) has failed to find effective (see Rossi, 2013 for a recent compre-hensive review) – in fact carry valuable information for out-of-sample forecasting even atshort-term horizons1. We obtain similar success with predicting the direction of changeof exchange rates and several economic criteria, and also conduct various robustnesschecks.

These results have important implications both for economic theory and policy. Forexample, Alvarez et al. (2007) argue that if exchange rates are random walks, monetarypolicy concentrated on short-term interest rates does not have typically assumed nominalor real effects, but rather affects only risk premia. More generally, it has been frustratingfor researchers in international economics that forces stipulated by the simplest and mostintuitive models about one of the most important prices – exchange rates – did not seemto be empirically detectable.

Our findings are contrary to the consensus in the literature led by the famous Meeseand Rogoff (1983) result on the inability of short-term forecasts based on fundamentals tooutperform the hard-to-beat no-change/random walk predictions2. The improvements inforecasts using fundamentals presented here (up to 1.3 % in terms of RMSE) are not large– the fundamentals we consider do not help to add a lot of predictive content relative tothe one of “no change” from the previous observed exchange rate – but are substantial(and often statistically significant at conventionally accepted levels) with respect to theexisting literature. The observations of Engel and West (2005) may well be valid: cur-rent and past fundamentals may have low correlations with future exchange rate values.Fundamentals may have little predictive power against an exchange rate that can beapproximated by a random walk, though they may add some useful information.

Why should we use these methods and not others? A common feature of the methodswe use is the following: they attempt to avoid a problem known as overfitting past data,which means being able to reconstruct well – using the forecasting equations – previ-ous data (having a good “in-sample” prediction error) but with poor future predictions(delivering a poor “out-of-sample” prediction error). Such problems have permeated theexchange-rate prediction literature for the last 35 years, from Meese and Rogoff (1983)to Fratzscher et al. (2015).

These methods do not estimate the coefficients of some underlying model, but rather,treat the coefficients as numbers to be chosen (not estimated) over time in order to formgood predictions. Indeed, they do not rely on stochastic modeling of exchange rates(e.g., in terms of a linear combination of the fundamentals plus stochastic errors). Theyhave appealing features (described in Section 3.2) as theoretical guarantees (bounds) ontheir performance can be proven. In particular, their RMSEs converge, as the number

1Forecastability of exchange rates over longer time periods – such as 2-year forward forecasts andbeyond – has been established in the literature, see Section 6. Forecastability is not the same thing aspredictability, which has also been studied; see Footnote 25.

2The rare exceptions are that of Clark and West (2006) that demonstrate the predictability of a UIRP-based model or Wright (2008) that uses Bayesian model averaging methods. See a longer discussion inSection 6.

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of instances to be predicted increases, to a quantity that is (always) not larger than theRMSE of the no-change/random walk prediction, and even not larger than the RMSEof the best fixed linear forecasting equation (for certain methods). The latter value isin general strictly smaller than the RMSE of the no-change/random walk prediction,so asymptotically such methods should beat the random walk if fundamentals indeeddrive the exchange rate. It can also be shown that no such guarantee can be achievedfor conventional (rolling or recursive) OLS regressions. Moreover, we apply our meth-ods to fundamentals precisely identified by classic economic models of exchange rates,and do not run “kitchen-sink” regressions where many different and not necessarily re-lated fundamentals are mixed together. The following are crucial differences between ourwork and that of other studies like Li et al. (2015) and Plakandaras et al. (2015), whichforecast exchange rates with different machine learning techniques: the methods we usecome with relevant performance guarantees, solve outright a statistical problem encoun-tered by many previous studies, and use only one (very limited) set of theory-motivatedfundamentals at a time.

We stress that both the discount factors and (hyper)parameters of our methods (alearning rate for the exponentially weighted average strategy with discount factors, aregularization factor for sequential ridge regression with discount factors) are not imposedfrom above or set arbitrarily. In particular, our methods do not rely on educated guessesabout important parameters, as for example in Wright, 2008 for Bayesian model averagingmethods and in other papers, where typically the performance of methods at hand arereported for several, hand-picked, parameter values). Note finally that the methodswe implement are general forecasting methods that were not specifically designed forexchange rate forecasting, but proved to work well in other problems (e.g., air qualityand electricity consumption forecasting; see Cesa-Bianchi and Lugosi, 2006, Mauricetteet al., 2009, and Stoltz, 2010).

Machine learning methods. The first method – the exponentially weighted average strat-egy with discount factors – allows us to weigh fundamentals by their past performance.Coefficients chosen over time are convex and are proportional to the exponential of theaverage past performance of each fundamental. These vary over time as the performanceof each fundamental evolves. If certain fundamentals always perform poorly, they arerapidly (almost) discarded, though by construction all coefficients are non-zero even ifsome may be close to zero. The success of this algorithm, which identifies the best pre-dictors in any period, may be due to the channel exhibited in the scapegoat exchangerates model of Bacchetta and van Wincoop (2004) (tested for in Fratzscher et al., 2015):traders consider some of the “most fashionable” fundamentals (in some periods) thateventually drive the exchange rates. When we look closer at the weights we see thatquite often there is one prevailing fundamental that carries most of the information forforecasting exchange rates; details are provided in Appendix D, Figures D.1–D.3 of theonline supplementary material (Amat et al., 2018b).

The second method – sequential ridge regression with discount factors3 – resemblesOLS regression, but prevents the in-sample overfitting issue encountered by adding a

3Ridge regression was introduced by Hoerl and Kennard (1970) in a stochastic and non-sequentialsetting. However, it turns out that the exact same method can be analyzed in a sequential non-stochasticsetting, leading to guarantees of another nature, namely in the form (7). We will discuss these machine-learning analyses in Section 3.2.

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regularization term to an otherwise standard OLS regression. The regularization termprevents the smallest possible in-sample error being obtained, which typically leads tobetter generalization (smaller out-of-sample error). The elastic net method consideredby Li et al. (2015) proceeds in a similar fashion, with a combination of Euclidean andabsolute-value regularization. However, it comes with no clear theoretical performanceguarantees like those which can be shown for ridge regression.

For both methods, we consider variants of the basic machine-learning formulationswhich include discount factors. Discounting puts more weight on recent forecasting errors.This allows us to accommodate structural breaks that may be present in the data (e.g.,changes in the conduct of monetary policy, crisis times, etc.) better than the originalformulations. This is also true for any changes in the behavior of market participants. Itshould be noted that our discounts are of a polynomial and not geometric form (as usedin most of the economic literature). This is because it can be proven that theoreticalguarantees can be associated with the former (see Section 3.2) but not the latter in ourcontext (see Theorems 2.7 and 2.8 of Cesa-Bianchi and Lugosi, 2006 for a discussion ofthe performance of different discounting methods).

The data used and the scope of our study. In our study, aside from the specific machine-learning methods, we use data and forecast evaluation methods that are standard in theliterature, as discussed in Rossi (2013). We use lagged fundamentals and do not detrend,filter, or seasonally adjust the data in any way. We use single-equation methods and ourbenchmark for comparison is the no change in the exchange rate prediction. We studythe floating currency pairs for major industrialized countries, (i) for the entire periodsince the breakup of the Bretton Woods fixed exchange rate system (1973–2014), and(ii) for the period 1999–2017 for a subset of currencies using “real-time” data. We useend-of-month exchange rates (with one reading of the exchange rate at the end of themonth).

We conduct several robustness checks, such as trying out different (sub)samples,variations on the exact form of the fundamentals included, etc., and our results remainvalid. Furthermore, an investigation of statistical properties of the errors shows that ourimprovement is uniform over time, rather than driven by a few well-predicted instances.There are several themes discussed by Rossi (2013) that are also relevant to our study.Traditional linear estimation methods (rolling or recursive OLS regressions) with ourfundamentals fail to forecast exchange rates better than the no change in exchange ratesprediction. With the addition of several fundamentals (aggregating information) we donot get gains in prediction precision – that is, parsimonious forecasting equations workbest. Fundamentals from the same time period as the forecast (true observed values)perform poorly.

Our study is not comprehensive nor exhaustive in nature. Our goal is not to providethe best possible forecasts of exchange rates by finding fundamentals from the large setconsidered in the vast literature that have the strongest predictive power (or aggregatingthe information contained therein for example in hundreds of different time series). Weare interested in showing that we can detect fundamentals driving exchange rates evenat short horizons, and argue for the usefulness of machine learning methods in applica-tions to well-known and hard-to-crack economic problems such as predicting inflation oroutput.

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1.1. Organization of the paper

In Section 2, we present the fundamentals that we shall consider. Then, in Section 3,we discuss the analysis methods and put them in perspective with respect to standardlinear regressions. Next, in Section 4 we present the data, while in Section 5 we discuss theresults. Section 6 contains a literature review and Section 7 concludes. Supplementaryonline material (Amat et al., 2018b) contains further details regarding the methods andresults tables.

2. Fundamentals considered

We study “classic” fundamentals stemming from the simple exchange rate models ofthe 1970s and Taylor-rule based models. As described in Rossi (2013), there are four mainreasons for this: (i) these fundamentals come from exchange rate models that involvebasic relationships in standard international economics, (ii) they have been extensivelyused in the literature, (iii) studies have shown that at short-time horizons, forecastingmethods based on them were not successful with respect to the no-change forecast4

benchmark, and (iv) data is widely available for long periods for a large set of countriesand does not require any transformations (as for example, productivity data does).

2.1. General framework

In the following, “home” or “domestic” refers to the U.S., while “foreign” refers tosome other country whose exchange rate w.r.t. the U.S. dollar is studied.

The general forecasting equation for exchange rate change for a currency pair is givenby

st+1 − st = αt +

N∑j=1

βj,tfj,t , (1)

where st is the logarithm of the exchange rate (home currency units per unit of the foreigncurrency) at time t, the intercept αt and slope coefficients βj,t are to be chosen basedon information available up to time t, and the fj,t are the N fundamentals considered attime t. Throughout the paper we will impose5 that αt = 0. The time unit is months.

By the exchange rate, we here mean the end-of-the month value.The coefficients βj,t (and αt when intercepts are considered, which is not the case

here) are usually picked through rolling or recursive OLS regression. Rossi (2013) statesthat “the literature has been focusing mainly on rolling or recursive window forecastingschemes (see West, 2006), where parameters are reestimated over time using a windowof recent data”, and Cheung et al. (2005) mention that “the convention in the empiricalexchange rate modeling literature of implementing rolling regressions [was] established

4We remind readers that no other set of fundamentals has been as yet shown to beat such forecastsin a consistent manner either. See Footnote 25 on the predictability of exchange rates with Taylor-rulemodel-based fundamentals.

5Affine forecasting equations can be handled via only one of our two forecasting methods, namelysequential ridge regression with discount factors, and do not offer an improvement upon the resultsshown in the paper. Details are available upon request. The other method – the exponentially weightedaverage strategy with discount factors – is not applicable to affine forecasting equations.

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by Meese and Rogoff (1983).” This is because these gave the best performance andno other method consistently beat them. In this paper, we revisit this convention andconsider other methods for picking these coefficients. We do not write “estimate thesecoefficients”, as this would imply some underlying model with true parameters α and β.We will be interested solely in forecasting performance, and not at all in the existence oruse of a model6.

Before recalling how rolling and recursive OLS regressions proceed to that end, andpresenting alternative methods stemming from the machine learning field, we discuss thefundamentals of interest.

2.2. Fundamentals from PPP, UIRP, monetary, and Taylor-rule models

We are only interested in the predictive performance of the forecasting methods weuse. We allow forecasting equations with both different (which is going to be our basescenario) and equal (we shall call them “coupled”) coefficients for home and foreignfundamentals relating to the same measured quantity. There are good reasons to allowfor different coefficients. Identical series, even if pertaining to a similar economic concept,may be either unavailable or measured completely differently in any two countries (giventhat they are provided by independent institutions using a range of methods). Hence,the best weights picked may differ because the elasticities of response of investors topurportedly similar fundamentals in each country may and indeed should differ7. Afurther issue that is pertinent while analyzing data series over such long periods is thatcentral banks sometimes change their ways of conducting monetary policy – for exampleby following inflation targets or Taylor rules – at different times across countries which –as is well known – may also change the way in which exchange rates react to fundamentals.

The first time-series of fundamentals is formed by inflation differentials, which arealso used for instance in the relative purchasing power parity model. The associatedforecasting equation is given by

st+1 − st = β1,tπt − β2,tπ?t , (2)

where πt and π?t are respectively the home and foreign measures of 12-month inflationrates available (known) at time t. In what follows, a “?” will denote the variables for the

6We avoid using the word “model” here, as we will not have to assume that some true underlyingmodel of the differences exists like

st+1 − st = αt +

N∑j=1

βjfj,t + εt+1 ,

from which, by estimation of the unknown coefficients βj some prediction st+1 could be obtained.7The most flagrant example is perhaps money stocks: for the United Kingdom and Sweden only M0

aggregates, and for Italy and the Netherlands only M2, are available for the entire period considered,and not M1 as for other countries. M0, M1 and M2 money aggregates are typically correlated, but theyobviously measure different things, and the relationship between those within the same country may notbe stable. Even for countries for which M1 measures are available, these contain different componentsdepending on the country. Such issues unfortunately exist to some extent for every fundamental consid-ered, and is a general problem in the literature. Due to the asymptotic properties of the methods, wewanted to obtain the longest series possible for the largest number of currencies and had to – given thedata available – sacrifice standardization to some extent.

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foreign country8.The forecasting equations using the fundamentals from the uncovered interest rate

parity model are of the form

st+1 − st = β1,tit→t+1 − β2,ti?t→t+1 , (3)

where it→t+1 and i?t→t+1 are the short run (money-market) interest rates at home andin the foreign country respectively. These would be the interest rates at which investorscould place money at time t for the period t to t + 1 and be observed by them in realtime.

The third time-series of fundamentals consists of changes in money stocks and out-puts. The simplest monetary model (flexible price, Frenkel-Bilson model) states thatexchanges rates can be modeled as linear combinations of the form

st = α+ φ(mt −m?

t

)− ω

(yt − y?t

),

where mt and m?t are the logarithms of the money stocks at time t, and yt and y?t the

logarithms of outputs. Here, α, φ and ω denote some true underlying parameters for themodel. As indicated several times, we will not rely on the existence of such a monetarymodel, and only consider it to extract the fundamentals it proposes and substitute theminto our forecasting equations. Indeed, after lagging the said fundamentals by a month (toaccount for known and present but not forward values), differencing the above equation9,and allowing for decoupled fundamentals, we consider10 the forecasting equations

st+1 − st = β1,t∆mt − β2,t∆m?t − β3,t∆yt + β4,t∆y

?t , (4)

where ∆xt = xt − xt−1 denotes the change between periods t− 1 and t for a variable x.In (4), the parameters β1,t, . . . , β4,t are to be picked over time as in (2) and (3).

We also investigate whether all of the fundamentals mentioned above taken togetherare able to predict changes in the exchange rate11.

Lastly, we also consider whether the fundamentals extracted from a Taylor-rule basedexchange-rate model (which have been found successful in establishing the predictability

8The relative PPP forecasting equation would actually stipulate st+1 − st = β1,t∆pt+1 − β2,t∆p?t+1,where ∆pt+1 is the change in the price level between periods t and t+ 1. However, as the change in thefuture price level is unknown at time t, we use past price changes (inflation) in our forecasting exercise.

9We proceeded this way as some of the methods we use – in particular the exponentially weightedaverage strategy with discount factors – require direct predictions of the exchange rate change. Analternative is to find some “equilibrium” exchange rate driven by the aforementioned fundamentals anduse the deviation of the current exchange rate from that “theoretical” exchange rate as the fundamental.Doing this does not qualitatively change our results.

10Recall that the model does not need to hold – it merely helps us decide which fundamentals toconsider in our forecasting equations.

11The form in which the fundamentals are included does not appear to matter, at least for sequentialridge regression with discount factors. We produced forecasts using fundamentals from the PPP andmonetary models for our main sample also using price or output indices in levels or the monetary stock:our qualitative conclusions do not change for sequential ridge regression with discount factors. We alsoinvestigated fundamentals extracted from the other popular monetary flexible-price model that alsoinvolves differences in interest rates, which in our forecasting equations would add terms it − it−1 andi?t − i?t−1 as predictors. We do not show the results as they were very close to the performance of theforecasting equation (4). We included these fundamentals, however, when considering forecasts with allfundamentals at hand to see whether aggregating information from many different time-series helps.

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of exchange rate changes, as discussed in Section 6) are also useful in forecasting. We usethe fundamentals from the most parsimonious version of equation (7) from Molodtsovaand Papell (2009) for our forecasting equation:

st+1 − st = β1,tπ?t − β2,tπt + β3,ty

?t − β4,tyt + β5,ti

?t−1→t − β6,tit−1→t , (5)

where yt and y?t are present measures of the output gaps. Therefore, our Taylor-rulefundamentals are the inflation, output gaps, and lagged interest rates.

The actual data time-series used are further discussed in Section 4.

Coupled fundamentals. In forecasting equations (2)–(5), fundamentals of home and for-eign countries pertaining to the same concept can also have the same weights. Forexample, the forecasting equation inspired by the PPP model in (2) is then st+1 − st =β1,t(πt − π?t

). We will refer to this situation as coupled fundamentals, and perform ro-

bustness checks using this forecasting equation. Note that it still falls under the generalumbrella of (1), only with N/2 coefficients to pick instead of N , and with the fj,t nowreferring to the differences between home and foreign fundamentals.

3. Methodology

As indicated above, forecasts st+1 of 1-month ahead exchange rates st+1 are based onthe forecasting equations (2)–(5). A forecasting method consists of a rule for picking thecoefficients βj,t over time based on past and present information. Several such methodsare presented in Sections 3.1 and 3.2, namely conventional rolling and recursive OLSregressions (in brief – Section 3.1), as well as two other methods stemming from the fieldof sequential learning (in detail – Section 3.2).

As discussed in detail in Appendix B of the online supplementary material (Amatet al., 2018b), we evaluate the forecasting ability of these methods through their (out-of-sample) root mean square error, which is computed by working with a training periodof t0 = 120 months, as is standard in the literature. (This training period is not takeninto account in the evaluation of performance.) To determine whether the improvementsin RMSE of one method over another are statistically significant, we use the Dieboldand Mariano (1995) test, referred to as the DM test in the following; we in fact do soon bootstrapped data. For the sake of completeness (and though not fully applicablein our setting, see Appendix B.3 of the online supplementary material) we also reportthe results of the tests by West (1996) and Clark and West (2006, 2007) for determiningsignificant improvements over the no-change prediction only (denoted CW hereafter).Last, we compute the Theil ratio of the RMSE predicted by the forecasting method ofinterest over that of the no-change prediction. A ratio below 1 means that the givenmethod gave a lower RMSE than the no-change prediction.

3.1. Forecasting methods (part 1): classical methods

We now present the various forecasting methods considered in this paper. We willdo so in some generality, encompassing all of the forecasting equations (2)–(5) under theumbrella of (1):

st+1 − st =

N∑j=1

βj,tfj,t .

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We recall that in our view, the coefficients βj,t are to be picked according to some rule;they do not need to be understood as estimating some unknown underlying true value.

The no-change forecasting method. The first strategy consists of choosing βj,t = 0 forall j at each round t, that is, of forecasting st+1 with st. We call this the no-changeforecasting method.

Rolling OLS regression. This is the most standard technique in the literature, “the con-vention” as Cheung et al. (2005) state. The idea is to truncate the available informationto account for the most recent relationships between variables that can change over timebecause of policy changes (for example, a change to Taylor-rule based monetary policy),structural changes in the economy (such as shifting relationships between the moneystock and inflation), etc. For this forecasting method, we need to choose the length h ofthe estimation window, which we also use for the training period, as is standard in theliterature (see Molodtsova and Papell, 2009): h = t0 = 120 months. The rolling OLSregression picks, for months t > h,

(β1,t, . . . , βN,t) = arg minβ1,...,βN∈R

t∑τ=t−h+1

sτ − sτ−1 − N∑j=1

βjfj,τ−1

2 .Recursive OLS regression. This is another standard technique in the literature. It con-sists of choosing, for each month t,

(β1,t, . . . , βN,t) = arg minβ1,...,βN∈R

t∑τ=1

sτ − sτ−1 − N∑j=1

βjfj,τ−1

2 ,i.e., all past time instances, and not only the h most recent ones, are used to form theprediction.

3.2. Forecasting methods (part 2): simple machine learning algorithms

The forecasting methods considered in this study stem from the field of machinelearning, where they are already standard methods for the robust online prediction ofquantitative phenomena by aggregation of basic predictors (fundamentals). The bookby Cesa-Bianchi and Lugosi (2006) summarizes research performed on and around themover the period 1989–2006.

These methods have been applied in the following fields, among others: forecastingair quality (see, e.g., Mauricette et al., 2009; Mallet, 2010; Debry and Mallet, 2014) andelectricity consumption (see, e.g., Devaine et al., 2013; Gaillard and Goude, 2015). Weuse them below “by the book”, i.e., we provide no tweaking and apply them “as are”in the references above. We underline that therefore these methods are not ad hoc onesconstructed solely for the problem of predicting exchanges rates.

The theoretical out-of-sample guarantees that they come with are of the followingform: for all possible bounded sequences of exchange rates and fundamentals, their

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predictions are such that

T∑t=1

st − st−1 − N∑j=1

βj,t−1fj,t−1

2− infβ†∈F

T∑t=1

st − st−1 − N∑j=1

β†jfj,t−1

2 6 B(F , T ) ,

(6)with B(F , T )� T and where F ⊂ RN is some comparison class (the forecasting methodspresented below are independent of the choice of F , only the bound B(F , T ) is sensitiveto it). We explain below what we mean by a comparison class, i.e., a set of candidatesfor the underlying expression of the exchange rates in terms of linear combinations offundamentals. With the algorithms presented below, such linear combinations can begiven by F equal to the set of all point mass combinations (weights that equal 1 for onefundamental and are null for all others), or equal to some bounded Euclidean ball of RN(e.g., all linear weights with Euclidean norm bounded by some constant U , whereby Uappears in the bound).

In particular, from (6), dealing separately with the training period of size t0 (for whichan error bounded by something of the order of t0 is made in the worst case), dividing byT − t0 and taking square roots, we get the following guarantee on out-of-sample RMSEs:

lim supT→∞

√√√√√ 1

T − t0

T∑t=t0+1

st − st−1 − N∑j=1

βj,t−1fj,t−1

2

− infβ†∈F

√√√√√ 1

T − t0

T∑t=t0+1

st − st−1 − N∑j=1

β†jfj,t−1

2 6 0 . (7)

Let us comment on these out-of-sample guarantees. First, they rely on no stochasticmodeling; they are achieved for all possible bounded sequences of exchange rates andfundamentals, and are thus deterministic. In fact, the bound B(F , T ) also depends onthe range in which the exchange rates and the fundamentals lie, but is a uniform bound.Second, what these out-of-sample guarantees truly ensure is that the forecasting methodhas asymptotically an average performance as good as or better than that of the bestconstant linear combination in F ⊂ RN , i.e., the best fixed model with coefficients in F .However, of course, as mentioned above, such a model does not need to exist, since nostochastic modeling assumption is required. It just turns out that the methods presentedbelow mimic the performance of the best model if applicable (or of the best fixed linearforecasting equation otherwise). Put differently, we can interpret the infimum in (6) (theerror suffered by the best pick in the comparison class) as measuring some approximationerror (how well in hindsight the fundamentals can predict the exchange rates), while theB(F , T ) term is a sequential estimation error (the price to pay for facing a sequentialrather than batch problem).

One needs to note that getting close to or slightly better than the performance of thebest fixed linear forecast is not necessarily good enough for obtaining great forecasts; thebest fixed linear forecast can have poor performance. This may be true in the exchangerates setting where it was hard until now to obtain forecastability using fundamentals

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(see Rossi, 2013). In other fields of application mentioned above – such as forecastingof electricity consumption or air quality – such forecasts do well. Lastly, we note thatit can be shown that out-of-sample guarantees like (7) cannot be achieved by rolling orrecursive OLS regression (details available upon request). Performance guarantees on thelatter (if any exist) thus would require at minimum stochastic modeling of the sequenceof exchange rates.

Is it better to use coupled or decoupled fundamentals? The short answer is: there isno theoretical reason to prefer one over the other. Indeed, while the approximationerrors of the decoupled version are always smaller than those of coupled versions (simplybecause they correspond to an infimum taken on a larger set), the bounds B(F , T ) onthe sequential estimator errors in (6) always increase with the number N of independentfundamentals, and are thus larger with decoupled than with coupled fundamentals. Asthe total errors suffered by our forecasting methods are the sums of these two errors,their empirical behavior as N increases is unclear, since theoretically they can eitherincrease or decrease.

EWA: the exponentially weighted average strategy with discount factors. This was intro-duced in the early 90s by Vovk (1990) and Littlestone and Warmuth (1994), and furtherstudied and developed by – among others – Cesa-Bianchi et al. (1997); Cesa-Bianchi(1999); Auer et al. (2002). Here, we present a minor generalization – already considered,e.g., by Mauricette et al. (2009) – in which past prediction instances get slightly moreweight when they are more recent. This forecasting method involves different parame-ters: a sequence (ηt) of positive numbers referred to as learning rates, a non-negativenumber γ called the discount factor, and a positive number κ > 0 called the discountpower. It picks the weights according to

βj,t =1

Ztexp

(−ηt

t∑τ=1

(1 +

γ

(t+ 1− τ)κ

)(sτ − sτ−1 − fj,τ−1

)2), (8)

where Zt is a normalization factor:12

Zt = 2 exp

(−ηt

t∑τ=1

(1 +

γ

(t+ 1− τ)κ

)(sτ − sτ−1

)2)

+

N∑j=1

exp

(−ηt

t∑τ=1

(1 +

γ

(t+ 1− τ)κ

)(sτ − sτ−1 − fj,τ−1

)2). (9)

Because of this factor and the exponent function in the equations defining the βj,t,this strategy is referred to as the exponentially weighted average strategy (the EWA

12This is the version for decoupled fundamentals. With coupled fundamentals, the number of sum-mands is reduced by a factor of 2 in the defining sum over j and concomitantly the factor 2 in the firstterm in the definition of Zt is replaced by 1. This is because this first term accounts for the no-changeprediction, which in its decoupled version as in (9) corresponds to the difference of 2 zero variations (onefor the value of the currency of each country), instead of 1 zero variation (of the exchange rate) in thecoupled case.

11

strategy in short). Note that the weights obtained form a sub-convex weight vector: thecomponents βj,t are non-negative and sum up to something smaller than or equal to 1.(The missing mass with respect to 1 can be interpreted as a measure of the confidencethat no change will take place between st and st+1.)

A study of the theoretical out-of-sample guarantees of EWA in the presence of dis-count factors was undertaken in Stoltz (2010, Theorem 3), see also Cesa-Bianchi andLugosi (2006, § 2.11), and can be interpreted as follows in our context. For all choices ofγ, κ, and non-increasing sequences (ηt) such that, as t→ +∞,

t ηt −→ +∞ and ηt

t∑τ=1

1

τκ−→ 0 , (10)

the desired guarantee (7) holds for the set F =M of all point-mass vectors. For instance,ηt = 1/

√t and κ = 2 would be suitable choices but many other choices associated with

the desired theoretical guarantees exist.More precisely13 the bound B(M, T ) of (6) equals

B(M, T ) =2 lnN

ηT+

T∑t=1

ηt2L2 + L

T∑t=1

(exp(2LηtBt−1)− 1

),

where

Bt−1 =

t−1∑τ=1

γ

τκ

and L is a bound on the quadratic errors (st − st−1)2 and (st − st−1 − fj,t−1)2 as t andj vary.

SRidge: sequential ridge regression with discount factors. The issue with (recursive orrolling) OLS regressions is that they tend to overfit past data, i.e., they lead to goodin-sample predictions but poor out-of-sample ones. To prevent this, one can add whatis called a regularization term to the squared error to help control (reduce) the rangeof components βj,t of the linear vector picked. The smallest in-sample error is typicallyachieved with large-valued coefficients, which are exactly the ones that the includedregularization term tries to avoid. Focusing solely on the in-sample error tends to bedetrimental when generalizing results, i.e., when making out-of-sample predictions. Weimplement below this regularization strategy for a forecasting method called sequentialridge regression with discount factors.

Ridge regression was introduced by Hoerl and Kennard (1970) in a stochastic (non-sequential) setting. What follows relies on recent new analyses of ridge regression in themachine learning community – see the papers by Vovk (2001) and Azoury and Warmuth(2001), as well as the survey in the book by Cesa-Bianchi and Lugosi (2006). Sequential

13See Chapter 6 of the technical report by Mallet et al. (2007), which is cited as the main ingredientin the proof of Theorem 3 of Stoltz (2010). Also, readers aware of the bounds that can be proven forthe square loss in our context may note that we do not exploit its exp-concavity with the help of awell-chosen fixed learning rate; this is for practical performance purposes.

12

ridge regression (without discount factors and for a constant regularization factor λ > 0)picks the weights

(β1,t, . . . , βN,t) = arg minβ1,...,βN∈R

λN∑j=1

β2j +

t∑τ=1

sτ − sτ−1 − N∑j=1

βjfj,τ−1

2 . (11)

We now state the associated performance bound as in (6). This comes in terms of theclasses FU of linear weights with Euclidean norm bounded by U > 0 (where the boundholds for all U > 0 simultaneously). We denote by L a bound on the exchange rates andthe fundamentals, whose value is such that st ∈ [−L,L] and fj,t ∈ [−L,L]. Then, for allU > 0, the bound B

(FU , T

)of (6) equals14

B(FU , T

)= λU2 + 4NL2

(1 +

NTL2

λ

)ln

(1 +

TL2

Nλ

).

In particular, when λ is well chosen (e.g., of the order of√T ), one has

B(FU , T

)= O

(√T lnT

)� T .

Recursive OLS regression corresponds to the special case when λ = 0, but no theoreticalbound is offered in this case.

As announced above, we have used a common variant of the classical ridge regressionpresented above so as to focus more on recent observations. This is obtained via discount-ing: sequential ridge regression (with discount factor γ and for a constant regularizationfactor λ > 0) picks the weights

(β1,t, . . . , βN,t)

= arg minβ1,...,βN∈R

λN∑j=1

β2j +

t∑τ=1

(1 +

γ

(t+ 1− τ)κ

)sτ − sτ−1 − N∑j=1

βjfj,τ−1

2 .

EWA and SRidge in practice: how to choose their (hyper)parameters. Following machinelearning terminology, we call hyperparameters the parameters (learning rate, regulariza-tion factor, discount factor, etc.) of the learning method at hand. In our simulation

14The original bound of Azoury and Warmuth (2001, Theorem 4.6), as cited by Cesa-Bianchi andLugosi (2006, Section 11.7) in the special case λ = 1/2, reads

λ

N∑j=1

β2j +N ln

(1 +

TL2

Nλ

)maxt6T

(st − st

)2.

Nevertheless, this does not imply a logarithmic bound as the term maxt6T

(st − st

)2may be large, as

pointed out by Gerchinovitz (2011, page 66). One can show (details upon request) that this maximumis however never larger than

4 max

{L2,

NTL4

λ

},

hence our stated bound.

13

study we will not report the performance of the EWA and SRidge methods for sev-eral well-chosen sets of hyperparameters, as is usual in the literature (as for example inWright, 2008 for Bayesian model averaging methods, or in other papers, where typicallythe performance of the methods at hand are reported for several, hand-picked values ofthe hyperparameters). We instead use a sequential grid search (i.e., perform a grid searchat each step, which is a natural approach already considered in Devaine et al., 2013) andthus only report the results obtained for a single instance of the method. This instancedoes not require any previous knowledge since the hyperparameters are set online anddo not have to be determined in advance. Details are provided in Appendix C.1 of theonline supplementary material (Amat et al., 2018b).

4. Data and fundamentals used

Our main sample includes 12 floating exchange rates for major industrial economiesfrom March 1973 to December 2014 (we have at most 502 data points per currency).We considered the same currencies as Molodtsova and Papell (2009). Machine learningmethods should have superior performance, especially over long time periods, given theguarantees on their performance discussed in Section 3 – hence our desire to use thelongest time-series possible. We use end-of-month exchange rates which are taken fromthe IMF’s IFS database. We tried to extend the same data series for fundamentals asused in Molodtsova and Papell (2009), but some of them were discontinued. As a result,we tried to find the closest substitutes possible for the entire period 1973–2014 fromsimilar sources (IMF, OECD) through Datastream. For robustness checks we also usedreal-time data obtained from the OECD for the period February 1999 to April 2017.A detailed description of the data is given in Appendix A of the online supplementarymaterial (Amat et al., 2018b).

We principally study the behavior of 12 major floating currencies that are activethroughout the 1973–2014 period. The introduction of the Euro in 1999 constrainsthe sample for some continental Europe currencies (FRF/USD, DEM/USD, ITL/USD,NLG/USD, PTE/USD). Even for the active currencies (USD/GBP, JPY/USD, CHF/USD,CAD/USD, SEK/USD, USD/AUD, DNK/USD), however, it was not possible to obtainall fundamentals time-series for the entire 1973-2014 period (see details for each in Ap-pendix A of the online supplementary material).

Our fundamentals are formed as follows. The inflation differentials are calculated as12-month changes in consumer price indexes (CPI). We use a money market rate or 3-month interest rate differentials for the interest rate-based fundamentals. For differencesin money stock growth and output growth we use the preceding 12-month trends inthese variables. We do not detrend, filter or seasonally adjust the data. The outputgaps for the Taylor-rule fundamentals are percentage deviations from “potential” outputthat were computed including (i) a linear trend, (ii) a quadratic trend, (iii) a linear andquadratic trend, and (iv) a Hodrick-Prescott filter using the data available prior to thedate for which the output gap was calculated.

The data set thus created is freely available for download (Amat et al., 2018a).

14

5. Results

5.1. Results for “classic” fundamentals

Our baseline results are shown in Tables 1 and 2, and include the RMSE×100 ofthe no-change prediction15 (column 1) and the Theil ratios (columns 2, 6, 10 and 14)of predictions, as well as the corresponding p-values of the CW and DM tests. Theformer are referred to as “CW p-values” and are found in columns 3, 7, 11 and 15. Asexplained in Appendix B.3 of the online supplementary material (Amat et al., 2018b),these should be used with extreme caution in our context. Indeed, they correspondto fundamentals being useful (H ′1) or not (H ′0) in some existing underlying model – amatter of predictability not necessarily associated with better forecasting ability. Onthe other hand, the p-values for the DM tests are more reliable in our context, as theDM test is a general test for comparing forecasting abilities (see Appendix B.2 of theonline supplementary material for details16). These p-values, are called “DM p-values”(columns 5, 9, 13 and 17) and evaluate the hypothesis H0 that the difference in theforecasting performance of the method under scrutiny is not significantly better thanthat of the no-change prediction, against the alternative hypothesis H1 that it is.

The sets of columns correspond to the forecasting methods: rolling OLS regression,recursive OLS regression, sequential ridge regression with discount factors and the expo-nentially weighted average strategy with discount factors respectively, while the sets ofrows correspond to the sets of fundamentals discussed in Section 2.2: PPP fundamentals,UIRP fundamentals, monetary model fundamentals, and all of these combined.

Results allowing different coefficients on the same fundamentals across countries. Ta-ble 1 shows that we obtain better predictions than the no-change prediction for sequentialridge regression with discount factors and the exponentially weighted average strategywith discount factors methods using the PPP or UIRP fundamentals, allowing for differ-ent coefficients for fundamentals across countries. The improvements, however, are small(not higher than 1.3 % in terms of RMSE) and most of the time we cannot reject thehypothesis that the forecasting methods being compared perform similarly. The expo-nentially weighted average strategy with discount factors is more successful as a method– both for the PPP and the UIRP it can improve upon the no-change prediction for 10of the 12 currency pairs. In the PPP fundamentals case, this performance is statisticallysuperior at the 10 % level in 8 of the 12 cases according to the DM tests (and in 9 ofthe 12 according to the CW tests). Sequential ridge regression with discount factors isless successful, being able to outperform the no-change prediction in 7 of the 12 cases forPPP fundamentals, and 6 of the 12 for UIRP fundamentals. There is no evidence thatby using monetary fundamentals or all fundamentals one could improve broadly uponthe no-change prediction with these methods.

15These may differ for the same currency pair for different sets of fundamentals given that the latter areavailable for different periods (see the data Tables A.8–A.9 in Appendix A of the online supplementarymaterial), so the forecasted periods may vary.

16In particular, for the DM tests, we compute p-values based on bootstrapping the time series at hand,as is commonly done in the literature, see references cited in Appendix B.2 of the online supplementarymaterial.

15

Results with coupled fundamentals. Table 2 presents the results in which we enforcesymmetric coefficients across countries for the same fundamentals. The results are weakerthan those where different coefficients were allowed. We are now able to beat randomwalk predictions (as signaled by Theil ratios < 1) only for the exponentially weightedaverage strategy with discount factors for PPP (8 of the 12 currency pairs) and UIRP(10 of the 12), but very few of these are statistically significant at conventional levels.

Remarks on the performance of our methods. The conclusions we can make from Tables 1and 2 are quite interesting. While allowing for different coefficients across countries, wecan claim forecastability of exchange rates for PPP and UIRP fundamentals using theexponentially weighted average strategy with discount factors, which has not been pre-viously demonstrated for such a wide range of currencies at such short range. The gainsin RMSE compared to the random walk are small and often not statistically significantas judged by the DM test. However, comparing to existing literature – as surveyed byRossi (2013) – these gains are substantial.

The theoretical tradeoff signalled in Section 3.2 between parsimony and adding fun-damentals is not clear, which is visible in our results. For the decoupled case, when wecombine all fundamentals – in the spirit of the information aggregation ability of ma-chine learning techniques – we do not gain in terms of forecasting accuracy. Sometimes,the forecasts even turn out to be less precise in terms of RMSE. On the other hand,constraining the coefficients on fundamentals to be equal across countries leads to worseforecasting results for PPP- and UIRP-based fundamentals. However, when we use allfundamentals together (our version of “all but-the-kitchen-sink” regression) in the lastpanel of Table 2, we obtain better results for the coupled (constrained coefficients) case.Indeed, we can beat the no-change prediction in 8 of the 12 cases for the exponentiallyweighted average strategy with discount factors, while only in 5 of the 12 in the decoupledcase (cf. the corresponding panel in Table 1).

There is a cost suffered by our methods when adding fundamentals that do not per-form well. To rephrase this in the terms used in Section 3.2, the potential decrease inapproximation error when considering all fundamentals is too small to be compensatedby a larger sequential estimation error linked to dealing with more fundamentals. Theimportant criterion is then the (a priori, of course, unknown) information content of thefundamentals used for forecasting that, themselves, by construction are predicted proba-bly with errors based on their past values. Our results therefore show that parsimonious(PPP- and UIRP-based) forecasting equations seem to be preferable in our setting, asalso seen in other studies of exchange rate predictability using different methods. Thismay be an indication that the different types of fundamentals we study contain essen-tially the same information17. However, the better performance of the settings with onefundamental allowing for differing coefficients across countries, rather than restrictingthem to be equal, shows that the concern that different monetary policy conducts orimperfectly comparable economic time-series pose additional difficulties in exchange rate

17Theoretically, with additional assumptions, there are links between the theories from which our“classic” fundamentals are taken. Interest rates could be high in a high inflation environment, and withthe same risk premium both UIRP and PPP would predict a depreciation of such a currency. If inflationis primarily a monetary phenomenon, high monetary growth would be correlated with high inflation andinterest rates.

16

forecasting is warranted. One should underline that the performance of OLS methodsworsens with the number of fundamentals, but no difference in forecasting performancebetween equations with constrained/unconstrained coefficients could be determined forthese methods.

Performance of the rolling and recursive OLS strategies. The same fundamentals seemnot to have much forecasting power when evaluated using the classical methods consid-ered in the literature – either rolling or recursive OLS regression – no matter whetherwe allow for asymmetry in coefficients on the same fundamentals or not. For none ofthe currency pairs do we obtain a similar number of improvements in terms of RMSEover the no-change prediction18. These conclusions are in line with the existing empiricalliterature documenting that fundamentals do not allow for systematic improvements inforecasting (see Rossi, 2013).

Direct comparison of the machine learning and OLS strategies. Instead of making anindirect comparison between our forecasting methods and those typically used in the lit-erature, we can directly compare their performance by comparing the Theil ratios (of themachine learning method’s RMSE with respect to the OLS methods’ ones) and performan appropriate (bootstrapped) DM test. The results of this exercise are shown in Ta-ble 3, as well as in Table D.10 of the online supplementary material (Amat et al., 2018b).The first conclusion is that both methods do globally better than the OLS methods whilecomparing the Theil ratios, no matter what type of fundamentals are considered (even formonetary-model fundamentals and all fundamentals when the machine learning meth-ods were not successful in beating the no-change prediction) or what assumption on the(a)symmetry of coefficients on fundamentals across countries is made. Interestingly, thelargest gains are obtained when all fundamentals are used (last sub-table of each table)– up to 15.6 % in terms of RMSE (for the SEK/USD pair in the coupled version of theexponentially weighted average strategy with discount factors vs. rolling OLS regression).This is partly because of the weak performance of the OLS methods as the number offundamentals used grows, which has been signaled in the literature (Rossi, 2013), while

18 In Table 1 the p-value 4.1% of the DM test for USD/GBP prediction, using a rolling OLS regressionbased on PPP fundamentals, may seem odd given that the Theil ratio equals 1.0025 and the DM statistictakes the value −0.2934. This is a consequence of the evaluation of the p-value through bootstrapping:it turns out that on the bootstrapped samples, the performance of the rolling OLS regressions are evenworse (much higher Theil ratios, more negative DM statistics). The distribution of bootstrapped DMstatistics is thus highly shifted to the left, which makes the value −0.2934 observed on the original dataunlikely under the distribution of the bootstrapped DM statistic. (A detailed analysis of this can beprovided upon request.) Such patterns occur in the analyzed data for OLS methods, as can be seen inother entries in the same table. This does not mean that a statistically significant gain in forecastingaccuracy is achieved, as no gain in forecasting accuracy is achieved in the first place. The originalTheil ratio is still > 1 or, equivalently, the DM statistic is negative. This only illustrates the fact thatthe bootstrapping correction is (only) of help to better evaluate potential improvements in forecastingaccuracy; it is meaningless in cases where no improvement exists in the first place. Similar observationsare documented in the literature: see, e.g., Rogoff and Stavrakeva (2008, Table 2) and Mark and Sul(2001, Tables 6 and 7). The bootstrapping in the latter concerns Theil ratios (not DM statistics) butthe issue is similar, of course; they always check whether, in discussing the results, the Theil ratios onthe original data are < 1. As for Rogoff and Stavrakeva (2008), they write (in their Appendix A): “Asignificant and positive DMW test implies that the structural model outperforms the random walk.”Hence, positivity of the DM statistic is not to be forgotten.

17

the effect of this increasing number is milder in the case of our machine learning methods(see, e.g., Mauricette et al., 2009).

Finally, we note that some descriptive statistics show that this on average good perfor-mance of our new methods does not come at the cost of local disasters. On the contrary,the forecasting errors seem to be uniformly better over time. Indeed, when computingthe quantiles of the difference between the forecasting error of the no change minus theone of the methods under scrutiny (see Appendix B.1 of the online supplementary ma-terial), as shown in Table D.14 of Appendix D of the online supplementary material, wesee that these differences are uniformly much larger for sequential ridge regression withdiscount factors and the exponentially weighted average strategy with discount factorsthan for the recursive or rolling OLS regressions when the methods exhibit forecastabilityfor a given currency pair. By “uniformly” we mean here that quantiles of the same orderfor two sequences of differences are almost always ranked in the same manner, thosecorresponding to sequential ridge regression with discount factors and the exponentiallyweighted average strategy with discount factors (first sequence) being larger than thosefor rolling or recursive OLS regressions (second sequence).

5.2. Robustness checks

We conducted several robustness checks to verify whether our results were not due tosome quirk related to the samples that we constructed. First, we decided to also consider6-month inflation, money stock or output changes in the decoupled version to see whethera different way of constructing fundamentals matters. This did not qualitatively changeour results.

Next, we ran again the basic combinations of fundamentals and methods from Tables 1and 2 on our data trimmed to shorter samples: 1980–2014, the post-Plaza accord period1985–2014, the post ERM-crisis period 1992–2014, and 1973–2006, so as to see whetherthe inclusion of a particular period (e.g., the high inflation period of 1970s or the postfinancial crisis period) drives the results. We show the results for the 1980–2014 samplein Table D.15 of the online supplementary material. What we observe in general is thatthe forecasting properties of our methods still hold, though the observed gains againstthe no-change prediction are typically smaller, but – especially for the forecasts basedon parsimonious sets of fundamentals (PPP, UIRP) – still often statistically significant.This may indicate that in practice, longer data series are in general beneficial for the(asymptotic) guarantees to kick in to minimize RMSE, as indicated in Section 3.2.

We also tried to work with what we call “absolute” fundamentals. This means thatinstead of inflation differentials we directly used price, money and output levels as substi-tutes for the PPP and monetary model fundamentals19. The results here are weaker. Wecan still obtain improvements with sequential ridge regression with discount factors. Aproblem arises, though, with the exponentially weighted average strategy with discountfactors in that the forecast errors involving deviations of the exchange rates from theabsolute fundamentals considered are much larger that those for relative fundamentalsused in our original exercise. Also, the weights assigned to the absolute fundamentals

19The UIRP-model inspired forecasting equation has no such counterpart. “Absolute” fundamentalsAt were created by adding price, money or output changes to an initial exchange rate in our data sets.Then the deviation of the actual nominal rate from its thus calculated “fundamental” value At − st wasused to predict st+1 − st.

18

by the method tend to be small (i.e., the predictions output by the method are close tothe no-change ones). This is a documented fact: this method does not easily allow forfundamentals that err much and are consistently dominated by a fundamental with sys-tematically good performance (a situation which arises in our case, the latter being theno-change prediction). This is in contrast with sequential ridge regression with discountfactors, which can correct the fundamentals for proper scaling.

Last, we worked with the actual values of the fundamentals. That is, instead of“predicting” the next period’s change in fundamentals using past values, we fed in theactual values that occurred in the course of economic activity. As stated in the literaturestarting with Meese and Rogoff (1983), these do not lead to good or better forecasts.

5.3. Directional tests

A different, though secondary, measure of forecast quality that has been employedin the literature is “directional tests”, i.e., tests of whether forecasting methods are ableto predict the direction of change of the exchange rate better than a fair coin toss.Although the methods considered in this paper are not constructed to be efficient in thisregard – as their focus is on controlling RMSE (see Section 3.2) – we have tested howthe predictions obtained by these methods fare in this case. In Table 4, as well as inTable D.11 of the online supplementary material, we show the percentage of successes ofeach method (OLS regressions as well as sequential ridge regression with discount factorsand the exponentially weighted average strategy with discount factors) in forecasting thedirection of change of exchange rates with the corresponding (bootstrapped) DM p-valueof the test of the difference with a fair coin toss.

The patterns that emerge are interesting. Machine learning methods are able topredict the correct direction in several cases 55 % of the time (with a maximum of 60.8 %)– improvements which are often statistically significant. For machine-learning methods,the success in predicting RMSE goes typically hand in hand with the success of directionalpredictions for the decoupled fundamentals. This is not true for the forecasts made whileconstraining the coefficients to be equal, nor by OLS-based methods; in these cases, asthe number of fundamentals included grows, they do better in “directional tests” whilefailing in terms of RMSE.

5.4. Economic evaluation

In Table 5 we show various economic criteria used to evaluate the economic benefitsone would obtain using our forecasting methods as opposed to a random walk. Thisinvolves calculating investment returns, and we do this for the exponentially weightedaverage strategy with discount factors, generally obtaining improvements in terms of theRMSE, as shown in Table 1, which shows that detected improvements would also leadto superior performance from an investor’s perspective.

The idea is to assess the utility of the forecasts from an investor’s point of view,who builds one-period portfolios using information known at that given period understandard mean-variance preferences with a target standard deviation (see West et al.,1993, Della Corte et al., 2012 and Della Corte and Tsiakas, 2012). It is assumed in thiscontext that investors make decisions on investing in money market rates of differentcountries, taking into consideration the forecasted movements of the exchange rates.Then various performance measures of such portfolios are compared to see whether some

19

forecasting method generates predictions that can be profitably used by investors overthose of a no-change model. (See Appendix C of the online supplementary material forfurther details.)

We consider the standard measures in the literature: a direct comparison of annual-ized returns for a portfolio using forecasts and a random walk, an annualized performancefee (in bps), annualized premium return (in bps), and annualized Sharpe and Sortinoratios. Given the oft-discussed concern that optimal portfolio selection applied to invest-ment problems with exchange rates frequently leads to large portfolio weights (implyinginvestors are taking very risky bets), we also study portfolio allocation when we restrictthe weights w1,t, . . . , wK,t on the K = 7 currencies20 so that w2

1,t+. . .+w2K,t 6 c, which is

a standard constraint in the machine-learning literature. (See Appendix C of the onlinesupplementary material for the technical details of how this can be achieved. When weuse such a constraint, we provide the corresponding value for c; otherwise, we write “noc” in our tables.)

We see that in line with the results shown in Table 1 obtained from RMSE compar-isons of forecasts, our methods would add value to investors using them over strategiesbased on no-change predictions for PPP or UIRP fundamentals. For the PPP, annualizedreturns are 13.48 % for the EWA-based strategy as opposed to 11.83 % for the randomwalk. However, investment risk is also an important factor in decision-making and in-vestors would be willing to pay a performance fee of 229 bps, or would obtain a premiumreturn of 277 bps, by sticking to our forecasts. Both the Sharpe and the Sortino ratioswould be higher for the EWA-forecast based portfolios versus no-change ones (0.84 vs.0.64 and 1.08 vs. 0.68 respectively). The same holds for UIRP-fundamentals. Returns,the Sharpe ratios, and the Sortino ratios, for a portfolio based on EWA-algorithm fore-casts are higher (12.13 % vs. 11.44 %, 0.78 vs. 0.62, and 1.38 vs. 0.65 respectively) andinvestors would be willing to pay a performance fee of 192 bps or obtain a premiumreturn of 268 bps. This no longer holds true for predictions based on “monetary model”fundamentals or when we use all fundamentals together. It can be seen that constrainingthe weights (with c = 1 or c = 2) greatly improves the results in every relevant way forall types of fundamentals. We achieve a premium return larger than 360 bps with theUIRP fundamental and constraint c = 2.

In Table D.16 of Appendix D of the online supplementary material we repeat theexercise, dropping the DNK/USD exchange rate, which is highly correlated with theSEK/USD and CHF/USD ones. This correlation creates some instability in the weightsand leads to, in our calculations, optimally shorting one of these currencies with a veryhigh weight and going long in the others. We see that the results of the economic crite-rion evaluation greatly improve when dropping the DNK/USD exchange rate, especiallyfor “monetary model” fundamentals and when we use all fundamentals together. InTables D.17 and D.18 of the online supplementary material we impose an additionalconstraint on the weights, requiring that their sum on the currencies is smaller than 1,i.e., w1,t + . . .+ wK,t 6 1.

We therefore observe that overall, investors would benefit from using our forecastsinstead of the random walk prediction, and would be willing to pay up to slightly morethan 360 bps for our best predictions in our basic scenarios.

20We consider: GBP, JPY, CHF, CAD, SEK, DNK and AUD, but not the currencies that later mergedto become the Euro as they would mean a shorter overall time horizon.

20

5.5. Real-time data results

We re-ran our methods using real-time data. The use of real-time data can help toalleviate the concern that forecasting performance is improperly evaluated because theinformation known at the time by market participants is not taken into account. Futurerevisions of the data that are reported by statistical agencies and available to researchersmay blur the information that investors actually possessed while making decisions inthe past, and hence bias a proper evaluation of the forecasts. We note in passing thatall UIRP-fundamentals forecasts shown for example in Tables 1 and 2 are in this senseconducted with real-time data all along, and as such give a useful benchmark.

Unfortunately for us, the real-time data we obtained from the OECD21 starts onlyfrom February 1999, and we thus conducted our study on a shorter22 data set: February1999–April 2017, data permitting. The results are given in Table 6 for the case wherewe allow asymmetric coefficients on the fundamentals. Compared to Tables 1 and 2,this points to a deterioration of the forecasting abilities of both of our machine learningmethods, though we still obtain improvements in RMSE versus the random walk in themajority of the cases for PPP- and UIRP-based fundamentals. This is further confirmedin the economic evaluation criterion in Table 7, as well as in the directional resultsin Table D.13 of the online supplementary material. In a separate study, shown inTables D.19–D.24 in Appendix D of the online supplementary material, we re-ran thereal-time data and revised-fundamentals forecasts on the same data period of February1999–December 2014 (not all time-series from our original data set could be extended to2017), with similar conclusions.

Several points need to be made here. First, as explained earlier in Section 3.2, shorten-ing time-series in general leads to worse predictions for these machine-learning methods.The UIRP-based fundamentals are always real-time, and although we see a similar num-ber of Theil values < 1 for forecasts using them in comparison to what was shown inTable 1 (for the same set of currencies), these improvements are smaller (close to 1),though there is also an across-the-board performance drop in Tables 4 and 5. Second,OLS-based methods do not provide different (better) forecasting performance either. Infact, as shown in Table D.12 of the online supplementary material, machine-learningmethods beat OLS-based ones in terms of RMSE improvement in a similar fashion asbefore (i.e., Table 3), at times spectacularly (cf. the 25.1 % gain in terms of RMSE forCHF/USD for sequential ridge regression with discount factors and all fundamentals con-sidered). Third, using same time-period comparisons for 1999–2014 (Tables D.19–D.24of the online supplementary material), there is no difference in performance when usingreal-time or revised data. This is in line with the literature, e.g., Ince (2014), whichbroadly concludes that the lack of use of real-time and actual data is not related to thefailure of OLS-based methods. The conclusions based on these exercises have to be takenwith caution, however, as they may be sample-specific. It is clear from Figures D.1–D.3of the online supplementary material, based on the original revised-data forecasting, thatafter the 2008 crisis (ca. the last 80 observations) the pattern of the weights picked bythe exponentially weighted average strategy with discount factors changes dramatically

21Available at http://stats.oecd.org/mei/default.asp?rev=122Because the period is essentially half as long as before, we also set t0 = 60 months as a training

period for the algorithms and as a window for rolling OLS regression, instead of the original valuet0 = 120 months.

21

for the three shown currencies (also true for the others analyzed, but not shown), whichcould point to an important “regime” change, perhaps related to the central bank ormarket reactions to the crisis.

5.6. Taylor-rule fundamentals

A recent strand in the literature has identified Taylor-rule fundamentals as useful inachieving exchange rate predictability (which is not the same as forecastability; see Foot-note 25). As Taylor-rule fundamentals can be created from the data in our possession,we also re-ran forecasting equations based on (5) with these fundamentals for our basicdecoupled sample. The results are shown in Table D.25 of Appendix D of the onlinesupplementary material.

Forecasts based on Taylor-rule model fundamentals beat the random walk in 9 of the12 cases, no matter how we calculate the output gaps. In 6 of the 12 cases for fundamen-tals including output gaps – constructed either using deviations from both a linear anda quadratic trend, or from a Hodrick-Prescott filtered trend – such improvements arestatistically significant at the 10 % level according to the bootstrapped DM-test p-values.In this respect, globally they fare better than forecasts based on UIRP fundamentals(more statistically significant results) but slightly worse than predictions based on PPPfundamentals. Indeed, the Taylor-rule fundamentals as shown in (5) do include inflationdifferentials, precisely those used in PPP-based models – see (2) –, whereas interest ratedifferentials are lagged by one period. As we do not estimate models, we do not inves-tigate further what signs the coefficients on the inflation rates have in our forecastingequations.

Also in these cases, OLS methods do not23 allow us to consistently beat the randomwalk. The rolling OLS regression actually achieves a forecasting accuracy that is consis-tently worse than the random walk (all Theil rations are > 1, all DM test statistics arenegative). As for recursive OLS regression, gains in forecasting accuracy are observedonly for 3 of the 4 × 12 cases considered (but these 3 occurrences are all statisticallysignificant).

6. Related literature

Exchange rate forecasting literature. Rossi (2013) and Della Corte and Tsiakas (2012)provide comprehensive recent reviews of the exchange-rate forecasting literature so wekeep ours to a bare minimum. There have been different ways in which researchers havetried to cope with the negative result of Meese and Rogoff (1983) that showed that thesimple exchange rate models from the 1970s (i.e., those that proposed the fundamentalswe study here) did poorly in comparison to a random walk without drift (a no-changeprediction) in forecasting exchange rates in the floating period after 1973 for short fore-cast horizons. One way to cope, that we have pursued here, has been to use better

23A similar remark needs to be made as in Section 5.1, see Footnote 18 therein: in many cases,especially for the rolling OLS regression, the Theil ratios are > 1 (equivalently, the DM statistics arenegative) while the bootstrapped DM p-values would otherwise indicate statistical significance. However,this merely means that there is no gain in forecasting accuracy in the first place, so determining whetherit is significant or not is irrelevant. Technical details on why such seemingly paradoxical situations ariseare provided in the mentioned footnote. Indeed, this is a documented fact in the literature.

22

tools to extract information from the data (better forecasting tools, not necessarily sta-tistical/econometric tools). Some of the other solutions proposed involve, for example,cointegration techniques (in Mark, 1995) combined with the use of panel data (in Markand Sul, 2001), long samples in panels as in Rapach and Wohar (2002), and including alarge set of countries (in Cerra and Saxena, 2010). These techniques are typically used forlonger-horizon (over 1 year) forecasts when it is believed that the long-run relationship, asmodeled by the cointegrating equations, kicks in. The cited studies obtain some successin demonstrating predictability and/or forecastability at longer horizons (typically morethan 2 years). We have focused on short-term forecasts and use single-equation methods– a setup in which forecastability has not been ascertained until relatively recently. Forthe short-run, Greenaway-Mcgrevy et al. (2012) obtained considerable success in outpre-dicting the no-change prediction using factor analysis, extracting the factors from theexchange rates themselves (but not the economic fundamentals). Dal Bianco et al. (2012)obtained forecastability for the Euro/U.S. dollar rate at weekly to monthly horizons us-ing a stylized econometric model that mixes information from different fundamentalsarriving at different frequencies. Our method is more general, directly geared for predic-tion, and importantly, comes with certain theoretical guarantees on convergence of theRMSEs (see Section 3.2).

Sequential combination of fundamentals. The methodology used for forecasting in thispaper consists of sequentially combining (aggregating) fundamentals. The distinguish-ing feature of our methodology is that it is sequential in nature; in contrast, all otherapproaches we know of that forecast exchange rates use batch estimation or learningmethods that need to be run in an incremental way (which usually leads to the loss ofthe theoretical guarantees associated with the batch case). These approaches can bedivided into two groups: (i) the learning methods of the machine-learning literature, and(ii) the estimation methods of the statistics literature.

The first group contains, for instance, the studies of Li et al. (2015) and Wright (2008)for exchange rates, and Bajari et al. (2015) for demand estimation. The first two refer-ences are perhaps the closest efforts to ours insofar as the idea of extracting informationfrom fundamentals to forecast exchange rates is concerned. In the introduction, we havealready compared our approach and results to those of Li et al. (2015): the final improve-ment in performance of end-of-month forecasts with respect to no change are comparable,though ours are slightly better. More importantly however, we obtain this improvementby considering only well-identified fundamentals (PPP or UIRP fundamentals, instead ofusing “kitchen-sink” aggregation) and by implementing machine-learning methods thatcome with theoretical performance guarantees (unlike the elastic net method used by Liet al. (2015)). Our approach thus conveys economic meaning and could therefore beconsidered “safer”. Wright (2008) used a method called Bayesian model averaging. Thebuilding blocks for prediction he uses are a large number of predictors, many of whichdo not come from the standard models considered in this paper (not only classic fun-damentals are used). His approach does not give improved forecasts compared to theno-change prediction in a statistically significant way in most cases he considers. More-over, he does not explain how to properly choose the “shrinkage” parameter that retainsthe informativeness of priors without knowing the properties of the data ex ante. Our

23

methods, in contrast, are entirely data-driven24.As for estimation methods, two approaches related to ours are estimation of nonlinear

models, and models with time-varying parameters. Rossi (2013) states that variousnonlinear methods were not particularly successful in forecasting exchange rates, whileRossi (2006) questions the robustness of time-varying parameter models. Bacchetta et al.(2010) argue that the gain from using such an approach would be minimal in practice.On simulated data, these authors find that the benefits from using such models in termsof greater explanatory power are in practice outweighed by additional estimation errorsin the time-varying parameters. Schinasi and Swamy (1989) reassess the study of Meeseand Rogoff (1983) using various nonlinear methods, including an early version of ridgeregression. Engel (1994) documents the failure of a Markov-switching model to beat theno-change prediction in forecasting.

Other issues considered in exchange rate forecasting. Another way that researchers havetried to improve the ability to forecast exchange rates from fundamentals is to con-sider different economic models with other fundamentals. It has been seen recentlythat exchange rate models based on Taylor-rule fundamentals perform well in ascer-taining the predictability25 of exchange rates at short horizons (see Engel and West,2006, Molodtsova and Papell, 2009, Molodtsova et al., 2008, Giacomini and Rossi, 2010,Molodtsova et al., 2011, Rossi and Inoue, 2012; though Rogoff and Stavrakeva, 2008disagree) but do not find that these perform much better in terms of forecasting. Forthis reason we test our methods on Taylor-rule fundamentals, obtaining good forecast-ing results – only the plain vanilla inflation rates used as fundamentals work better (seeSection 5.6).

Another successful fundamental was the behavior of net foreign assets, as in Gourin-chas and Rey (2007) and Della Corte et al. (2012). The fundamentals to conduct thesetests are available at 3-month frequencies, resulting in fewer observations that can beused, so we did not investigate them here. Other studies have assessed the forecastingability of exchange rate models of the 1990s, including Cheung et al. (2005), differencesin the term structures of forward premia (Clarida et al., 2003), and the scapegoat model(Fratzscher et al., 2015). Given the scope of our exercise and the gaps in some necessarydata, we did not evaluate these models with our machine-learning methods, but this maybe a useful research agenda for the future.

24In fact, some inspiration could be taken from them to perform a data-driven choice of the “shrinkage”parameter of Wright (2008).

25 Predictability is a different concept to forecastability. In Molodtsova and Papell (2009), it meanstesting whether the estimated coefficients of a model are jointly significantly different from zero whenexplaining changes in the exchange rate. It does not mean that a model that exhibits predictabilitynecessarily provides better forecasts (in the literature, typically it does not). In general, the focus ofthese and many other attempts is rather to assess whether fundamentals play a role in exchange ratedetermination, since it can be motivated theoretically why the forecasts they produce in terms of anevaluation criterion such as RMSE may fare worse than those of a forecast based on no change in theexchange rate. (See Rossi, 2013 for a discussion of this. Special tests were designed by West, 1996and Clark and West, 2006, 2007 for this purpose. See also Appendix B.3 of the online supplementarymaterial.) In this paper, however, we are not focused on predictability – we actually are interested inwhether we can produce better forecasts of exchange rates than the no-change one. We are also notinterested in validating a particular model – that is, trying to fit coefficients and check whether the signsand magnitudes are those posited by the theory; we simply try to extract from the fundamentals inquestion information useful for the behavior of exchange rates.

24

7. Conclusions

In this paper we have applied methods stemming from the field of machine learning –sequential ridge regression with discount factors and the exponentially weighted averagestrategy with discount factors – to the perennial problem of exchange rate forecasting.In doing so, we obtain gains in forecasting in terms of the standard RMSE criterion forPPP or UIRP fundamentals that were not found using traditionally applied estimationmethods based on OLS regressions. The key is to use these machine-learning forecastingschemes to do what they are good for: produce forecasts – and not try to estimatesome underlying model (if any such model exists) as has traditionally been the casewith more statistical methods. We conclude thus that a major problem in internationaleconomics – whether there is a short-term relationship between “classic” fundamentalsand exchange rates that can be detected and that beats the random walk – is answeredin the affirmative under the condition that proper machine-learning techniques, e.g.,sequential ridge regression with discount factors, or the exponentially weighted averagestrategy with discount factors, are applied. Our success points to the potential of suchtechniques for improving the evaluation of economic problems.

Machine learning techniques also serve to effectively aggregate information from manysources. A tempting exercise, beyond the scope of this paper, is to evaluate the fore-casting performance when including many more fundamentals than the “classic” onesconsidered here that come from suggestions in the literature – for example those basedon productivity, interest rate yield curves, net foreign assets, etc. One set of fundamen-tals – based on the UIRP relationship – is especially promising since interest rates (forvarious maturities) can be obtained for long-time periods at high frequencies. Venturingfurther, one could consider many more time-series that are not typically associated withexchange rate forecasting, in the true spirit of machine learning.

As with any new method applied to exchange rate forecasting, it remains to be seenwhether our results can be replicated for other currencies, samples, forecasting periodsand fundamentals. However, given the robustness of the results shown in this paper, wehope that the application of these methods to exchange rate forecasting will stand thetest of time and allow for better predictions and decision-making in the future.

Acknowledgements

This work was supported by Investissements d’Avenir (grant number ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047).

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29

Table 1: 1-month ahead forecasts for the PPP, UIRP, monetary model and all fundamentals: decoupled formulation.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:EO

M -

De

cou

ple

d -

RM

SE

No

ch

ange

RM

SE x

10

0Th

eil r

atio

CW

p-v

alu

eD

M s

tati

stic

DM

p-v

alu

eTh

eil r

atio

CW

p-v

alu

eD

M s

tati

stic

DM

p-v

alu

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eil r

atio

CW

p-v

alu

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M s

tati

stic

DM

p-v

alu

eTh

eil r

atio

CW

p-v

alu

eD

M s

tati

stic

DM

p-v

alu

e

U

SD/G

BP

2

.905

1

1.0

025

0.2

12 -

0.29

34 0

.041

**

1.

0032

0.9

12 -

1.21

31 0

.626

0.

9998

0.3

65 0

.121

2 0

.110

0.

9975

0.1

35 0

.42

69

0.1

38

JP

Y/U

SD

3.1

023

1

.014

4 0

.890

-1.

7054

0.7

48

1.00

10 0

.880

-1.

1087

0.6

89

0.99

99 0

.169

0.9

024

0.0

38 *

*

0.99

53 0

.021

**

1.2

41

7 0

.04

7 *

*

C

HF/

USD

3

.342

8

1.0

117

0.9

19 -

1.86

47 0

.828

1.

0009

0.5

43 -

0.33

48 0

.265

0.

9995

0.1

67 0

.824

6 0

.035

**

0.

9947

0.0

20 *

* 1

.53

13

0.0

19

**

C

AD

/USD

1

.998

2

0.9

988

0.1

33 0

.155

7 0

.011

**

0.

9991

0.2

04 0

.382

0 0

.029

**

1.

0003

0.7

50 -

0.51

27

0.3

44

1.00

11 0

.997

-2.

212

1 0

.98

7

SE

K/U

SD

3.2

290

1

.008

9 0

.445

-0.

9745

0.2

97

1.00

09 0

.299

-0.

2268

0.1

82

1.00

01 0

.426

-0.

067

6 0

.226

0.

9969

0.0

50 *

* 0

.82

66

0.0

68

*

D

NK

/USD

3

.120

0

1.0

112

0.8

90 -

2.03

26 0

.809

1.

0008

0.3

57 -

0.20

10 0

.124

0.

9997

0.2

32 0

.650

0 0

.035

**

0.

9944

0.0

12 *

* 1

.63

77

0.0

08

**

*

U

SD/A

UD

3

.378

6

0.9

984

0.0

75 *

0.1

468

0.0

25 *

*

1.00

09 0

.455

-0.

2530

0.1

91

1.00

07 0

.722

-0.

677

7 0

.466

1.

0001

0.3

43 -

0.03

02

0.3

32

FR

F/U

SD

3.1

978

1

.012

0 0

.631

-1.

0735

0.5

56

1.00

36 0

.442

-0.

4357

0.3

01

1.00

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257

4 0

.294

0.

9957

0.0

93 *

0.8

51

4 0

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5 *

D

EM

/USD

3

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6

1.0

139

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1.48

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0.

9994

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1.7

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1 0

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*

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L/U

SD

3.1

907

1

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0 0

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6030

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1870

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9 0

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0.0

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**

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LG/U

SD

3.3

319

1

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6 0

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-1.

3393

0.6

67

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470

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* 1

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3

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7

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018

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36 *

* -

0.06

28 0

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0.

9936

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23 *

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7 0

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0.

9916

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31 *

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9886

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0.5

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SD/G

BP

2

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1

1.0

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1.20

86 0

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1.

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00

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7

JP

Y/U

SD

3.1

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1

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0 0

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-0.

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83 0

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612

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0.

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8 0

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9918

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**

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HF/

USD

3

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1

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K/U

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3.2

263

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SD/A

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3

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3

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71 0

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80 0

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F/U

SD

3.1

978

1

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3.1

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7 0

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6032

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92

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364

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9867

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* 0

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0.2

62

N

LG/U

SD

3.3

319

1

.038

8 0

.583

-1.

0340

0.5

76

1.00

36 0

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-0.

5149

0.3

90

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458

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0.8

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TE/U

SD

2.8

024

0

.997

0 0

.278

0.1

557

0.1

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0.99

68 0

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0.1

662

0.1

38

0.99

65 0

.342

0.2

454

0.1

52

0.99

32 0

.290

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78

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0

U

SD/G

BP

2

.905

1

1.0

350

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10 -

2.75

05 0

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1.

0165

0.7

11 -

1.49

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.681

1.

0004

0.1

36 -

0.04

20

0.1

61

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.091

* -

0.07

79

0.2

23

JP

Y/U

SD

3.1

046

1

.010

6 0

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**

-0.

7277

0.0

91 *

1.

0046

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38 *

* -

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56 0

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0.1

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0.28

67

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* 0

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0.0

81

*

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HF/

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3

.388

7

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249

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1.68

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1.

0103

0.2

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0.78

97 0

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0.4

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0.49

50

0.3

60

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492

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3

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AD

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1

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4

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396

0.4

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0.8

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58

0.5

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K/U

SD

3.2

263

1

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0686

0.9

69

1.02

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47

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402

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0076

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49

0.5

89

D

NK

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3

.167

5

1.0

435

0.9

99 -

2.72

80 0

.896

1.

0186

0.9

99 -

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0041

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29

0.5

93

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797

0 0

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1

U

SD/A

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3

.392

6

1.0

284

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33 0

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13

0.5

79

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725

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7

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F/U

SD

3.1

041

1

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2343

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48

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177

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0100

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0.69

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49

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EM

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3

.328

6

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192

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90 0

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94 -

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IT

L/U

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3.2

797

1

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.538

-1.

1557

0.4

96

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.528

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9153

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29

1.00

03 0

.388

-0.

038

0 0

.287

1.

0035

0.2

65 -

0.23

18

0.4

45

N

LG/U

SD

3.3

319

1

.030

5 0

.698

-1.

6010

0.7

17

1.00

85 0

.406

-0.

6856

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29

0.99

78 0

.107

0.7

269

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44 *

*

0.99

88 0

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63

0.1

33

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TE/U

SD

2.7

703

1

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.200

-0.

1213

0.1

51

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04 0

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2511

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77

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71 0

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269

8 0

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81 -

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25

0.6

07

U

SD/G

BP

2

.905

1

1.0

715

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03 -

3.06

98 0

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9984

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279

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5

JP

Y/U

SD

3.1

046

1

.047

8 0

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**

-0.

5941

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0.1

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0.14

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6

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HF/

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3

.388

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1.0

784

0.3

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42 0

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1

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K/U

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3.2

263

1

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72

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3

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3

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SD

3.1

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1

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96

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3287

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0.4

22

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EM

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3

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6

1.0

411

0.4

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1.20

77 0

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1.

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0.3

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98 0

.246

1.

0037

0.5

73 -

0.59

85

0.4

83

0.99

52 0

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0.1

41

IT

L/U

SD

3.2

797

1

.002

5 0

.032

**

-0.

0742

0.0

08 *

**

1.

0392

0.3

30 -

1.19

08 0

.414

1.

0024

0.5

19 -

0.28

47

0.4

80

1.00

43 0

.181

-0.

223

3 0

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4

N

LG/U

SD

3.3

319

1

.044

9 0

.042

**

-1.

4244

0.1

78

1.02

92 0

.280

-1.

2863

0.4

21

0.99

82 0

.134

0.6

858

0.0

50 *

*

0.99

20 0

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**

0.5

95

4 0

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7 *

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TE/U

SD

2.7

703

1

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8 0

.139

-0.

3345

0.1

22

1.05

80 0

.863

-1.

8471

0.8

43

1.00

47 0

.534

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260

7 0

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0.

9955

0.3

19 0

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19

0.3

25

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

All

fun

dam

enta

ls

Cu

rren

cy p

air

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSR

idge

EW

A

Notes: This table presents the results of forecasting end-of-month exchange rate 1-month ahead by the no-change prediction, the

rolling OLS, recursive OLS, SRidge and EWA methods based on decoupled fundamentals. Column 1 shows the RMSE values for

the no-change prediction. For each method, the Theil ratio (RMSE of the given method / RMSE of the no-change prediction), the

CW p-values, the DM statistic and its bootstrapped p-value are shown. The CW and DM tests are one-sided tests of equal out-

of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) for the methods considered compared

to the no-change prediction. ***, **, and * denote statistical significance at the 1 %, 5 %, and 10 % levels. Theil ratios < 1 are

indicated in bold. Sample period for the exchange rates: March 1973 – December 2014 unless shorter (indicated in Appendix A of

the online supplementary material).

30

Table 2: 1-month ahead forecasts for the PPP, UIRP, monetary model and all fundamentals: coupled formulation.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:EO

M -

Co

up

led

- R

MSE

No

ch

ange

RM

SE x

100

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

2.

905

1

1.00

50 0

.913

-1.

8180

0.8

97

1.00

35 0

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-1.

3838

0.8

26

1.00

06 0

.525

-0

.29

92 0

.20

2

1.0

005

0.5

31 -

0.2

765

0.5

42

JP

Y/U

SD

3.1

023

1.

0058

0.7

23 -

0.93

98 0

.532

1.

0010

0.9

94 -

1.98

33 0

.966

0.

9999

0.4

07 0

.15

97 0

.13

7

0.9

973

0.0

60 *

1.0

661

0.1

01

C

HF/

USD

3.

342

8

1.00

27 0

.735

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0253

0.5

76

1.00

09 0

.807

-0.

9049

0.6

01

1.00

00 0

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.85

15 0

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9

0.9

996

0.2

78 0

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C

AD

/USD

1.

998

2

0.99

64 0

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* 0

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9 0

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6 0

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1.

0000

0.5

18 -

0.0

751

0.1

44

1

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29 0

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378

0 0

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SE

K/U

SD

3.2

290

1.

0015

0.1

76 -

0.24

95 0

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0033

0.7

34 -

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20 0

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1.

0016

0.6

87 -

0.7

800

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31

1

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00 0

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65

D

NK

/USD

3.

120

0

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01 0

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0.0

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0.

9992

0.1

54 0

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0.

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82 0

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89 0

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8 *

0

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89 0

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0.6

908

0.0

99 *

U

SD/A

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3.

378

6

0.99

76 0

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* 0

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9 0

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**

1.

0032

0.8

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1.10

63 0

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0006

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1.0

451

0.6

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1

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368

1 0

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FR

F/U

SD

3.1

978

1.

0072

0.4

86 -

0.67

25 0

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0.3

46 -

0.29

07 0

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0005

0.3

08 -

0.0

669

0.1

83

0

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81 0

.176

0.4

701

0.2

39

D

EM

/USD

3.

313

6

1.00

63 0

.976

-1.

2304

0.7

78

1.00

09 0

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7742

0.5

70

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00 0

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24 0

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0.9

968

0.0

53 *

1.2

832

0.0

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*

IT

L/U

SD

3.1

907

1.

0026

0.2

97 -

0.22

58 0

.326

1.

0018

0.3

12 -

0.16

62 0

.359

1.

0011

0.2

96 -

0.1

181

0.2

94

0

.99

59 0

.073

* 0

.563

1 0

.136

N

LG/U

SD

3.3

319

1.

0056

0.9

93 -

2.15

69 0

.980

1.

0017

0.9

69 -

1.46

01 0

.877

1.

0000

0.9

71 -

1.5

584

0.9

27

0

.99

76 0

.115

0.9

444

0.0

92 *

P

TE/U

SD

3.2

207

0.

9961

0.0

28 *

* 0

.128

5 0

.506

0.

9930

0.0

23 *

* 0

.236

7 0

.543

0.

9925

0.0

23 *

* 0

.26

02 0

.52

1

0.9

821

0.0

10 *

**

0.6

295

0.4

09

U

SD/G

BP

2.

905

1

1.01

32 0

.344

-0.

8874

0.4

57

1.00

12 0

.323

-0.

1565

0.1

82

1.00

45 0

.338

-0

.34

13 0

.22

9

0.9

976

0.1

89 0

.496

9 0

.128

JP

Y/U

SD

3.1

023

1.

0062

0.7

07 -

0.89

98 0

.498

1.

0020

0.6

04 -

0.63

03 0

.462

1.

0012

0.9

19 -

0.8

999

0.5

35

0

.99

93 0

.277

0.2

532

0.2

34

C

HF/

USD

3.

374

7

1.00

09 0

.387

-0.

3609

0.2

49

1.00

05 0

.709

-0.

5991

0.4

29

1.00

00 0

.709

-0

.49

95 0

.33

5

0.9

990

0.1

91 0

.546

3 0

.107

C

AD

/USD

1.

998

2

1.00

27 0

.331

-0.

4322

0.2

42

1.00

06 0

.749

-0.

6664

0.4

72

1.00

01 0

.894

-1

.09

38 0

.70

1

1.0

020

0.8

63 -

1.4

258

0.8

00

SE

K/U

SD

3.2

263

1.

0242

0.9

12 -

1.95

21 0

.937

1.

0037

0.6

02 -

0.61

85 0

.445

0.

9907

0.0

77 *

0.8

021

0.0

36

**

0.9

868

0.0

71 *

0.9

468

0.1

11

D

NK

/USD

3.

120

0

1.01

19 0

.912

-1.

4864

0.7

88

1.00

06 0

.578

-0.

3062

0.2

85

1.00

02 0

.847

-0

.99

17 0

.59

8

1.0

012

0.5

31 -

0.4

593

0.5

31

U

SD/A

UD

3.

379

3

1.00

97 0

.471

-0.

8183

0.3

96

1.00

67 0

.740

-0.

9751

0.6

13

1.00

75 0

.763

-0

.93

82 0

.53

3

0.9

999

0.3

28 0

.012

7 0

.305

FR

F/U

SD

3.1

978

1.

0085

0.9

08 -

1.07

15 0

.687

1.

0039

0.6

85 -

0.61

46 0

.447

1.

0019

0.4

13 -

0.3

082

0.2

60

0

.99

84 0

.195

0.5

071

0.1

06

D

EM

/USD

3.

313

6

1.00

36 0

.849

-1.

1067

0.6

95

1.00

26 0

.778

-0.

6834

0.4

92

1.00

01 0

.841

-1

.00

00 0

.61

5

0.9

962

0.0

69 *

1.0

409

0.0

54 *

IT

L/U

SD

3.1

907

1.

0060

0.4

88 -

0.46

77 0

.633

0.

9996

0.2

11 0

.031

3 0

.487

1.

0055

0.4

87 -

0.4

164

0.5

42

0

.98

69 0

.023

**

1.0

302

0.1

82

N

LG/U

SD

3.3

319

1.

0044

0.6

41 -

0.69

58 0

.487

1.

0014

0.6

58 -

0.52

65 0

.405

1.

0022

0.2

81 -

0.2

382

0.2

29

0

.99

78 0

.140

0.7

264

0.1

16

PTE

/USD

2.

802

4

0.98

74 0

.117

0.7

617

0.0

87 *

0.

9917

0.1

96 0

.382

1 0

.182

0.

9918

0.2

73 0

.39

43 0

.15

1

0.9

899

0.2

12 0

.559

2 0

.221

U

SD/G

BP

2.

905

1

1.02

01 0

.853

-1.

6787

0.6

71

1.00

21 0

.486

-0.

3922

0.1

97

1.00

50 0

.309

-0

.51

18 0

.25

2

1.0

034

0.3

09 -

0.4

235

0.2

67

JP

Y/U

SD

3.1

046

1.

0042

0.0

71 *

-0.

4363

0.1

16

1.00

42 0

.220

-0.

4880

0.2

73

1.00

65 0

.523

-0

.77

13 0

.44

8

0.9

965

0.0

29 *

* 0

.390

9 0

.063

*

C

HF/

USD

3.

388

7

1.00

29 0

.104

-0.

3098

0.0

84 *

1.

0063

0.1

52 -

0.59

19 0

.299

1.

0015

0.2

04 -

0.2

288

0.2

00

1

.00

05 0

.046

**

-0.

049

0 0

.199

C

AD

/USD

1.

999

4

1.02

54 0

.591

-0.

9221

0.3

18

1.00

32 0

.697

-0.

6773

0.4

11

1.00

65 0

.839

-1

.03

35 0

.61

1

1.0

074

0.9

39 -

1.3

492

0.6

23

SE

K/U

SD

3.2

263

1.

0283

0.9

56 -

2.06

75 0

.860

1.

0047

0.7

25 -

1.00

71 0

.573

1.

0138

0.4

24 -

0.9

989

0.5

94

1

.00

40 0

.357

-0.

628

5 0

.482

D

NK

/USD

3.

167

5

1.01

51 0

.805

-1.

6233

0.7

10

1.00

59 0

.923

-1.

1256

0.6

01

1.00

08 0

.796

-0

.84

20 0

.47

3

1.0

086

0.9

61 -

1.4

739

0.7

75

U

SD/A

UD

3.

392

6

1.01

80 0

.692

-1.

0149

0.3

47

1.00

44 0

.891

-1.

5495

0.8

01

1.00

08 0

.683

-0

.44

79 0

.27

0

1.0

016

0.1

36 -

0.1

644

0.1

88

FR

F/U

SD

3.1

041

1.

0460

0.9

17 -

2.11

78 0

.975

1.

0439

0.9

29 -

2.04

09 0

.970

1.

0280

0.9

52 -

2.0

982

0.9

95

1

.01

17 0

.524

-0.

744

3 0

.515

D

EM

/USD

3.

328

6

1.03

12 0

.935

-2.

5345

0.9

88

1.01

69 0

.970

-1.

8124

0.9

23

1.00

01 0

.089

* -

0.0

246

0.1

74

0

.98

27 0

.024

**

1.1

118

0.0

13 *

*

IT

L/U

SD

3.2

797

1.

0051

0.2

38 -

0.25

94 0

.308

1.

0050

0.4

45 -

0.44

61 0

.468

0.

9976

0.2

04 0

.35

89 0

.12

8

0.9

951

0.1

73 0

.442

1 0

.289

N

LG/U

SD

3.3

319

0.

9969

0.1

00 0

.235

5 0

.143

0.

9984

0.1

88 0

.160

3 0

.271

0.

9945

0.0

30 *

* 0

.84

62 0

.07

5 *

1

.00

22 0

.359

-0.

434

5 0

.140

P

TE/U

SD

2.7

703

0.

9864

0.1

23 0

.433

2 0

.079

*

0.99

65 0

.247

0.2

174

0.1

63

1.00

45 0

.635

-0

.51

38 0

.40

4

0.9

991

0.2

85 0

.033

4 0

.271

U

SD/G

BP

2.

905

1

1.05

15 0

.895

-3.

0570

0.9

11

1.01

54 0

.787

-2.

0536

0.8

43

1.00

00 0

.188

-0

.00

08 0

.11

2

0.9

977

0.0

78 *

0.2

483

0.1

24

JP

Y/U

SD

3.1

046

1.

0190

0.0

70 *

-1.

0114

0.1

05

1.00

72 0

.285

-0.

8846

0.3

50

1.00

39 0

.522

-0

.64

07 0

.39

6

0.9

951

0.0

20 *

* 0

.568

5 0

.136

C

HF/

USD

3.

388

7

1.01

90 0

.429

-1.

7321

0.5

44

1.01

00 0

.194

-0.

8816

0.5

48

1.00

20 0

.334

-0

.36

69 0

.26

4

0.9

975

0.0

29 *

* 0

.248

1 0

.198

C

AD

/USD

1.

999

4

1.02

74 0

.183

-1.

0814

0.1

51

1.00

43 0

.560

-0.

8502

0.4

56

1.00

60 0

.822

-0

.95

93 0

.54

7

1.0

072

0.9

58 -

1.5

766

0.7

69

SE

K/U

SD

3.2

263

1.

1676

0.9

02 -

1.29

51 0

.342

1.

0753

0.8

60 -

1.07

44 0

.506

1.

0060

0.8

47 -

1.0

243

0.5

95

0

.98

52 0

.021

**

0.9

585

0.1

01

D

NK

/USD

3.

167

5

1.03

07 0

.439

-1.

9982

0.5

44

1.00

81 0

.945

-1.

6280

0.6

45

1.00

05 0

.780

-0

.79

88 0

.43

3

1.0

051

0.9

94 -

2.2

930

0.9

89

U

SD/A

UD

3.

392

6

1.03

10 0

.266

-1.

3592

0.2

04

1.01

38 0

.846

-1.

5433

0.6

75

1.00

21 0

.835

-0

.99

51 0

.54

5

1.0

014

0.1

21 -

0.1

374

0.2

31

FR

F/U

SD

3.1

041

1.

0761

0.8

13 -

2.01

10 0

.922

1.

0516

0.7

81 -

1.44

44 0

.843

1.

0291

0.8

70 -

1.6

421

0.9

51

1

.00

61 0

.382

-0.

383

3 0

.372

D

EM

/USD

3.

328

6

1.05

01 0

.984

-2.

6413

0.9

52

1.02

65 0

.970

-2.

1314

0.9

39

0.99

95 0

.193

0.2

950

0.1

07

0

.98

15 0

.019

**

1.2

739

0.0

24 *

*

IT

L/U

SD

3.2

797

1.

0226

0.2

12 -

0.73

26 0

.308

1.

0306

0.7

53 -

1.21

94 0

.714

0.

9988

0.3

32 0

.11

39 0

.20

4

0.9

955

0.1

84 0

.349

5 0

.367

N

LG/U

SD

3.3

319

1.

0125

0.2

55 -

0.59

97 0

.205

1.

0025

0.2

55 -

0.20

52 0

.220

0.

9939

0.0

25 *

* 0

.91

94 0

.04

1 **

0

.99

78 0

.108

0.8

387

0.0

40 *

*

PTE

/USD

2.

770

3

1.01

70 0

.314

-0.

6179

0.3

85

1.04

56 0

.977

-1.

9926

0.9

53

1.00

23 0

.424

-0

.11

25 0

.24

3

0.9

990

0.2

02 0

.037

4 0

.421

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

All

fun

dam

enta

ls

Cu

rren

cy p

air

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSR

idge

EW

A

Notes: This table presents the results of forecasting end-of-month exchange rate 1-month ahead by the no-change prediction, the

rolling OLS, recursive OLS, SRidge and EWA methods based on coupled fundamentals. Column 1 shows the RMSE values for the

no-change prediction. For each method, the Theil ratio (RMSE of the given method / RMSE of the no-change prediction), the

CW p-values, the DM statistic and its bootstrapped p-value are shown. The CW and DM tests are one-sided tests of equal out-

of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) for the methods considered compared

to the no-change prediction. ***, **, and * denote statistical significance at the 1 %, 5 %, and 10 % levels. Theil ratios < 1 are

indicated in bold. Sample period for the exchange rates: March 1973 – December 2014 unless shorter (indicated in Appendix A of

the online supplementary material).

31

Table 3: Relative forecasting performance of machine learning methods vs. rolling and recursive regressions based on decoupled fundamentals.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:EO

M -

Dec

ou

ple

d -

Ver

sus

OLS

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

0

.99

73

0.3

35

1 0

.92

5

0.9

96

6 1

.33

39

0.2

19

0

.99

50

0.5

10

0 0

.89

3

0.9

94

3 0

.92

35

0.4

03

JP

Y/U

SD

0.9

85

7 1

.71

99

0.2

32

0

.99

89

1.2

21

6 0

.22

5

0.9

81

2 2

.19

35

0.0

86

*

0.9

94

3 1

.55

76

0.1

27

C

HF/

USD

0

.98

79

1.9

63

0 0

.12

4

0.9

98

6 0

.54

68

0.5

42

0

.98

32

2.4

57

7 0

.03

1 *

*

0.9

93

8 1

.59

93

0.0

89

*

C

AD

/USD

1

.00

16

-0

.19

58

0.9

86

1

.00

12

-0

.58

23

0.9

73

1

.00

23

-0

.28

63

0.9

82

1

.00

19

-0

.87

05

0.9

80

SE

K/U

SD

0.9

91

3 0

.90

44

0.7

10

0

.99

92

0.3

07

0 0

.74

7

0.9

88

1 1

.03

53

0.6

05

0

.99

60

0.8

15

3 0

.43

2

D

NK

/USD

0

.98

87

2.0

55

3 0

.13

2

0.9

98

9 0

.28

48

0.7

63

0

.98

34

2.3

86

7 0

.06

0 *

0

.99

36

1.1

72

1 0

.29

9

U

SD/A

UD

1

.00

23

-0

.20

98

0.9

74

0

.99

98

0.0

74

1 0

.83

7

1.0

01

7 -

0.1

64

9 0

.97

7

0.9

99

3 0

.18

50

0.7

49

FR

F/U

SD

0.9

88

8 1

.23

86

0.3

26

0

.99

71

0.4

88

6 0

.62

2

0.9

83

9 1

.41

43

0.2

30

0

.99

21

0.9

31

1 0

.38

0

D

EM/U

SD

0.9

85

8 1

.07

99

0.3

67

0

.99

60

1.7

91

3 0

.03

6 *

*

0.9

77

9 1

.52

32

0.1

51

0

.98

80

2.4

89

7 0

.00

3 *

**

IT

L/U

SD

0.9

93

3 1

.15

87

0.2

62

0

.99

83

0.8

53

3 0

.37

1

0.9

82

2 1

.52

45

0.1

14

0

.98

72

1.5

09

9 0

.08

4 *

N

LG/U

SD

0.9

90

0 1

.43

42

0.2

70

0

.99

72

2.3

32

7 0

.00

7 *

**

0.9

80

6 2

.52

78

0.0

16

**

0

.98

78

2.4

51

7 0

.00

7 *

**

PTE

/USD

0

.98

98

1.3

95

2 0

.10

6

0.9

98

0 0

.57

14

0.3

36

0

.98

68

1.0

65

3 0

.16

1

0.9

95

0 0

.46

14

0.2

75

U

SD/G

BP

0

.98

10

1.0

15

7 0

.61

7

0.9

98

4 0

.25

10

0.7

48

0

.98

06

0.8

31

8 0

.66

4

0.9

98

0 0

.15

27

0.7

23

JP

Y/U

SD

0.9

93

4 0

.52

74

0.8

42

1

.00

10

-0

.22

15

0.8

91

0

.98

59

1.1

18

1 0

.55

4

0.9

93

5 0

.71

95

0.5

24

C

HF/

USD

0

.98

89

0.8

84

2 0

.61

6

1.0

00

8 -

0.0

87

9 0

.80

3

0.9

82

1 1

.25

64

0.3

61

0

.99

40

0.5

40

3 0

.46

9

C

AD

/USD

0

.99

48

0.7

47

5 0

.77

4

0.9

99

3 0

.36

03

0.7

00

0

.99

55

0.6

37

1 0

.78

1

1.0

00

1 -

0.0

40

6 0

.82

0

SE

K/U

SD

0.9

46

2 1

.84

66

0.1

46

0

.98

40

1.8

45

3 0

.04

6 *

*

0.9

40

7 1

.61

85

0.2

29

0

.97

82

1.3

10

4 0

.20

1

D

NK

/USD

0

.98

22

1.8

02

1 0

.21

1

0.9

96

3 1

.01

08

0.3

78

0

.97

72

1.9

07

6 0

.16

4

0.9

91

2 1

.08

23

0.3

06

U

SD/A

UD

0

.99

17

0.6

93

7 0

.77

2

0.9

99

0 0

.27

88

0.7

55

0

.98

50

1.0

16

3 0

.59

7

0.9

92

2 0

.79

82

0.4

44

FR

F/U

SD

0.9

81

2 2

.02

03

0.0

61

*

0.9

90

3 1

.10

41

0.2

13

0

.97

50

1.6

02

1 0

.11

7

0.9

84

1 1

.12

25

0.1

78

D

EM/U

SD

0.9

78

2 1

.41

14

0.2

01

0

.99

35

0.7

49

0 0

.40

2

0.9

68

5 1

.54

33

0.1

18

0

.98

37

1.1

47

0 0

.16

4

IT

L/U

SD

0.9

94

1 0

.56

34

0.5

48

0

.99

41

1.0

17

0 0

.16

6

0.9

75

3 1

.41

51

0.1

16

0

.97

54

1.4

23

9 0

.06

1 *

N

LG/U

SD

0.9

62

3 1

.03

75

0.3

84

0

.99

61

0.5

54

6 0

.53

2

0.9

53

3 1

.17

07

0.2

74

0

.98

68

0.9

72

3 0

.28

5

PTE

/USD

0

.99

95

0.0

29

9 0

.77

0

0.9

99

6 0

.03

21

0.7

27

0

.99

62

0.2

61

3 0

.69

9

0.9

96

4 0

.33

99

0.6

21

U

SD/G

BP

0

.96

65

2.1

22

2 0

.35

1

0.9

84

1 1

.35

94

0.3

29

0

.96

69

1.7

98

5 0

.46

3

0.9

84

6 1

.06

61

0.4

00

JP

Y/U

SD

0.9

92

0 0

.56

24

0.9

54

0

.99

79

0.1

96

6 0

.85

7

0.9

85

0 1

.26

23

0.6

73

0

.99

09

0.8

24

4 0

.43

2

C

HF/

USD

0

.97

80

1.8

38

2 0

.44

3

0.9

92

1 0

.74

15

0.6

44

0

.98

57

1.0

73

5 0

.79

2

0.9

99

9 0

.00

92

0.8

86

C

AD

/USD

0

.97

38

1.1

96

7 0

.86

3

1.0

02

0 -

0.1

39

2 0

.95

1

0.9

68

6 1

.46

00

0.6

56

0

.99

66

0.3

92

3 0

.74

7

SE

K/U

SD

0.9

51

7 3

.44

73

0.0

08

***

0

.98

28

1.5

70

6 0

.23

5

0.9

54

5 2

.95

52

0.0

27

**

0

.98

57

1.0

99

0 0

.36

7

D

NK

/USD

0

.96

22

2.7

97

7 0

.08

1 *

0

.98

58

2.1

99

7 0

.05

3 *

0

.96

43

2.4

48

8 0

.12

4

0.9

87

9 1

.50

07

0.2

07

U

SD/A

UD

0

.97

40

1.7

09

7 0

.55

4

0.9

91

0 1

.64

92

0.2

09

0

.97

82

1.4

05

0 0

.65

3

0.9

95

3 0

.54

84

0.6

73

FR

F/U

SD

0.9

46

6 2

.35

30

0.0

36

**

0

.95

32

2.2

82

3 0

.02

5 *

*

0.9

55

1 1

.95

39

0.0

86

*

0.9

61

7 2

.08

34

0.0

35

**

D

EM/U

SD

0.9

83

7 1

.23

37

0.5

04

0

.99

06

0.9

88

2 0

.43

7

0.9

82

1 1

.11

40

0.4

78

0

.98

90

0.7

25

4 0

.46

7

IT

L/U

SD

0.9

77

5 1

.35

32

0.3

97

0

.98

55

0.7

50

7 0

.56

3

0.9

80

6 1

.03

58

0.5

23

0

.98

87

0.7

56

4 0

.50

3

N

LG/U

SD

0.9

68

3 1

.74

49

0.1

91

0

.98

94

0.8

73

5 0

.40

4

0.9

69

3 1

.61

92

0.1

76

0

.99

04

0.6

93

7 0

.38

8

PTE

/USD

1

.00

18

-0

.04

43

0.8

66

0

.95

88

2.1

12

4 0

.02

9 *

*

1.0

09

8 -

0.2

49

1 0

.90

4

0.9

66

5 1

.11

43

0.2

78

U

SD/G

BP

0

.93

18

3.2

88

4 0

.46

7

0.9

71

1 1

.75

67

0.6

16

0

.93

64

3.0

92

7 0

.54

3

0.9

75

9 1

.28

17

0.8

05

JP

Y/U

SD

0.9

55

3 1

.87

31

0.9

74

0

.99

09

0.6

35

9 0

.96

2

0.9

51

6 2

.13

68

0.9

23

0

.98

71

1.0

04

3 0

.84

7

C

HF/

USD

0

.93

03

2.4

10

5 0

.69

0

0.9

79

3 1

.26

36

0.5

41

0

.93

12

2.4

74

6 0

.63

3

0.9

80

3 1

.25

55

0.5

18

C

AD

/USD

0

.96

46

1.6

79

3 0

.99

2

0.9

93

0 0

.95

97

0.9

11

0

.96

30

1.6

45

7 0

.98

4

0.9

91

3 1

.01

33

0.8

70

SE

K/U

SD

0.8

66

0 1

.85

52

0.7

48

0

.94

31

1.1

20

5 0

.74

5

0.8

55

7 1

.73

70

0.7

91

0

.93

19

1.1

62

3 0

.71

8

D

NK

/USD

0

.94

30

2.8

91

3 0

.60

1

0.9

84

4 1

.37

89

0.7

62

0

.94

46

2.8

21

5 0

.61

4

0.9

86

1 1

.21

68

0.8

06

U

SD/A

UD

0

.96

17

1.6

08

3 0

.98

5

0.9

82

8 1

.37

30

0.7

09

0

.96

25

1.7

09

4 0

.97

0

0.9

83

6 1

.20

23

0.7

38

FR

F/U

SD

0.9

21

3 1

.93

51

0.2

46

0

.96

79

1.0

77

8 0

.47

0

0.9

20

6 2

.11

61

0.1

68

0

.96

71

1.2

72

1 0

.30

5

D

EM/U

SD

0.9

64

1 1

.21

47

0.9

13

0

.97

63

1.0

37

1 0

.76

0

0.9

55

9 1

.30

14

0.8

68

0

.96

80

1.1

40

9 0

.68

3

IT

L/U

SD

0.9

99

9 0

.00

36

0.9

97

0

.96

46

1.1

17

0 0

.65

6

1.0

01

8 -

0.0

59

6 0

.99

6

0.9

66

5 1

.21

37

0.6

04

N

LG/U

SD

0.9

55

2 1

.49

28

0.7

89

0

.96

99

1.4

07

8 0

.51

8

0.9

49

3 1

.69

03

0.6

74

0

.96

39

1.7

50

3 0

.31

1

PTE

/USD

0

.99

10

0.2

47

0 0

.89

2

0.9

49

6 1

.67

60

0.2

17

0

.98

19

0.4

87

0 0

.83

0

0.9

40

9 1

.69

83

0.2

10

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

All

fun

dam

enta

ls

Cu

rren

cy p

air

SRid

ge

vs.

EWA

vs

.

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nR

olli

ng

regr

essi

on

Rec

urs

ive

regr

essi

on

Notes: This table presents the comparison of forecasting performance between, respectively, SRidge and EWA vs. the rolling and

recursive OLS based on decoupled fundamentals. Columns 1-6 show the comparison between SRidge and OLS methods, while

columns 7-12 for EWA vs. OLS. For each comparison, the Theil ratio (RMSE of the given method / RMSE of the compared)

and the DM statistic and its bootstrapped p-value are shown. The DM test is a one-sided test of equal out-of-sample prediction

accuracy (H0) against superior out-of-sample prediction accuracy (H1) of the machine-learning method considered against the

OLS methods. ***, **, and * denote statistical significance at the 1 %, 5 %, and 10 % levels. Theil ratios < 1 are indicated in

bold. Sample period for the exchange rates: March 1973 – December 2014 unless shorter (indicated in Appendix A of the online

supplementary material).

32

Table 4: Directional predictions of the exchange rates based on decoupled fundamentals.

What do we read? EOM - FRED - RT? Coupled or decoupled? RMSE / versus OLS / directional changes?

Answer: EOM - Decoupled - Directional

Proportion of

changes

predicted

DM statistic DM p-value

Proportion of

changes

predicted

DM statistic DM p-value

Proportion of

changes

predicted

DM statistic DM p-value

Proportion of

changes

predicted

DM statistic DM p-value

USD/GBP 0.512 0.4294 0.300 0.493 -0.2498 0.593 0.470 -0.9744 0.829 0.522 0.6041 0.249

JPY/USD 0.470 -1.1431 0.860 0.462 -1.3166 0.906 0.546 1.7192 0.017 ** 0.543 1.4995 0.027 **

CHF/USD 0.525 0.8142 0.165 0.488 -0.3462 0.637 0.556 1.9711 0.011 ** 0.541 1.3711 0.064 *

CAD/USD 0.528 0.9373 0.138 0.496 -0.1207 0.554 0.496 -0.1153 0.516 0.441 -2.2202 0.989

SEK/USD 0.501 0.0410 0.459 0.480 -0.5793 0.718 0.493 -0.1838 0.547 0.512 0.4047 0.314

DNK/USD 0.499 -0.0424 0.495 0.525 0.6536 0.223 0.564 2.1772 0.010 *** 0.559 2.2801 0.004 ***

USD/AUD 0.494 -0.1572 0.491 0.497 -0.0741 0.475 0.483 -0.5116 0.657 0.480 -0.6659 0.725

FRF/USD 0.481 -0.3854 0.655 0.540 0.6175 0.248 0.481 -0.4106 0.679 0.571 1.8923 0.028 **

DEM/USD 0.455 -1.0227 0.885 0.481 -0.4448 0.728 0.603 2.3448 0.020 ** 0.608 2.5567 0.008 ***

ITL/USD 0.487 -0.2303 0.597 0.466 -0.6356 0.745 0.466 -0.6356 0.755 0.577 1.3190 0.089 *

NLG/USD 0.519 0.3670 0.397 0.455 -0.9079 0.833 0.587 1.9547 0.040 ** 0.593 1.9582 0.055 *

PTE/USD 0.481 -0.2859 0.777 0.497 -0.0432 0.680 0.497 -0.0432 0.702 0.550 0.9484 0.290

USD/GBP 0.496 -0.1160 0.531 0.522 0.7841 0.177 0.480 -0.6037 0.733 0.517 0.5310 0.268

JPY/USD 0.546 1.4607 0.037 ** 0.522 0.7917 0.168 0.562 2.4264 0.003 *** 0.570 2.7419 0.001 ***

CHF/USD 0.566 2.4290 0.011 ** 0.547 1.7035 0.044 ** 0.550 1.7495 0.033 ** 0.563 2.0034 0.006 ***

CAD/USD 0.551 1.9736 0.015 ** 0.522 0.7710 0.202 0.493 -0.1882 0.583 0.483 -0.5147 0.651

SEK/USD 0.497 -0.0855 0.504 0.471 -0.9192 0.839 0.508 0.2847 0.346 0.539 1.4367 0.062 *

DNK/USD 0.501 0.0427 0.486 0.483 -0.4942 0.685 0.512 0.4001 0.328 0.564 2.5218 0.002 ***

USD/AUD 0.500 0.0000 0.424 0.514 0.4352 0.287 0.470 -0.9811 0.792 0.492 -0.2896 0.397

FRF/USD 0.497 -0.0413 0.494 0.460 -0.7051 0.772 0.561 1.6855 0.029 ** 0.593 2.5907 0.002 ***

DEM/USD 0.508 0.1344 0.599 0.540 0.8111 0.368 0.593 1.9107 0.095 * 0.608 2.5974 0.007 ***

ITL/USD 0.497 -0.0470 0.506 0.466 -0.6481 0.762 0.450 -0.9429 0.855 0.603 2.1527 0.012 **

NLG/USD 0.556 1.2824 0.126 0.561 1.4899 0.100 * 0.593 2.1242 0.032 ** 0.593 2.1898 0.013 **

PTE/USD 0.479 -0.3564 0.612 0.437 -0.8302 0.772 0.437 -1.0768 0.841 0.507 0.1187 0.383

USD/GBP 0.501 0.0512 0.461 0.535 1.0881 0.128 0.509 0.2975 0.366 0.517 0.4324 0.247

JPY/USD 0.532 1.1515 0.093 * 0.524 0.7503 0.208 0.542 1.3905 0.067 * 0.558 2.1103 0.008 ***

CHF/USD 0.497 -0.0913 0.536 0.541 1.4121 0.060 * 0.541 1.3316 0.073 * 0.533 1.2019 0.084 *

CAD/USD 0.513 0.5132 0.274 0.489 -0.3737 0.626 0.487 -0.4104 0.642 0.487 -0.4902 0.674

SEK/USD 0.432 -2.4520 0.992 0.474 -0.7033 0.772 0.537 1.4292 0.062 * 0.524 0.7452 0.326

DNK/USD 0.437 -2.0040 0.984 0.480 -0.5277 0.687 0.480 -0.5528 0.733 0.489 -0.3536 0.696

USD/AUD 0.526 0.8000 0.154 0.489 -0.4265 0.644 0.477 -0.6859 0.732 0.497 -0.0933 0.478

FRF/USD 0.525 0.4293 0.300 0.467 -0.5844 0.748 0.558 1.2868 0.052 * 0.550 1.1010 0.085 *

DEM/USD 0.559 1.5897 0.072 * 0.576 1.8096 0.058 * 0.559 1.0695 0.217 0.548 1.2069 0.057 *

ITL/USD 0.449 -0.8892 0.841 0.442 -1.1587 0.897 0.513 0.2405 0.406 0.538 0.8534 0.118

NLG/USD 0.434 -1.2346 0.945 0.513 0.2685 0.461 0.582 1.7186 0.078 * 0.534 0.9146 0.140

PTE/USD 0.457 -0.5744 0.748 0.386 -1.9096 0.978 0.371 -2.1989 0.982 0.471 -0.4383 0.468

USD/GBP 0.556 2.0805 0.011 ** 0.491 -0.3228 0.637 0.496 -0.1258 0.514 0.488 -0.3126 0.596

JPY/USD 0.537 1.4172 0.079 * 0.537 1.0926 0.147 0.566 2.1856 0.010 *** 0.558 2.2724 0.006 ***

CHF/USD 0.525 0.8156 0.341 0.514 0.4455 0.535 0.541 1.3797 0.054 * 0.547 1.5987 0.027 **

CAD/USD 0.558 1.8536 0.036 ** 0.503 0.0918 0.534 0.495 -0.1529 0.570 0.455 -1.4481 0.924

SEK/USD 0.505 0.1815 0.465 0.495 -0.1603 0.582 0.487 -0.3850 0.620 0.542 1.6474 0.072 *

DNK/USD 0.529 0.8728 0.169 0.511 0.3648 0.352 0.443 -1.6257 0.957 0.483 -0.5244 0.718

USD/AUD 0.517 0.5688 0.283 0.520 0.6296 0.293 0.472 -0.8429 0.758 0.500 0.0000 0.310

FRF/USD 0.567 1.4214 0.092 * 0.550 0.9472 0.217 0.500 0.0000 0.484 0.533 0.7098 0.235

DEM/USD 0.537 0.7360 0.310 0.508 0.1919 0.562 0.542 1.0706 0.267 0.576 1.5195 0.062 *

ITL/USD 0.513 0.2581 0.424 0.513 0.2700 0.408 0.429 -1.2882 0.915 0.558 1.4508 0.027 **

NLG/USD 0.513 0.2924 0.460 0.540 0.9389 0.213 0.582 1.7087 0.072 * 0.561 1.2465 0.098 *

PTE/USD 0.486 -0.1967 0.607 0.414 -1.3264 0.925 0.471 -0.4789 0.658 0.471 -0.4730 0.666

UIRP fundamental

Monetary model fundamentals

All fundamentals

Rolling regression Recursive regression Sridge EWA

Currency pair

PPP fundamental

Notes: This table presents the results of forecasting the direction of change of the end-of-month exchange rate 1-month ahead

by the rolling OLS, recursive OLS, SRidge and EWA methods based on decoupled fundamentals. For each method, the share of

correctly forecasted changes, the DM statistic and its bootstrapped p-value are shown. The DM tests is a one-sided test of equal

out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) for the methods considered against

the benchmark of a 50 % success rate. ***, **, and * denote statistical significance at the 1 %, 5 %, and 10 % levels. Shares > 0.5

are indicated in bold. Sample period for the exchange rates: March 1973 – December 2014 unless shorter (indicated in Appendix A

of the online supplementary material).

33

Table 5: Economic criterion for evaluating forecasts for the EWA algorithm based on decoupled fundamentals.

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

EWA

/ c

= 0

.511

.07

12.0

31

53

19

00.

650.

800.

671

.13

EWA

/ c

= 1

12.0

613

.24

21

32

69

0.65

0.83

0.67

1.0

9

EWA

/ c

= 2

11.7

613

.67

27

03

19

0.63

0.85

0.64

1.0

5

EWA

/ c

= 5

11.8

613

.44

22

42

71

0.64

0.83

0.68

1.0

6

EWA

/ c

= 1

011

.83

13.4

82

29

27

70.

640.

840.

681

.08

EWA

/ n

o c

11.8

313

.48

22

92

77

0.64

0.84

0.68

1.0

8

EWA

/ c

= 0

.510

.62

12.1

22

02

25

10.

610.

810.

641

.31

EWA

/ c

= 1

11.6

713

.18

24

93

32

0.63

0.84

0.65

1.3

6

EWA

/ c

= 2

11.3

713

.04

27

03

62

0.60

0.83

0.62

1.4

4

EWA

/ c

= 5

11.4

712

.16

18

42

63

0.62

0.77

0.65

1.3

5

EWA

/ c

= 1

011

.44

12.1

31

92

26

80.

620.

780.

651

.38

EWA

/ n

o c

11.4

412

.13

19

22

68

0.62

0.78

0.65

1.3

8

EWA

/ c

= 0

.511

.02

8.89

-124

-89

0.64

0.51

0.67

0.6

7

EWA

/ c

= 1

11.9

59.

59-8

2-2

30.

640.

540.

660

.70

EWA

/ c

= 2

11.6

88.

95-1

31-6

60.

620.

470.

640

.63

EWA

/ c

= 5

11.7

38.

54-1

82-1

140.

630.

440.

670

.62

EWA

/ c

= 1

011

.70

8.49

-189

-120

0.63

0.43

0.67

0.6

2

EWA

/ n

o c

11.7

08.

49-1

89-1

200.

630.

430.

670

.62

EWA

/ c

= 0

.511

.02

8.99

-80

-37

0.64

0.55

0.67

0.8

9

EWA

/ c

= 1

11.9

59.

76-4

33

00.

640.

580.

660

.93

EWA

/ c

= 2

11.6

89.

37-8

32

0.62

0.52

0.64

0.9

1

EWA

/ c

= 5

11.7

38.

70-1

69-8

50.

630.

450.

670

.83

EWA

/ c

= 1

011

.70

8.62

-179

-94

0.63

0.44

0.67

0.8

2

EWA

/ n

o c

11.7

08.

62-1

79-9

40.

630.

440.

670

.82

Po

rtfo

lio /

c co

nst

rain

t

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

s fu

nd

amen

tals

All

fun

dam

enta

ls

An

nu

aliz

ed

Ret

urn

s (i

n %

)P

erfo

rman

ce f

ee

of

po

rtfo

lio w

.r.t

.

no

ch

ange

(in

bp

s)

Pre

miu

m r

etu

rn

of

po

rtfo

lio w

.r.t

.

no

ch

ange

(in

bp

s)

Shar

pe

rati

os

Sort

ino

rat

ios

Notes: This table presents the comparison of performance of portfolios formed using forecasts from the EWA algorithm and the

no-change prediction based on decoupled fundamentals, using the procedure described in Section 5.4. Column 1 gives the conditions

under which the portfolios were constructed. In columns 2 and 3 the annualized returns are given for the EWA and no-change

prediction-based portfolios respectively. Column 4 shows the performance fee in basis points (bps) while column 5 the premium

return (in bps) of the EWA-based portfolios relative to the ones formed based on the no-change forecast. The Sharpe and Sortino

ratios for portfolios created using the two competing forecasting methods are reported in columns 6-10. For details on how these

portfolios were created please consult Appendix C.3 of the online supplementary material. The results for the portfolios based on

the no-change exchange rate predictions vary because of the constraints on weights and because for different sets of fundamentals

different time periods are considered depending on the availability of data.

34

Table 6: 1-month ahead forecasts for the PPP, UIRP, monetary model and all fundamentals on the real-time data set: decoupled formulation.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:R

T -

De

cou

ple

d -

RM

SE

Dat

e:27

-fév

r.-1

8

No

ch

ange

RM

SE x

100

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

2.

611

5

1.00

39 0

.090

* -

0.24

04 0

.096

*

0.99

52 0

.044

**

0.3

509

0.0

72 *

0.

9909

0.1

35 0

.40

42 0

.081

*

0.9

998

0.4

18 0

.043

2 0

.34

3

JP

Y/U

SD

2.78

59

1.

0208

0.5

22 -

1.24

81 0

.537

1.

0120

0.8

25 -

1.38

33 0

.802

0.

9952

0.1

07 0

.67

65 0

.046

**

0

.99

65 0

.072

* 0

.976

3 0

.08

6 *

C

HF/

USD

3.

022

3

1.03

96 0

.851

-1.

6468

0.7

35

1.01

91 0

.926

-1.

6482

0.8

78

1.01

12

0.7

24 -

0.80

77 0

.537

1

.00

17

0.5

56 -

0.3

238

0.7

16

C

AD

/USD

2.

848

3

1.01

12 0

.259

-0.

5617

0.1

82

1.00

42 0

.300

-0.

2625

0.1

83

1.00

67

0.7

19 -

1.10

26 0

.674

1

.00

43

0.8

86 -

1.0

826

0.7

80

SE

K/U

SD

3.40

12

1.

0238

0.3

22 -

1.30

77 0

.591

1.

0117

0.4

44 -

0.59

84 0

.382

0.

9988

0.2

13 0

.04

94 0

.136

0

.99

66 0

.152

0.6

014

0.0

97

*

D

NK

/USD

2.

993

7

1.03

00 0

.706

-0.

9464

0.3

07

1.00

76 0

.784

-0.

8155

0.3

93

1.00

60

0.3

99 -

0.35

11 0

.253

0

.99

95 0

.351

0.0

809

0.3

67

U

SD/A

UD

3.

690

1

1.00

88 0

.210

-0.

5004

0.2

12

0.99

43 0

.091

* 0

.320

4 0

.106

1.

009

5 0

.388

-0.

9589

0.6

35

1.0

01

7 0

.581

-0

.31

35 0

.41

3

U

SD/G

BP

2.

611

5

1.03

23 0

.351

-1.

1367

0.3

77

1.01

52 0

.899

-1.

8572

0.9

14

0.99

84 0

.134

0.9

613

0.0

21 *

*

0.9

978

0.0

29 *

* 1

.118

7 0

.05

9 *

JP

Y/U

SD

2.78

59

1.

0863

0.7

49 -

1.58

38 0

.601

1.

0233

0.6

10 -

1.28

56 0

.687

1.

004

3 0

.947

-1.

4109

0.8

75

0.9

997

0.3

42 0

.262

4 0

.30

0

C

HF/

USD

3.

022

3

1.02

83 0

.482

-1.

4124

0.5

69

1.00

93 0

.575

-0.

7673

0.4

30

1.00

07

0.5

53 -

0.28

95 0

.273

0

.99

96 0

.334

0.1

066

0.4

18

C

AD

/USD

2.

848

3

1.02

74 0

.798

-1.

8536

0.7

32

1.00

83 0

.892

-1.

3748

0.6

77

1.00

53

0.8

28 -

0.97

34 0

.552

1

.00

08

0.6

69 -

0.4

735

0.5

41

SE

K/U

SD

3.40

12

1.

0329

0.6

21 -

1.59

81 0

.596

1.

0241

0.9

52 -

1.48

75 0

.763

0.

9999

0.3

06 0

.01

06 0

.120

0

.99

77 0

.223

0.3

608

0.2

30

D

NK

/USD

2.

993

7

1.03

68 0

.783

-1.

3103

0.4

15

1.02

93 0

.909

-1.

1967

0.6

06

1.00

39

0.6

09 -

0.42

76 0

.264

0

.99

61 0

.158

0.5

846

0.1

73

U

SD/A

UD

3.

690

1

1.05

17 0

.975

-2.

4870

0.9

51

1.02

95 0

.944

-1.

4942

0.8

07

1.00

63

0.6

22 -

1.15

01 0

.770

0

.99

95 0

.394

0.1

212

0.3

29

U

SD/G

BP

2.

611

5

1.11

35 0

.251

-1.

3190

0.1

91

1.03

23 0

.955

-2.

0623

0.8

83

0.99

09 0

.063

* 0

.43

18 0

.058

*

0.9

971

0.2

05 0

.237

3 0

.13

2

JP

Y/U

SD

2.78

59

1.

0916

0.4

60 -

1.78

69 0

.492

1.

0131

0.1

73 -

0.57

08 0

.189

1.

010

2 0

.793

-1.

1883

0.7

22

0.9

999

0.1

61 0

.013

7 0

.21

9

C

HF/

USD

3.

022

3

1.06

72 0

.860

-1.

6963

0.4

22

1.01

53 0

.654

-1.

2301

0.5

21

1.00

41

0.9

22 -

1.46

68 0

.905

1

.01

14

0.9

09 -

1.5

369

0.9

03

C

AD

/USD

2.

848

3

1.09

73 0

.933

-1.

8609

0.4

91

1.02

84 0

.976

-1.

5809

0.7

03

1.01

89

0.7

05 -

0.78

54 0

.435

1

.00

68

0.7

79 -

0.8

941

0.6

86

SE

K/U

SD

3.40

12

1.

1253

0.7

06 -

1.67

05 0

.398

1.

0378

0.8

96 -

1.65

62 0

.747

1.

022

0 0

.643

-0.

7055

0.4

03

0.9

960

0.1

48 0

.234

3 0

.10

7

D

NK

/USD

2.

993

7

1.07

67 0

.765

-1.

6678

0.4

26

1.02

16 0

.408

-0.

9262

0.2

67

1.01

11

0.7

82 -

0.69

94 0

.224

1

.01

50

0.5

70 -

0.7

942

0.7

43

U

SD/A

UD

3.

690

1

1.09

66 0

.598

-2.

0506

0.6

73

1.02

50 0

.308

-0.

8842

0.4

19

1.00

56

0.2

10 -

0.38

68 0

.271

0

.99

29 0

.178

0.4

240

0.0

69

*

U

SD/G

BP

2.

611

5

1.13

58 0

.226

-2.

0000

0.1

08

1.03

74 0

.095

* -

0.76

55 0

.064

*

0.98

87 0

.053

* 0

.55

47 0

.034

**

0

.99

40 0

.126

0.4

545

0.0

99

*

JP

Y/U

SD

2.78

59

1.

2555

0.5

42 -

2.76

25 0

.377

1.

0792

0.7

66 -

1.54

60 0

.399

1.

007

9 0

.701

-0.

9289

0.5

02

1.0

01

1 0

.239

-0

.13

94 0

.27

9

C

HF/

USD

3.

022

3

1.34

19 0

.751

-2.

1973

0.2

62

1.12

81 0

.884

-2.

6520

0.9

08

1.00

48

0.9

06 -

1.27

18 0

.814

1

.00

94

0.8

66 -

1.3

346

0.9

22

C

AD

/USD

2.

848

3

1.15

76 0

.413

-2.

9164

0.5

39

1.05

81 0

.336

-2.

1299

0.7

31

1.01

95

0.7

78 -

0.93

64 0

.515

1

.00

72

0.7

91 -

0.9

214

0.6

76

SE

K/U

SD

3.40

12

1.

1733

0.4

89 -

1.74

11 0

.051

*

1.10

11 0

.722

-2.

4776

0.8

52

1.01

82

0.5

48 -

0.56

83 0

.302

0

.99

65 0

.162

0.2

059

0.1

27

D

NK

/USD

2.

993

7

1.19

64 0

.341

-2.

9683

0.4

90

1.11

97 0

.951

-2.

4245

0.7

61

1.01

32

0.8

35 -

0.77

00 0

.273

1

.01

00

0.4

08 -

0.6

101

0.6

35

U

SD/A

UD

3.

690

1

1.21

90 0

.167

-2.

5173

0.3

28

1.06

57 0

.312

-1.

8644

0.5

70

1.00

48

0.2

27 -

0.35

60 0

.311

0

.99

38 0

.189

0.4

476

0.1

25

All

fun

dam

enta

ls

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

Cu

rren

cy p

air

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSR

idge

EW

A

Notes: This table presents the results of forecasting end-of-month exchange rate 1-month ahead by the no-change prediction, the

rolling OLS, recursive OLS, SRidge and EWA methods for decoupled fundamentals on real-time data. Column 1 shows the RMSE

values for the no-change prediction. For each method, the Theil ratio (RMSE of the given method / RMSE of the no-change

prediction), the CW-p-values, the DM statistic and its bootstrapped p-value are shown. The CW and DM tests are one-sided tests

of equal out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) for the methods considered

compared to the no-change prediction. ***, **, and * denote statistical significance at the 1 %, 5 %, and 10 % levels. Theil ratios

< 1 are indicated in bold. Sample period for the exchange rates: February 1999 – April 2017, data permitting (more details in

Appendix A of the online supplementary material).

35

Table 7: Economic criterion for evaluating forecasts on the real-time data set for the EWA algorithm based on decoupled fundamentals.

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

EWA

/ c

= 0

.52.

051

.30

49

12

20.

060.

000.

07-0

.01

EWA

/ c

= 1

3.91

3.4

74

01

02

0.22

0.20

0.24

0.28

EWA

/ c

= 2

4.56

6.0

71

55

20

80.

270.

400.

310.

57

EWA

/ c

= 5

4.78

6.2

91

32

19

20.

280.

400.

340.

61

EWA

/ c

= 1

05.

236

.39

10

01

61

0.31

0.40

0.40

0.64

EWA

/ n

o c

5.23

0.7

1-1

37-1

040.

31-0

.09

0.40

-0.1

2

EWA

/ c

= 0

.52.

050

.84

14

86

0.06

-0.0

50.

07-0

.08

EWA

/ c

= 1

3.91

1.3

5-1

62-1

140.

220.

000.

240.

00

EWA

/ c

= 2

4.56

2.1

0-2

43-2

360.

270.

060.

310.

08

EWA

/ c

= 5

4.78

2.0

6-2

65-2

620.

280.

060.

340.

08

EWA

/ c

= 1

05.

232

.14

-277

-288

0.31

0.07

0.40

0.09

EWA

/ n

o c

5.23

0.7

1-1

37-1

040.

31-0

.09

0.40

-0.1

2

EWA

/ c

= 0

.52.

053

.87

23

13

20

0.06

0.24

0.07

0.38

EWA

/ c

= 1

3.91

3.7

42

71

09

0.22

0.22

0.24

0.36

EWA

/ c

= 2

4.56

1.7

7-2

50-1

890.

270.

040.

310.

06

EWA

/ c

= 5

4.78

1.1

4-3

54-3

170.

28-0

.02

0.34

-0.0

2

EWA

/ c

= 1

05.

230

.87

-462

-493

0.31

-0.0

40.

40-0

.05

EWA

/ n

o c

5.23

0.7

1-1

37-1

040.

31-0

.09

0.40

-0.1

2

EWA

/ c

= 0

.52.

053

.58

22

23

11

0.06

0.22

0.07

0.34

EWA

/ c

= 1

3.91

4.2

78

51

68

0.22

0.27

0.24

0.46

EWA

/ c

= 2

4.56

3.6

9-1

02-2

70.

270.

190.

310.

35

EWA

/ c

= 5

4.78

2.7

0-2

48-1

970.

280.

110.

340.

17

EWA

/ c

= 1

05.

232

.09

-382

-391

0.31

0.06

0.40

0.08

EWA

/ n

o c

5.23

0.7

1-1

37-1

040.

31-0

.09

0.40

-0.1

2

Po

rtfo

lio /

c co

nst

rain

t

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

s fu

nd

amen

tals

All

fun

dam

enta

ls

An

nu

aliz

ed

Ret

urn

s (i

n %

)P

erfo

rman

ce f

ee

of

po

rtfo

lio w

.r.t

.

no

ch

ange

(in

bp

s)

Pre

miu

m r

etu

rn

of

po

rtfo

lio w

.r.t

.

no

ch

ange

(in

bp

s)

Shar

pe

rati

os

Sort

ino

rat

ios

Notes: This table presents the comparison of performance of portfolios formed using forecasts from the EWA algorithm and the

no-change prediction for the real-time data set based on decoupled fundamentals, using the procedure described in Section 5.4.

Column 1 gives the conditions under which the portfolios were constructed. In columns 2 and 3 the annualized returns are given for

the EWA and no-change prediction-based portfolios respectively. Column 4 shows the performance fee in basis points (bps) while

column 5 the premium return (in bps) of the EWA-based portfolios relative to the ones formed based on the no-change forecast.

The Sharpe and Sortino ratios for portfolios created using the two competing forecasting methods are reported in columns 6-10.

For details on how these portfolios were created please consult Appendix C.3 of the online supplementary material. The results for

the portfolios based on the no-change exchange rate predictions vary because of the constraints on weights and because for different

sets of fundamentals different time periods are considered depending on the availability of data.

36

Fundamentals and exchange rate forecastability with simplemachine learning methodsI

Christophe Amata, Tomasz Michalskib,∗, Gilles Stoltzc

aEcole Polytechnique, Palaiseau, FrancebHEC Paris – GREGHEC, Jouy-en-Josas, France

cHEC Paris – CNRS, Jouy-en-Josas, France

Abstract

Using methods from machine learning we show that fundamentals from simple exchangerate models (PPP or UIRP) or Taylor-rule based models lead to improved exchange rateforecasts for major currencies over the floating period era 1973–2014 at a 1-month forecasthorizon which beat the no-change forecast. Fundamentals thus contain useful informationand exchange rates are forecastable even for short horizons. Such conclusions cannot beobtained when using rolling or recursive OLS regressions as used in the literature. Themethods we use – sequential ridge regression and the exponentially weighted averagestrategy, both with discount factors – do not estimate an underlying model but combinethe fundamentals to directly output forecasts.

JEL Classification: C53, F31, F37

Keywords: exchange rates, forecasting, machine learning, purchasing power parity,uncovered interest rate parity, Taylor-rule exchange rate models

1. Introduction

We show that fundamentals from “classic” exchange rate models and Taylor-rule ex-change rate models are useful in forecasting short-term changes in exchange rates formajor currencies. Using prediction techniques borrowed from machine learning thatcome with performance guarantees – sequential ridge regression with discount factorsand the exponentially weighted average strategy with discount factors – we are broadlyable to improve forecasting forward the 1-month exchange rate for the floating rate pe-riod 1973–2014 in terms of the usual root mean square error (RMSE) criterion, and

IWe would like to thank an anonymous referee, Philippe Bacchetta, Charles Engel, Refet Gurkaynak,Robert Kollmann, Jose Lopez, Evren Ors, Cavit Pakel, Martin Puhl, Romain Ranciere, Helene Rey,Barbara Rossi, Micha l Rubaszek, Kenneth D. West and the seminar participants at Bilkent University,HEC Paris, the National Bank of Poland, OECD, UN, EEA 2015 conference, Royal Economic Society2015, 8th Financial Risks International Forum 2015 (Institut Louis Bachelier) for helpful comments anddiscussions. All remaining errors are ours.

∗Corresponding authorEmail addresses: christophe.amat@polytechnique.edu (Christophe Amat), michalski@hec.fr

(Tomasz Michalski), stoltz@hec.fr (Gilles Stoltz)

Preprint submitted to Elsevier May 18, 2018

Appendices published online

Posted at https://halshs.archives-ouvertes.fr/halshs-01003914

A. Detailed data description

B. Assessing forecast quality

C. Additional technical details (notes on computations and forecasts)

D. Additional graphs and tables

37

Appendix A. Detailed data description

We recall that, as indicated in Section 4, the data used is freely available for download(Amat et al., 2018a).

The original fundamentals dataset of Molodtsova and Papell (2009) was extendedto the period 1973–2014 whenever possible. For the discontinued time-series we usedinstead Datastream as the principal source. For those, as there were typically moretime-series available without a clear advantage of one over the others, only those thatwere finally used to generate predictions presented in the paper are listed (we did notget qualitatively different results using the others). Data was obtained from Datastreamon 30/01/2014. When the code from the original data series was known, it was giveninstead. All series are monthly unless noted.

Quarterly time-series (there were only three of them) were transformed into monthlyones by local quadratic interpolation, i.e., a local quadratic polynomial was fit for eachset of three adjacent quarterly observations; then, the monthly observations right beforeand after the central quarterly observation (the second one in the set) were imputedusing the values of the quadratic polynomial. The output gap was estimated for eachcountry with at least 24 data points. For the Hodrick-Prescott filter we used λ = 129 600as advocated by Ravn and Uhlig (2002).

The exact data series (and the periods for which they could be used) are describedin detail in Tables A.8–A.9 below. In particular, the three aforementioned quarterlytime-series are indicated therein by a “quarterly” comment in their descriptions.

Real-time data on CPI, money stocks and industrial production for the period Febru-ary 1999 to April 2017 were obtained from the OECD MEI data base (http://stats.oecd.org/mei/default.asp?rev=1) for countries whose currencies were not supersededby the euro (i.e., the British pound, Japanese yen, Swiss franc, Canadian dollar, Swedishkrona, Danish kronor, Australian dollar and the comparison currency – the U.S. dol-lar). The time-series for M2 U.S. money supply until March 2001 (as there is a break inOECD reporting of M3/M2 series) were directly taken from the Federal Reserve historicalreleases website (https://www.federalreserve.gov/releases/h6/).

38

Table A.8: Data series description, part I.

Des

crip

tio

nSe

ries

nam

eIn

term

ed

iate

so

urc

e (

if a

ny)

Ori

gin

al s

ou

rce

Earl

iest

dat

e af

ter

Mar

ch 1

97

3 w

he

n t

he

seri

es is

ava

ilab

leEn

d d

ate

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, U

.S.-

U.K

UK

I..A

G.

Dat

astr

eam

IFS

Mar

-73

Dec

-14

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, J

apan

--

U.S

.JP

I..A

E.D

atas

trea

mIF

SM

ar-7

3D

ec-1

4

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, S

wit

zerl

and

--

U.S

.SW

I..A

E.D

atas

trea

mIF

SM

ar-7

3D

ec-1

4

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, C

anad

a --

U.S

.C

NI.

.AE.

Dat

astr

eam

IFS

Mar

-73

Dec

-14

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, S

wed

en

--

U.S

.SD

I..A

E.D

atas

trea

mIF

SM

ar-7

3D

ec-1

4

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, D

enm

ark

-- U

.S.

DK

I..A

E.D

atas

trea

mIF

SM

ar-7

3D

ec-1

4

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, U

.S.-

- A

ust

ralia

AU

I..A

G.

Dat

astr

eam

IFS

Mar

-73

Dec

-14

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, F

ran

ce -

- U

.S.

FRI.

.AE.

Dat

astr

eam

IFS

Mar

-73

Dec

-14

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, G

erm

any

-- U

.S.

BD

I..A

E.D

atas

trea

mIF

SM

ar-7

3D

ec-1

4

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, It

aly

-- U

.S.

ITI.

.AE.

Dat

astr

eam

IFS

Mar

-73

Dec

-14

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, N

eth

erla

nd

s --

U.S

.N

LI..

AE.

Dat

astr

eam

IFS

Mar

-73

Dec

-14

End

-of-

mo

nth

no

min

al e

xch

ange

rat

e, P

ort

uga

l --

U.S

.P

TI..

AE.

Dat

astr

eam

IFS

Mar

-73

Dec

-14

M1

mo

ney

su

pp

ly, n

.s.a

., U

.S.

USI

59

MA

.AD

atas

trea

mIF

SM

ar-7

3N

ov-

14

No

tes

and

co

ins

in c

ircu

lati

on

ou

tsid

e th

e B

ank

of

Engl

and

, n.s

.a.,

U.K

.U

KA

VA

A..

Dat

astr

eam

Ban

k o

f En

glan

dM

ar-7

3D

ec-1

4

M1

mo

ney

su

pp

ly, b

illio

ns

of

yen

s, n

.s.a

, Jap

anJP

I59

MA

.AD

atas

trea

mIF

SM

ar-7

3N

ov-

14

Nar

row

mo

ney

, bill

ion

s o

f Sw

iss

fran

ks, n

.s.a

., Sw

itze

rlan

dSW

I34

...A

Dat

astr

eam

IFS

Mar

-73

Oct

-14

M1

mo

ney

su

pp

ly, b

illio

ns

of

Can

adia

n d

olla

rs, n

.s.a

., C

anad

aC

NI5

9M

AD

AD

atas

trea

mIF

SM

ar-7

3O

ct-1

4

M0

mo

ney

su

pp

ly, m

illio

ns

of

Swed

ish

kro

no

rs, n

.s.a

., Sw

ede

nSD

M0

....

AD

atas

trea

mSt

atis

tics

Sw

ede

nM

ar-7

3D

ec-1

4

M1

mo

ney

su

pp

ly, m

illio

ns

of

Dan

ish

kro

ner

s, n

.s.a

., D

enm

ark

DK

OM

A0

27

AD

atas

trea

mM

EI, O

ECD

Mar

-73

Feb

-14

M1

mo

ney

su

pp

ly, m

illio

ns

of

Au

stra

lian

do

llars

, n.s

.a.,

Au

stra

liaA

UO

MA

02

7A

Dat

astr

eam

MEI

, OEC

DM

ar-7

3Fe

b-1

4

M1

mo

ney

su

pp

ly, b

illio

ns

of

Fren

ch f

ran

ks, n

.s.a

., Fr

ance

13

25

9M

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Mo

lod

tso

va a

nd

Pap

ell (

20

09

)IF

SD

ec-7

7D

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ney

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pp

ly, b

illio

ns

of

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tsch

e m

arks

, s.a

., G

erm

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13

45

9M

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olo

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ova

an

d P

apel

l (2

00

9)

IFS

Mar

-73

Dec

-98

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mo

ney

su

pp

ly, b

illio

ns

of

Ital

ian

lira

s, n

.s.a

., It

aly

13

65

9M

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F...

Mo

lod

tso

va a

nd

Pap

ell (

20

09

)IF

SD

ec-7

4D

ec-9

8

M2

mo

ney

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pp

ly, b

illio

ns

of

Du

tch

gu

ilde

rs, n

.s.a

., N

eth

erla

nd

s 1

38

59

MB

.ZF.

..M

olo

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ova

an

d P

apel

l (2

00

9)

IFS

Mar

-73

Dec

-98

M1

mo

ney

su

pp

ly, b

illio

ns

of

Po

rtu

gue

se e

scu

do

s, n

.s.a

., P

ort

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l 1

82

59

MA

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olo

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an

d P

apel

l (2

00

9)

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Dec

-79

Dec

-98

Fed

eral

Fu

nd

s R

ate,

Un

ited

Sta

tes

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60

B..

Dat

astr

eam

IFS

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Dec

-14

Mo

ney

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ket

Rat

e, U

.K.

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Dat

astr

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IFS

Mar

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v-1

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ney

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ket

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e, J

apan

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atas

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mIF

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ar-7

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uro

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atas

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mM

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4

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s ra

te, C

anad

aC

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atas

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mIF

SM

ar-7

3D

ec-1

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ney

Mar

ket

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e, S

wed

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atas

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e, A

ust

ralia

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Dat

astr

eam

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-14

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ney

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e, F

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olo

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-73

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ney

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ket

Rat

e, G

erm

any

13

46

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Mo

lod

tso

va a

nd

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ell (

20

09

)IF

SM

ar-7

3D

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8

Mo

ney

Mar

ket

Rat

e, It

aly

13

66

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..ZF

...

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lod

tso

va a

nd

Pap

ell (

20

09

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SM

ar-7

3D

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8

Mo

ney

Mar

ket

Rat

e, N

eth

erla

nd

s1

38

60

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ZF..

Mo

lod

tso

va a

nd

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ell (

20

09

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ar-7

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ney

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ket

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e, P

ort

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l1

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ZF..

Mo

lod

tso

va a

nd

Pap

ell (

20

09

)IF

SJa

n-8

3D

ec-9

8

39

Table A.9: Data series description, part II.

Des

crip

tio

nSe

ries

nam

eIn

term

ed

iate

so

urc

e (

if a

ny)

Ori

gin

al s

ou

rce

Earl

iest

dat

e af

ter

Mar

ch 1

97

3 w

he

n t

he

seri

es is

ava

ilab

leEn

d d

ate

Ind

ust

rial

pro

du

ctio

n, s

.a.,

U.S

.U

SI6

6..

CE

Dat

astr

eam

IFS

Mar

-73

Dec

-14

Ind

ust

rial

pro

du

ctio

n, s

.a.,

U.K

.U

KI6

6..

CE

Dat

astr

eam

IFS

Mar

-73

No

v-1

4

Ind

ust

rial

pro

du

ctio

n, s

.a.,

Jap

anJP

I66

..C

ED

atas

trea

mIF

SM

ar-7

3O

ct-1

4

Ind

ust

rial

pro

du

ctio

n, s

.a.,

Swit

zerl

and

, qu

arte

rly

SWQ

66

..B

HD

atas

trea

mIF

S

Ind

ust

rial

pro

du

ctio

n, s

.a.,

Can

ada

CN

I66

..C

ED

atas

trea

mIF

SM

ar-7

3O

ct-1

4

Ind

ust

rial

pro

du

ctio

n e

xclu

din

g co

nst

ruct

ion

, s.a

., Sw

ede

nSD

OP

RI3

5G

Dat

astr

eam

MEI

, OEC

DM

ar-7

3O

ct-1

4

Ind

ust

rial

pro

du

ctio

n, s

.a.,

Den

mar

kD

KI6

6..

BH

Dat

astr

eam

IFS

Jan

-74

Oct

-14

Ind

ust

rial

pro

du

ctio

n, s

.a.,

Au

stra

lia, q

uar

terl

yA

UQ

66

..C

ED

atas

trea

mIF

S

Ind

ust

rial

pro

du

ctio

n, s

.a.,

Fran

ce1

32

66

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ZF..

.M

olo

dts

ova

an

d P

apel

l (2

00

9)

IFS

Mar

-73

Dec

-98

Ind

ust

rial

pro

du

ctio

n, s

.a.,

Ger

man

y1

34

66

..C

ZF..

. M

olo

dts

ova

an

d P

apel

l (2

00

9)

IFS

Mar

-73

Dec

-98

Ind

ust

rial

pro

du

ctio

n, s

.a.,

Ital

y1

36

66

..C

ZF..

. M

olo

dts

ova

an

d P

apel

l (2

00

9)

IFS

Mar

-73

Dec

-98

Ind

ust

rial

pro

du

ctio

n, s

.a.,

Net

her

lan

ds

13

86

6..

CZF

...

Mo

lod

tso

va a

nd

Pap

ell (

20

09

)IF

SM

ar-7

3D

ec-9

8

Ind

ust

rial

pro

du

ctio

n, s

.a.,

Po

rtu

gal

18

26

6..

BZF

...

Mo

lod

tso

va a

nd

Pap

ell (

20

09

)IF

SM

ar-7

3D

ec-9

8

Co

nsu

mer

pri

ce in

de

x, U

.S.

USI

64

...F

Dat

astr

eam

IFS

Mar

-73

Dec

-14

Co

nsu

mer

pri

ce in

de

x (r

etai

l pri

ce in

de

x), U

.K.

UK

I64

B..

FD

atas

trea

mIF

SM

ar-7

3N

ov-

14

Co

nsu

mer

pri

ce in

de

x, J

apan

JPI6

4..

.FD

atas

trea

mIF

SM

ar-7

3N

ov-

14

Co

nsu

mer

pri

ce in

de

x, S

wit

zerl

and

SWI6

4..

.FD

atas

trea

mIF

SM

ar-7

3D

ec-1

4

Co

nsu

mer

pri

ce in

de

x, C

anad

aC

NI6

4..

.FD

atas

trea

mIF

SM

ar-7

3N

ov-

14

Co

nsu

mer

pri

ce in

de

x, S

wed

en

SDI6

4..

.FD

atas

trea

mIF

SM

ar-7

3D

ec-1

4

Co

nsu

mer

pri

ce in

de

x, D

enm

ark

DK

I64

...F

Dat

astr

eam

IFS

Mar

-73

Dec

-14

Co

nsu

mer

pri

ce in

de

x, q

uar

terl

y, A

ust

ralia

AU

I64

...F

Dat

astr

eam

IFS

Mar

-73

Au

g-1

4

Co

nsu

mer

pri

ce in

de

x, F

ran

ce1

32

64

...Z

F...

Mo

lod

tso

va a

nd

Pap

ell (

20

09

)IF

SM

ar-7

3D

ec-9

8

Co

nsu

mer

pri

ce in

de

x, G

erm

any

Co

nsu

mer

pri

ces:

all

item

s, 2

00

0=1

00

Mo

lod

tso

va a

nd

Pap

ell (

20

09

)M

EI, O

ECD

Mar

-73

Dec

-98

Co

nsu

mer

pri

ce in

de

x, It

aly

13

66

4..

.ZF.

..M

olo

dts

ova

an

d P

apel

l (2

00

9)

IFS

Mar

-73

Dec

-98

Co

nsu

mer

pri

ce in

de

x, N

eth

erla

nd

s1

38

64

...Z

F...

Mo

lod

tso

va a

nd

Pap

ell (

20

09

)IF

SM

ar-7

3D

ec-9

8

Co

nsu

mer

pri

ce in

de

x, P

ort

uga

l1

82

64

...Z

F...

Mo

lod

tso

va a

nd

Pap

ell (

20

09

)IF

SM

ar-7

3D

ec-9

8

40

Appendix B. Assessing forecast quality

We denote by T the total number of monthly values to be forecast, from months 1to T . As is standard in the literature (see Molodtsova and Papell, 2009 and Rossi, 2013)and for essentially the same reasons, namely, the consideration of rolling OLS regressions,we allow for a training period of length t0 = 120 months (10 years) and only evaluatethe accuracy of the forecasts in months t0 + 1 = 121 to T . To that end, we consider theroot mean square error,

rmse =

√√√√ 1

T − t0

T∑t=t0+1

(st − st

)2(B.1)

=

√√√√ 1

T − t0

T∑t=t0+1

((st − st−1

)−(st − st−1

))2,

and note that (by subtracting and adding the pivotal values st−1) this root mean squareerror is for both the logarithms of exchange rates st and for changes in logarithms ofexchange rates st − st−1.

We want to investigate whether the improvements in RMSE of one method overanother are statistically significant. Denote by s′t and st the respective forecasts of twomethods of interest. We aim to test the hypothesis H0 that the difference in forecastingaccuracy is not significant against the alternative hypothesis H1 that the second method– the one outputting the forecasts st – is significantly better, on average. To that end,the standard practice (see the two tests presented below) is to consider the instantaneousdifferences in accuracy

dt =(s′t − st

)2 − (st − st)2 .We denote by

dT =1

T − t0

T∑t=t0+1

dt and σ2T =

1

T − t0

T∑t=t0+1

(dt − dT

)2their empirical average and variance.

Appendix B.1. Descriptive statistics

In this section we consider the no-change prediction as the benchmark; it providesthe forecasts s′t = st−1.

To get an initial feeling of whether H0 should be rejected or not in this case, wecan take a look at the quantiles of the sequence of dt. For example, in Table D.14of Appendix D, we see that in the sequential ridge regression with discount factors andthe exponentially weighted average strategy with discount factors cases, the distributionof the differences dt is shifted toward positive values: the 75% and 90% quantiles arelarger in absolute value than the 25% and 10% ones. This is not the case for rolling orrecursive OLS regressions. Additional comments on these matters are provided at theend of Section 5.1.

41

Appendix B.2. A general test for comparing predictive accuracies

In this section we sometimes still consider the no-change prediction as the benchmark:s′t = st−1, but also consider other benchmarks, like rolling or recursive OLS regression,e.g., in Table 3. We recall that we aim to test the hypothesis H0 that the differencein forecasting accuracy between the method under scrutiny and the benchmark is notsignificant against the alternative hypothesis H1 that the method – the one outputtingthe forecasts st – is significantly better, on average.

The Diebold-Mariano test with relevant assumptions. Diebold and Mariano (1995) havepresented a test relying directly on the differences dt in the forecasting errors, eventhough these may be serially correlated. Only mild and direct assumptions on the sta-tistical behavior of these differences dt need to made. Here, we state their results with arectangular lag, as they advocate.

More precisely, they showed that under assumptions of covariance stationarity andshort memory of the differences dt, for a properly chosen truncation lag denoted byH > 0, the test statistics

SDM,H =√T − t0

dT√σH,2T

,

where

σH,2T =1

T − t0

T∑t=t0+1

(dt − dT

)2+

2

T − t0

H∑τ=1

T∑t=τ+t0+1

(dt − dT

)(dt−τ − dT

),

converge to a N (0, 1) distribution under H0 and to +∞ under H1. Diebold (2012) em-phasized the generality of the method to compare the predictive accuracy of forecastsbetween any two methods, as long as the mild assumption stated above, namely, covari-ance stationarity of the differences in accuracy dt, holds. He explains in Section 2.2 ofthe mentioned reference why this assumption is natural and often met in practice.

The choice of the truncation lag H was partially left open; Diebold and Mariano sug-gested choosing it as a function of the length of the short memory (of the autocovariance

degree). Estimating by H a proper H based on our data, then substituting its value,would lead to considering the test statistic SDM,H , which would no longer be guaranteed

to converge to a N (0, 1) distribution under H0. We instead take a more robust approachto reject H0, that is, we build a more conservative test than the original test based on agood a priori value of H. Namely, we consider

SDM =√T − t0

dT√max

H∈{0,1,...,20}σH,2T

, (B.2)

which is smaller than any of the corresponding original statistics SDM,H . The limitingdistribution under H0 has a cumulative distribution function that is uniformly smallerthan that of the N (0, 1) distribution on positive numbers (while convergence to +∞ isstill achieved under H1, at a slower rate). Note that for our data set a maximal value of20 for H was set because it corresponds roughly to the maximal value of

√T − t0 on our

data. In the following, the test described above will be referred to as the DM test.

42

We do not comment on the direct computation of the p-values for this DM testusing quantiles of the normal distribution (which would be a conservative approach, i.e.,the size and power of the test would be smaller than usual), as we rather resort to abootstrapped approach to compute them.

Bootstrapped version. We follow here, as others have, the method and choices madeby Mark and Sul (2001), Rogoff and Stavrakeva (2008), and Ince (2014). They all pointout that the DM test is undersized and has less power than it should have when theconditions for its applications are not met, in particular, in the presence of forecast biases.They recommend computing its p-values based on a bootstrap procedure. Briefly, M =1 000 bootstrapped simulations of the exchange rate and fundamentals time-series arecomputed, the relevant forecasting procedure (e.g., the exponentially weighted averagestrategy with discount factors) and its benchmark (e.g., the no-change predictor or rollingOLS regression) are applied to each simulation, and the resulting DM test statistic iscomputed (on each). This gives rise to an (empirical) distribution for the DM teststatistic under H0. Then, the DM test statistic is computed on the original data andthe bootstrapped DM p-value is obtained, in a one-sided way, based on this distribution(it equals the empirical frequency of simulated DM test statistics that are above theoriginal DM test statistic). These bootstrapped DM p-values are found in various tablesof this article in the columns labeled “DM p-value” (without any specific reference totheir bootstrap origin).

We now describe the method to simulate the time-series at hand. There are slight vari-ations between the methods considered by Mark and Sul (2001), Rogoff and Stavrakeva(2008) and Ince (2014); we chose to stick to Rogoff and Stavrakeva (2008) here. For eachcountry and each fundamental j, the data generating process is assumed to be

∆st = εt,

∆fj,t = αj + βjt+ γjfj,t−1 +

d∑k=1

δj,k∆st−k +∑k=1

ζj,k∆fj,t−k + ε′j,t,

where st and fj,t are defined as in Section 2.1, and ∆ is shorthand for the unit variations

∆st = st − st−1 and ∆fj,t = fj,t − fj,t−1 ,

with εt and ε′j,t denoting the residual terms, which are modeled by random variables.That is, an autoregressive model on the evolution of the fundamentals is considered, withsome external input given by the evolution of the exchange rate. For each country andeach fundamental j, the parameters d and `, as well as the coefficients δj,k and ζj,k ofthe autoregressive equation, are chosen separately. Given d and `, the coefficients areestimated using OLS regression. The parameters d and ` selected minimize Akaike’sinformation criterion (AIC) over the choices considered, respectively, integers 1-10 for dand 1-30 for ` (we set these intervals so that in our simulations, d and ` rarely reachthe upper values 10 and 30). These two parameters may differ across countries andfundamentals. AIC is also used to decide whether to include no constant and no trend, aconstant and no trend, or a constant and a trend. Note in passing that the residuals ε′j,tare also estimated by quantities denoted ε′j,t. Once the data generating processes havebeen all estimated, the exchange rates st and fundamentals fj,t are simulated recursivelyas follows.

43

First, we need two inputs for this recursive simulation: (i) seeds to initiate the au-toregressive simulations, and (ii) values for the stochastic residuals ε′j,t and εt. The latterwill be sampled with replacement among the ε′j,τ and the ∆sτ , where τ ∈ {1, . . . , T}. Asfor the seeds, they need to be in number max{d, `}; we obtain them by sampling withreplacement among the original vectors (∆sτ , f1,τ , . . . , fN,τ ), where τ ∈ {1, . . . , T}. Notethat we use here the case-by-case resampling also considered in Rogoff and Stavrakeva(2008); we do not consider blocks of vectors.

Last, we note that we actually simulate the process until round T + 100 and discardthe 100 first observations thus simulated, to obtain time-series of the same length T asthe original series. We do so, following Rogoff and Stavrakeva (2008), to minimize theinfluence of the choice of seeds and make sure that the re-simulated times series are in asteady state.

Appendix B.3. A test for the case of nested forecasting equations?

The tests by West (1996), Clark and West (2006, 2007) are designed for comparingmodels, while we are interested here only in comparing forecasts, and not in assumingthe existence of models – see Diebold (2012) and Rossi (2013) for related discussions.Furthermore, the tests by Clark and West have been designed and studied for only therolling and recursive OLS regression cases, while here we are obviously interested inalternative methods. This is underlined in the original papers and pointed out againby Rogoff and Stavrakeva (2008).

Nonetheless, for the sake of completeness, we computed p-values associated with thetest developed by Clark and West for nested models (even though we consider “fore-casting equations” only and not models). These p-values should merely be considered asstatistical indicators of predictability. We believe that the most significant tests beingperformed in our study are the DM tests mentioned above. The readers will note thatin our tables, the test by Clark and West is typically less conservative than the DM one(as seen in other empirical and theoretical studies).

Mathematical description of the test. For this test we restrict our attention to the no-change prediction as the benchmark method: s′t+1 = st.

This no-change prediction is nested into the larger forecasting equation (1). In thiscase, assuming the existence of underlying models, Clark and West (2006) argued thatthe differences in forecasting abilities dt were an unfair measure, and advocated insteadto use adjusted differences, which in our context would be

at = dt +(st − st−1

)2,

where st is the prediction associated with the larger forecasting equation (1). We denoteby

aT =1

T − t0 + 1

T∑t=t0+1

at (B.3)

the empirical average of the adjusted differences at and

S2

T =1

T − t0 + 1

T∑t=t0+1

(at − aT

)2(B.4)

44

their empirical variance. The test statistic is equal to

SCW =√T − t0 + 1

aT√S2

T

. (B.5)

The hypotheses tested are in terms of an underlying model (if we are ready to assumethat one exists). Namely, H ′0 : the true underlying model is the no-change model versusH ′1 : the true underlying linear model uses at least one fundamental of (1). Here again,we see that the test concerns predictability, not forecastability. Under the assumptionthat residuals of the models form a martingale difference sequence, and provided thatrolling and recursive OLS regressions are considered when selecting coefficients in (1),Clark and West (2006) showed that SCW converges to a N (0, 1) distribution under H ′0and to +∞ under H ′1. The p-values associated with the test thus constructed are foundin the tables in the columns labeled “CW p-value”; they are based on quantiles of thenormal distribution. In this article, this test will be referred to as the CW test.

We recall that as indicated several times above, these p-values are merely indicators,and of predictability rather than of forecastability. We include them here for the sake ofcompleteness.

45

Appendix C. Additional technical details

In this section, we provide

• explanations on how to run a single instance of EWA and SRidge in practice, bychoosing their hyperparameters (Appendix C.1) well,

• some additional details on the numerical implementation of our machine-learningmethods (Appendix C.2),

• a more complete description of the economic evaluation of our forecasting strategies,including a presentation of how to include constraints on the currency weights(Appendix C.3).

Appendix C.1. EWA and SRidge in practice: how to choose their hyperparameters

We recall that in our simulation study we do not report the performance of the EWAand SRidge methods for several well-chosen (hand-picked) sets of hyperparameters, as isusual in the literature. We instead re-use a more operational approach already consideredin Devaine et al. (2013), based on a sequential grid search. We thus only report the resultsobtained for one run of the method. The aim of this section is to state all details of thegrid-search approach followed.

For EWA, the single run selects at each time instance t, based on available data, thehyperparameters ηt > 0 and γt > 0 (rather than the fixed γ initially considered) to beused in (8). We chose κ = 2, which is common in the literature and allows us to have atheoretical bound given condition (10). More precisely, the parameters ηt and γt for theprediction of instance t+ 1 are respectively selected in the grids E/

√t and G, where

E ={m× 10k, m ∈ {1, 2, 5} and k ∈ {−4,−3,−2,−1}

}∪ {1} ,

and G = {0} ∪{m× 10k, m ∈ {1, 2, 5} and k ∈ {0, 1, 2}

}∪ {1 000}

={

0, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000}.

These grids are (somewhat) evenly spaced on a logarithmic scale. To predict instance t+1,we resort to (8), with the pair of parameters (η, γ) in the grids E and G, whose associatedEWA strategy with sequence of learning parameters η, η/

√1, . . . , η/

√t− 1 and fixed

discount factor γ performed best overall on instances 1 to t.We proceed similarly for SRidge: we take the same discount power κ = 2 as in the

EWA case and calibrate the hyperparameters λt and γt to be used for month t+ 1 basedon past data, using the same grid G for the γ parameters, and the grid for λ given by

Λ = {0} ∪{m× 10k, m ∈ {1, 2, 5} and k ∈ {1, 2, 3}

}∪ {10 000}

={

0, 1, 2, 5, 10, 20, 50, 100, 200, 500, 1 000, 2 000, 5 000, 10 000}.

Appendix C.2. Notes on computations and forecasts

Calculations were performed with the Scilab 5.5.0 software; the code is available uponrequest.

We recall that the training period for our methods was set at 120 months, which isstandard in the literature, and that, correspondingly, the rolling OLS regressions were

46

estimated with 120 months of past data at each instance. The computed coefficients of ourforecasts vary greatly over time and it is difficult to discern any temporal patterns – alsoa feature known in the literature. This is not a surprise as our forecasting methods do notaim to estimate some model – they merely output efficient forecasts. The regularizationand discount terms that are computed from past data are non-zero most of the time,and differ substantially between currencies, fundamentals used, and time periods. Mostof the time the best discount factors are high – above 10 – which means that short-termtrends are weighted most strongly.

We tried different grids from those in Appendix C.1. All grids were logarithmic, asrecommended by machine learning theory. We found that a larger grid typically yieldedsmall improvements in terms of RMSE with respect to the initial grid – but not always,as the RMSE of predictions can get worse. With a denser grid we may overfit theregularization and the discounting terms themselves. Also, smaller obtained RMSEs donot guarantee better DM tests being obtained; all depends on the errors of individualpredictions. It appears that there is a large set of said parameters where the quality ofpredictions (measured by the RMSE) are qualitatively very similar, all corresponding toan improvement over the OLS methods.

Appendix C.3. Economic evaluation of forecasting strategies

We recall here the methodology introduced by Della Corte et al. (2009, 2011) andwell-summarized in Li et al. (2015). We followed it scrupulously to produce our maineconomic evaluation results, namely, those reported in Table 5 in the “no c” lines. Wealso considered two additional approaches that deal with known problems in economiccriterion evaluation – by constraining the weights, which led to the lines with “c = . . .”in the main and certain other tables. These: Tables D.17 and D.18, show results undera summation-to-less-than-1 constraint for the weights. Below, we recall only the basicelements of the methodology for economic evaluation, so as to be able to describe thetwo modified approaches.

Brief description of the basic methodology for economic evaluation. We consider onedomestic (US) bond, with risk-free return rate rfree,t, and K foreign bonds, indexedby k = 1, . . . ,K, with risk-free return rates in their local currencies (that would beused as UIRP fundamentals) but with risky returns rk,t in US dollars because (andonly because) of the exchange-rate risk. The subscripts t index the time instances,namely, months in our case. At each month t − 1, an investor forecasts the evolutionof the exchange rates using our methods, and based on these forecasts, picks weightswt = (w1,t, . . . , wK,t)

T ∈ RK determining how to invest their money in the foreign bondsfor month t; these weights are real numbers. Their return in month t equals

K∑k=1

wk,trk,t +

(1−

K∑k=1

wk,t

)rfree,t = rfree,t +

K∑k=1

wk,t(rk,t − rfree,t

).

Next, the returns rk,t are not yet known in month t − 1, but their conditional meansrk,t and variance-covariance matrix Σt can be assumed to be (where conditioning is withrespect to the rk,t−1 values for month t − 1). Indeed, the covariance matrix Σt usedfollows from an empirical estimation based on the past covariances, while the means rk,t

47

are given precisely by our forecasts. A target conditional volatility σ is then set and theweights are picked to optimize the following mean-variance problem:

maxwt∈RK

rfree,t +

K∑k=1

wk,t(rk,t − rfree,t

),

subject to wTt Σt wt 6 σ2.

A closed-form solution exists when no further constraint is imposed on the weights.

Imposing further constraints on the weights. As indicated in the main text, the weightschosen as above often have large components – especially if the returns of some currenciesare highly correlated – and are unstable over time. A good way to avoid such issues isto add a squared regularization term; given a parameter c > 0, we want to perform theoptimization

maxwt∈RK

rfree,t +

K∑k=1

wk,t(rk,t − rfree,t

),

subject to wTt Σt wt 6 σ2

and

K∑k=1

w2k,t 6 c.

We provide no closed-form solution to this problem, and instead solve it numericallyto obtain our results. As this constrained maximization problem is a particular caseof what is called a quadratically constrained quadratic program (QCQP), its solution isequal to that of its Lagrangian dual problem, which is of the form: minimize a (nonlinear)function of finitely many variables λ1, λ2, . . . , subject to positivity constraints on onlythe λi. See Boyd and Vandenberghe (2009, Section 5.2.4) for details on this equivalence.Now, the dual problem is easily solved numerically with the help of toolboxes from varioussoftware packages; in our case, we resorted to the optim routine of Scilab.

Further constraining the problem as

maxwt∈RK

rfree,t +

K∑k=1

wk,t(rk,t − rfree,t

),

subject to wTt Σt wt 6 σ2

and

K∑k=1

w2k,t 6 c

and

K∑k=1

wk,t 6 1,

it remains a QCQP, so the methodology above still applies.Details on these calculations (exact statements of the dual problems, code written to

solve them, etc.) are available upon request.

48

Appendix D. Additional graphs and tables

This section provides complementary results as well as tables omitted from the maintext.

Tables with results discussed in the text.

• Tables D.10–D.13 give supplementary information to the basic results presented inSection 5.

• Table D.14 gives the quantiles of the differences in forecasting error between theno-change prediction and the methods we look at.

• Table D.15 gives the results for a subsample spanning the whole range of observa-tions from January 1980 until the end of our sample. Note that some results arenot available for the PTE/USD pair as data on interest rates starts in 1983.

• Tables D.16–D.18 show robustness results for the economic evaluation of the pre-diction performance of the exponentially weighted average strategy with discountfactors: with or without the Danish Krone, and with or without a summation-to-less-than-1 constraint on the currency weights.

• Tables D.19–D.24 show our results for the shorter period discussed in Section 5.5,namely, February 1999–December 2014.

• Table D.25 shows the results for the Taylor-rule based exchange rate fundamentalsin their “decoupled” (heterogenous) form.

Comments on the weights seen in the exponentially weighted average strategy with dis-count factors for forecasts with all fundamentals. By construction, the exponentiallyweighted average strategy with discount factors has a natural interpretation as beingable to identify “scapegoat” fundamentals that could have been considered by tradersin some periods as in the model of Bacchetta and van Wincoop (2004). This is becausebased on past data and discounting, the fundamentals that are best at predicting ex-change rates in a given period are given more weight while forming predictions. It isinsightful therefore to investigate the behavior of these weights with time. For the sakeof brevity, we show the results in Figures D.1–D.3 only for the case of predictions withall fundamentals for JPY/USD, USD/GBP and CHF/USD. Note that these were thesetups which did not necessarily perform well in comparison to parsimonious models suchas those based solely on PPP or UIRP; however, they did much better overall than theOLS methods as measured by Theil ratios and in direct comparisons.

The codes for the various fundamentals are the following: PPP refers to inflation rates,UIRP to money-market interest rates; MONEY to money stocks, and Y to industrialproduction, while INTCH denotes changes in money market interest rates. Lastly, “no-change” refers to the no-change forecast that also is assigned a weight in the exponentiallyweighted average strategy with discount factors.

The conclusion from these figures is striking: typically only one fundamental is given apredominant weight in a given period, and most of the time for very short time intervals.In most cases, the fundamentals that are given the most weight pertain to money market

49

interest rates and money supply, and for one country only. Rarely do we observe thisbehavior for inflation or industrial production. It would be consistent with the scapegoatmodel as well as the fickle nature of expectations about the fundamentals driving theexchange rate. This is, however, open to yet another interpretation. It also may pointout that different shocks to different fundamentals (news) could be driving exchangerates at different times, and – given their prevalence in our data – monetary policy andfinancial market shocks are the most important.

50

Table D.10: Relative forecasting performance of machine learning methods vs. rolling and recursive regressions based on coupled fundamentals.

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.78

3

1.0

00

8 -

0.1

70

0 0

.71

6

0.9

90

3 0

.71

77

0.5

66

0

.99

33

0.9

60

8 0

.21

3

FR

F/U

SD

0.9

93

4 1

.02

11

0.2

14

0

.99

80

0.3

00

7 0

.48

8

0.9

89

9 1

.17

51

0.1

62

0

.99

45

0.6

35

2 0

.36

2

D

EM/U

SD

0.9

96

5 1

.08

15

0.1

96

0

.99

75

0.6

60

3 0

.28

5

0.9

92

6 1

.19

11

0.1

87

0

.99

36

0.8

94

8 0

.25

1

IT

L/U

SD

0.9

99

5 0

.07

96

0.5

35

1

.00

59

-0

.64

30

0.6

86

0

.98

10

1.7

67

7 0

.04

4 *

*

0.9

87

3 1

.66

76

0.0

49

**

N

LG/U

SD

0.9

97

8 0

.24

26

0.6

27

1

.00

09

-0

.10

21

0.6

75

0

.99

35

0.7

80

7 0

.38

0

0.9

96

5 0

.70

81

0.3

44

P

TE/U

SD

1.0

04

5 -

0.3

61

1 0

.78

0

1.0

00

1 -

0.0

06

9 0

.59

6

1.0

02

5 -

0.2

31

5 0

.75

3

0.9

98

1 0

.15

78

0.5

50

U

SD/G

BP

0

.98

52

0.9

50

4 0

.66

2

1.0

02

8 -

0.2

40

3 0

.89

9

0.9

83

7 1

.14

65

0.4

15

1

.00

13

-0

.19

06

0.7

80

JP

Y/U

SD

1.0

02

3 -

0.2

94

1 0

.97

8

1.0

02

3 -

0.3

49

5 0

.89

8

0.9

92

3 0

.94

46

0.4

71

0

.99

23

0.9

46

0 0

.21

0

C

HF/

USD

0

.99

86

0.2

43

7 0

.88

5

0.9

95

2 0

.74

18

0.4

60

0

.99

76

0.2

58

9 0

.83

6

0.9

94

2 0

.64

98

0.3

23

C

AD

/USD

0

.98

16

0.8

23

3 0

.66

4

1.0

03

3 -

0.4

51

6 0

.88

3

0.9

82

4 0

.73

88

0.5

07

1

.00

42

-0

.67

74

0.8

17

SE

K/U

SD

0.9

85

9 0

.91

82

0.6

00

1

.00

90

-0

.74

31

0.9

56

0

.97

64

1.8

59

6 0

.12

0

0.9

99

3 0

.08

69

0.6

48

D

NK

/USD

0

.98

60

1.5

67

1 0

.23

8

0.9

95

0 1

.12

39

0.2

60

0

.99

37

0.6

45

0 0

.55

2

1.0

02

7 -

0.5

25

4 0

.79

8

U

SD/A

UD

0

.98

31

0.9

60

8 0

.60

1

0.9

96

4 1

.11

61

0.2

75

0

.98

39

0.7

78

1 0

.57

3

0.9

97

2 0

.34

62

0.4

67

FR

F/U

SD

0.9

82

7 1

.36

14

0.1

57

0

.98

47

1.0

01

8 0

.24

7

0.9

67

2 1

.61

77

0.0

73

*

0.9

69

2 1

.67

77

0.0

35

**

D

EM/U

SD

0.9

69

8 2

.59

38

0.0

04

***

0

.98

35

1.9

43

1 0

.04

0 *

*

0.9

53

0 2

.33

38

0.0

10

***

0

.96

64

1.4

95

5 0

.05

2 *

IT

L/U

SD

0.9

92

5 0

.46

79

0.5

45

0

.99

26

0.8

15

0 0

.28

0

0.9

90

0 0

.67

96

0.5

36

0

.99

02

0.9

25

0 0

.29

5

N

LG/U

SD

0.9

97

5 0

.25

24

0.6

30

0

.99

61

0.4

85

5 0

.33

5

1.0

05

3 -

0.3

38

1 0

.76

7

1.0

03

8 -

0.2

80

2 0

.57

0

PTE

/USD

1

.01

84

-0

.60

59

0.9

27

1

.00

81

-0

.54

46

0.8

81

1

.01

29

-0

.75

68

0.8

88

1

.00

27

-0

.19

71

0.6

60

U

SD/G

BP

0

.95

10

2.4

70

4 0

.25

8

0.9

84

8 1

.38

41

0.4

18

0

.94

88

2.6

54

1 0

.14

4

0.9

82

5 1

.37

37

0.3

49

JP

Y/U

SD

0.9

85

2 0

.82

67

0.9

38

0

.99

68

0.5

58

6 0

.77

0

0.9

76

5 1

.43

80

0.7

81

0

.98

80

1.2

91

8 0

.40

3

C

HF/

USD

0

.98

33

1.9

80

2 0

.32

0

0.9

92

1 1

.13

89

0.2

71

0

.97

88

2.0

20

0 0

.29

9

0.9

87

6 1

.31

77

0.2

01

C

AD

/USD

0

.97

91

0.9

81

8 0

.87

8

1.0

01

6 -

0.2

22

2 0

.91

6

0.9

80

3 0

.88

55

0.8

55

1

.00

29

-0

.47

81

0.9

25

SE

K/U

SD

0.8

61

6 1

.30

58

0.6

42

0

.93

55

1.0

77

0 0

.46

7

0.8

43

8 1

.29

01

0.6

83

0

.91

62

1.0

95

9 0

.47

9

D

NK

/USD

0

.97

07

1.9

68

9 0

.45

2

0.9

92

5 1

.61

96

0.2

91

0

.97

51

1.6

27

0 0

.61

5

0.9

97

1 0

.57

58

0.7

74

U

SD/A

UD

0

.97

19

1.2

85

4 0

.81

0

0.9

88

5 1

.50

11

0.3

08

0

.97

12

1.2

57

9 0

.79

1

0.9

87

8 1

.11

58

0.3

94

FR

F/U

SD

0.9

56

3 1

.97

26

0.0

83

*

0.9

78

6 0

.98

04

0.2

88

0

.93

49

1.9

59

7 0

.06

6 *

0

.95

67

1.3

08

7 0

.14

0

D

EM/U

SD

0.9

51

9 2

.69

61

0.0

32

**

0

.97

37

2.2

04

9 0

.03

9 *

*

0.9

34

7 2

.43

42

0.0

65

*

0.9

56

2 1

.85

37

0.0

92

*

IT

L/U

SD

0.9

76

7 0

.94

64

0.6

08

0

.96

91

1.2

01

0 0

.28

8

0.9

73

5 1

.01

42

0.6

26

0

.96

59

1.1

45

8 0

.36

9

N

LG/U

SD

0.9

81

6 1

.02

00

0.6

20

0

.99

14

0.8

21

3 0

.49

6

0.9

85

4 0

.71

95

0.7

05

0

.99

53

0.3

71

3 0

.65

1

PTE

/USD

0

.98

56

0.5

71

8 0

.61

1

0.9

58

6 1

.44

98

0.1

48

0

.98

23

0.5

86

7 0

.62

0

0.9

55

4 1

.15

61

0.2

73

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

All

fun

dam

enta

ls

Cu

rren

cy p

air

SRid

ge

vs.

EWA

vs

.

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nR

olli

ng

regr

essi

on

Rec

urs

ive

regr

essi

on

Notes: This table presents the comparison of forecasting performance between, respectively, SRidge and EWA vs. the rolling and

recursive OLS for coupled fundamentals. Columns 1-6 show the comparison between SRidge and OLS methods, while columns 7-12

for EWA vs. OLS. For each comparison, the Theil ratio (RMSE of the given method / RMSE of the compared) and the DM statistic

and its bootstrapped p-value are shown. The DM test is a one-sided test of equal out-of-sample prediction accuracy (H0) against

superior out-of-sample prediction accuracy (H1) of the machine-learning method considered against the OLS methods. ***, **,

and * denote statistical significance at the 1 %, 5 %, and 10 % levels. Theil ratios < 1 are indicated in bold. Sample period for the

exchange rates: March 1973 – December 2014 unless shorter (indicated in Appendix A).

51

Table D.11: Directional predictions of the exchange rates based on coupled fundamentals.

What do we read? EOM - FRED - RT? Coupled or decoupled? RMSE / versus OLS / directional changes?

Answer: EOM - Coupled - Directional

Proportion of

changes

predicted

DM statistic DM p-value

Proportion of

changes

predicted

DM statistic DM p-value

Proportion of

changes

predicted

DM statistic DM p-value

Proportion of

changes

predicted

DM statistic DM p-value

USD/GBP 0.501 0.0385 0.491 0.493 -0.2215 0.565 0.507 0.1924 0.392 0.493 -0.2215 0.561

JPY/USD 0.467 -1.1724 0.878 0.478 -0.7410 0.757 0.554 2.1120 0.008 *** 0.509 0.3016 0.267

CHF/USD 0.504 0.1330 0.441 0.486 -0.4875 0.697 0.486 -0.4875 0.667 0.486 -0.4875 0.686

CAD/USD 0.517 0.6664 0.215 0.491 -0.3508 0.627 0.491 -0.3508 0.637 0.478 -0.8586 0.861

SEK/USD 0.514 0.3831 0.313 0.486 -0.3768 0.656 0.507 0.2160 0.401 0.441 -1.8029 0.981

DNK/USD 0.504 0.1227 0.485 0.507 0.1916 0.424 0.564 2.2004 0.005 *** 0.491 -0.2737 0.553

USD/AUD 0.522 0.5909 0.228 0.447 -1.6911 0.956 0.447 -1.6911 0.954 0.447 -1.5597 0.944

FRF/USD 0.466 -0.5013 0.708 0.545 0.6949 0.218 0.513 0.2284 0.377 0.545 0.6949 0.250

DEM/USD 0.487 -0.3266 0.692 0.529 0.6309 0.345 0.529 0.6309 0.316 0.540 0.8492 0.337

ITL/USD 0.534 0.5861 0.283 0.455 -0.8170 0.863 0.450 -0.9118 0.874 0.455 -0.8170 0.872

NLG/USD 0.471 -0.5985 0.783 0.471 -0.5328 0.783 0.471 -0.5328 0.771 0.492 -0.1520 0.751

PTE/USD 0.524 0.4381 0.492 0.481 -0.3120 0.812 0.481 -0.3120 0.805 0.481 -0.3120 0.809

USD/GBP 0.514 0.5527 0.259 0.533 1.2562 0.089 * 0.504 0.1455 0.423 0.465 -1.3720 0.919

JPY/USD 0.491 -0.2988 0.600 0.520 0.6265 0.219 0.517 0.5416 0.250 0.478 -0.7171 0.704

CHF/USD 0.512 0.4190 0.323 0.482 -0.5189 0.712 0.482 -0.5189 0.687 0.512 0.3727 0.249

CAD/USD 0.535 1.1080 0.124 0.507 0.1946 0.426 0.507 0.1946 0.418 0.483 -0.4852 0.553

SEK/USD 0.500 0.0000 0.422 0.489 -0.3343 0.579 0.542 1.5619 0.044 ** 0.466 -1.0618 0.787

DNK/USD 0.538 1.2535 0.094 * 0.530 0.9022 0.164 0.530 0.9022 0.163 0.467 -0.9952 0.783

USD/AUD 0.508 0.3073 0.328 0.445 -2.1153 0.984 0.533 1.2218 0.079 * 0.436 -2.4375 0.981

FRF/USD 0.550 1.0082 0.141 0.571 1.4934 0.056 * 0.571 1.7470 0.028 ** 0.423 -1.6341 0.924

DEM/USD 0.471 -0.6314 0.760 0.466 -0.7603 0.801 0.466 -0.7603 0.801 0.524 0.5189 0.389

ITL/USD 0.492 -0.1391 0.550 0.487 -0.2371 0.604 0.481 -0.3022 0.602 0.487 -0.2371 0.576

NLG/USD 0.455 -0.8298 0.838 0.476 -0.4540 0.689 0.561 1.5501 0.054 * 0.524 0.4540 0.446

PTE/USD 0.465 -0.4644 0.695 0.479 -0.2975 0.617 0.479 -0.2975 0.611 0.479 -0.2975 0.659

USD/GBP 0.514 0.4750 0.285 0.541 1.2704 0.073 * 0.535 1.1756 0.086 * 0.525 0.9346 0.080 *

JPY/USD 0.476 -0.7489 0.798 0.479 -0.6835 0.795 0.516 0.5121 0.305 0.568 2.4519 0.007 ***

CHF/USD 0.516 0.5688 0.250 0.536 1.2759 0.067 * 0.538 1.3542 0.056 * 0.530 0.9899 0.136

CAD/USD 0.482 -0.6747 0.772 0.500 0.0000 0.453 0.503 0.0925 0.458 0.482 -0.6720 0.733

SEK/USD 0.476 -0.8706 0.810 0.482 -0.5908 0.707 0.487 -0.4972 0.690 0.518 0.6926 0.282

DNK/USD 0.469 -0.7879 0.814 0.471 -0.7988 0.822 0.497 -0.0917 0.512 0.466 -1.0449 0.886

USD/AUD 0.511 0.4179 0.295 0.511 0.4168 0.270 0.514 0.4535 0.297 0.540 1.3903 0.048 **

FRF/USD 0.467 -0.6763 0.797 0.475 -0.5034 0.752 0.517 0.3654 0.359 0.525 0.5484 0.249

DEM/USD 0.435 -1.3965 0.937 0.452 -0.9529 0.858 0.548 1.2480 0.113 0.548 0.8565 0.091 *

ITL/USD 0.474 -0.4744 0.690 0.481 -0.4249 0.649 0.513 0.2732 0.390 0.538 0.8854 0.124

NLG/USD 0.492 -0.1438 0.688 0.513 0.2017 0.568 0.577 1.6343 0.094 * 0.466 -0.8248 0.632

PTE/USD 0.529 0.3462 0.354 0.486 -0.1644 0.521 0.471 -0.4729 0.696 0.429 -1.0750 0.787

USD/GBP 0.496 -0.1469 0.516 0.493 -0.2330 0.546 0.535 1.0098 0.135 0.522 0.6154 0.180

JPY/USD 0.537 1.2802 0.083 * 0.495 -0.1728 0.564 0.534 1.2103 0.093 * 0.571 2.7985 0.002 ***

CHF/USD 0.481 -0.6283 0.847 0.536 1.2186 0.244 0.538 1.3417 0.068 * 0.541 1.4025 0.056 *

CAD/USD 0.534 1.3369 0.113 0.500 0.0000 0.643 0.484 -0.5375 0.701 0.450 -1.7858 0.931

SEK/USD 0.487 -0.4661 0.651 0.503 0.0776 0.453 0.503 0.0778 0.429 0.505 0.1947 0.460

DNK/USD 0.506 0.1976 0.406 0.500 0.0000 0.464 0.489 -0.3858 0.684 0.446 -1.6878 0.943

USD/AUD 0.514 0.4478 0.333 0.500 0.0000 0.496 0.480 -0.6106 0.714 0.520 0.7277 0.110

FRF/USD 0.500 0.0000 0.530 0.517 0.3507 0.439 0.542 0.8541 0.233 0.583 1.8516 0.025 **

DEM/USD 0.429 -1.4965 0.961 0.446 -0.9310 0.895 0.514 0.3155 0.480 0.610 2.4179 0.015 **

ITL/USD 0.538 0.8281 0.166 0.500 0.0000 0.448 0.423 -1.4450 0.949 0.487 -0.2305 0.558

NLG/USD 0.519 0.3282 0.453 0.529 0.6907 0.342 0.593 2.1752 0.019 ** 0.524 0.6554 0.258

PTE/USD 0.443 -0.7467 0.809 0.357 -2.1654 0.984 0.443 -0.9625 0.827 0.500 0.0000 0.487

UIRP fundamental

Monetary model fundamentals

All fundamentals

Rolling regression Recursive regression Sridge EWA

Currency pair

PPP fundamental

Notes: This table presents the results of forecasting the direction of change of the end-of-month exchange rate 1-month ahead by

the rolling OLS, recursive OLS, SRidge and EWA methods for based on coupled fundamentals. For each method, the share of

correctly forecasted changes, the DM statistic and its bootstrapped p-value are shown. The DM tests is a one-sided test of equal

out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) for the methods considered against

the benchmark of a 50 % success rate. ***, **, and * denote statistical significance at the 1 %, 5 %, and 10 % levels. Shares > 0.5

are indicated in bold. Sample period for the exchange rates: March 1973 – December 2014 unless shorter (indicated in Appendix

A).

52

Table D.12: Relative forecasting performance on the real-time data set of machine-learning methods vs. rolling and recursive regressions basedon coupled fundamentals.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:R

T -

Dec

ou

ple

d -

Ver

sus

OLS

Dat

e:2

7/0

2/2

01

8

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

0

.98

71

0.5

72

3 0

.78

1

0.9

95

7 0

.18

64

0.7

01

0

.99

60

0.2

50

3 0

.91

7

1.0

04

7 -

0.3

03

8 0

.91

8

JP

Y/U

SD

0.9

74

9 1

.66

93

0.1

99

0

.98

34

1.9

59

1 0

.02

2 *

*

0.9

76

2 1

.60

26

0.2

65

0

.98

46

2.0

03

3 0

.02

0 *

*

C

HF/

USD

0

.97

27

1.2

84

0 0

.39

0

0.9

92

2 0

.65

11

0.5

31

0

.96

35

1.8

36

4 0

.18

8

0.9

82

9 1

.82

53

0.0

83

*

C

AD

/USD

0

.99

56

0.2

40

1 0

.88

2

1.0

02

4 -

0.1

28

9 0

.86

3

0.9

93

3 0

.35

10

0.8

76

1

.00

01

-0

.00

61

0.8

68

SE

K/U

SD

0.9

75

6 0

.75

10

0.6

75

0

.98

73

0.3

77

4 0

.57

9

0.9

73

4 1

.20

39

0.4

44

0

.98

51

0.6

26

0 0

.54

8

D

NK

/USD

0

.97

67

0.8

63

2 0

.69

5

0.9

98

4 0

.08

94

0.8

20

0

.97

04

0.9

29

2 0

.71

3

0.9

92

0 0

.66

98

0.6

73

U

SD/A

UD

1

.00

07

-0

.04

19

0.9

27

1

.01

53

-0

.72

11

0.9

38

0

.99

30

0.3

42

3 0

.84

3

1.0

07

5 -

0.4

08

0 0

.90

1

U

SD/G

BP

0

.96

71

1.2

02

2 0

.51

7

0.9

83

4 1

.90

48

0.0

42

**

0

.96

66

1.2

17

5 0

.56

7

0.9

82

8 1

.85

46

0.0

58

*

JP

Y/U

SD

0.9

24

4 1

.50

55

0.3

67

0

.98

14

1.1

04

8 0

.29

1

0.9

20

3 1

.60

42

0.3

85

0

.97

69

1.2

85

3 0

.30

2

C

HF/

USD

0

.97

32

1.4

92

7 0

.32

7

0.9

91

5 0

.74

55

0.4

69

0

.97

21

1.4

89

1 0

.39

1

0.9

90

4 0

.78

68

0.5

64

C

AD

/USD

0

.97

85

1.5

81

0 0

.35

1

0.9

97

1 0

.45

22

0.7

00

0

.97

41

1.8

67

7 0

.24

3

0.9

92

6 1

.24

98

0.3

29

SE

K/U

SD

0.9

68

0 1

.98

63

0.1

75

0

.97

63

1.1

81

1 0

.28

4

0.9

65

9 1

.80

16

0.3

01

0

.97

42

1.2

55

7 0

.34

8

D

NK

/USD

0

.96

82

1.4

45

2 0

.44

0

0.9

75

3 1

.24

00

0.2

35

0

.96

07

1.4

33

3 0

.51

4

0.9

67

7 1

.24

69

0.3

42

U

SD/A

UD

0

.95

69

2.1

86

0 0

.10

7

0.9

77

5 1

.19

90

0.3

00

0

.95

04

2.4

31

3 0

.05

4 *

0

.97

08

1.2

87

7 0

.29

1

U

SD/G

BP

0

.88

99

1.2

89

5 0

.82

4

0.9

60

0 1

.33

46

0.4

08

0

.89

54

1.2

87

6 0

.77

6

0.9

65

9 1

.43

62

0.2

92

JP

Y/U

SD

0.9

25

4 1

.72

43

0.5

24

0

.99

71

0.1

47

0 0

.90

5

0.9

16

0 1

.90

44

0.3

76

0

.98

70

0.6

58

7 0

.68

1

C

HF/

USD

0

.94

09

1.6

31

0 0

.63

6

0.9

89

0 0

.98

57

0.5

77

0

.94

77

1.5

35

4 0

.65

0

0.9

96

1 0

.35

36

0.8

32

C

AD

/USD

0

.92

86

1.9

99

8 0

.39

9

0.9

90

8 0

.42

83

0.8

45

0

.91

76

1.9

25

9 0

.43

7

0.9

79

0 1

.54

16

0.2

29

SE

K/U

SD

0.9

08

2 2

.11

05

0.3

57

0

.98

48

0.5

04

0 0

.80

5

0.8

85

1 1

.70

55

0.5

10

0

.95

97

1.1

85

3 0

.37

8

D

NK

/USD

0

.93

91

1.8

68

3 0

.27

4

0.9

89

7 0

.69

40

0.5

63

0

.94

27

1.5

68

6 0

.64

3

0.9

93

5 0

.26

11

0.9

29

U

SD/A

UD

0

.91

71

2.1

48

5 0

.27

3

0.9

81

1 0

.89

50

0.5

21

0

.90

55

2.0

10

8 0

.30

0

0.9

68

7 1

.24

67

0.2

78

U

SD/G

BP

0

.87

05

2.2

93

8 0

.80

8

0.9

53

1 1

.29

33

0.7

80

0

.87

52

2.1

89

4 0

.83

0

0.9

58

2 1

.09

60

0.8

43

JP

Y/U

SD

0.8

02

8 2

.74

86

0.6

29

0

.93

39

1.5

57

8 0

.58

6

0.7

97

4 2

.80

75

0.5

80

0

.92

77

1.6

33

1 0

.50

3

C

HF/

USD

0

.74

88

2.1

95

1 0

.75

4

0.8

90

7 2

.69

71

0.0

80

*

0.7

52

2 2

.17

47

0.7

54

0

.89

47

2.6

42

0 0

.10

0 *

C

AD

/USD

0

.88

07

3.1

08

0 0

.37

6

0.9

63

5 1

.48

50

0.6

38

0

.87

01

3.0

71

6 0

.39

8

0.9

51

9 1

.93

54

0.3

47

SE

K/U

SD

0.8

67

8 1

.92

90

0.9

04

0

.92

47

1.8

04

8 0

.46

5

0.8

49

3 1

.83

51

0.9

26

0

.90

50

2.2

19

9 0

.18

7

D

NK

/USD

0

.84

69

2.8

74

0 0

.47

9

0.9

04

9 2

.91

10

0.0

42

**

0

.84

42

3.2

01

2 0

.38

4

0.9

02

0 2

.17

31

0.3

98

U

SD/A

UD

0

.82

43

2.3

79

2 0

.76

6

0.9

42

8 2

.00

04

0.3

42

0

.81

53

2.3

78

2 0

.75

7

0.9

32

5 2

.41

30

0.1

29

All

fun

dam

enta

ls

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

Cu

rren

cy p

air

SRid

ge

vs.

EWA

vs

.

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nR

olli

ng

regr

essi

on

Rec

urs

ive

regr

essi

on

Notes: This table presents the comparison of forecasting performance between, respectively, SRidge and EWA vs. the rolling and

recursive OLS for the real-time data set, based on decoupled fundamentals. Columns 1-6 show the comparison between SRidge

and OLS methods, while columns 7-12 for EWA vs. OLS. For each comparison, the Theil ratio (RMSE of the given method /

RMSE of the compared) and the DM statistic and its bootstrapped p-value are shown. The DM test is a one-sided test of equal

out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) of the machine-learning method

considered against the OLS methods. ***, **, and * denote statistical significance at the 1 %, 5 %, and 10 % levels. Theil ratios

< 1 are indicated in bold. Sample period for the exchange rates: February 1999 – December 2014.

53

Table D.13: Directional predictions of the exchange rates for the real-time data set based on decoupled fundamentals.W

hat

do

we

read

? EO

M -

FR

ED -

RT?

Co

up

led

or

dec

ou

ple

d?

RM

SE /

ver

sus

OLS

/ d

irec

tio

nal

ch

ange

s?

An

swer

:R

T -

Dec

ou

ple

d -

Dir

ecti

on

al

Dat

e:2

7-f

évr.

-18

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

0

.54

4 1

.11

82

0.1

07

0

.56

3 1

.60

40

0.0

41

**

0

.53

2 0

.68

38

0.2

36

0

.42

4 -

1.8

53

9 0

.97

4

JP

Y/U

SD

0.4

75

-0

.51

14

0.7

32

0

.48

1 -

0.4

61

1 0

.71

4

0.5

19

0.4

28

5 0

.32

2

0.5

44

0.9

95

4 0

.19

7

C

HF/

USD

0

.51

3 0

.26

31

0.3

68

0

.48

7 -

0.2

18

9 0

.56

7

0.4

68

-0

.72

92

0.7

61

0

.46

8 -

0.5

60

3 0

.61

9

C

AD

/USD

0

.52

5 0

.56

14

0.2

50

0

.46

8 -

0.7

19

9 0

.79

1

0.4

94

-0

.15

62

0.5

78

0

.52

5 0

.63

73

0.2

73

SE

K/U

SD

0.5

63

1.5

60

0 0

.05

4 *

0

.54

4 0

.89

97

0.2

14

0

.53

2 0

.77

79

0.2

07

0

.47

5 -

0.5

52

4 0

.60

5

D

NK

/USD

0

.52

9 0

.72

41

0.1

68

0

.51

6 0

.32

75

0.3

18

0

.49

0 -

0.1

83

7 0

.53

7

0.4

65

-0

.74

12

0.7

97

U

SD/A

UD

0

.48

1 -

0.4

18

3 0

.70

8

0.5

19

0.3

33

8 0

.45

6

0.5

00

0.0

00

0 0

.58

9

0.3

92

-2

.43

17

0.9

87

U

SD/G

BP

0

.49

4 -

0.1

59

1 0

.59

0

0.4

94

-0

.15

91

0.5

85

0

.51

9 0

.46

34

0.2

92

0

.53

8 0

.90

26

0.1

61

JP

Y/U

SD

0.5

06

0.0

93

9 0

.47

3

0.4

94

-0

.12

65

0.5

44

0

.55

1 0

.96

73

0.1

57

0

.51

3 0

.25

98

0.4

75

C

HF/

USD

0

.51

9 0

.44

24

0.2

58

0

.50

6 0

.14

44

0.3

92

0

.46

8 -

0.6

74

6 0

.71

9

0.4

75

-0

.43

56

0.6

26

C

AD

/USD

0

.44

9 -

1.2

79

5 0

.90

6

0.4

81

-0

.38

08

0.6

57

0

.50

0 0

.00

00

0.4

88

0

.52

5 0

.59

91

0.2

70

SE

K/U

SD

0.5

13

0.2

52

5 0

.38

1

0.4

43

-1

.37

94

0.9

28

0

.50

6 0

.13

03

0.4

43

0

.51

3 0

.28

02

0.3

27

D

NK

/USD

0

.49

0 -

0.1

83

7 0

.57

1

0.4

77

-0

.50

38

0.7

18

0

.49

7 -

0.0

64

7 0

.48

3

0.5

42

0.8

88

0 0

.12

8

U

SD/A

UD

0

.39

9 -

1.8

34

5 0

.97

4

0.4

87

-0

.31

83

0.6

15

0

.46

2 -

0.8

30

5 0

.80

3

0.4

75

-0

.60

06

0.7

15

U

SD/G

BP

0

.46

8 -

0.7

17

9 0

.76

9

0.4

37

-1

.18

40

0.9

11

0

.50

0 0

.00

00

0.4

82

0

.46

2 -

0.7

76

0 0

.81

9

JP

Y/U

SD

0.5

44

1.1

18

2 0

.11

2

0.6

14

2.9

18

1 0

.00

0 *

**

0.5

57

1.4

41

4 0

.04

3 *

*

0.5

51

1.1

27

9 0

.06

8 *

C

HF/

USD

0

.51

3 0

.27

50

0.3

19

0

.48

7 -

0.1

82

7 0

.50

2

0.4

49

-1

.03

00

0.8

08

0

.42

4 -

1.9

19

0 0

.95

1

C

AD

/USD

0

.47

5 -

0.5

90

2 0

.77

0

0.4

18

-2

.09

70

0.9

93

0

.44

9 -

1.2

16

6 0

.91

8

0.4

62

-0

.71

59

0.8

43

SE

K/U

SD

0.4

81

-0

.37

09

0.6

62

0

.46

2 -

0.6

95

9 0

.80

1

0.5

00

0.0

00

0 0

.49

8

0.5

38

0.9

25

4 0

.11

3

D

NK

/USD

0

.54

2 0

.90

22

0.1

53

0

.51

0 0

.21

61

0.4

07

0

.51

0 0

.15

07

0.3

71

0

.52

3 0

.42

24

0.3

25

U

SD/A

UD

0

.50

6 0

.15

75

0.4

81

0

.49

4 -

0.1

39

3 0

.64

4

0.4

62

-0

.94

27

0.8

49

0

.44

9 -

1.1

82

4 0

.83

8

U

SD/G

BP

0

.51

9 0

.47

77

0.2

91

0

.57

0 1

.12

77

0.1

13

0

.50

0 0

.00

00

0.4

74

0

.48

7 -

0.2

79

0 0

.60

7

JP

Y/U

SD

0.5

06

0.1

34

9 0

.53

3

0.5

38

0.8

01

2 0

.27

8

0.5

51

1.2

41

9 0

.09

6 *

0

.55

7 1

.40

67

0.0

56

*

C

HF/

USD

0

.55

1 1

.09

02

0.1

01

0

.50

6 0

.11

49

0.4

36

0

.48

1 -

0.3

71

5 0

.60

6

0.4

18

-2

.03

99

0.9

76

C

AD

/USD

0

.47

5 -

0.4

99

4 0

.81

6

0.4

87

-0

.30

58

0.7

92

0

.44

3 -

1.4

25

8 0

.94

2

0.4

37

-1

.37

36

0.9

46

SE

K/U

SD

0.4

49

-1

.10

06

0.8

98

0

.43

0 -

1.7

67

4 0

.98

1

0.5

19

0.3

97

0 0

.32

9

0.5

44

1.1

12

2 0

.07

2 *

D

NK

/USD

0

.49

0 -

0.2

06

7 0

.58

3

0.4

39

-1

.00

02

0.8

56

0

.51

0 0

.14

93

0.3

96

0

.52

3 0

.38

04

0.3

99

U

SD/A

UD

0

.48

7 -

0.3

18

3 0

.77

0

0.5

38

0.6

92

2 0

.37

0

0.4

62

-0

.94

27

0.8

75

0

.45

6 -

1.1

18

2 0

.85

0

All

fun

dam

enta

ls

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSr

idge

EWA

Cu

rren

cy p

air

PP

P f

un

dam

enta

l

Notes: This table presents the results of forecasting the direction of change of the end-of-month exchange rate 1-month ahead

by the rolling OLS, recursive OLS, SRidge and EWA methods for the real-time data set, based on decoupled fundamentals. For

each method, the share of correctly forecasted changes, the DM statistic and its bootstrapped p-value are shown. The DM tests

is a one-sided test of equal out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) of the

methods considered against the benchmark of a 50 % success rate. ***, **, and * denote statistical significance at the 1 %, 5 %,

and 10 % levels. Shares > 0.5 are indicated in bold.

54

Table D.14: Distributions of the differences in forecasting error between the no-change prediction and the method under scrutiny based ondecoupled fundamentals.

10

%2

5%

50

%7

5%

90

%1

0%

25

%5

0%

75

%9

0%

10

%2

5%

50

%7

5%

90

%1

0%

25

%5

0%

75

%9

0%

U

SD/G

BP

-1.3

2-0

.45

-0.0

10

.41

1.0

0-0

.55

-0.1

40

.00

0.1

20

.50

-0.2

1-0

.06

0.0

00

.04

0.2

5-1

.47

-0.3

80

.00

0.3

71

.65

JP

Y/U

SD-2

.32

-0.7

8-0

.06

0.5

31

.75

-0.3

0-0

.13

-0.0

10

.10

0.2

6-0

.02

0.0

00

.00

0.0

10

.02

-0.9

0-0

.11

0.0

00

.24

1.0

6

C

HF/

USD

-1.8

1-0

.51

0.0

00

.43

1.4

8-0

.78

-0.2

50

.00

0.2

60

.85

-0.1

4-0

.03

0.0

00

.06

0.2

1-1

.34

-0.2

90

.00

0.4

21

.76

C

AD

/USD

-0.6

5-0

.21

0.0

00

.19

0.7

6-0

.26

-0.1

1-0

.01

0.1

00

.27

-0.0

6-0

.02

0.0

00

.02

0.0

5-0

.05

-0.0

10

.00

0.0

10

.04

SE

K/U

SD-2

.84

-1.0

2-0

.07

0.9

02

.78

-1.1

1-0

.49

-0.0

30

.42

1.2

8-0

.33

-0.0

90

.00

0.0

80

.37

-1.1

9-0

.23

0.0

00

.32

1.4

1

D

NK

/USD

-2.2

8-0

.82

-0.0

20

.63

1.5

6-1

.27

-0.6

00

.01

0.4

51

.24

-0.0

8-0

.01

0.0

00

.03

0.1

0-1

.19

-0.3

00

.03

0.4

61

.57

U

SD/A

UD

-2.2

5-1

.13

-0.0

30

.92

2.6

6-0

.92

-0.3

6-0

.01

0.2

80

.71

-0.2

4-0

.08

0.0

00

.04

0.1

6-1

.00

-0.0

50

.00

0.0

30

.48

FR

F/U

SD-2

.39

-0.8

1-0

.04

0.6

32

.06

-1.1

7-0

.44

0.0

10

.31

1.1

1-0

.26

-0.0

40

.00

0.0

40

.30

-1.6

3-0

.55

0.1

00

.68

1.4

8

D

EM/U

SD-2

.05

-0.7

1-0

.04

0.4

11

.50

-0.4

4-0

.18

0.0

00

.09

0.2

7-0

.11

-0.0

20

.01

0.0

60

.16

-1.5

3-0

.33

0.0

90

.98

1.7

6

IT

L/U

SD-2

.38

-0.8

3-0

.04

0.4

72

.28

-2.1

7-0

.76

-0.1

60

.66

2.3

9-1

.67

-0.5

7-0

.11

0.6

01

.86

-1.4

6-0

.35

0.0

30

.83

3.1

5

N

LG/U

SD-2

.24

-0.5

3-0

.01

0.3

21

.30

-0.2

9-0

.11

-0.0

10

.06

0.1

5-0

.11

-0.0

20

.00

0.0

50

.14

-1.4

6-0

.39

0.0

50

.90

1.8

5

P

TE/U

SD-5

.02

-2.0

9-0

.21

1.5

44

.88

-4.4

3-1

.38

-0.0

90

.99

4.4

5-3

.72

-1.2

4-0

.13

1.1

24

.33

-2.7

2-0

.70

0.0

00

.75

3.9

4

U

SD/G

BP

-2.5

8-0

.64

-0.0

30

.54

2.5

9-0

.92

-0.2

50

.00

0.2

61

.36

-0.2

9-0

.06

0.0

00

.05

0.4

2-2

.26

-0.4

90

.00

0.3

52

.46

JP

Y/U

SD-2

.52

-0.9

20

.03

0.9

73

.19

-1.1

4-0

.27

0.0

00

.39

1.6

4-0

.23

-0.0

20

.00

0.0

40

.25

-1.5

7-0

.08

0.0

00

.16

1.7

8

C

HF/

USD

-3.4

1-0

.68

0.0

20

.79

2.7

6-1

.76

-0.3

20

.01

0.6

52

.44

-0.4

1-0

.06

0.0

00

.12

0.5

4-1

.67

-0.0

20

.00

0.0

92

.45

C

AD

/USD

-0.8

3-0

.26

0.0

20

.29

0.6

9-0

.28

-0.1

00

.00

0.1

00

.24

-0.0

20

.00

0.0

00

.00

0.0

1-0

.05

-0.0

10

.00

0.0

00

.02

SE

K/U

SD-3

.89

-1.4

0-0

.05

0.7

62

.76

-2.0

9-0

.55

-0.0

30

.21

1.5

5-0

.68

-0.1

20

.00

0.1

00

.72

-2.2

1-0

.57

0.0

00

.38

2.0

6

D

NK

/USD

-2.6

6-0

.68

-0.0

20

.47

1.9

8-1

.49

-0.3

9-0

.01

0.3

91

.37

-0.0

5-0

.01

0.0

00

.01

0.0

8-1

.92

-0.2

80

.00

0.6

22

.33

U

SD/A

UD

-3.8

4-1

.13

-0.0

31

.04

3.1

3-1

.60

-0.5

80

.00

0.4

91

.23

-0.6

3-0

.19

0.0

00

.12

0.3

6-0

.26

0.0

00

.00

0.0

00

.08

FR

F/U

SD-3

.36

-1.3

8-0

.01

0.6

22

.26

-2.6

4-1

.02

-0.0

90

.93

2.3

7-0

.92

-0.1

50

.02

0.2

50

.94

-3.5

7-1

.19

0.2

11

.29

3.2

1

D

EM/U

SD-2

.15

-0.7

4-0

.04

0.5

41

.82

-1.1

0-0

.23

0.0

20

.45

1.2

6-0

.46

-0.1

00

.03

0.2

50

.69

-3.4

7-1

.06

0.1

21

.72

3.3

8

IT

L/U

SD-4

.11

-1.6

9-0

.16

1.0

63

.89

-3.5

1-1

.28

-0.2

00

.88

2.8

2-2

.75

-0.8

2-0

.05

0.5

32

.35

-3.9

0-1

.20

0.0

01

.69

5.5

6

N

LG/U

SD-4

.07

-1.0

20

.00

1.0

43

.35

-0.8

3-0

.21

0.0

40

.39

1.0

7-0

.28

-0.0

50

.01

0.1

50

.31

-3.1

8-1

.01

0.0

71

.61

3.6

4

P

TE/U

SD-2

.10

-0.9

2-0

.11

0.6

72

.60

-1.4

5-0

.78

-0.0

80

.45

1.3

5-0

.70

-0.3

0-0

.02

0.2

60

.74

-0.8

0-0

.28

0.0

00

.22

0.5

7

U

SD/G

BP

-4.2

9-1

.28

-0.0

90

.66

2.7

6-2

.29

-0.7

10

.01

0.4

52

.22

-1.3

2-0

.27

0.0

00

.25

1.2

8-2

.49

-0.8

70

.00

0.8

12

.58

JP

Y/U

SD-4

.37

-1.0

4-0

.01

1.1

24

.69

-3.8

5-1

.12

-0.0

11

.00

4.2

5-2

.31

-0.5

40

.01

0.6

72

.47

-3.2

3-0

.90

0.0

01

.18

3.6

1

C

HF/

USD

-5.6

1-1

.74

-0.0

91

.44

4.8

2-4

.71

-1.2

50

.01

1.4

34

.21

-1.3

3-0

.37

0.0

10

.50

1.3

6-1

.76

-0.4

80

.01

0.5

81

.45

C

AD

/USD

-1.8

6-0

.64

-0.0

10

.36

1.0

9-0

.71

-0.2

6-0

.02

0.1

80

.59

-0.1

1-0

.01

0.0

00

.01

0.0

9-0

.09

-0.0

30

.00

0.0

20

.06

SE

K/U

SD-6

.77

-2.3

7-0

.35

0.6

52

.93

-3.0

5-0

.83

-0.0

50

.42

1.3

5-0

.85

-0.1

10

.00

0.1

70

.72

-2.4

1-0

.79

0.0

00

.71

2.4

5

D

NK

/USD

-5.1

5-1

.51

-0.1

40

.52

1.7

6-2

.26

-0.6

6-0

.01

0.3

70

.99

-0.0

2-0

.01

0.0

00

.01

0.0

1-0

.80

-0.2

8-0

.01

0.2

10

.67

U

SD/A

UD

-4.7

2-1

.57

-0.0

11

.03

3.6

6-2

.00

-0.6

7-0

.02

0.4

51

.50

-0.2

4-0

.02

0.0

00

.02

0.0

8-1

.67

-0.4

6-0

.01

0.4

31

.49

FR

F/U

SD-5

.48

-2.5

7-0

.12

1.1

83

.26

-5.3

1-2

.22

-0.1

70

.73

3.5

8-1

.30

-0.3

80

.02

0.3

30

.99

-2.9

7-1

.16

0.0

10

.75

2.7

1

D

EM/U

SD-5

.79

-1.4

00

.08

1.7

33

.64

-4.1

8-0

.94

0.0

81

.47

3.1

5-1

.63

-0.2

20

.01

0.4

51

.18

-3.9

6-1

.69

0.0

11

.47

4.3

9

IT

L/U

SD-4

.61

-1.9

8-0

.40

0.9

53

.87

-3.2

3-1

.41

-0.1

60

.83

3.4

2-1

.20

-0.4

10

.00

0.2

60

.83

-4.1

0-1

.15

0.0

01

.09

3.5

0

N

LG/U

SD-6

.38

-2.1

1-0

.31

0.8

93

.88

-3.8

1-1

.03

0.0

00

.82

3.1

0-0

.57

-0.0

90

.02

0.2

40

.62

-3.0

0-0

.96

0.0

01

.05

3.3

3

P

TE/U

SD-6

.51

-1.6

0-0

.21

1.3

84

.32

-3.8

0-1

.54

-0.2

90

.42

1.3

0-1

.77

-0.4

1-0

.06

0.1

40

.87

-2.0

8-0

.54

-0.0

10

.66

1.7

6

U

SD/G

BP

-9.1

8-2

.66

-0.1

61

.94

5.5

4-3

.11

-0.8

4-0

.07

0.6

62

.94

-0.6

4-0

.16

0.0

00

.19

0.8

4-3

.09

-0.9

3-0

.03

0.6

93

.44

JP

Y/U

SD-7

.77

-2.9

5-0

.08

2.0

17

.04

-5.7

8-2

.35

-0.0

51

.99

5.7

1-1

.58

-0.3

80

.02

0.5

52

.03

-3.6

2-1

.08

0.0

11

.21

3.4

6

C

HF/

USD

-11

.91

-4.0

6-0

.32

2.1

96

.89

-7.1

1-2

.06

-0.0

41

.68

5.7

4-0

.70

-0.1

40

.01

0.1

50

.52

-3.8

7-1

.23

0.0

11

.39

4.1

8

C

AD

/USD

-2.6

5-0

.88

0.0

00

.64

1.8

8-0

.82

-0.2

8-0

.01

0.2

20

.65

-0.0

6-0

.01

0.0

00

.01

0.0

5-0

.09

-0.0

30

.00

0.0

10

.05

SE

K/U

SD-9

.47

-3.4

9-0

.20

1.7

05

.17

-4.7

1-1

.63

-0.1

41

.18

4.0

7-0

.35

-0.0

80

.00

0.0

60

.34

-3.3

9-1

.40

0.0

11

.37

3.7

2

D

NK

/USD

-7.6

5-2

.45

-0.1

31

.43

5.4

7-3

.15

-0.9

5-0

.01

0.7

32

.79

-0.0

4-0

.01

0.0

00

.01

0.0

2-0

.53

-0.1

8-0

.01

0.1

20

.33

U

SD/A

UD

-9.1

4-3

.09

-0.1

52

.00

5.7

9-4

.24

-1.3

10

.00

0.8

62

.52

-0.4

6-0

.11

0.0

00

.07

0.2

6-2

.80

-0.2

60

.00

0.1

71

.44

FR

F/U

SD-8

.52

-3.1

00

.06

1.4

85

.26

-6.5

8-2

.19

0.0

01

.80

4.4

4-1

.53

-0.2

70

.00

0.1

80

.92

-4.6

4-1

.59

0.0

11

.47

4.5

2

D

EM/U

SD-7

.86

-2.3

8-0

.02

1.8

06

.15

-5.6

8-1

.89

-0.0

51

.75

5.0

6-1

.01

-0.0

90

.00

0.1

10

.83

-4.9

9-1

.86

0.0

51

.95

4.7

8

IT

L/U

SD-6

.21

-2.4

7-0

.16

1.2

46

.67

-5.7

7-1

.98

-0.1

61

.25

6.0

8-1

.55

-0.5

4-0

.06

0.3

31

.33

-5.3

9-1

.67

0.0

31

.38

6.0

5

N

LG/U

SD-9

.23

-3.0

9-0

.45

2.3

68

.71

-8.0

9-2

.47

0.0

01

.49

5.6

7-0

.47

-0.0

80

.02

0.2

30

.73

-4.1

1-1

.01

0.0

21

.67

4.2

3

P

TE/U

SD-4

.84

-2.5

3-0

.25

1.6

34

.40

-4.2

8-1

.87

-0.3

10

.51

2.0

7-0

.68

-0.2

6-0

.02

0.1

50

.57

-0.8

3-0

.36

-0.0

10

.22

0.9

2

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

All

fun

dam

enta

ls

Cu

rren

cy p

air

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSR

idge

EW

A

PP

P f

un

dam

enta

l

Notes: This table presents the results of forecasting end-of-month exchange rate 1-month ahead by the no-change prediction, the

rolling OLS, recursive OLS, SRidge and EWA methods based on decoupled fundamentals. For each method, the time-series of

the differences between the monthly forecasting errors of the no-change predictions and the ones of the method under scrutiny are

computed. The table reports the quantiles of these time-series at the 10%, 25%, 50%, 75% and 90% levels, to give an idea of the

distributions of the monthly forecasting errors. Distributions of errors that are shifted towards positive values indicate methods

that outperfom the no-change prediction. Sample period for the exchange rates: March 1973 – December 2014 unless shorter

(indicated in Appendix A).

55

Table D.15: 1-month ahead forecasts for the PPP, UIRP, monetary model and all fundamentals: shorter-period sample (1980 onwards,decoupled fundamentals).

Wh

at d

o w

e re

ad?

EOM

or

FRED

? C

ou

ple

d o

r d

eco

up

led

? R

MSE

or

dir

ecti

on

al c

han

ges?

An

swer

:EO

M -

De

cou

ple

d -

RM

SE S

ho

rter

Dat

e:m

ars-

18

Sho

rter

per

iod

198

0 o

nw

ard

s

No

ch

ange

RM

SE x

100

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

2.

634

3

0.99

99

0.1

32 0

.007

3 0

.063

*

0.99

81 0

.056

* 0

.140

5 0

.114

0.

9981

0.0

78 *

0.8

516

0.0

32 *

*

0.99

81 0

.202

0.2

321

0.2

61

JP

Y/U

SD

3.05

08

1.

0098

0.7

18 -

1.0

566

0.5

27

1.00

36 0

.843

-1.

0451

0.6

79

0.99

99 0

.338

0.3

785

0.1

42

0.99

80 0

.125

0.7

678

0.1

54

C

HF/

USD

3.

201

6

1.01

40 0

.891

-1.

729

7 0

.805

1.

0036

0.5

50 -

0.57

57 0

.404

0.

9992

0.1

98 0

.606

1 0

.066

*

0.99

51 0

.039

**

1.3

309

0.0

67

*

C

AD

/USD

2.

168

9

0.99

79

0.1

22 0

.244

8 0

.018

**

0.

9975

0.0

71 *

0.9

960

0.0

06 *

**

1.

0000

0.5

84 -

0.1

745

0.2

03

1.00

15 0

.990

-1.

8477

0.9

61

SE

K/U

SD

3.39

55

1.

0067

0.3

94 -

0.6

855

0.2

41

0.99

64 0

.057

* 0

.572

4 0

.029

**

1.

0006

0.4

62 -

0.1

691

0.2

12

0.99

51 0

.037

**

1.0

407

0.0

56

*

D

NK

/USD

3.

000

6

1.00

97 0

.729

-1.

444

2 0

.634

1.

0004

0.3

58 -

0.10

56 0

.164

0.

9985

0.1

06 0

.927

6 0

.018

**

0.

9918

0.0

05 *

**

1.9

845

0.0

04

**

*

U

SD/A

UD

3.

203

6

0.99

52

0.0

44 *

* 0

.334

8 0

.028

**

0.

9962

0.0

38 *

* 0

.915

1 0

.015

**

0.

9977

0.2

16 0

.471

3 0

.067

*

0.99

77 0

.183

0.6

150

0.0

90

*

FR

F/U

SD

3.03

52

1.

0136

0.7

02 -

1.1

107

0.6

63

1.00

56 0

.961

-1.

5651

0.8

63

1.00

45 0

.865

-1.

007

4 0

.642

0.

9944

0.1

23 0

.780

5 0

.11

2

D

EM

/USD

3.

048

2

1.02

90 0

.978

-1.

317

5 0

.736

1.

0147

0.9

19 -

1.22

86 0

.756

0.

9983

0.1

82 0

.822

2 0

.036

**

0.

9897

0.0

30 *

* 1

.410

6 0

.02

4 *

*

IT

L/U

SD

3.21

41

1.

0025

0.3

70 -

0.3

244

0.2

91

0.99

94 0

.301

0.0

590

0.1

98

0.99

86 0

.292

0.2

960

0.1

62

0.99

59 0

.214

0.4

274

0.1

69

N

LG/U

SD

3.06

34

1.

0143

0.8

85 -

1.3

641

0.7

81

1.01

56 0

.736

-0.

9276

0.6

05

0.99

78 0

.240

0.5

537

0.0

74 *

0.

9912

0.0

50 *

1.2

584

0.0

40

**

P

TE/U

SD

3.10

25

1.

0164

0.5

49 -

0.9

713

0.7

47

1.00

93 0

.518

-0.

5240

0.6

18

1.00

63 0

.483

-0.

404

5 0

.496

0.

9986

0.2

62 0

.135

6 0

.06

4 *

U

SD/G

BP

2.

634

3

1.04

00 0

.804

-1.

445

6 0

.683

1.

0344

0.7

60 -

1.39

49 0

.797

1.

0184

0.6

88 -

0.9

752

0.6

87

1.00

25 0

.377

-0.

2823

0.4

77

JP

Y/U

SD

3.05

08

1.

0151

0.3

17 -

0.9

629

0.3

80

1.00

56 0

.134

-0.

4047

0.2

85

1.00

04 0

.471

-0.

197

8 0

.298

0.

9951

0.0

59 *

0.9

300

0.1

42

C

HF/

USD

3.

201

6

1.01

83 0

.307

-1.

010

0 0

.456

1.

0170

0.3

00 -

0.89

07 0

.609

0.

9980

0.0

86 *

1.1

408

0.0

24 *

*

0.99

33 0

.022

**

1.3

955

0.0

27

**

C

AD

/USD

2.

168

9

1.00

66 0

.530

-0.

914

4 0

.337

1.

0031

0.4

01 -

0.72

83 0

.463

1.

0000

0.7

17 -

0.5

477

0.4

35

1.00

03 0

.621

-0.

3582

0.4

53

SE

K/U

SD

3.39

27

1.

0504

0.8

86 -

1.5

555

0.7

93

1.02

98 0

.786

-0.

9937

0.6

26

0.99

10 0

.058

* 0

.992

8 0

.031

**

0.

9833

0.0

50 *

0.8

884

0.1

23

D

NK

/USD

3.

000

6

1.02

08 0

.902

-1.

584

5 0

.740

1.

0068

0.6

77 -

0.89

11 0

.563

0.

9986

0.1

00 *

1.1

028

0.0

20 *

*

0.99

38 0

.035

**

0.8

994

0.0

79

*

U

SD/A

UD

3.

208

1

1.01

10 0

.259

-0.

716

6 0

.221

1.

0002

0.3

66 -

0.06

92 0

.137

0.

9997

0.3

66 0

.168

0 0

.137

0.

9986

0.2

27 0

.338

2 0

.18

4

FR

F/U

SD

3.03

52

1.

0234

0.9

07 -

1.4

493

0.8

19

1.02

07 0

.745

-0.

9752

0.6

62

1.00

01 0

.395

-0.

012

7 0

.241

0.

9902

0.0

88 *

0.6

242

0.1

29

D

EM

/USD

3.

048

2

1.03

58 0

.742

-1.

114

3 0

.673

1.

0226

0.8

07 -

1.02

70 0

.664

1.

0000

0.4

05 -

0.0

003

0.2

61

0.98

53 0

.039

**

1.0

974

0.0

35

**

IT

L/U

SD

3.21

41

1.

0044

0.2

83 -

0.1

777

0.2

58

1.01

75 0

.771

-1.

0810

0.7

31

0.99

09 0

.111

0.7

981

0.0

99 *

0.

9789

0.0

31 *

* 0

.952

5 0

.06

1 *

N

LG/U

SD

3.06

34

1.

0884

0.6

78 -

1.1

892

0.7

32

1.05

24 0

.646

-1.

0759

0.7

35

0.99

61 0

.177

0.6

878

0.0

94 *

0.

9877

0.0

63 *

0.9

017

0.0

61

*

PTE

/USD

U

SD/G

BP

2.

634

3

1.03

35 0

.752

-2.

356

6 0

.829

1.

0154

0.6

15 -

1.33

15 0

.584

0.

9900

0.0

33 *

* 0

.631

5 0

.035

**

1.

0058

0.3

10 -

0.49

96 0

.34

8

JP

Y/U

SD

3.05

36

1.

0216

0.2

34 -

1.3

911

0.4

06

1.01

80 0

.296

-1.

2058

0.5

88

1.00

58 0

.287

-0.

545

7 0

.423

0.

9999

0.0

32 *

* 0

.009

1 0

.18

6

C

HF/

USD

3.

216

9

1.03

05 0

.644

-1.

618

6 0

.515

1.

0137

0.5

98 -

1.34

70 0

.640

1.

0046

0.6

13 -

0.9

298

0.5

96

1.00

91 0

.414

-0.

8724

0.5

77

C

AD

/USD

2.

170

8

1.03

95 0

.472

-1.

405

0 0

.299

1.

0100

0.6

62 -

0.88

86 0

.374

1.

0131

0.8

27 -

0.9

607

0.5

68

1.00

90 0

.919

-1.

4540

0.7

91

SE

K/U

SD

3.39

27

1.

0467

0.9

41 -

2.4

612

0.8

72

1.01

95 0

.888

-2.

0119

0.8

87

1.00

87 0

.376

-0.

434

7 0

.307

1.

0057

0.3

33 -

0.53

62 0

.51

3

D

NK

/USD

3.

025

2

1.04

07 0

.979

-2.

351

4 0

.879

1.

0087

0.9

44 -

1.75

72 0

.845

1.

0010

0.7

40 -

0.6

841

0.4

59

1.00

62 0

.499

-0.

7246

0.4

49

U

SD/A

UD

3.

219

1

1.02

77 0

.512

-1.

393

0 0

.351

1.

0109

0.9

52 -

2.11

54 0

.922

1.

0014

0.6

12 -

0.3

317

0.2

53

1.00

28 0

.426

-0.

2994

0.2

69

FR

F/U

SD

3.03

52

1.

0488

0.7

12 -

1.6

674

0.8

42

1.04

53 0

.810

-1.

7522

0.9

04

1.00

06 0

.399

-0.

095

1 0

.260

1.

0116

0.5

30 -

0.72

74 0

.53

7

D

EM

/USD

3.

048

2

1.02

20 0

.370

-0.

784

6 0

.424

1.

0132

0.2

74 -

0.51

95 0

.381

0.

9993

0.1

09 0

.090

7 0

.221

1.

0019

0.1

56 -

0.09

93 0

.21

7

IT

L/U

SD

3.21

41

1.

0179

0.3

81 -

0.6

209

0.3

10

1.01

89 0

.467

-0.

7490

0.4

35

0.99

37 0

.148

0.5

077

0.0

81 *

1.

0004

0.3

07 -

0.02

11 0

.36

2

N

LG/U

SD

3.06

34

1.

0459

0.7

71 -

1.4

259

0.8

74

1.00

76 0

.514

-0.

5150

0.6

23

0.99

48 0

.140

0.7

099

0.1

11

1.00

12 0

.308

-0.

0773

0.2

15

P

TE/U

SD

U

SD/G

BP

2.

634

3

1.08

81 0

.287

-2.

995

4 0

.588

1.

0389

0.5

92 -

1.47

56 0

.341

0.

9934

0.0

39 *

* 0

.831

4 0

.027

**

1.

0050

0.3

68 -

0.51

08 0

.42

5

JP

Y/U

SD

3.05

36

1.

0604

0.1

83 -

2.0

425

0.1

59

1.02

81 0

.164

-1.

4731

0.3

62

1.00

29 0

.332

-0.

390

8 0

.368

0.

9978

0.0

20 *

* 0

.216

4 0

.20

2

C

HF/

USD

3.

216

9

1.10

53 0

.678

-2.

653

8 0

.518

1.

0301

0.3

13 -

1.69

70 0

.543

1.

0034

0.6

02 -

0.7

146

0.5

27

1.00

93 0

.402

-0.

8452

0.6

82

C

AD

/USD

2.

170

8

1.03

77 0

.164

-1.

519

2 0

.021

**

1.

0090

0.3

18 -

0.51

39 0

.033

**

1.

0035

0.8

45 -

1.0

152

0.6

40

1.00

23 0

.931

-1.

5095

0.8

79

SE

K/U

SD

3.39

27

1.

1693

0.8

93 -

1.6

888

0.2

73

1.11

65 0

.823

-1.

1405

0.3

24

1.01

04 0

.906

-1.

362

7 0

.841

0.

9895

0.0

44 *

* 0

.531

1 0

.22

6

D

NK

/USD

3.

025

2

1.07

63 0

.669

-2.

872

0 0

.592

1.

0293

0.6

31 -

1.55

07 0

.390

0.

9999

0.3

89 0

.060

9 0

.180

1.

0055

0.5

23 -

0.71

25 0

.51

8

U

SD/A

UD

3.

219

1

1.04

05 0

.087

* -

1.3

170

0.0

27 *

*

1.02

24 0

.731

-1.

5560

0.4

17

1.00

02 0

.441

-0.

049

5 0

.182

0.

9981

0.1

30 0

.151

0 0

.17

5

FR

F/U

SD

3.03

52

1.

1011

0.5

95 -

1.8

930

0.7

05

1.04

75 0

.643

-1.

3648

0.6

38

0.99

92 0

.336

0.1

564

0.1

65

1.01

27 0

.420

-0.

6667

0.5

56

D

EM

/USD

3.

048

2

1.07

21 0

.571

-1.

345

8 0

.295

1.

0496

0.5

91 -

1.10

94 0

.359

0.

9948

0.0

81 *

0.7

301

0.0

64 *

0.

9938

0.1

00 *

0.3

100

0.1

82

IT

L/U

SD

3.21

41

0.

996

7 0

.063

* 0

.065

5 0

.030

**

1.

0259

0.1

85 -

0.64

82 0

.299

0.

9911

0.1

34 0

.776

4 0

.056

*

0.99

05 0

.126

0.4

129

0.1

70

N

LG/U

SD

3.06

34

1.

0656

0.3

48 -

1.5

904

0.7

19

1.05

79 0

.760

-1.

7240

0.8

78

0.99

47 0

.138

0.7

694

0.1

08

0.99

42 0

.152

0.3

015

0.1

79

P

TE/U

SD

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

All

fun

dam

enta

ls

Cu

rren

cy p

air

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSR

idge

EW

A

PP

P f

un

dam

enta

l

Notes: This table presents the results of forecasting end-of-month exchange rate 1-month ahead by the no-change prediction, the

rolling OLS, recursive OLS, SRidge and EWA methods for decoupled fundamentals for the time period 1980–2014. Column 1

shows the RMSE values for the no-change prediction. For each method, the Theil ratio (RMSE of the given method / RMSE of

the no-change prediction), the CW-p-values, the DM statistic and its bootstrapped p-value are shown. The CW and DM tests are

one-sided tests of equal out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) for the

methods considered compared to the no-change prediction. ***, **, and * denote statistical significance at the 1 %, 5 %, and 10 %

levels. Theil ratios < 1 are indicated in bold.

56

Table D.16: Economic criterion for evaluating forecasts for the EWA algorithm without the DNK/USD exchange rate.

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

EWA

/ c

= 0

.510

.36

11.3

81

46

17

70.

600.

750.

631

.06

EWA

/ c

= 1

11.4

312

.83

24

12

89

0.61

0.81

0.63

1.0

4

EWA

/ c

= 2

10.8

712

.36

22

62

56

0.56

0.75

0.61

0.9

4

EWA

/ c

= 5

11.1

112

.33

19

32

23

0.58

0.74

0.62

0.9

0

EWA

/ c

= 1

011

.11

12.3

31

93

22

30.

580.

740.

620

.90

EWA

/ n

o c

11.1

112

.33

19

32

23

0.58

0.74

0.62

0.9

0

EWA

/ c

= 0

.59

.92

11.8

42

46

28

80.

570.

820.

601

.38

EWA

/ c

= 1

11.0

413

.32

32

64

09

0.58

0.86

0.60

1.5

1

EWA

/ c

= 2

10.4

912

.70

32

44

03

0.54

0.82

0.59

1.5

7

EWA

/ c

= 5

10.7

312

.26

27

63

61

0.55

0.79

0.60

1.4

8

EWA

/ c

= 1

010

.73

12.2

62

76

36

10.

550.

790.

601

.48

EWA

/ n

o c

10.7

312

.26

27

63

61

0.55

0.79

0.60

1.4

8

EWA

/ c

= 0

.510

.33

9.38

-36

-80.

590.

550.

620

.72

EWA

/ c

= 1

11.4

010

.75

73

12

10.

600.

640.

620

.81

EWA

/ c

= 2

10.8

59.

701

24

10.

560.

540.

600

.67

EWA

/ c

= 5

11.0

89.

82-2

27

0.57

0.54

0.61

0.6

6

EWA

/ c

= 1

011

.08

9.82

-22

70.

570.

540.

610

.66

EWA

/ n

o c

11.0

89.

82-2

27

0.57

0.54

0.61

0.6

6

EWA

/ c

= 0

.510

.33

9.71

20

61

0.59

0.61

0.62

1.0

7

EWA

/ c

= 1

11.4

010

.61

59

13

30.

600.

630.

621

.03

EWA

/ c

= 2

10.8

510

.47

75

15

30.

560.

600.

601

.04

EWA

/ c

= 5

11.0

810

.27

38

12

80.

570.

580.

611

.00

EWA

/ c

= 1

011

.08

10.2

73

81

28

0.57

0.58

0.61

1.0

0

EWA

/ n

o c

11.0

810

.27

38

12

80.

570.

580.

611

.00

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

s fu

nd

amen

tals

All

fun

dam

enta

ls

Po

rtfo

lio /

c co

nst

rain

t

An

nu

aliz

ed

Ret

urn

s (i

n %

)P

erfo

rman

ce f

ee

of

po

rtfo

lio w

.r.t

.

no

ch

ange

(in

bp

s)

Pre

miu

m r

etu

rn

of

po

rtfo

lio w

.r.t

.

no

ch

ange

(in

bp

s)

Shar

pe

rati

os

Sort

ino

rat

ios

Notes: This table presents the comparison of performance of portfolios formed using forecasts from the EWA algorithm and the

no-change prediction based on decoupled fundamentals without the DNK/USD exchange rate, using the procedure described in

Section 5.4. Column 1 gives the conditions under which the portfolios were constructed. In columns 2 and 3 the annualized returns

are given for the EWA and no-change prediction-based portfolios respectively. Column 4 shows the performance fee in basis points

(bps) while column 5 the premium return (in bps) of the EWA-based portfolios relative to the ones formed based on the no-change

forecast. The Sharpe and Sortino ratios for portfolios created using the two competing forecasting methods are reported in columns

6-10. For details on how these portfolios were created please consult Appendix C.3. The results for the portfolios based on the

no-change exchange rate predictions vary because of the constraints on weights and because for different sets of fundamentals

different time periods are considered depending on the availability of data.

57

Table D.17: Economic criterion for evaluating forecasts for the EWA algorithm with an additional summation-to-less-than-1 constraint on theweights.

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

EWA

/ c

= 0

.51

0.7

611

.77

15

51

85

0.64

0.80

0.67

1.1

3

EWA

/ c

= 1

11.

55

13.0

72

44

30

10.

610.

820.

631

.08

EWA

/ c

= 2

11.

58

13.5

42

76

32

50.

610.

840.

631

.05

EWA

/ c

= 5

11.

72

13.3

52

31

27

80.

630.

820.

671

.06

EWA

/ c

= 1

01

1.6

913

.39

23

62

84

0.63

0.83

0.67

1.0

8

EWA

/ n

o c

11.

69

13.3

92

36

28

40.

630.

830.

671

.08

EWA

/ c

= 0

.51

0.3

211

.89

19

62

39

0.60

0.79

0.64

1.2

9

EWA

/ c

= 1

11.

16

13.0

22

79

36

50.

590.

820.

611

.34

EWA

/ c

= 2

11.

19

12.9

92

83

37

50.

590.

820.

611

.43

EWA

/ c

= 5

11.

33

12.1

41

97

27

50.

600.

770.

651

.35

EWA

/ c

= 1

01

1.3

012

.11

20

42

80

0.60

0.78

0.65

1.3

8

EWA

/ n

o c

11.

30

12.1

12

04

28

00.

600.

780.

651

.38

EWA

/ c

= 0

.51

0.7

08.

54-1

17-8

70.

630.

500.

670

.67

EWA

/ c

= 1

11.

46

9.34

-56

60.

600.

520.

630

.68

EWA

/ c

= 2

11.

51

8.89

-114

-50

0.60

0.47

0.63

0.6

2

EWA

/ c

= 5

11.

59

8.46

-168

-101

0.62

0.43

0.66

0.6

2

EWA

/ c

= 1

01

1.5

58.

41-1

75-1

080.

620.

430.

660

.61

EWA

/ n

o c

11.

55

8.41

-175

-108

0.62

0.43

0.66

0.6

1

EWA

/ c

= 0

.51

0.7

08.

52-1

00-6

30.

630.

510.

670

.84

EWA

/ c

= 1

11.

46

9.56

-12

62

0.60

0.56

0.63

0.9

1

EWA

/ c

= 2

11.

51

9.20

-81

30.

600.

500.

630

.89

EWA

/ c

= 5

11.

59

8.57

-166

-82

0.62

0.44

0.66

0.8

0

EWA

/ c

= 1

01

1.5

58.

48-1

75-9

10.

620.

430.

660

.79

EWA

/ n

o c

11.

55

8.48

-175

-91

0.62

0.43

0.66

0.7

9

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

s fu

nd

amen

tals

All

fun

dam

enta

ls

Po

rtfo

lio /

c co

nst

rain

t

An

nu

aliz

ed

Ret

urn

s (i

n %

)P

erfo

rman

ce f

ee

of

po

rtfo

lio w

.r.t

.

no

ch

ange

(in

bp

s)

Pre

miu

m r

etu

rn

of

po

rtfo

lio w

.r.t

.

no

ch

ange

(in

bp

s)

Shar

pe

rati

os

Sort

ino

rat

ios

Notes: This table presents the comparison of performance of portfolios formed using forecasts from the EWA algorithm and the

no-change prediction based on decoupled fundamentals with an additional summation-to-less-than-1 constraint on the weights,

using the procedure described in Section 5.4. Column 1 gives the conditions under which the portfolios were constructed. In

columns 2 and 3 the annualized returns are given for the EWA and no-change prediction-based portfolios respectively. Column 4

shows the performance fee in basis points (bps) while column 5 the premium return (in bps) of the EWA-based portfolios relative

to the ones formed based on the no-change forecast. The Sharpe and Sortino ratios for portfolios created using the two competing

forecasting methods are reported in columns 6-10. For details on how these portfolios were created please consult Appendix C.3.

The results for the portfolios based on the no-change exchange rate predictions vary because of the constraints on weights and

because for different sets of fundamentals different time periods are considered depending on the availability of data.

58

Table D.18: Economic criterion for evaluating forecasts for the EWA algorithm without the DNK/USD exchange rate but with an additionalsummation-to-less-than-1 constraint on the weights.

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

No

ch

ange

(w

ith

c co

nst

rain

t)P

ort

folio

EWA

/ c

= 0

.51

0.2

41

1.3

01

58

18

50

.60

0.7

60

.64

1.0

8

EWA

/ c

= 1

10

.89

12

.71

28

83

40

0.5

60

.80

0.5

81

.05

EWA

/ c

= 2

10

.57

12

.28

25

22

82

0.5

40

.74

0.5

80

.94

EWA

/ c

= 5

10

.73

12

.20

22

32

53

0.5

40

.72

0.5

90

.89

EWA

/ c

= 1

01

0.7

31

2.2

02

23

25

30

.54

0.7

20

.59

0.8

9

EWA

/ n

o c

10

.73

12

.20

22

32

53

0.5

40

.72

0.5

90

.89

EWA

/ c

= 0

.59

.81

11

.66

23

72

75

0.5

60

.81

0.6

01

.37

EWA

/ c

= 1

10

.51

13

.17

36

74

52

0.5

30

.84

0.5

61

.48

EWA

/ c

= 2

10

.19

12

.49

33

84

18

0.5

10

.80

0.5

61

.53

EWA

/ c

= 5

10

.35

12

.13

30

43

91

0.5

20

.77

0.5

71

.46

EWA

/ c

= 1

01

0.3

51

2.1

33

04

39

10

.52

0.7

70

.57

1.4

6

EWA

/ n

o c

10

.35

12

.13

30

43

91

0.5

20

.77

0.5

71

.46

EWA

/ c

= 0

.51

0.2

29

.09

-35

-90

.59

0.5

50

.63

0.7

4

EWA

/ c

= 1

10

.86

10

.61

11

71

68

0.5

50

.63

0.5

80

.80

EWA

/ c

= 2

10

.54

9.6

84

87

70

.53

0.5

40

.58

0.6

7

EWA

/ c

= 5

10

.70

9.6

83

16

10

.54

0.5

30

.58

0.6

5

EWA

/ c

= 1

01

0.7

09

.68

31

61

0.5

40

.53

0.5

80

.65

EWA

/ n

o c

10

.70

9.6

83

16

10

.54

0.5

30

.58

0.6

5

EWA

/ c

= 0

.51

0.2

29

.26

-11

28

0.5

90

.58

0.6

31

.02

EWA

/ c

= 1

10

.86

10

.39

93

17

10

.55

0.6

10

.58

1.0

2

EWA

/ c

= 2

10

.54

10

.28

90

16

80

.53

0.5

80

.58

1.0

1

EWA

/ c

= 5

10

.70

10

.12

64

15

60

.54

0.5

70

.58

0.9

8

EWA

/ c

= 1

01

0.7

01

0.1

26

41

56

0.5

40

.57

0.5

80

.98

EWA

/ n

o c

10

.70

10

.12

64

15

60

.54

0.5

70

.58

0.9

8

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

s fu

nd

amen

tals

All

fun

dam

enta

ls

Po

rtfo

lio /

c co

nst

rain

t

An

nu

aliz

ed

Ret

urn

s (i

n %

)P

erfo

rman

ce f

ee

of

po

rtfo

lio w

.r.t

.

no

ch

ange

(in

bp

s)

Pre

miu

m r

etu

rn

of

po

rtfo

lio w

.r.t

.

no

ch

ange

(in

bp

s)

Shar

pe

rati

os

Sort

ino

rat

ios

This table presents the comparison of performance of portfolios formed using forecasts from the EWA algorithm and the no-change

prediction based on decoupled fundamentals with an additional summation-to-less-than-1 constraint on the weights and without

the DNK/USD exchange rate, using the procedure described in Section 5.4. Column 1 gives the conditions under which the

portfolios were constructed. In columns 2 and 3 the annualized returns are given for the EWA and no-change prediction-based

portfolios respectively. Column 4 shows the performance fee in basis points (bps) while column 5 the premium return (in bps) of

the EWA-based portfolios relative to the ones formed based on the no-change forecast. The Sharpe and Sortino ratios for portfolios

created using the two competing forecasting methods are reported in columns 6-10. For details on how these portfolios were created

please consult Appendix C.3. The results for the portfolios based on the no-change exchange rate predictions vary because of

the constraints on weights and because for different sets of fundamentals different time periods are considered depending on the

availability of data.

59

Table D.19: 1-month ahead forecasts for the PPP, UIRP, monetary model and all fundamentals based on decoupled fundamentals, for a sampleFebruary 1999 – December 2014.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:EO

M -

De

cou

ple

d -

RM

SE99

-14

per

iod

Dat

e:2-

mar

s-18

No

ch

ange

RM

SE x

100

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

2.

5258

1.

0085

0.1

70 -

0.5

437

0.1

61

1.00

68 0

.286

-0.

6949

0.3

57

0.99

70 0

.259

0.1

330

0.1

01

1.00

16

0.6

05 -

0.24

54 0

.41

2

JP

Y/U

SD

2.76

10

1.02

23 0

.543

-0

.854

0 0

.353

1.

0139

0.8

26 -

0.99

46 0

.596

0.

9975

0.1

50 0

.55

66 0

.065

*

0.99

81 0

.179

0.6

756

0.0

86

*

C

HF/

USD

3.

1193

1.

0376

0.9

20 -

1.8

890

0.8

73

1.02

47 0

.933

-2.

0651

0.9

70

1.00

82 0

.673

-0.

822

6 0

.568

0.

9999

0.4

10 0

.022

2 0

.61

2

C

AD

/USD

2.

7942

1.

0126

0.3

63 -

0.6

230

0.2

12

1.00

05 0

.276

-0.

0411

0.1

04

1.00

84 0

.732

-1.

072

1 0

.702

1.

004

8 0

.882

-1.

2080

0.8

59

SE

K/U

SD

3.51

00

1.02

45 0

.295

-1

.173

4 0

.488

1.

0137

0.3

98 -

0.62

53 0

.357

0.

9939

0.2

06 0

.20

98 0

.086

*

0.99

50 0

.104

0.7

234

0.1

08

D

NK

/USD

3.

0702

1.

0356

0.7

10 -

0.8

647

0.3

58

1.00

52 0

.448

-0.

3771

0.2

43

1.00

16 0

.289

-0.

090

5 0

.178

0.

9972

0.2

24 0

.389

5 0

.27

6

USD

/AU

D

3.84

42

0.99

93 0

.083

* 0

.028

4 0

.139

0.

9866

0.0

41 *

* 0

.739

4 0

.069

*

0.99

99 0

.301

0.0

030

0.2

82

1.00

17

0.5

61 -

0.26

65 0

.46

1

U

SD/G

BP

2.

5258

1.

0115

0.3

85 -

0.5

488

0.1

34

1.02

02 0

.910

-1.

9441

0.9

45

0.99

79 0

.134

0.9

643

0.0

23 *

*

0.99

72 0

.034

**

1.0

874

0.0

79

*

JP

Y/U

SD

2.76

10

1.06

30 0

.747

-1

.064

4 0

.353

1.

0290

0.6

18 -

1.32

38 0

.691

1.

0053

0.9

47 -

1.4

617

0.9

02

0.99

97 0

.361

0.2

149

0.2

57

C

HF/

USD

3.

1193

1.

0229

0.4

08 -

1.1

126

0.4

60

1.01

00 0

.582

-0.

7324

0.4

16

1.00

07 0

.538

-0.

252

5 0

.285

0.

9988

0.2

59 0

.283

2 0

.32

3

C

AD

/USD

2.

7942

1.

0182

0.6

81 -

1.4

171

0.5

94

1.01

00 0

.878

-1.

3278

0.7

09

1.00

67 0

.828

-0.

973

8 0

.573

1.

001

2 0

.703

-0.

5686

0.5

90

SE

K/U

SD

3.50

47

1.02

66 0

.509

-1

.174

0 0

.439

1.

0278

0.9

52 -

1.51

56 0

.808

0.

9999

0.3

05 0

.01

25 0

.147

0.

9972

0.2

15 0

.380

6 0

.25

2

D

NK

/USD

3.

0702

1.

0354

0.7

13 -

1.0

895

0.3

49

1.03

38 0

.908

-1.

2070

0.6

20

1.00

45 0

.609

-0.

427

4 0

.276

0.

9954

0.1

54 0

.600

1 0

.16

9

USD

/AU

D

3.82

57

1.04

17 0

.943

-2

.013

0 0

.867

1.

0326

0.9

39 -

1.47

54 0

.814

1.

0071

0.6

21 -

1.1

503

0.7

33

0.99

96 0

.413

0.0

782

0.2

96

U

SD/G

BP

2.

5258

1.

1494

0.1

77 -

1.3

749

0.3

79

1.05

17 0

.926

-1.

7443

0.8

09

0.99

82 0

.067

* 0

.05

95 0

.095

*

0.99

95 0

.312

0.0

349

0.1

65

JP

Y/U

SD

2.76

59

1.07

86 0

.409

-1

.466

2 0

.318

1.

0320

0.6

20 -

1.17

96 0

.426

1.

0153

0.7

37 -

1.0

185

0.5

81

1.00

98

0.5

09 -

0.65

69 0

.44

5

C

HF/

USD

3.

1525

1.

0950

0.7

77 -

1.4

593

0.5

38

1.05

03 0

.898

-1.

5718

0.8

57

1.00

24 0

.787

-0.

801

0 0

.642

1.

012

5 0

.825

-1.

1172

0.8

76

C

AD

/USD

2.

8019

1.

1202

0.2

55 -

1.3

916

0.3

28

1.04

32 0

.399

-1.

0928

0.4

50

1.03

55 0

.781

-1.

093

7 0

.670

1.

011

0 0

.847

-1.

1910

0.7

98

SE

K/U

SD

3.50

47

1.11

82 0

.632

-1

.586

1 0

.456

1.

0672

0.8

99 -

2.09

26 0

.919

1.

0456

0.8

67 -

1.0

169

0.6

23

1.01

71

0.8

83 -

1.10

29 0

.82

5

D

NK

/USD

3.

1342

1.

1042

0.4

71 -

1.4

754

0.5

39

1.03

52 0

.774

-1.

7878

0.8

81

1.00

90 0

.694

-0.

886

2 0

.656

1.

015

2 0

.822

-1.

1638

0.7

81

U

SD/A

UD

3.

8947

1.

1123

0.2

56 -

1.4

525

0.5

19

1.04

44 0

.637

-1.

5648

0.7

81

1.00

47 0

.460

-0.

370

9 0

.377

1.

003

8 0

.386

-0.

2533

0.5

02

U

SD/G

BP

2.

5258

1.

1026

0.0

16 *

* -

1.0

425

0.0

39 *

*

1.02

96 0

.191

-1.

0780

0.2

45

0.99

32 0

.080

* 0

.21

18 0

.066

*

0.99

95 0

.306

0.0

464

0.2

01

JP

Y/U

SD

2.76

59

1.14

23 0

.059

* -

1.9

922

0.1

86

1.10

92 0

.711

-1.

9643

0.7

09

1.01

43 0

.768

-0.

970

2 0

.535

1.

010

9 0

.580

-0.

7225

0.5

35

C

HF/

USD

3.

1525

1.

3213

0.8

52 -

1.9

938

0.3

56

1.14

86 0

.969

-2.

4218

0.9

14

1.00

09 0

.678

-0.

538

9 0

.527

1.

005

9 0

.885

-1.

2726

0.9

23

C

AD

/USD

2.

8019

1.

2338

0.4

30 -

1.6

283

0.1

05

1.05

47 0

.395

-1.

7072

0.5

78

1.02

17 0

.759

-1.

017

9 0

.637

1.

010

0 0

.879

-1.

2133

0.8

42

SE

K/U

SD

3.50

47

1.13

06 0

.369

-1

.300

5 0

.041

**

1.

0977

0.4

52 -

1.45

93 0

.355

1.

0322

0.6

21 -

0.7

284

0.4

48

1.01

10

0.5

13 -

0.55

62 0

.61

8

D

NK

/USD

3.

1342

1.

2111

0.5

40 -

2.8

525

0.7

09

1.05

68 0

.398

-1.

1348

0.2

18

1.01

49 0

.736

-0.

910

2 0

.645

1.

012

4 0

.746

-0.

8549

0.6

83

U

SD/A

UD

3.

8947

1.

2327

0.2

05 -

1.5

940

0.1

01

1.03

91 0

.134

-0.

8867

0.1

49

1.00

63 0

.446

-0.

444

5 0

.428

1.

004

5 0

.395

-0.

2805

0.4

64

All

fun

dam

enta

ls

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

Cu

rren

cy p

air

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSR

idge

EW

A

Notes: This table presents the results of forecasting end-of-month exchange rate 1-month ahead by the no-change prediction, the

rolling OLS, recursive OLS, SRidge and EWA methods for decoupled fundamentals on a sample February 1999 – December 2014.

Column 1 shows the RMSE values for the no-change prediction. For each method, the Theil ratio (RMSE of the given method /

RMSE of the no-change prediction), the CW-p-values, the DM statistic and its bootstrapped p-value are shown. The CW and DM

tests are one-sided tests of equal out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1)

for the methods considered compared to the no-change prediction. ***, **, and * denote statistical significance at the 1 %, 5 %,

and 10 % levels. Theil ratios < 1 are indicated in bold.

60

Table D.20: 1-month ahead forecasts on the real-time data set for the PPP, UIRP, monetary model and all fundamentals based on decoupledfundamentals, for a sample February 1999 – December 2014.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:R

T -

De

cou

ple

d -

RM

SE99

-14

per

iod

Dat

e:2-

mar

s-18

No

ch

ange

RM

SE x

10

0Th

eil r

atio

CW

p-v

alu

eD

M s

tati

stic

DM

p-v

alu

eTh

eil r

atio

CW

p-v

alu

eD

M s

tati

stic

DM

p-v

alu

eTh

eil r

atio

CW

p-v

alu

eD

M s

tati

stic

DM

p-v

alu

eTh

eil r

atio

CW

p-v

alu

eD

M s

tati

stic

DM

p-v

alu

e

U

SD/G

BP

2.

5258

1.

0002

0.0

70 *

-0.

0118

0.0

60 *

0.

9957

0.0

55 *

0.2

563

0.0

68 *

0.

9881

0.1

34 0

.407

0 0

.062

*

0.99

96 0

.401

0.0

751

0.2

77

JP

Y/U

SD

2.76

10

1.01

88 0

.482

-0.

8736

0.3

51

1.01

41 0

.823

-1.

3547

0.7

71

0.99

35 0

.093

* 0

.697

8 0

.040

**

0.

9956

0.0

69 *

0.9

662

0.0

51

*

C

HF/

USD

3.

1193

1.

0403

0.8

63 -

1.53

31 0

.689

1.

0208

0.9

21 -

1.60

44 0

.858

1.

0127

0.7

24 -

0.8

081

0.5

19

1.00

15 0

.530

-0.

2597

0.6

36

C

AD

/USD

2.

7942

1.

0048

0.1

66 -

0.19

70 0

.096

*

0.99

32 0

.116

0.4

592

0.0

44 *

*

1.00

85 0

.722

-1.

119

7 0

.735

1.

0057

0.8

96 -

1.15

90 0

.82

0

SE

K/U

SD

3.51

00

1.02

69 0

.333

-1.

3824

0.6

32

1.01

39 0

.468

-0.

6405

0.3

79

0.99

86 0

.214

0.0

478

0.1

59

0.99

60 0

.147

0.6

227

0.0

80

*

D

NK

/USD

3.

0702

1.

0343

0.7

82 -

0.99

13 0

.422

1.

0092

0.8

40 -

0.91

07 0

.543

1.

0070

0.4

01 -

0.3

545

0.2

79

0.99

88 0

.319

0.1

613

0.3

94

U

SD/A

UD

3.

8442

1.

0052

0.1

50 -

0.24

99 0

.248

0.

9867

0.0

45 *

* 0

.736

9 0

.088

*

1.01

13 0

.396

-0.

977

2 0

.739

1.

0024

0.6

23 -

0.38

18 0

.48

4

U

SD/G

BP

2.

5258

1.

0117

0.3

87 -

0.55

60 0

.135

1.

0202

0.9

10 -

1.94

39 0

.926

0.

9979

0.1

34 0

.964

3 0

.017

**

0.

9972

0.0

34 *

* 1

.086

3 0

.06

6 *

JP

Y/U

SD

2.76

10

1.06

30 0

.747

-1.

0644

0.3

78

1.02

90 0

.618

-1.

3241

0.7

26

1.00

53 0

.947

-1.

462

7 0

.914

0.

9997

0.3

61 0

.213

7 0

.28

8

C

HF/

USD

3.

1193

1.

0229

0.4

07 -

1.11

34 0

.467

1.

0100

0.5

81 -

0.73

07 0

.455

1.

0007

0.5

38 -

0.2

520

0.3

19

0.99

88 0

.259

0.2

838

0.3

44

C

AD

/USD

2.

7942

1.

0183

0.6

81 -

1.41

86 0

.614

1.

0100

0.8

78 -

1.32

79 0

.729

1.

0067

0.8

28 -

0.9

738

0.6

14

1.00

12 0

.703

-0.

5674

0.5

97

SE

K/U

SD

3.50

47

1.02

66 0

.509

-1.

1752

0.4

79

1.02

78 0

.952

-1.

5146

0.8

28

0.99

99 0

.305

0.0

122

0.1

85

0.99

72 0

.215

0.3

808

0.2

71

D

NK

/USD

3.

0702

1.

0354

0.7

13 -

1.08

99 0

.355

1.

0338

0.9

08 -

1.20

61 0

.622

1.

0045

0.6

09 -

0.4

273

0.2

80

0.99

54 0

.154

0.6

002

0.1

76

U

SD/A

UD

3.

8257

1.

0416

0.9

43 -

2.01

29 0

.874

1.

0326

0.9

39 -

1.47

49 0

.810

1.

0071

0.6

21 -

1.1

503

0.7

48

0.99

96 0

.413

0.0

777

0.3

18

U

SD/G

BP

2.

5258

1.

1412

0.2

42 -

1.28

63 0

.222

1.

0432

0.9

80 -

2.63

39 0

.980

0.

9847

0.0

48 *

* 0

.562

2 0

.038

**

0.

9932

0.1

36 0

.435

8 0

.08

5 *

JP

Y/U

SD

2.76

59

1.07

52 0

.243

-1.

2117

0.2

55

1.01

39 0

.154

-0.

4874

0.1

66

1.01

27 0

.794

-1.

195

4 0

.718

1.

0003

0.1

79 -

0.03

86 0

.23

9

C

HF/

USD

3.

1525

1.

0634

0.8

96 -

1.40

40 0

.423

1.

0169

0.6

52 -

1.19

13 0

.632

1.

0048

0.9

21 -

1.4

640

0.9

05

1.01

34 0

.912

-1.

5559

0.9

08

C

AD

/USD

2.

8019

1.

1077

0.9

66 -

1.67

15 0

.507

1.

0336

0.9

72 -

1.52

07 0

.728

1.

0238

0.7

05 -

0.7

849

0.5

02

1.00

94 0

.803

-0.

9744

0.7

52

SE

K/U

SD

3.50

47

1.13

12 0

.656

-1.

5253

0.4

03

1.04

34 0

.904

-1.

6997

0.7

67

1.02

53 0

.642

-0.

703

5 0

.411

0.

9916

0.1

26 0

.439

4 0

.10

0 *

D

NK

/USD

3.

1342

1.

0795

0.8

43 -

1.49

30 0

.552

1.

0255

0.4

41 -

0.92

38 0

.540

1.

0131

0.7

76 -

0.6

939

0.5

11

1.01

80 0

.657

-0.

8030

0.7

08

U

SD/A

UD

3.

8947

1.

1073

0.6

21 -

1.99

13 0

.777

1.

0326

0.3

65 -

0.99

59 0

.520

1.

0064

0.2

11 -

0.3

726

0.3

22

0.99

18 0

.183

0.4

194

0.1

28

U

SD/G

BP

2.

5258

1.

0983

0.1

85 -

1.30

71 0

.045

**

1.

0160

0.0

42 *

* -

0.28

01 0

.017

**

0.

9838

0.0

46 *

* 0

.615

8 0

.026

**

0.

9897

0.0

87 *

0.6

106

0.0

71

*

JP

Y/U

SD

2.76

59

1.18

96 0

.440

-2.

0695

0.2

07

1.09

28 0

.792

-1.

5691

0.4

81

1.01

02 0

.719

-0.

977

3 0

.563

1.

0018

0.2

63 -

0.17

55 0

.30

5

C

HF/

USD

3.

1525

1.

3392

0.7

64 -

1.89

14 0

.264

1.

1393

0.8

89 -

2.58

19 0

.908

1.

0056

0.9

05 -

1.2

996

0.8

51

1.01

09 0

.865

-1.

3334

0.8

77

C

AD

/USD

2.

8019

1.

1503

0.7

31 -

2.34

59 0

.401

1.

0570

0.1

97 -

1.70

03 0

.574

1.

0246

0.7

77 -

0.9

360

0.5

85

1.00

96 0

.807

-0.

9836

0.7

65

SE

K/U

SD

3.50

47

1.15

84 0

.490

-1.

3902

0.0

50 *

*

1.10

25 0

.649

-2.

1765

0.7

69

1.02

09 0

.547

-0.

565

8 0

.337

0.

9921

0.1

38 0

.416

2 0

.09

9 *

D

NK

/USD

3.

1342

1.

1709

0.4

80 -

2.72

54 0

.536

1.

1167

0.8

85 -

1.99

97 0

.617

1.

0155

0.8

26 -

0.7

561

0.5

55

1.01

14 0

.455

-0.

5986

0.6

68

U

SD/A

UD

3.

8947

1.

2219

0.3

18 -

2.23

26 0

.420

1.

0677

0.2

74 -

1.63

88 0

.563

1.

0054

0.2

29 -

0.3

417

0.4

53

0.99

29 0

.196

0.4

334

0.2

07

All

fun

dam

enta

ls

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

Cu

rren

cy p

air

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSR

idge

EW

A

Notes: This table presents the results of forecasting end-of-month exchange rate 1-month ahead by the no-change prediction, the

rolling OLS, recursive OLS, SRidge and EWA methods for decoupled fundamentals on a sample February 1999 – December 2014

using real time data. Column 1 shows the RMSE values for the no-change prediction. For each method, the Theil ratio (RMSE

of the given method / RMSE of the no-change prediction), the CW-p-values, the DM statistic and its bootstrapped p-value are

shown. The CW and DM tests are one-sided tests of equal out-of-sample prediction accuracy (H0) against superior out-of-sample

prediction accuracy (H1) for the methods considered compared to the no-change prediction. ***, **, and * denote statistical

significance at the 1 %, 5 %, and 10 % levels. Theil ratios < 1 are indicated in bold.

61

Table D.21: Relative forecasting performance of the machine learning methods vs. rolling and recursive regressions based on decoupledfundamentals, for a sample February 1999 – December 2014.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:EO

M -

Dec

ou

ple

d -

Ver

sus

OLS

Dat

e:0

2/0

3/2

01

89

9-1

4 p

erio

d

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

0

.98

86

0.5

36

2 0

.78

7

0.9

90

2 0

.39

86

0.6

63

0

.99

32

0.4

56

0 0

.86

0

0.9

94

8 0

.47

30

0.7

09

JP

Y/U

SD

0.9

75

8 1

.05

66

0.4

87

0

.98

38

1.3

02

4 0

.18

1

0.9

76

3 0

.99

44

0.5

52

0

.98

43

1.2

20

6 0

.22

7

C

HF/

USD

0

.97

17

1.9

81

6 0

.08

4 *

0

.98

39

1.8

92

4 0

.03

8 *

*

0.9

63

7 2

.27

34

0.0

49

**

0

.97

58

2.2

72

7 0

.01

7 *

*

C

AD

/USD

0

.99

59

0.2

30

5 0

.86

3

1.0

07

9 -

0.6

98

5 0

.97

2

0.9

92

3 0

.40

35

0.8

74

1

.00

43

-0

.39

24

0.9

62

SE

K/U

SD

0.9

70

1 0

.70

74

0.6

82

0

.98

05

0.4

54

5 0

.60

0

0.9

71

2 1

.10

70

0.5

48

0

.98

16

0.6

75

1 0

.61

8

D

NK

/USD

0

.96

72

0.7

90

2 0

.62

2

0.9

96

4 0

.17

48

0.7

37

0

.96

29

0.8

79

9 0

.65

6

0.9

92

0 0

.45

64

0.7

30

U

SD/A

UD

1

.00

06

-0

.01

82

0.8

67

1

.01

35

-0

.55

33

0.8

77

1

.00

24

-0

.08

59

0.9

01

1

.01

53

-0

.86

47

0.9

52

U

SD/G

BP

0

.98

65

0.7

14

9 0

.76

5

0.9

78

1 2

.12

47

0.0

22

**

0

.98

58

0.7

43

2 0

.78

9

0.9

77

4 1

.92

11

0.0

42

**

JP

Y/U

SD

0.9

45

7 0

.97

93

0.6

43

0

.97

70

1.1

32

4 0

.29

5

0.9

40

5 1

.08

30

0.6

30

0

.97

15

1.3

19

6 0

.27

5

C

HF/

USD

0

.97

83

1.1

92

9 0

.45

8

0.9

90

8 0

.71

68

0.5

23

0

.97

65

1.2

33

6 0

.46

9

0.9

88

9 0

.80

88

0.5

51

C

AD

/USD

0

.98

87

0.8

97

0 0

.65

2

0.9

96

8 0

.39

85

0.6

82

0

.98

33

1.3

70

6 0

.41

6

0.9

91

3 1

.17

48

0.3

28

SE

K/U

SD

0.9

74

0 1

.50

61

0.3

24

0

.97

28

1.1

92

8 0

.28

1

0.9

71

4 1

.37

57

0.4

44

0

.97

02

1.2

76

2 0

.28

9

D

NK

/USD

0

.97

02

1.1

74

2 0

.52

5

0.9

71

7 1

.25

12

0.2

32

0

.96

14

1.2

20

0 0

.57

0

0.9

62

9 1

.26

41

0.3

19

U

SD/A

UD

0

.96

68

1.6

11

7 0

.25

5

0.9

75

2 1

.17

70

0.2

68

0

.95

96

1.8

53

6 0

.16

3

0.9

68

0 1

.26

11

0.2

56

U

SD/G

BP

0

.86

85

1.2

11

6 0

.66

8

0.9

49

2 1

.09

32

0.4

47

0

.86

96

1.2

52

2 0

.60

7

0.9

50

4 1

.33

31

0.2

76

JP

Y/U

SD

0.9

41

4 1

.28

63

0.7

38

0

.98

39

0.7

90

8 0

.71

8

0.9

36

3 1

.47

44

0.5

71

0

.97

85

1.1

62

5 0

.41

6

C

HF/

USD

0

.91

55

1.4

43

8 0

.52

4

0.9

54

4 1

.57

01

0.1

64

0

.92

47

1.3

70

0 0

.55

5

0.9

64

0 1

.50

93

0.1

80

C

AD

/USD

0

.92

44

1.3

62

7 0

.69

0

0.9

92

6 0

.19

91

0.9

21

0

.90

25

1.3

60

0 0

.64

4

0.9

69

2 0

.90

32

0.5

75

SE

K/U

SD

0.9

35

1 1

.46

02

0.5

96

0

.97

98

0.7

11

5 0

.69

3

0.9

09

6 1

.53

77

0.5

68

0

.95

31

2.1

66

6 0

.04

8 *

*

D

NK

/USD

0

.91

38

1.3

72

0 0

.55

1

0.9

74

7 1

.80

77

0.1

06

0

.91

94

1.2

07

6 0

.52

8

0.9

80

7 1

.02

38

0.3

29

U

SD/A

UD

0

.90

33

1.4

02

6 0

.53

5

0.9

62

0 1

.34

81

0.2

79

0

.90

25

1.4

12

9 0

.54

0

0.9

61

1 1

.46

85

0.2

14

U

SD/G

BP

0

.90

07

1.3

06

5 0

.91

6

0.9

64

6 1

.47

44

0.5

15

0

.90

65

1.1

26

3 0

.94

4

0.9

70

7 1

.23

68

0.6

46

JP

Y/U

SD

0.8

87

9 2

.02

37

0.7

98

0

.91

44

1.9

60

9 0

.28

0

0.8

85

0 2

.13

89

0.7

38

0

.91

14

2.0

81

2 0

.19

0

C

HF/

USD

0

.75

75

1.9

88

3 0

.66

8

0.8

71

4 2

.40

17

0.0

90

*

0.7

61

3 1

.99

73

0.6

61

0

.87

58

2.4

44

4 0

.08

0 *

C

AD

/USD

0

.82

81

1.6

46

0 0

.89

9

0.9

68

7 1

.18

90

0.7

17

0

.81

86

1.6

34

2 0

.89

0

0.9

57

6 1

.53

27

0.5

05

SE

K/U

SD

0.9

12

9 1

.39

03

0.9

46

0

.94

03

1.6

70

4 0

.49

7

0.8

94

2 1

.26

41

0.9

67

0

.92

11

1.3

80

2 0

.68

8

D

NK

/USD

0

.83

80

2.9

58

6 0

.25

6

0.9

60

3 0

.89

85

0.8

90

0

.83

59

2.9

59

3 0

.24

3

0.9

57

9 0

.97

16

0.8

26

U

SD/A

UD

0

.81

64

1.5

58

5 0

.92

9

0.9

68

4 0

.75

68

0.9

11

0

.81

49

1.5

65

1 0

.92

6

0.9

66

7 0

.81

67

0.8

88

All

fun

dam

enta

ls

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

Cu

rren

cy p

air

SRid

ge

vs.

EWA

vs

.

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nR

olli

ng

regr

essi

on

Rec

urs

ive

regr

essi

on

Notes: This table presents the comparison of forecasting performance between, respectively, SRidge and EWA vs. the rolling and

recursive OLS for decoupled fundamentals for the period February 1999–December 2014. Columns 1-6 show the comparison between

SRidge and OLS methods, while columns 7-12 for EWA vs. OLS. For each comparison, the Theil ratio (RMSE of the given method

/ RMSE of the compared) and the DM statistic and its bootstrapped p-value are shown. The DM test is a one-sided test of equal

out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) of the machine-learning method

considered against the OLS methods. ***, **, and * denote statistical significance at the 1 %, 5 %, and 10 % levels. Theil ratios

< 1 are indicated in bold.

62

Table D.22: Relative forecasting performance on the real-time data set (February 1999 – December 2014) of the machine learning methods vs.rolling and recursive regressions based on decoupled fundamentals.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:R

T -

Dec

ou

ple

d -

Ver

sus

OLS

Dat

e:0

2/0

3/2

01

89

9-1

4 p

erio

d

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

Rat

io o

f

RM

SEs

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

0

.98

79

0.4

42

0 0

.80

3

0.9

92

4 0

.25

65

0.7

23

0

.99

94

0.0

37

7 0

.92

1

1.0

03

9 -

0.2

06

7 0

.90

0

JP

Y/U

SD

0.9

75

2 1

.41

89

0.2

84

0

.97

97

1.8

70

5 0

.04

3 *

*

0.9

77

3 1

.21

69

0.4

31

0

.98

18

1.9

62

4 0

.03

8 *

*

C

HF/

USD

0

.97

35

1.1

15

7 0

.49

0

0.9

92

1 0

.59

04

0.5

86

0

.96

27

1.7

48

5 0

.22

0

0.9

81

1 1

.82

43

0.0

84

*

C

AD

/USD

1

.00

37

-0

.16

24

0.9

46

1

.01

54

-0

.98

17

0.9

89

1

.00

09

-0

.03

72

0.9

48

1

.01

26

-0

.72

17

0.9

77

SE

K/U

SD

0.9

72

4 0

.76

32

0.6

49

0

.98

50

0.3

93

6 0

.62

6

0.9

69

8 1

.27

96

0.4

02

0

.98

23

0.6

63

7 0

.56

1

D

NK

/USD

0

.97

36

0.9

03

1 0

.57

5

0.9

97

8 0

.10

70

0.7

60

0

.96

57

0.9

88

8 0

.59

9

0.9

89

7 0

.78

30

0.5

53

U

SD/A

UD

1

.00

61

-0

.31

41

0.9

41

1

.02

50

-1

.17

95

0.9

80

0

.99

72

0.1

14

7 0

.81

5

1.0

15

9 -

0.7

81

5 0

.93

3

U

SD/G

BP

0

.98

64

0.7

23

4 0

.76

5

0.9

78

1 2

.12

31

0.0

25

**

0

.98

57

0.7

51

5 0

.79

1

0.9

77

4 1

.92

00

0.0

60

*

JP

Y/U

SD

0.9

45

7 0

.97

95

0.5

93

0

.97

70

1.1

32

7 0

.25

5

0.9

40

4 1

.08

30

0.5

99

0

.97

15

1.3

19

8 0

.25

2

C

HF/

USD

0

.97

83

1.1

93

6 0

.44

8

0.9

90

8 0

.71

51

0.4

69

0

.97

64

1.2

34

2 0

.45

0

0.9

88

9 0

.80

70

0.4

98

C

AD

/USD

0

.98

86

0.8

99

0 0

.64

5

0.9

96

8 0

.39

81

0.6

82

0

.98

32

1.3

72

2 0

.40

3

0.9

91

3 1

.17

48

0.3

25

SE

K/U

SD

0.9

73

9 1

.50

75

0.2

94

0

.97

28

1.1

92

6 0

.24

8

0.9

71

4 1

.37

69

0.4

03

0

.97

02

1.2

76

0 0

.28

2

D

NK

/USD

0

.97

01

1.1

74

6 0

.52

8

0.9

71

7 1

.25

01

0.2

58

0

.96

13

1.2

20

2 0

.57

6

0.9

62

9 1

.26

33

0.3

23

U

SD/A

UD

0

.96

68

1.6

11

5 0

.23

2

0.9

75

2 1

.17

66

0.2

89

0

.95

96

1.8

52

9 0

.14

7

0.9

68

0 1

.26

06

0.2

62

U

SD/G

BP

0

.86

29

1.2

89

1 0

.76

2

0.9

43

9 1

.56

64

0.2

73

0

.87

03

1.2

84

4 0

.73

4

0.9

52

0 1

.76

07

0.1

59

JP

Y/U

SD

0.9

41

8 1

.11

46

0.7

76

0

.99

88

0.0

49

0 0

.91

2

0.9

30

3 1

.29

84

0.6

48

0

.98

66

0.5

49

5 0

.74

5

C

HF/

USD

0

.94

49

1.3

19

9 0

.66

0

0.9

88

2 0

.93

63

0.5

16

0

.95

30

1.2

17

9 0

.68

5

0.9

96

6 0

.27

79

0.8

27

C

AD

/USD

0

.92

43

1.7

26

7 0

.47

9

0.9

90

6 0

.35

06

0.8

55

0

.91

12

1.6

91

6 0

.45

8

0.9

76

6 1

.38

57

0.2

86

SE

K/U

SD

0.9

06

4 1

.89

67

0.3

73

0

.98

26

0.5

04

5 0

.81

7

0.8

76

6 1

.63

12

0.4

80

0

.95

04

1.3

11

3 0

.33

5

D

NK

/USD

0

.93

85

1.6

31

0 0

.36

3

0.9

87

9 0

.70

62

0.5

50

0

.94

31

1.3

43

5 0

.52

9

0.9

92

7 0

.25

14

0.7

73

U

SD/A

UD

0

.90

88

2.1

35

1 0

.14

8

0.9

74

6 1

.04

01

0.4

04

0

.89

57

1.9

97

7 0

.18

8

0.9

60

5 1

.38

24

0.2

17

U

SD/G

BP

0

.89

57

1.5

64

6 0

.91

5

0.9

68

3 0

.64

92

0.9

19

0

.90

11

1.4

78

1 0

.93

0

0.9

74

1 0

.50

58

0.9

51

JP

Y/U

SD

0.8

49

2 2

.09

08

0.7

95

0

.92

44

1.5

69

2 0

.51

5

0.8

42

1 2

.14

97

0.7

60

0

.91

67

1.6

61

7 0

.41

8

C

HF/

USD

0

.75

09

1.8

85

8 0

.75

9

0.8

82

6 2

.62

33

0.0

89

*

0.7

54

9 1

.86

43

0.7

65

0

.88

73

2.5

94

6 0

.08

6 *

C

AD

/USD

0

.89

08

2.4

99

2 0

.53

3

0.9

69

4 1

.01

98

0.7

87

0

.87

77

2.4

18

2 0

.57

8

0.9

55

2 1

.47

65

0.5

53

SE

K/U

SD

0.8

81

3 1

.50

78

0.9

25

0

.92

60

1.5

59

8 0

.57

5

0.8

56

5 1

.50

39

0.9

27

0

.89

99

2.0

74

3 0

.25

5

D

NK

/USD

0

.86

73

2.6

38

3 0

.55

0

0.9

09

4 2

.35

60

0.1

96

0

.86

38

3.1

63

5 0

.26

8

0.9

05

8 1

.76

17

0.5

64

U

SD/A

UD

0

.82

28

2.2

13

2 0

.64

3

0.9

41

7 1

.76

59

0.3

86

0

.81

26

2.2

32

8 0

.62

4

0.9

30

0 2

.16

63

0.1

68

All

fun

dam

enta

ls

PP

P f

un

dam

enta

l

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

Cu

rren

cy p

air

SRid

ge

vs.

EWA

vs

.

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nR

olli

ng

regr

essi

on

Rec

urs

ive

regr

essi

on

Notes: This table presents the comparison of forecasting performance between, respectively, SRidge and EWA vs. the rolling and

recursive OLS for decoupled fundamentals for the period February 1999–December 2014 using real-time data. Columns 1-6 show

the comparison between SRidge and OLS methods, while columns 7-12 for EWA vs. OLS. For each comparison, the Theil ratio

(RMSE of the given method / RMSE of the compared) and the DM statistic and its bootstrapped p-value are shown. The DM

test is a one-sided test of equal out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) of

the machine-learning method considered against the OLS methods. ***, **, and * denote statistical significance at the 1 %, 5 %,

and 10 % levels. Theil ratios < 1 are indicated in bold.

63

Table D.23: Directional predictions of the exchange rates based on decoupled fundamentals, for a sample February 1999 – December 2014.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:EO

M -

Dec

ou

ple

d -

Dir

ecti

on

al9

9-1

4 p

erio

d

Dat

e:2

-mar

s-1

8

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

0

.52

3 0

.51

61

0.2

33

0

.54

6 0

.97

81

0.1

17

0

.53

1 0

.59

66

0.2

32

0

.42

3 -

1.3

96

5 0

.91

4

JP

Y/U

SD

0.5

15

0.3

12

5 0

.37

6

0.4

31

-1

.47

99

0.9

54

0

.50

8 0

.14

75

0.4

52

0

.53

8 0

.81

66

0.1

66

C

HF/

USD

0

.50

0 0

.00

00

0.4

28

0

.46

2 -

0.7

66

7 0

.76

9

0.5

00

0.0

00

0 0

.42

7

0.4

77

-0

.40

22

0.6

32

C

AD

/USD

0

.48

5 -

0.2

48

0 0

.62

0

0.4

62

-0

.83

90

0.8

22

0

.45

4 -

1.0

11

8 0

.86

7

0.5

15

0.3

43

7 0

.37

0

SE

K/U

SD

0.5

85

1.8

76

5 0

.01

7 *

*

0.5

92

1.8

79

2 0

.01

8 *

*

0.5

46

1.0

09

1 0

.11

8

0.4

62

-0

.66

48

0.7

68

D

NK

/USD

0

.60

6 1

.81

15

0.0

32

**

0

.53

5 0

.60

90

0.2

74

0

.53

5 0

.57

52

0.3

08

0

.47

2 -

0.5

13

5 0

.81

2

U

SD/A

UD

0

.53

2 0

.52

49

0.4

79

0

.53

2 0

.44

97

0.5

65

0

.50

8 0

.13

67

0.7

07

0

.44

4 -

0.9

44

8 0

.81

4

U

SD/G

BP

0

.48

5 -

0.3

51

0 0

.62

9

0.4

69

-0

.68

04

0.7

37

0

.53

8 0

.87

97

0.1

49

0

.52

3 0

.45

62

0.3

17

JP

Y/U

SD

0.4

92

-0

.09

93

0.5

40

0

.50

8 0

.13

36

0.4

65

0

.56

2 0

.96

45

0.1

47

0

.49

2 -

0.1

40

5 0

.58

8

C

HF/

USD

0

.51

5 0

.31

84

0.3

17

0

.49

2 -

0.1

53

1 0

.47

2

0.4

92

-0

.15

05

0.4

72

0

.48

5 -

0.2

32

6 0

.58

4

C

AD

/USD

0

.46

9 -

0.6

72

5 0

.79

5

0.5

00

0.0

00

0 0

.50

0

0.5

15

0.3

23

3 0

.41

1

0.5

15

0.3

14

9 0

.38

4

SE

K/U

SD

0.5

19

0.3

75

9 0

.34

6

0.4

11

-1

.74

38

0.9

78

0

.51

2 0

.21

40

0.4

31

0

.51

2 0

.23

62

0.4

06

D

NK

/USD

0

.49

6 -

0.0

61

6 0

.51

6

0.4

72

-0

.49

92

0.7

15

0

.52

8 0

.46

57

0.2

99

0

.55

9 1

.04

93

0.1

34

U

SD/A

UD

0

.40

8 -

1.4

82

6 0

.93

8

0.4

92

-0

.17

39

0.6

06

0

.47

7 -

0.4

23

5 0

.69

7

0.4

62

-0

.78

18

0.7

27

U

SD/G

BP

0

.48

5 -

0.2

78

8 0

.65

6

0.4

46

-1

.11

49

0.8

82

0

.50

8 0

.17

04

0.4

34

0

.49

2 -

0.1

59

3 0

.60

3

JP

Y/U

SD

0.5

50

1.0

00

4 0

.15

7

0.5

74

1.6

91

3 0

.03

9 *

*

0.5

66

1.5

09

9 0

.06

4 *

0

.57

4 1

.69

13

0.0

35

**

C

HF/

USD

0

.52

4 0

.48

66

0.3

49

0

.46

8 -

0.5

74

3 0

.82

2

0.5

08

0.1

62

4 0

.49

9

0.4

68

-0

.64

99

0.8

15

C

AD

/USD

0

.55

0 1

.01

72

0.1

83

0

.54

3 0

.95

84

0.2

20

0

.51

9 0

.42

42

0.4

37

0

.48

8 -

0.2

64

2 0

.77

7

SE

K/U

SD

0.4

57

-0

.82

25

0.8

44

0

.45

0 -

1.0

79

4 0

.90

7

0.4

11

-1

.87

15

0.9

83

0

.43

4 -

1.2

72

3 0

.96

1

D

NK

/USD

0

.53

4 0

.71

86

0.3

70

0

.53

4 0

.60

08

0.4

74

0

.50

0 0

.00

00

0.6

77

0

.48

3 -

0.3

64

4 0

.77

5

U

SD/A

UD

0

.50

8 0

.17

14

0.5

60

0

.47

5 -

0.4

02

2 0

.78

9

0.4

58

-0

.77

47

0.8

93

0

.45

8 -

0.8

71

0 0

.96

9

U

SD/G

BP

0

.51

5 0

.22

12

0.4

81

0

.55

4 1

.23

51

0.1

25

0

.50

8 0

.15

71

0.4

42

0

.46

2 -

0.7

90

6 0

.81

5

JP

Y/U

SD

0.6

20

2.7

91

7 0

.00

8 *

**

0.4

96

-0

.07

48

0.6

12

0

.56

6 1

.50

99

0.0

53

*

0.5

66

1.0

85

1 0

.12

9

C

HF/

USD

0

.51

6 0

.32

57

0.3

73

0

.46

8 -

0.7

19

9 0

.81

7

0.5

00

0.0

00

0 0

.56

0

0.4

44

-0

.81

23

0.8

94

C

AD

/USD

0

.54

3 0

.97

20

0.2

47

0

.49

6 -

0.0

87

4 0

.76

0

0.5

19

0.4

37

0 0

.45

5

0.5

04

0.0

88

0 0

.60

8

SE

K/U

SD

0.5

89

1.5

22

6 0

.05

4 *

0

.55

0 1

.07

94

0.1

62

0

.44

2 -

1.1

72

7 0

.93

2

0.4

57

-0

.75

05

0.8

86

D

NK

/USD

0

.49

2 -

0.1

17

9 0

.65

0

0.5

85

1.3

43

3 0

.15

5

0.5

08

0.1

74

1 0

.63

5

0.5

17

0.3

68

4 0

.53

9

U

SD/A

UD

0

.47

5 -

0.4

56

3 0

.82

4

0.4

83

-0

.31

08

0.7

87

0

.47

5 -

0.5

31

2 0

.86

1

0.4

42

-1

.07

88

0.9

76

All

fun

dam

enta

ls

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSr

idge

EWA

Cu

rren

cy p

air

PP

P f

un

dam

enta

l

Notes: This table presents the results of forecasting the direction of change of the end-of-month exchange rate 1-month ahead

by the rolling OLS, recursive OLS, SRidge and EWA methods for a sample February 1999 – December 2014, based on decoupled

fundamentals. For each method, the share of correctly forecasted changes, the DM statistic and its bootstrapped p-value are shown.

The DM tests is a one-sided test of equal out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy

(H1) for the methods considered against the benchmark of a 50 % success rate. ***, **, and * denote statistical significance at the

1 %, 5 %, and 10 % levels. Shares > 0.5 are indicated in bold.

64

Table D.24: Directional predictions of the exchange rates on the real-time data set (February 1999 – December 2014) based on decoupledfundamentals.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:R

T -

Dec

ou

ple

d -

Dir

ecti

on

al9

9-1

4 p

erio

d

Dat

e:2

-mar

s-1

8

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

Pro

po

rtio

n o

f

chan

ges

pre

dic

ted

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

0

.54

6 1

.05

70

0.1

28

0

.56

9 1

.59

41

0.0

41

**

0

.52

3 0

.48

83

0.3

01

0

.43

1 -

1.4

63

8 0

.92

1

JP

Y/U

SD

0.5

00

0.0

00

0 0

.47

2

0.4

77

-0

.48

04

0.7

18

0

.51

5 0

.31

13

0.3

73

0

.53

1 0

.60

79

0.2

54

C

HF/

USD

0

.50

0 0

.00

00

0.4

08

0

.46

9 -

0.4

70

7 0

.64

9

0.4

54

-0

.90

66

0.8

07

0

.47

7 -

0.3

57

4 0

.50

4

C

AD

/USD

0

.52

3 0

.45

35

0.3

41

0

.47

7 -

0.4

26

8 0

.72

9

0.4

85

-0

.32

51

0.6

83

0

.51

5 0

.35

10

0.3

75

SE

K/U

SD

0.5

85

1.7

24

7 0

.05

1 *

0

.56

9 1

.30

36

0.0

89

*

0.5

31

0.6

69

0 0

.26

4

0.4

69

-0

.57

83

0.6

96

D

NK

/USD

0

.54

3 0

.96

96

0.1

49

0

.52

8 0

.47

53

0.3

34

0

.52

8 0

.46

57

0.3

39

0

.48

0 -

0.3

65

0 0

.72

3

U

SD/A

UD

0

.50

0 0

.00

00

0.6

60

0

.53

2 0

.48

00

0.5

85

0

.52

4 0

.49

50

0.5

85

0

.38

7 -

2.0

93

1 0

.94

7

U

SD/G

BP

0

.48

5 -

0.3

51

0 0

.62

4

0.4

69

-0

.68

04

0.7

30

0

.53

8 0

.87

97

0.1

59

0

.52

3 0

.45

62

0.3

05

JP

Y/U

SD

0.4

92

-0

.09

93

0.5

15

0

.50

8 0

.13

36

0.4

58

0

.56

2 0

.96

45

0.1

52

0

.49

2 -

0.1

40

5 0

.54

2

C

HF/

USD

0

.51

5 0

.31

84

0.3

09

0

.49

2 -

0.1

53

1 0

.52

3

0.4

92

-0

.15

05

0.5

00

0

.48

5 -

0.2

32

6 0

.57

7

C

AD

/USD

0

.46

9 -

0.6

72

5 0

.81

1

0.5

00

0.0

00

0 0

.54

4

0.5

15

0.3

23

3 0

.43

7

0.5

15

0.3

14

9 0

.41

4

SE

K/U

SD

0.5

19

0.3

75

9 0

.37

2

0.4

11

-1

.74

38

0.9

74

0

.51

2 0

.21

40

0.4

30

0

.51

2 0

.23

62

0.3

91

D

NK

/USD

0

.49

6 -

0.0

61

6 0

.49

3

0.4

72

-0

.49

92

0.6

89

0

.52

8 0

.46

57

0.2

59

0

.55

9 1

.04

93

0.1

18

U

SD/A

UD

0

.40

8 -

1.4

82

6 0

.96

3

0.4

92

-0

.17

39

0.5

98

0

.47

7 -

0.4

23

5 0

.72

2

0.4

62

-0

.78

18

0.7

29

U

SD/G

BP

0

.46

9 -

0.6

48

9 0

.74

3

0.4

00

-1

.93

19

0.9

76

0

.51

5 0

.34

17

0.3

35

0

.46

9 -

0.6

69

2 0

.74

0

JP

Y/U

SD

0.5

43

0.9

72

0 0

.13

0

0.6

12

2.3

68

8 0

.00

7 *

**

0.5

58

1.2

51

2 0

.07

9 *

0

.55

0 1

.01

69

0.1

04

C

HF/

USD

0

.48

4 -

0.3

55

4 0

.62

2

0.4

35

-1

.01

45

0.8

43

0

.47

6 -

0.3

92

6 0

.66

9

0.4

03

-2

.05

81

0.9

85

C

AD

/USD

0

.45

7 -

0.8

89

3 0

.88

0

0.4

26

-1

.69

13

0.9

74

0

.47

3 -

0.5

64

8 0

.81

6

0.4

57

-0

.72

14

0.8

80

SE

K/U

SD

0.5

04

0.0

75

5 0

.50

0

0.4

50

-0

.77

21

0.8

10

0

.50

4 0

.06

98

0.4

77

0

.54

3 0

.93

20

0.1

18

D

NK

/USD

0

.53

4 0

.62

54

0.3

52

0

.51

7 0

.29

73

0.5

65

0

.55

9 1

.00

93

0.2

26

0

.50

0 0

.00

00

0.5

92

U

SD/A

UD

0

.51

7 0

.36

54

0.3

98

0

.50

0 0

.00

00

0.5

71

0

.47

5 -

0.4

76

7 0

.74

3

0.4

42

-1

.16

05

0.8

50

U

SD/G

BP

0

.51

5 0

.35

10

0.3

51

0

.61

5 2

.70

42

0.0

03

***

0

.51

5 0

.34

17

0.3

38

0

.50

8 0

.14

41

0.4

25

JP

Y/U

SD

0.5

04

0.0

68

6 0

.57

2

0.5

35

0.6

55

1 0

.34

1

0.5

50

1.0

36

6 0

.12

3

0.5

43

0.9

22

5 0

.13

8

C

HF/

USD

0

.53

2 0

.64

75

0.2

47

0

.48

4 -

0.2

58

9 0

.61

4

0.5

16

0.2

80

5 0

.38

0

0.4

03

-2

.05

81

0.9

73

C

AD

/USD

0

.48

1 -

0.3

08

7 0

.77

5

0.5

12

0.2

59

5 0

.59

4

0.4

65

-0

.77

11

0.8

61

0

.43

4 -

1.4

60

4 0

.95

3

SE

K/U

SD

0.4

65

-0

.78

60

0.8

47

0

.45

0 -

1.1

44

4 0

.92

3

0.5

27

0.5

00

9 0

.31

0

0.5

58

1.3

27

9 0

.05

2 *

D

NK

/USD

0

.50

0 0

.00

00

0.5

68

0

.49

2 -

0.1

41

6 0

.64

3

0.5

59

0.9

98

5 0

.26

1

0.5

08

0.1

12

2 0

.63

3

U

SD/A

UD

0

.50

8 0

.18

26

0.5

80

0

.55

0 0

.76

89

0.3

54

0

.47

5 -

0.4

76

7 0

.87

2

0.4

50

-1

.04

93

0.8

52

All

fun

dam

enta

ls

UIR

P f

un

dam

enta

l

Mo

net

ary

mo

del

fu

nd

amen

tals

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSr

idge

EWA

Cu

rren

cy p

air

PP

P f

un

dam

enta

l

Notes: This table presents the results of forecasting the direction of change of the end-of-month exchange rate 1-month ahead by the

rolling OLS, recursive OLS, SRidge and EWA methods based on decoupled fundamentals for the period February 1999–December

2014 using real-time data. For each method, the share of correctly forecasted changes, the DM statistic and its bootstrapped

p-value are shown. The DM tests is a one-sided test of equal out-of-sample prediction accuracy (H0) against superior out-of-sample

prediction accuracy (H1) for the methods considered against the benchmark of a 50 % success rate. ***, **, and * denote statistical

significance at the 1 %, 5 %, and 10 % levels. Shares > 0.5 are indicated in bold.

65

Table D.25: 1-month ahead forecasts using decoupled Taylor-rule fundamentals.

Wh

at d

o w

e re

ad?

EOM

- F

RED

- R

T? C

ou

ple

d o

r d

eco

up

led

? R

MSE

/ v

ersu

s O

LS /

dir

ecti

on

al c

han

ges?

An

swer

:EO

M -

De

cou

ple

d -

RM

SED

ate:

04/0

3/20

18

Tayl

or

No

ch

ange

RM

SE x

100

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

Thei

l rat

ioC

W p

-val

ue

DM

sta

tist

icD

M p

-val

ue

U

SD/G

BP

2.

9051

1.

0430

0.1

92

-2.

1392

0.2

11

1.02

76 0

.566

-1.

6732

0.4

89

1.00

11 0

.493

-0.

2706

0.2

37

1

.00

30 0

.18

5 -

0.3

110

0.4

61

JP

Y/U

SD

3.10

46

1.02

97 0

.09

8 *

-1.

3703

0.0

61 *

1.

0048

0.0

25 *

* -

0.31

62 0

.060

*

0.99

96 0

.204

0.7

441

0.0

90

*

0.9

937

0.0

16

**

0.8

87

8 0

.077

*

C

HF/

USD

3.

3887

1.

0644

0.5

35

-2.

7086

0.6

39

1.02

16 0

.508

-1.

5669

0.5

53

0.99

86 0

.097

* 1

.074

8 0

.05

6 *

0

.994

1 0

.02

8 *

* 1

.06

64

0.0

78 *

C

AD

/USD

1.

9994

1.

0145

0.2

87

-0.

7791

0.0

05 *

**

0.

9988

0.1

37 0

.112

3 0

.006

**

*

1.00

02 0

.640

-0.

3738

0.3

03

0

.999

5 0

.20

1 0

.48

14

0.1

61

SE

K/U

SD

3.22

63

1.10

08 0

.83

7 -

1.81

30 0

.503

1.

0241

0.6

50 -

2.39

86 0

.927

0.

9971

0.0

97 *

0.2

861

0.1

90

0

.995

4 0

.01

2 *

* 0

.80

34

0.0

60 *

D

NK

/USD

3.

1037

1.

0323

0.2

65

-1.

5297

0.0

92 *

1.

0136

0.3

61 -

1.22

82 0

.305

1.

0023

0.8

95 -

1.24

69 0

.79

2

0.9

939

0.0

28

**

0.7

76

6 0

.137

U

SD/A

UD

3.

3926

1.

0073

0.0

07

***

-0.

3638

0.0

02 *

**

1.

0126

0.5

33 -

1.04

60 0

.198

1.

0043

0.7

52 -

0.91

65 0

.61

2

1.0

009

0.3

75

-0

.157

1 0

.346

FR

F/U

SD

3.19

78

1.03

59 0

.43

0 -

1.37

33 0

.196

1.

0277

0.5

10 -

1.33

70 0

.429

1.

0005

0.4

43 -

0.09

83 0

.20

5

0.9

961

0.0

85

* 0

.30

01

0.2

12

D

EM

/USD

3.

3136

1.

0444

0.6

58

-1.

5401

0.3

14

1.03

03 0

.612

-1.

4812

0.5

72

0.99

87 0

.213

0.5

630

0.0

93

*

0.9

914

0.0

42

**

0.8

05

8 0

.053

*

IT

L/U

SD

3.19

07

1.00

82 0

.11

6 -

0.33

47 0

.031

**

1.

0216

0.8

14 -

1.59

07 0

.705

1.

0024

0.3

39 -

0.19

58 0

.54

7

1.0

067

0.2

76

-0

.427

4 0

.780

N

LG/U

SD

3.33

19

1.03

41 0

.25

6 -

1.12

81 0

.152

1.

0212

0.5

51 -

1.28

31 0

.422

0.

9998

0.3

94 0

.141

2 0

.12

9

0.9

883

0.0

22

**

1.0

66

8 0

.030

**

P

TE/U

SD

2.77

03

1.06

06 0

.71

4 -

1.39

14 0

.604

1.

0976

0.7

24 -

1.98

37 0

.902

0.

9970

0.3

65 0

.226

4 0

.17

7

0.9

987

0.3

38

0.1

85

8 0

.320

U

SD/G

BP

2.

9051

1.

0387

0.1

57

-1.

9223

0.1

44

1.02

98 0

.681

-1.

9623

0.6

64

1.00

10 0

.482

-0.

2455

0.2

50

1

.00

27 0

.15

5 -

0.2

798

0.4

99

JP

Y/U

SD

3.10

46

1.02

38 0

.05

6 *

-1.

1279

0.0

32 *

*

1.00

63 0

.048

**

-0.

4245

0.0

65 *

0.

9995

0.1

56 0

.926

0 0

.06

0 *

0

.994

3 0

.02

2 *

* 0

.79

11

0.1

36

C

HF/

USD

3.

3887

1.

0601

0.3

33

-2.

2897

0.4

02

1.02

23 0

.713

-1.

9219

0.7

27

0.99

86 0

.099

* 1

.064

9 0

.02

8 **

0

.992

7 0

.01

6 *

* 1

.30

13

0.0

53 *

C

AD

/USD

1.

9994

1.

0184

0.2

46

-0.

8391

0.0

05 *

**

0.

9991

0.1

72 0

.088

3 0

.004

**

*

1.00

02 0

.636

-0.

3663

0.3

08

0

.999

3 0

.13

6 0

.78

74

0.0

86 *

SE

K/U

SD

3.22

63

1.07

56 0

.75

7 -

1.58

23 0

.438

1.

0314

0.8

53 -

2.10

39 0

.849

0.

9986

0.1

32 0

.138

5 0

.24

3

0.9

951

0.0

10

**

0.8

49

5 0

.032

**

D

NK

/USD

3.

1037

1.

0270

0.1

52

-1.

1256

0.0

32 *

*

1.01

54 0

.723

-1.

8647

0.6

47

1.00

05 0

.892

-1.

0250

0.6

60

0

.993

3 0

.02

4 *

* 0

.84

76

0.1

19

U

SD/A

UD

3.

3926

1.

0063

0.0

03

***

-0.

2918

0.0

00 *

**

1.

0096

0.2

42 -

0.77

76 0

.081

*

1.00

43 0

.747

-0.

9038

0.6

00

1

.00

12 0

.42

9 -

0.2

275

0.4

19

FR

F/U

SD

3.19

78

1.03

33 0

.40

6 -

1.33

64 0

.199

1.

0279

0.5

66 -

1.36

93 0

.468

1.

0005

0.4

44 -

0.09

91 0

.21

2

0.9

964

0.0

87

* 0

.27

84

0.2

06

D

EM

/USD

3.

3136

1.

0456

0.6

74

-1.

6310

0.3

87

1.02

19 0

.599

-1.

2336

0.4

57

0.99

87 0

.213

0.5

639

0.1

01

0

.990

8 0

.03

7 *

* 0

.87

04

0.0

59 *

IT

L/U

SD

3.19

07

1.00

29 0

.08

3 *

-0.

1237

0.0

20 *

*

1.02

06 0

.851

-1.

5380

0.7

14

1.00

24 0

.339

-0.

1960

0.5

61

1

.00

47 0

.23

8 -

0.3

022

0.6

96

N

LG/U

SD

3.33

19

1.03

56 0

.27

2 -

1.18

89 0

.211

1.

0146

0.6

76 -

1.20

50 0

.468

0.

9998

0.3

95 0

.138

6 0

.15

7

0.9

883

0.0

22

**

1.0

62

3 0

.042

**

P

TE/U

SD

2.77

03

1.05

01 0

.60

5 -

0.92

18 0

.397

1.

0848

0.7

38 -

1.71

49 0

.848

0.

9970

0.3

66 0

.220

9 0

.17

2

0.9

988

0.3

51

0.1

78

4 0

.278

U

SD/G

BP

2.

9051

1.

0353

0.2

54

-1.

8864

0.1

51

1.02

62 0

.846

-2.

2301

0.8

05

1.00

10 0

.488

-0.

2580

0.2

38

1

.00

14 0

.12

0 -

0.1

427

0.4

18

JP

Y/U

SD

3.10

46

1.03

01 0

.08

1 *

-1.

5177

0.0

79 *

1.

0042

0.0

44 *

* -

0.32

60 0

.061

*

0.99

96 0

.207

0.7

351

0.0

85

*

0.9

943

0.0

21

**

0.8

01

0 0

.069

*

C

HF/

USD

3.

3887

1.

0461

0.2

57

-1.

8590

0.2

09

1.01

78 0

.557

-1.

4286

0.4

92

0.99

86 0

.099

* 1

.061

9 0

.04

4 **

0

.993

0 0

.01

8 *

* 1

.25

81

0.0

58 *

C

AD

/USD

1.

9994

1.

0167

0.2

11

-0.

7619

0.0

02 *

**

0.

9982

0.1

77 0

.176

7 0

.004

**

*

1.00

02 0

.643

-0.

3792

0.3

43

0

.999

0 0

.07

4 *

0.9

82

4 0

.060

*

SE

K/U

SD

3.22

63

1.06

39 0

.83

7 -

2.25

67 0

.663

1.

0272

0.8

68 -

2.35

81 0

.923

0.

9982

0.1

25 0

.178

9 0

.23

0

0.9

958

0.0

14

**

0.7

36

5 0

.060

*

D

NK

/USD

3.

1037

1.

0298

0.1

67

-1.

2546

0.0

50 *

*

1.01

36 0

.627

-1.

5159

0.4

90

1.00

04 0

.882

-0.

9723

0.6

65

0

.992

6 0

.02

0 *

* 0

.93

63

0.1

19

U

SD/A

UD

3.

3926

1.

0029

0.0

03

***

-0.

1332

0.0

00 *

**

1.

0052

0.1

81 -

0.41

99 0

.038

**

1.

0042

0.7

41 -

0.88

93 0

.60

2

1.0

021

0.4

72

-0

.354

3 0

.463

FR

F/U

SD

3.19

78

1.03

63 0

.24

3 -

1.23

55 0

.152

1.

0263

0.5

70 -

1.30

11 0

.415

1.

0005

0.4

43 -

0.09

83 0

.20

2

0.9

951

0.0

73

* 0

.37

33

0.1

74

D

EM

/USD

3.

3136

1.

0431

0.6

65

-1.

5888

0.3

98

1.01

74 0

.371

-0.

8289

0.2

70

0.99

87 0

.213

0.5

633

0.0

74

*

0.9

905

0.0

35

**

0.8

96

7 0

.054

*

IT

L/U

SD

3.19

07

1.01

60 0

.11

8 -

0.57

66 0

.084

*

1.01

82 0

.732

-1.

1898

0.5

71

1.00

25 0

.344

-0.

2033

0.6

09

1

.00

75 0

.29

4 -

0.4

786

0.8

18

N

LG/U

SD

3.33

19

1.03

52 0

.28

1 -

1.17

57 0

.183

1.

0171

0.6

18 -

1.09

07 0

.346

0.

9998

0.3

95 0

.138

2 0

.15

8

0.9

889

0.0

25

**

1.0

12

5 0

.048

**

P

TE/U

SD

2.77

03

1.04

89 0

.58

7 -

0.99

45 0

.424

1.

0811

0.7

27 -

1.76

52 0

.848

0.

9970

0.3

65 0

.226

4 0

.17

5

0.9

987

0.3

39

0.1

86

2 0

.300

U

SD/G

BP

2.

9051

1.

0427

0.5

55

-2.

2624

0.3

72

1.02

03 0

.750

-1.

6556

0.6

04

1.00

11 0

.495

-0.

2758

0.2

74

1

.00

26 0

.15

3 -

0.2

692

0.4

66

JP

Y/U

SD

3.10

46

1.02

33 0

.15

0 -

1.28

81 0

.064

*

1.00

34 0

.015

**

-0.

2266

0.0

60 *

0.

9996

0.1

75 0

.855

4 0

.06

6 *

0

.994

4 0

.02

2 *

* 0

.77

46

0.0

64 *

C

HF/

USD

3.

3887

1.

0504

0.6

73

-2.

9215

0.7

96

1.02

20 0

.757

-1.

7578

0.7

04

0.99

86 0

.094

* 1

.091

6 0

.03

7 **

0

.993

8 0

.02

5 *

* 1

.10

83

0.0

74 *

C

AD

/USD

1.

9994

1.

0167

0.2

74

-1.

2248

0.0

19 *

*

1.00

41 0

.650

-0.

9388

0.1

40

1.00

02 0

.643

-0.

3797

0.3

22

0

.999

2 0

.07

8 *

0.9

62

5 0

.060

*

SE

K/U

SD

3.22

63

1.06

79 0

.80

6 -

2.22

39 0

.671

1.

0300

0.8

41 -

2.06

96 0

.911

0.

9972

0.0

99 *

0.2

832

0.2

14

0

.995

4 0

.01

2 *

* 0

.80

77

0.0

37 *

*

D

NK

/USD

3.

1037

1.

0322

0.3

43

-2.

0045

0.3

45

1.00

86 0

.447

-0.

8599

0.2

12

1.00

04 0

.881

-0.

9637

0.6

73

0

.992

8 0

.02

2 *

* 0

.90

97

0.1

13

U

SD/A

UD

3.

3926

1.

0041

0.0

07

***

-0.

1774

0.0

00 *

**

1.

0121

0.5

20 -

0.99

78 0

.203

1.

0043

0.7

47 -

0.90

41 0

.60

3

1.0

019

0.4

68

-0

.340

8 0

.459

FR

F/U

SD

3.19

78

1.03

61 0

.37

4 -

1.41

69 0

.237

1.

0283

0.6

38 -

1.36

03 0

.478

1.

0005

0.4

44 -

0.09

87 0

.22

1

0.9

951

0.0

72

* 0

.37

38

0.1

80

D

EM

/USD

3.

3136

1.

0482

0.8

35

-2.

0596

0.6

85

1.02

39 0

.704

-1.

2549

0.5

08

0.99

87 0

.214

0.5

635

0.0

93

*

0.9

913

0.0

41

**

0.8

15

9 0

.058

*

IT

L/U

SD

3.19

07

1.01

90 0

.11

7 -

0.68

04 0

.115

1.

0218

0.8

44 -

1.44

35 0

.682

1.

0024

0.3

40 -

0.19

77 0

.58

9

1.0

077

0.2

96

-0

.486

4 0

.797

N

LG/U

SD

3.33

19

1.03

81 0

.39

3 -

1.33

68 0

.294

1.

0174

0.7

54 -

1.25

49 0

.488

0.

9998

0.3

96 0

.138

0 0

.15

5

0.9

889

0.0

25

**

1.0

13

5 0

.048

**

P

TE/U

SD

2.77

03

1.04

05 0

.36

5 -

0.88

04 0

.366

1.

0530

0.7

36 -

1.78

49 0

.850

0.

9970

0.3

65 0

.227

1 0

.18

0

0.9

987

0.3

43

0.1

77

9 0

.292

Ou

tpu

t ga

p f

un

dam

enta

ls: d

evia

tio

ns

fro

m a

lin

ear

tren

d

Ou

tpu

t ga

p f

un

dam

enta

ls: d

evia

tio

ns

fro

m a

qu

adra

tic

tren

d

Ou

tpu

t ga

p f

un

dam

enta

ls: d

evia

tio

ns

fro

m a

lin

ear

and

a q

uad

rati

c tr

end

Ou

tpu

t ga

p f

un

dam

enta

ls: d

evia

tio

ns

fro

m a

Ho

dri

ck-P

resc

ott

filt

ered

tre

nd

Cu

rren

cy p

air

Ro

llin

g re

gres

sio

nR

ecu

rsiv

e re

gres

sio

nSR

idge

EW

A

Notes: This table presents the results of forecasting end-of-month exchange rate 1-month ahead by the no-change prediction, the

rolling OLS, recursive OLS, SRidge and EWA methods for different sets of Taylor-rule fundamentals. Column 1 shows the RMSE

values for the no-change prediction. For each method, the Theil ratio (RMSE of the given method / RMSE of the no-change

prediction), the CW-p-values, the DM statistic and its bootstrapped p-value are shown. The CW and DM tests are one-sided tests

of equal out-of-sample prediction accuracy (H0) against superior out-of-sample prediction accuracy (H1) of the methods considered

compared to the no-change prediction. ***, **, and * denote statistical significance at the 1 %, 5 %, and 10 % levels. Theil ratios < 1

are indicated in bold. Sample period for the exchange rates: March 1973 – December 2014 unless shorter (indicated in Appendix

A).

66

PPP.GBP

MONEY.GBP

UIRP.GBP

INTC

H.GBP

Y.GBP

no.change

PPP.US

MONEY.US

UIRP.US

INTC

H.US

Y.US

0.00.20.40.60.81.0

Wei

ghts

ass

ocia

ted

with

the

fund

amen

tals

/ E

OM

+ G

BP

Weights

Mon

ths

200

300

400

500

0.00.20.40.60.81.0

Figure D.1: Weights of different fundamentals in prediction: the USD/GBP case, all fundamentals considered together.

67

PPP.JPY

MONEY.JPY

UIRP.JPY

INTC

H.JPY

Y.JPY

no.change

PPP.US

MONEY.US

UIRP.US

INTC

H.US

Y.US

0.00.20.40.60.81.0

Wei

ghts

ass

ocia

ted

with

the

fund

amen

tals

/ E

OM

+ J

PY

Weights

Mon

ths

200

300

400

500

0.00.20.40.60.81.0

Figure D.2: Weights of different fundamentals in prediction: the JPY/USD case, all fundamentals considered together.

68

PPP.CHF

MONEY.CHF

UIRP.CHF

INTC

H.CHF

Y.CHF

no.change

PPP.US

MONEY.US

UIRP.US

INTC

H.US

Y.US

0.00.20.40.60.81.0

Wei

ghts

ass

ocia

ted

with

the

fund

amen

tals

/ E

OM

+ C

HF

Weights

Mon

ths

200

300

400

0.00.20.40.60.81.0

Figure D.3: Weights of different fundamentals in prediction: the CHF/USD case, all fundamentals considered together.

69