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South African Reserve Bank Working Paper Series

WP/16/08

Modeling and Forecasting Daily Financial and Commodity Term Structures:

A Unified Global Approach

Shakill Hassan and Leonardo Morales-Arias

Authorised for distribution by Chris Loewald

June 2016

South African Reserve Bank Working Papers are written by staff members of the South African Reserve Bank and on occasion by consultants under the auspices of the Bank. The papers deal with topical issues and describe preliminary research findings, and develop new analytical or empirical approaches in their analyses. They are solely intended to elicit comments and stimulate debate.

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Modeling and Forecasting Daily Financial and Commodity Term

Structures: A Unified Global Approach

Shakill Hassan∗ and Leonardo Morales-Arias†

April 27, 2016

Abstract

In this article we propose a dynamic factor framework for modeling and forecasting

financial and commodity term structures in a unified global setting. The novelty of our

approach is that it exploits a large set of information (i.e. data properties, time and forward

dimensions, and cross-country, market, sector and weather dimensions) summarized in a

set of heteroskedastic components that have a clear time series interpretation and that

can be modeled dynamically to generate forecasts in real-time. The approach is motivated

by evidence of rising financial integration, and interdependence between commodity and

asset markets. We employ a battery of in-sample and out-of-sample techniques to evaluate

our framework and concentrate on relevant statistical and economic performance measures.

To preview our results with practical implications, we find that our framework provides

significant in-sample information in terms of product specific factors and commonalities

driving commodity and financial markets. Moreover, the specification proposed for modeling

the dynamics of financial and commodity term structures generates accurate out-of-sample

interval and point forecasts and leads to variance reduction when hedging a portfolio made

up of spot and futures contracts.

JEL Classification: C58, F7, G15

Keywords: Term structure forecasting, Fractional CVAR, Orthogonal GARCH, Regime-

Switching, Dynamic Hedging.

∗South African Reserve Bank and University of Cape Town. Email: shakill.hassan@resbank.co.za†Corresponding author. University of Kiel and South African Reserve Bank. Email: moralesl78@hotmail.com.

We would like to thank participants at the staff meetings of the South African Reserve Bank, Guilherme Moura,Thomas Lux for valuable comments and suggestions. Very special thanks to Elmarie Nel and Luchelle Soobyahwhose excellent research assistance made this article possible. This research was conducted while the secondauthor was a visiting fellow at the South African Reserve Bank whose excellent hospitality is also hereby ac-knowledged. The usual disclaimer applies.

1 Introduction

Modeling and forecasting term structures in financial and commodity markets is very impor-

tant for industry practitioners and policy makers. Appropriate methods that approximate term

structures -and thus the expected future path of financial and commodity prices- allow prac-

titioners to take better decisions, for instance, with respect to optimal portfolio holdings and

dynamic hedging strategies. Monetary authorities also benefit from information embedded in

financial and commodities term structures: forecasts of the future evolution of exchange rates,

asset prices, and commodity prices, are important inputs used by central banks when setting

policy interest rates (Rigobon and Sack, 2002; Bernanke, 2008).

In this article we propose a dynamic factor framework for modeling and forecasting financial

and commodity variables in a unified global setting. The novelty of our approach is that it

exploits a large set of information at the daily frequency (i.e. data properties, time and forward

dimensions as well as cross-country, market, sector and weather dimensions) summarized in

a set of heteroskedastic components that have a time series interpretation and that can be

modeled dynamically to generate forecasts on real time. We employ a battery of in-sample

and out-of-sample techniques to evaluate our framework and we concentrate on statistical and

economic performance measures relevant for decision makers (e.g. porfolio managers, central

bankers).

Our study is motivated by the fact that, to the best of our knowledge, no research has

been done so far that brings together a unified framework for modeling and forecasting financial

and commodity futures at the international level. Such a unified approach may be increasingly

important due to the degree of financial spillovers and interdependence, across borders, and

between asset classes (e.g., IMF (2016)). Commodity prices affect the demand for currencies

and equities of commodity exporters (e.g., Australia, Canada, Chile, Norway, South Africa);

and commodity currencies have been shown to help forecast commodity prices (Chen et al.,

2010). Rising commodity prices also raise the terms-of-trade and growth rates of commodity-

rich economies; the associated increase in aggregate demand induces monetary policy tightening

under inflation-targeting regimes, raising bond market yields. Moreover, international financial

integration has contributed to an erosion of monetary policy independence, and strengthened the

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responsiveness of bond yields (particularly along the long end of the yield curve, in advanced and

emerging economies) to the global financial cycle, which is largely driven by monetary conditions

in the US (Rey, 2014; Obstfeld, 2015). Last but not least, growth spillovers (e.g., from China)

cause a degree of international co-movement in rates of output growth, and short-term interest

rates.

One of the most popular term structure modeling benchmarks (albeit in the interest rate

literature) is the Nelson and Siegel (1987) model which decomposes the term structure of interest

rates into three factors, namely, the level, slope and curvature of the yield curve. Diebold

et al. (2008) and Diebold and Li (2006) have extended the NS approach to incorporate other

global factors and time-series structures and have demonstrated its good forecasting capabilities

and its applicability for macroeconomic analysis (Diebold et al., 2006). Recent work has also

highlighted the good fit of the NS structure for commodity markets (Karstanje et al., 2015)

vis-a-vis other important benchmarks in the commodities pricing literature such as the seminal

work by Schwartz (1997) and Schwartz and Smith (2000).

In general, factor models have shown to be a promising avenue for modeling futures and/or

yield curves and to explain the variation of the macroeconomy (Ang and Piazessi, 2003; Cochrane

and Piazessi, 2005, 2008). This is not surprising as term structures contain important informa-

tion along the time and forward dimensions which are difficult to account for with large scale

macro models. Nevertheless, research on term structures is still in its infancy, in particular

studies that account for both financial and commodity markets in a unified approach.

A handful of studies have recently put forward models for commodity products at the daily

frequency with ‘real world’ applications such as hedging and portfolio allocation (Boswijk et al.,

2015; Cavalier et al., 2015; Dolatabadi and Nielsen, 2015; Dolatabadi et al., 2015). What seems

to be a common finding is that accounting for fractional cointegration, fits well the data in-

sample and out-of-sample. Previous studies have also found evidence of fractional cointegration

in daily equity and exchange rate dynamics which imply a dissipation of shocks to equilib-

rium relations only at long horizons; thus hinting at the promising applicability of fractional

cointegrated models for forecasting with data at higher frequencies (de Truchis, 2013; Baillie

and Bollerslev, 1994). Moreover, as shown both empirically and theoretically in the behavioral

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finance literature, regime-switching mechanisms can help explain the stylized features found

in financial data as well as adapt to structural breaks (Grauwe and Grimaldi, 2006; Huisman,

2009). However, to what extend the commonalities of financial and commodity markets can be

modeled via fractional cointegration dynamics as well as heteroskedasticity and regime-switching

features and their contribution to in-sample and out-of-sample information is, to the best of our

knowledge, not known. In this study we contribute to the rising literature on term structure

modeling by bridging commodities and financial markets in a unified framework that accounts

for important commonalities between these markets with realistic time-series mechanisms.

The specific contributions of our study are threefold. First, we extend the NS-type structure

to account for stochastic seasonalities which are important determinants of some commodity

markets (e.g., gas, gasoline, livestock, grains, etc). Moreover, we adapt the model to incorporate

global, market, sector and idiosyncratic components by introducing commonalities that account

for the effects of different countries, financial and commodity markets, alternative sectors (i.e.

oil and gas, metals, foreign exchange, equity, bonds) and idiosyncratic (i.e. product) specific

shocks. Bond market yields have a strong ‘global’ commonality; and commodities, exchange

rates, and equities are driven by demand and supply conditions which have a strong global

spectrum. Thus the importance of modeling all the markets and sectors considered here in a

unified setting.

Second, we propose a Regime-Switching Fractional Cointegrated VAR with Orthogonal

GARCH-in-mean errors (RSFCVAR-OGARCH-M) to model the dynamics of the global, market,

sector and seasonalities at the daily frequency. The latter specification accounts for many ‘styl-

ized facts’ of financial and commodity markets data at the daily frequency, namely, jumps, lep-

tokurtosis, (conditional) heteroskedasticity, fractional integration, amongst others (Lux, 2009;

Huisman, 2009). This is important as market practitioners deal daily with hedging decisions

of financial and commodity products and policy authorities study the daily effects of monetary

policy on term structures (e.g. yield curves) so that an appropriate framework should account

for characteristics in data dynamics as close as possible.

Third, we adopt a battery of in-sample and out-of-sample analyses that should allow prac-

titioners not only to extract useful and relevant information in real-time (e.g., variance term

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structures, variance decompositions, macroeconomic mapping) but also to generate and evaluate

point and density forecasts of crucial financial and commodity variables.

The article is organized as follows. The next section describes the model. Section 3 discusses

the data and estimation approach and Section 4 the forecasting methodology. Section 5 and

Section 6 are devoted to discussing the results and our final remarks, respectively.

2 The model

The model is based on previous work by Nelson and Siegel (1987), Diebold et al. (2008), Diebold

and Li (2006), Karstanje et al. (2015) and Boswijk et al. (2015). We consider the NS structure

which is extended to incorporate seasonality components found in commodity markets such as

natural gas, gasoline, some grains and livestock, amongst others:

zit(τ) = lit + Λs(λit, τ)sit + Λc(λit, τ)cit + Λf (κit, cos(τ, η), sin(τ, η))fit + εit(τ). (1)

In the above expression zit(τ) = lnZit(τ) is used to denote the natural logarithm of the futures

price of product i = 1, ..., N at day t = 1, ..., T and forward month τ = 1, ..., T , lit is the so-

called level factor, sit is the slope factor, cit is the curvature factor, fit is the seasonality factor

introduced for our framework and εit(τ) is a ‘measurement error’.

The base loadings Λs(•),Λc(•),Λf (•) quantify the loads of the slope, curvature and seasonal-

ity along the forward dimension. In principle, the latter loadings can take alternative analytical

forms, for instance, cublic spline loadings (Wold, 1974; Suits et al., 1977). In our context, the

base loadings are given by:

Λs(λit, τ) =1− e−λitτ

λitτ, (2)

Λc(λit, τ) =1− e−λitτ

λitτ− e−λitτ , (3)

Λf (κit, cos(τ, η), sin(τ, η)) =κit [1− cos(ητ) + sin(ητ)]

2ητ, (4)

where λit is the so-called maturity parameter which quantifies the steepness and the shape of the

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term structure. More precisely, the latter parameter can be interpreted as the ‘mean reversion’

rate of the slope and curvature factors. Moreover, we introduce the trigonometric funtions cos(·)

and sin(·) which allow us to incorporate a cyclical effect along the forward dimension of the

term structure with time-varying amplitude given by κit and constant number of cycles with

respect to the forward month τ given by η = 2π3−1.1 To illustrate the latter functions, Figure 1

displays the behavior of the above base loadings for fixed (estimated) parameter values of λi

and κi for the products considered.

The commonalities of the level, slope, curvature and seasonality factors are modeled by

means of the following factor decomposition for each product i at day t:

lit = li + γgl,iLg,t + γml,iLm,t + γnl,iLn,t + εl,it, (5)

sit = si + γgs,iSg,t + γms,iSm,t + γns,iSn,t + εs,it, (6)

cit = ci + γgc,iCg,t + γmc,iCm,t + γnc,iCn,t + εc,it, (7)

fit = fi + γuf,iFu,t + γvf,iFv,t + γwf,iFw,t + εf,it, (8)

where Lg,t, Sg,t, Cg,t denote

the global components g = global =glb, the variables Lm,t, Sm,t, Cm,t denote the mar-

ket components m = commodities,financial = com,fin and Ln,t, Sn,t, Cn,t denote the sec-

tor components n = energy,metals, softs, grains, livestock, foreign exchange, bonds, equity =

ene,met, sof, gra, liv, forex,bon, eqt of the level, slope and curvature factors, respectively.2

Moreover, Fu,t, Fv,t, Fw,t denote the components driving the stochastic behavior of the season-

ality factors (e.g. weather). Finally, the components ε•,it for • = l, s, c, f are the idiosyncratic

shocks of the level, slope, curvature and seasonality factors. In the above decomposition, we

assume that the components per factor (global, market, sector and idiosyncratic) are uncorre-

lated.

The dynamics of Lj,t, Sj,t, Cj,t for j = g,m, n and Fj,t for j = u, v, w are modeled by

1We derive the seasonality specification in (4) by integrating the function f(τ, κi) = κ1,i sin(ητ) +κ2,i cos(ητ)forward over [0, τ ] and dividing the result by τ to remain along the lines of the original NS derivation. To reduceon the number of parameters to be estimated we assume κ1,i = w1κi, κ2,i = w2κi with w1 = w2 = 1/2 a weightingfactor and κi the amplitude parameter.

2‘Global’ component is taken here as a common component amongst all products and country specific markets.

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means of the following Regime-Switching Fractionally-Cointegrated VAR process with Orthog-

onal GARCH-in-mean (FCVAR-OGARCH-M) innovations:

∆dj,rtXj,t = αj,rtβ′j,rt∆

dj,rt−bj,rtΥj,bj,rtXj,t +

P∑p=1

Γpj,rt∆dj,rtΥp

j,bj,rtXj,t

+ ζj diag(Hj,t) + ξj,t, (9)

ξj,t ∼ N(0, Hj,t), (10)

Hj,t = BjEt[uj,tu

′j,t|It−1

]B′j = BjΩj,tB

′j , (11)

Ωj,t = (I3 − diag(θj)− diag(δj)) + diag(θj) Ωj,t−1 + diag(δj) uj,t−1u′j,t−1, (12)

where the vector Xj,t is given by Xj,t = [Lj,t, Sj,t, Cj,t]′ for j = g,m, n or Xj,t = [Fu,t, Fv,t, Fw,t]

′

for j = f . In (9), ∆d is the fractional difference operator (ignoring regime-switching) and

Υj,b = 1−∆b is the fractional lag operator. Following Johansen and Nielsen (2012) a time series

is said to be fractional of order d, denoted Xt ∈ I(d), if ∆dXt is fractional of order zero, that is if

∆dXt ∈ I(d). A k-dimensional time series Xt ∈ I(d) is said to be fractionally cointegrated when

one or more linear combinations are fractional of a lower order, that is, when a k × r matrix β

exists such that β′Xt ∈ I(d− b) with b > 0. In addition, the matrices α and Γ are the loadings

of equilibrium adjustment and of short-run dynamics, respectively. The reason for employing a

FCVAR structure here is that at higher frequency of the data, equilibrium relations may exhibit

long-memory and accounting for this feature can have important practical implications for, e.g.

hedging, as found in various studies (Brunetti and Gilbert, 2000; Dark, 2007; Coakley et al.,

2008). Although findings on fractional cointegration with daily data have not been categorized

as ‘stylized facts’ per se, they seem to be the rule rather than the exception when cointegration

is in fact present.

By introducing two regimes rt = 1, 2, we assume that the FCVAR definitions apply but at

different degrees over time. We opt for a specification with regime-switching for three main

reasons. First, regime-switching mechanisms can account for structural and/or random breaks

in the data due to, for instance, changes in investor preferences, which can affect the (parameter)

stability of a particular model. Second, given that the proposed unified framework accounts

for different common components (global, market, sector and weather), we can expect a certain

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degree of heterogeneity amongst the commonalities, so that a model with regime-switching

can potentially ‘absorb’ such heterogeneity. Third, regime switching features can approximate

jumps which are usually found in data of commodity and financial markets (see e.g., Lux (2009);

Huisman (2009)).

In the above specification, the covariance of the error process ξj,t is given by Hj,t for

j = g,m, n, f and is conditional on the information set It−1. We assume that the condi-

tional covariances Hj,t follow Orthogonal GARCH (OGARCH) processes. We also allow for the

variances in Hj,t to have an effect on the conditional mean of Xj,t so that the coefficients in

the 3 × 1 vector ζj can be interpreted as the ‘price’ of common components risk with the

element-by-element (Hadamard) multiplication operator. In addition, considering volatility-in-

mean is advantageous as it may price uncertainty and risk in the common components as well

as approximate arbitrage-free models with volatility-in-mean applicable to derivative pricing

(Duan, 1995; Ludvigson and Ng, 2007; Bollerslev et al., 2008).

Last but not least, another advantage of (10) is that it is general enough to account for other

popular specifications. For instance, with drt = brt = 1 and αrt = 0, the model is a VAR in

first differences, with drt = brt = 1 and αrt = α < 0 the model is a Cointegrated VAR (CVAR)

and with drt = d, brt = b and αrt = α < 0 the model becomes a Fractional Cointegrated VAR

(FCVAR). By considering a general model with various embedded specifications, we can vary

the structure on the set of components at hand (e.g. financial vs. commodities, vs. weather,

etc).

For the idiosyncratic components ε•,it, • = l, s, c, f , we assume simple heteroskedastic au-

toregressive models of order one, i.e.

ε•,it = φ•,iε•,it−1 + a•,it, a•,it ∼ N(0, ω•,it), (13)

where ω•,it follow GARCH(1,1) processes. Similarly, the measurement errors follow het-

eroskedastic autoregressive processes of order one:

εit(τ) = ϕi(τ)εit−1(τ) + %it(τ), %it(τ) ∼ N(0, υit(τ)), (14)

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υit(τ) follow simple GARCH(1,1) processes. Moreover, we assume that the shocks a•,it and %it(τ)

are uncorrelated across i’s, i.e. we assume a diagonal (co)variance structure for these quantities

are given by Dt = diag([ωl,1t, ωs,1t, ..., ωc,Nt, ωf,Nt]) and Vt = diag([υ1t(1), υ1t(2), ..., υNt(T −

1), υNt(T )]), respectively.3 The full model can be represented compactly in state-space form,

i.e.

Zt = K + ΠXt + Et, (15)

Xt|It−1 ∼ N(0, Qt), (16)

Et|It−1 ∼ N(0, Vt), (17)

where Zt = [z1t(1), ..., z1t(T ), z2t(1), ..., z2t(T ), ..., zNt(1), ..., zNt(T )]′,

Xt = [Lglb,t, Sglb,t, Cglb,t, ..., Lfin,t, Sfin,t, Cfin,t, ..., Lmet,t, Smet,t, Cmet,t, ..., Fu,t, Fv,t, Fw,t, ...,

εl,Nt, εs,Nt, εc,Nt, εf,Nt]′ and Et = [ε1t(1), ..., ε1t(T ), ε2t(1), ..., ε2t(T ), ..., εNt(1), ..., εNt(T )]′.

Moreover, K is a N · T vector of constants, Π is a N · T × 204 matrix of coefficients with

Qt and Vt the conditional covariances of Xt and Et, respectively. More details about the func-

tional form of the above state-space representation can be found in the Appendix.

3 Data and estimation

3.1 Data description

Term structure data is obtained mainly from Bloomberg and in some particular cases from

Datastream. Table 1 summarizes the data collected and the specific sources. We employ

monthly rollover futures series for 36 months at the daily frequency starting from 2010-10-

01 and ending on 2015-09-30.4 The sample period chosen was based on data availability and

covering up to eight (8) ‘seasonality years’.5 As it is usually the case, some products had the full

3While one could argue that the product and seasonality specific shocks or measurement errors might exhibitmore complex dynamic structures, previous studies have used similar specifications (albeit for monthly data) andhave demonstrated that such simple structures work well in-sample and out-of-sample (Diebold and Li, 2006).In our context, given our large scale model we opt for simple specifications for the idiosyncratic shocks to keepour estimations tractable.

4Note that 3 years × 12 months = 36 months ÷ 6 months = 6 cylces/seasons for the seasonality along theforward dimension τ

5A seasonal year is defined here as one that starts on October 1st and ends in September 30th, commonlyknown in the gas and power industry as a ‘gas year’.

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36 forward months available (e.g. gas and oil, softs and grains) while others had intermittent

data over the 36 forward months (e.g. exchange rates, bonds and some metals). We have

interpolated weekends and holidays for simplicity as weekend data were not available for many

of the products under consideration.6

The data employed for the analyses at the monthly frequency, i.e. the data used for the

mapping of the extracted components to macroeconomic factors (as will be explained below),

are mainly obtained from Datastream and the World Bank Database with some exceptions

which were obtained from the Haver. We collected macroeconomic data at the country level

for a balanced panel of 19 countries for the same period as for the futures data, i.e. 2010-10 and

ending on 2015-09. All groups of macro data contain the main industrialized economies (G7)

and the main emerging markets (BRICS) amongst others. Detailed information about the data

is provided in Table 1.

3.2 Model estimation

The model described in the preceding section considers time, cross-section and forward dimen-

sions along with a heteroskedastic and regime-switching dynamic specification of the common

components. While the model (or a restricted version of it) could in principle be estimated by

means of the so-called ‘first generation’ dynamic factor approaches (e.g., Kalman-Nelson-Kim

filter given its linear-Gaussian state-space representation) or Bayesian techniques, a large pa-

rameter space renders such estimation approaches computationally cumbersome in particular

when forecasting exercises are at hand. Given that we are dealing with quite a large-scale

model, we opt for the ‘third generation’ approach as detailed in Stock and Watson (2011)

whereby parameters and factors/components of the model are estimated in various steps, and

the state-space representation is subsequently employed for in-sample and out-of-sample anal-

yses. This approach reduces computational time for forecasting and allows for more general

dynamic specifications as opposed to (say) simpler autoregressive models as is the case in other

applications.7

6We have also employed business days as opposed to interpolated weekend data and the results do not differqualitatively. We decided for interpolation in order to smooth out any weekend effects out of the analysis.

7In fact, Diebold and Li (2006) show that k-step estimation approaches within the NS framework that arerelatively easy to implement have the advantage that they can be succesfully applied for forecasting without much

10

In what follows we describe the main steps of the base estimation approach considered here.

Additional details can be found in the Appendix of this article.

1. The first step consists of estimating a product’s level, slope, curvature and seasonality

factors lit, sit, cit, fit as well as the maturity and amplitude parameters λit and κit in

equation (1) by means Bayesian Averaging (BAV) based on various procedures which

are summarized in Table 2 along with their advantages and disadvantages. Note that

by employing BAV based on different estimation approaches (Non-linear Least Squares,

Ordinary Least Squares, Generalized Least Squares) and alternative specifications of the

maturity and amplitude parameters (e.g., time-varying product vs. constant product

vs. time-varying sector vs. constant sector) we should reduce uncertainty in the factor

estimates (Hoeting et al., 1999).

2. The second step consists of estimating the factor decomposition in (5)-(8). We start

by standardizing the BAV estimates of the level, slope, curvature and seasonality factors

denoted lBAVit , sBAVit , cBAVit and we employ principal component analysis (PCA) to identify

the common components by sequentially (i) extracting the estimated global components

Lg,t, Sg,t, Cg,t from the factors along time and cross-section dimensions, (ii) regrouping

the residuals into financial and commodity markets and extracting the corresponding

financial and commodity estimated components Lm,t, Sm,t, Cm,t from each market, (iii)

regrouping the residuals into sectors (energy, metals, etc) and extracting the estimated

sector components Ln,t, Sn,t, Cn,t from each sector. The seasonality components Fut,

Fvt, Fwt are the three first principal components of the seasonality factors fBAVit obtained

along both time and cross-section dimensions. Given the estimated common components

Lj,t, Sj,t, Cj,t for j = g,m, n and Fu,t, Fv,t, Fw,t we employ system Generalized Method of

Moments (GMM) to (re)estimate the parameters in (5)-(8) for all i = 1, ..., N . We employ

one lag of the component estimates as instruments and a Newey-West HAC covariance

as weighting matrix. The idiosyncratic component estimates ε•,it for • = l, s, c, f are the

computational burden. Similar findings on the out-of-sample applicability of multi-step estimation are providedby Caldeira et al. (2015, 2016). For a deeper discussion on the advantages and disadvantages of alternativedynamic factor modeling approaches, we refer the reader to Stock and Watson (2011) and leave the comparisonbetween alternative estimation methods for future research.

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residuals resulting from the GMM system regression.

3. The third step consists of estimating the RSFCVAR-OGARCH-M model in (10)-(12) (or

restricted versions) for the global, market, sector and seasonality common components

by means of (concentrated) Maximum Likelihood (ML). The likelihood function for j =

g,m, n, f is given by

Lj =∑t∈T

log[π

(1)j,t · N (Yj,t|rt = 1) + (1− π(1)

j,t ) · N (Yj,t|rt = 2)], (18)

where π(1)j,t = P(rt = 1|It−1) is the probability of regime 1 conditional on the information

set I at period t− 1, N (·|rt = r) is the conditional Normal distribution given that regime

r = 1, 2 occurs at time t for Yj,t = Xj,t|It−1. We employ the Hamilton Filter (HF) in order

to approximate the conditional probabilities π(1)j,t and the contribution of N (·|rt = r) to

the likelihood for each regime r = 1, 2. Note that we account for different versions of (9),

whose restrictions are found in Table 3 and for which the HF is not needed.

4. The fourth and last step consists of estimating (13) and (14) by employing the idiosyncratic

component estimates ε•,it obtained from the GMM residuals in step two and the measure-

ment error estimates εit(τ) obtained from Et = Zt − ΠXt in (15) and employing ML to

estimate the autoregressive and GARCH(1,1) parameters for each i and τ . In this case, the

likelihoods reduce to Li =∑

t∈T log [N (ε•,it|It−1)] and Li(τ) =∑

t∈T log [N (εit(τ)|It−1)],

respectively.

Once the parameters of the model have been estimated, we employ the state-space representation

in (15)-(17) to conduct various in-sample and out-of-sample analyses which are described in the

following sections.

4 In-sample analysis

4.1 Variance decompositions

The factor decomposition of our model presented in Section 2.1 provides an excellent platform

for analyzing the contribution of each of the common components to explaining the percentage

12

variation in the variance of the level, slope, curvature and seasonality factors of the term struc-

tures considered. Since our model accounts for conditional variances, we can decompose the

variance contributions of the components at every point in time. For the purpose of this study

we consider the following factor decompositions in percentage terms:

1 = Var%,iµ [Lg,µ] + Var%,iµ [Lm,µ] + Var%,iµ [Ln,µ] + Var%,iµ [εl,iµ] ,

+ Var%,iµ [Sg,µ] + Var%,µ [Sm,µ] + Var%,iµ [Sn,µ] + Var%,iµ [εs,iµ] ,

+ Var%,iµ [Cg,µ] + Var%,iµ [Cm,µ] + Var%,iµ [Cn,µ] + Var%,iµ [εc,iµ] ,

+ Var%,iµ [Fk,µ] + Var%,iµ [Fv,µ] + Var%,iµ [Fw,µ] + Var%,iµ [εf,iµ] , (19)

where µ = 1, ..., Tµ is used to denote a particular month whereby t ∈ µ and Var%,µ [•] is the

average percentage variance contribution of component • = Lg,µ, Lm,µ, ... over month µ. We

aggregate the daily decompositions at the monthly level for two main reasons. First, in practice

most macroeconomic analyses are performed at the monthly, quarterly or yearly frequencies.

Second, by aggregating at the monthly level we can reduce ‘noise’ that may occur at the daily

level due to volatility ‘jumps’ while conserving the overall trends in the percentage variance

contributions.

4.2 Macroeconomic mapping

As mentioned previously, several studies have documented the strong explanatory power of

term structure factors vis-a-vis the macroeconomy (Ang and Piazessi, 2003; Diebold et al.,

2006; Cochrane and Piazessi, 2008). Our study takes this type of analyses forward by mapping

the estimated global, market and sector components of the level, slope and curvature factors

obtained from commodities and financial term structures to the macroeconomy. Note, however,

that in the case of a macroeconomic analysis at the monthly level, the large dataset considered

here and relatively few data points at the monthly frequency (60) makes it impossible to use

all country specific data in single regression models. Thus, we aggregated all country informa-

tion in a set of common factors obtained from each of the categories of macroeconomic data

(consumption, consumer prices, industrial production, etc) as detailed in Table 1. The factors

13

are computed by means of PCA, and we chose the number of principal components based on

the percentage variance explained which we truncated to 95%.8 Specifically we consider the

following model:

∆12Xj,µ = Φj∆12Mµ + Ej,µ, (20)

where ∆12 denotes the year-on-year (yoy) difference operator Xjµ =[Ljµ, Sjµ, Cjµ

]′is the

vector of yoy changes of the estimated level, slope and curvature components for j = g,m, n

at month µ, Mµ =[CPI1µ, CP I2µ, ..., EMP 1µ, EMP 2µ

]′is the vector of yoy changes of the

estimated macroeconomic components, Φj is a matrix of macroeconomic loadings and Ej,µ is a

vector of measurement errors.9 To correct for possible endogeneity in the regressors, as well as

heteroskedasticity and autocorrelation in the measurement errors we employed system GMM

estimation with one lag of the factors as instruments and an estimate of the inverse of the

Newey-West HAC covariance as weighting matrix.

5 Forecasting Methodology

In the following subsections, we describe the forecasting strategy designed for this study. In

order to save on space, we concentrate on the most relevant issues. Specific details that are not

described here or in the Appendix to this article can be provided upon request.

5.1 Forecasting design

We employ a forecasting scheme whereby we estimate all parameters needed to ‘calibrate’ the

state-space representation in (15)-(17) with in-sample data up to time t = 1, ..., T and obtain

multi-step ahead forecasts T + h, T + h + 1, ... for horizons h = 1, 7, 30, i.e. daily, weekly and

monthly (the most frequently used horizons in practice). The chosen set of out-of-sample dates

8Nevertheless, in some cases the later procedure resulted in too many factors relative to the data pointsavailable. Thus, we truncated the number of factors to two (2) when more than two factors were needed to reachthe 95% cumulative variance threshold.

9The variables in Mµ are all in logs except for interest rate data and inventory changes.

14

run from 10/2014 to 09/2015. Formally, forecasts are computed as

Zt+h|t = K + ΠXt+h|t + Et+h|t, (21)

Σt+h|t = ΠQt+h|tΠ′ + Vt+h|t. (22)

In order to illustrate the forecasting procedure in the following discussion, we concentrate on

general concepts and refer the reader to the Appendix for other details. In our context, we

have two regimes rt = 1, 2 embedded in the full dynamic specification in (9). That is, forecasts

generated from (9) apply for each of the regimes with corresponding parameters dr, br, βr, αr,Γpr

for r = 1, 2. More precisely, forecasts Xj,t+h|t =[Lj,t+h|t, Sj,t+h|t, Cj,t+h|t

]′for j = g,m, n or

Xj,t+h|t =[Fu,t+h|t, Fv,t+h|t, Fw,t+h|t

]′for j = f are computed as

Xj,t+h|t = π(1)j,t+h|t · X

(1)j,t+h|t + (1− π(1)

j,t+h|t) · X(2)j,t+h|t, (23)

where X(r)j,t+h|t for r = 1, 2 are the forecasts corresponding to each regime and π

(r)j,t+h|t is an

estimate of the conditional regime-switching probabilities at horizon h. Note also that multi-

step ahead forecasts Hj,t+h|t can be obtained recursively so that they can be applied for the

GARCH-in-mean estimates or as input for the conditional covariance matrix Qt+h|t in (22).

In the case of the measurement errors in (14) and idiosyncratic components in (13) it is

relatively straightforward to obtain multi-step ahead forecasts ε•,it+h|t and εit+h|t(τ) as these

quantities follow simple autoregressive processes. The same applies for the conditional variance

forecasts ωit+h|t and υit+h|t(τ) which assume GARCH(1,1) processes and whose multi-step ahead

specifications are well-known (Tsay, 2010). The term structure forecasts that result from (21)

conditioned upon Xt+h|t and Et+h|t are given by:

zit+h|t(τ) = lit+h|t + Λs(λi, τ)sit+h|t + Λc(λi, τ)cit+h|t

+ Λf (κi, cos(τ, η), sin(τ, η))fit+h|t + εit+h|t(τ). (24)

Analogously, the volatility term structures that result from (22) conditioned upon Qt+h|t and

15

Vt+h|t are given by

Vart [zit+h(τ)] = Vart [lit+h] + Λs(λi, τ)2 · Vart [sit+h] + Λc(λi, τ)2 · Vart [cit+h] ,

+ Λf (κi, cos(τ, η), sin(τ, η))2 · Vart [fit+h]

+ 2Λs(λi, τ)Covt [lit+h, sit+h] + 2Λc(λi, τ)Covt [lit+h, cit+h]

+ 2Λf (κi, cos(τ, η), sin(τ, η))Covt [lit+h, fit+h]

+ 2Λs(λi, τ)Λc(λi, τ)Covt [sit+h, cit+h]

+ 2Λs(λi, τ)Λf (κi, cos(τ, η), sin(τ, η))Covt [sit+h, fit+h]

+ 2Λc(λi, τ)Λf (κi, cos(τ, η), sin(τ, η))Covt [cit+h, fit+h] + Vart [εit+h(τ)] ,(25)

where Vart [•] and Covt [•] are estimated (co)variances conditional on information available up

to period t.10

Ideally, we could estimate the model up to its forecasting origin and roll the estimation

to the next forecasting origin and so on, i.e., we could employ a rolling window (or recursive)

scheme for the estimation of parameters and subsequent forecasting. However, in our context,

rolling (or recursive) estimation is cumbersome due to the large scale model under consideration.

Instead, we have broken down the analysis into three sample periods for estimation (10/2010-

09/2012, 10/2012-09/2013, 10/2010-09/2014) and used a Jacknifing procedure first introduced

by Quenouille (1956) and employed empirically and in Monte Carlo simulation settings more

recently by other studies (Chiquoine and Hjalmarsson, 2009). The Jacknife estimator of our

model(s) is given by

Ψrt,Jack =SS − 1

· Ψrt,T−∑S

l=1 Ψrt,l

S2 − S. (26)

where, S is the number of consecutive subsamples and Ψrt,T, Ψrt,l are the vectors of estimated

parameters for the full sample T and the l-th subsample. The above estimator has been shown

to reduce the bias induced by estimating parameters when using a limited number of calibration

10Note that in the expressions in (24) and (25) we assume constant estimates λi and κi as opposed to their time-varying versions. This is done because the time-varying case would imply assuming and estimating a dynamicspecification for λit and κit which is out of the scope of this paper. Thus, for the purpose of this study we usethe mean of the BAV estimates λBAVit and κBAVit up to the forecasting origin for subsequent forecasting.

16

windows instead of, e.g., rolling (or recursive) estimation schemes.11 Moreover, in our context

we are assuming time variability (i.e. regime-switching) in some parameters of the model so

that, together with the Jacknifing approach, should help us circumvent the drawbacks of not

employing a rolling (or recursive) estimation scheme.

As mentioned previously, some restricted versions of (9) might fit some components better

than others, i.e. there could be a certain degree of heterogeneity with respect to global, market

or sector specific data dynamics. In order to reduce model uncertainty we employ Bayesian

averaging of the parameters obtained from the restricted versions considered of the dynamic

specification in (9).12

5.2 Forecast evaluation

We employ a battery of tools to evaluate the out-of-sample performance of the proposed frame-

work. We focus on statistical and economic performance measures that aim to uncover the

accuracy of point and interval forecasts of the dynamic specifications as well as the hedging

performance within a portfolio of spot and futures contracts.

5.2.1 Statistical performance measures

Let M and Mb indicate a particular competing model and the benchmark, respectively. Our

benchmark model is the random walk model for the factors. We chose this specific benchmark

since the random walk model is the most widely used benchmark in practice to forecast the evo-

lution of financial prices an other assets (Grauwe and Grimaldi, 2006). The average performance

Mc relative to Mb for each product i is computed as

dri(M) =di(Mc)

di(Mb), (27)

11We have also experimented with rolling-window and recursive schemes for estimation of parameters and sub-sequent forecasting with a smaller version of the model. However, rolling estimations of some of the specificationsestimated with the Hamilton filter were very time consuming and the results were qualitatively not better thanwith a few sample windows and the Jacknifing procedure. Indeed, the latter corroborates findings by Chiquoineand Hjalmarsson (2009)

12We experimented with combining forecasts directly with different forecast combination routines but theresults turned out to be qualitatively similar to Bayesian averaging of the parameters in some cases and notbetter statistically in other cases.

17

where di(Mb) and di(Mc) are defined as the average MSE of the benchmark and of the com-

peting model, respectively. There are several tests available to analyze, whether a particular

benchmark model Mb has the same predictive ability as a competing model Mc, against the

alternative that modelMb has a better predictive ability based on Mean Squared Errors (MSE)

(Diebold and Mariano, 1995; Harvey et al., 1997; Clark and West, 2007; McCracken, 2007). In

this study we employ the test proposed by Clark and West (2007), which corrects the non-

standard limiting distribution under the null of equal forecasting accuracy to a nested model.

Moreover, we test interval forecasts generated from the model by means of the three-step pro-

cedure proposed by Christoffersen (1998) which evaluates whether interval forecasts satisfy the

so-called (i) unconditional, (ii) conditional and (iii) independence hypothesis.13

5.2.2 Economic performance measures

We test the economic significance of the futures forecasts by ‘simulating’ a dynamic hedging

strategy whereby an agent enters into a spot position and into a futures contract position with

the aim to reduce the variability of his/her portfolio’s value. That is, the agent seeks to minimize

the variance of his/her portfolio by chosing an optimal amount of futures position per unit of

spot position. The set up is very similar to the one found in previous studies where conditional as

opposed to unconditional moments are treated (Kroner and Sultan, 1993; Brunetti and Gilbert,

2000; Moschini and Myers, 2003). In our context, the minimization problem reduces to the

following hedge ratio:

HRit,h =Covt [rit+h(1), rit+h(τ)]

Vart [rit+h(τ)], (28)

where rit(1) = zit(1)− zit−1(1) is the (log) return of the spot (month-ahead) product, rit(τ) =

zit(τ) − zit−1(τ) is the (log) return of the futures price at forward month τ for product i, and

Covt [·] and Vart [·] are the time-dependent (co)variances of rit(1) and rit(τ) conditional on

information available up to time t. Our benchmark model is a constant hedge ratio denoted

HRb,h obtained by replacing (28) with unconditional moments estimated up to the forecasting

13To save on space, we refer the interested reader to the Clark and West (2007) and Christoffersen (1998)articles for details about the respective tests.

18

origin.14 We evaluate the performance of the dynamic hedging strategy by means of the so-called

variance reduction measure:

VN i,h =

√Var[rh,p]

Var[rbh,p]− 1, (29)

where Var[rh,p] is the variance of the portfolio resulting from hedging with conditional moments

fitted from our model and Var[rbh,p] is the variance of the portfolio resulting from hedging with

the benchmark. Following Lee (2009a,b), in order to test the statistical significance of variance

reduction we use a test of predictive accuracy such as the Clark and West (2007) test.

6 Results

In what follows we discuss the in-sample results of our analysis and subsequently the out-of-

sample results. We opted to display our subsequent results in relevant figures that highlight the

main features of our modelling framework as far as possible as opposed to large tables for space

considerations. Detailed results not displayed here can be provided upon request.

6.1 In-sample results

Figures 5 and 6 display the results of the in-sample estimation for the parameters λit and κit

by means of BAV. The figures display the degree of heterogeneity for the maturity parameters

amongst the different products under inspection and this finding holds not only for commodity

markets but also for financial markets. This result indicates that when fitting such a NS-type

model with alternative markets, sectors and global data, it is advisable to estimate the maturity

parameter as opposed to ‘calibrate’ it as it is done in previous studies (Diebold and Li, 2006).

The latter result also confirms recent findings by Karstanje et al. (2015) who put forward a

non-trivial degree of heterogeneity in the maturity parameters of their commodities model. The

figures show not only that the maturity differs across products but that it varies with time in

14Note that we assume a ‘pair’ strategy for simplicity, i.e. we assume that the hedging is done with respectto one of the futures contract with forward dimension τ and do not consider cross-commodity hedges. This isindeed very interesting for practitioners but is out of the scope of this article.

19

most cases as proposed by Koopman et al. (2010). Since the λit’s can be interpreted as the

mean-reversion rate of the slope and curvatures of the term structures, the heterogeneity and

time-variability of this parameter suggests that the ‘velocity’ and shape of adjustment into, say,

contango or backwardation of terms structures differs amongst products and over time. The

amplitude parameters also seems to be time dependent but the level of heterogeneity is not as

strong.

Figure 7 displays selected examples of the estimated level lit, slope sit, curvature cit and sea-

sonality fit factors of the products considered with four of the different estimation approaches

employed (OLSPRD, GLSPTV, OLSSEC, BAV). As can be noted from the figures, the ap-

proaches differ quantitatively (as expected) in some time periods but overall they exhibit similar

path profiles. In terms of the Bayesian weights computed, however, it appears as if the OLSPTV

and GLSPTV approaches provide more useful information in terms of Bayesian Information Cri-

teria (see Table 2 for further details). Figure 2 displays the fitted smoothed futures prices over

the time dimension for all the products considered. Moreover, Figures 3 and 4 show futures

price and volatility term structures over time and forward dimensions for selected examples of

the products considered. The latter figures show an interesting implementation of the model as

the model-based term structure data over several forward months can be employed for approx-

imating expectations of future price and volatility paths of the products considered every day

and for every forward month wished. This application is important for practitioners as for some

products only intermittent data is readily available at the daily and forward dimensions (e.g.

foreign exchange, equity futures, aluminium, etc) and practitioners use these data as inputs for

taking decisions (i.e. option pricing, hedging, etc).

Tables 4 to 8 show the in-sample estimation results of the component loadings, the long-

run and short-run parameter estimates of the RSFCVAR-OGARCH-M and some diagnostics

of the alternative dynamic specifications considered, respectively.15 The results of a likelihood

ratio test favors the heteroskedastic specification (FCVAR-OGARCH) vis-a-vis the homoskedas-

tic specifications (VAR, CVAR, FCVAR) for most of the component estimates. Similarly, we

find that the likelihood ratio test favors the heteroskedastic and regime-switching specifica-

15We only present the results of the full specification RSFCVAR-OGARCH-M to save on space. Detailedestimation results for the restricted versions can be provided upon request.

20

tions (RSFCVAR-OGARCH, RSFCVAR-OGARCH-M) over the heteroskedastic but non-regime

switching counterpart (FCVAR-OGARCH) for most component estimates. Parameters of the

RSFCVAR-OGARCH-M model are statistically significant for the most part except for some

cases, for instance, the volatility-in-mean parameters of some of the components (Table 7). The

latter result suggests that risk is not significantly ‘priced’ in the conditional mean (in a statistical

sense) in most components.16 In fact, previous studies that analyze the risk-return relationship

have found mixed results with respect to direction or statistical significance of the effect of

risk variables in the (conditional) mean of asset pricing models (Campbell and Hentschel, 1992;

Ludvigson and Ng, 2007; Bollerslev et al., 2008). Overall, we find that the dynamic specifica-

tions that account for fractional cointegration, regime switching and heteroskedasticity fit well

the data in-sample and are statistically better than their restricted and ‘simpler’ counterparts

(VAR, CVAR, FCVAR) for most components according to the in-sample diagnostics considered

(LR, BIC).

Figure 9 displays the variance decomposition of the factors for each product at each point

in time aggregated at the monthly frequency while Table 11 presents results aggregated over

the full sample and over sectors. We find that the global components can explain on average

about 20% of the variance of the factors across all products considered while the market, sector

and idiosyncratic factors can explain up to 20%, 20% and 30% respectively. With respect to

the seasonality components, we find that these variables can explain up to 5% of the variance

of the term structures while the seasonality idiosyncratic component explains up to 5%. While

it is not possible here to compare these results to previous studies directly as there are, to the

best of our knowledge, no one-to-one comparable models, we find that other studies have found

similar results on the contributions of global, sector specific and idiosyncratic components to

total variation in term structures (Diebold et al., 2008; Diebold and Li, 2006; Karstanje et al.,

2015). Moreover, it is worth noting that the results on previous studies hold ‘on average’ while

we show that the variance decompositions appear to be strongly time dependent as depicted in

Figure 9.

16An earlier (experimental) version of our model accounted for regime changes in the GARCH-in-mean pa-rameters. However, the large parameter space made it very cumbersome computationally without some evidentforecasting gains. Thus, we decided for the simpler specification treated here.

21

Table 9 and 10 show the mapping of the extracted components to macroeconomic factors

as introduced in Section 3. We find that the global components are related to variables such

as exchange rates, wages, price-earnings ratios, leading economic indicators, employment and

house prices. In turn, the market component of commodity products is related to variables such

as output growth, exchange rates, volatility of exchange rates, business confidence, amongst

others while the market component of financial products is explained by inflation, dividend

yields, wages and terms of trade. The bond component is related to variable such as term

spreads, wages, employment and house prices amongst others. The foreign exchange component

is related to exchange rates, business confidence, leading economic indicator, term spreads,

amongst others. The equity component is related to leading economic indicators, PE ratios and

wages, amongst others.

The energy component is related to business confidence, exchange rates, volatility, house

prices, amongst others. The metals component is related to output growth, exchange rates,

leading economic indicators, consumption, amongst others. Other commodity sector compo-

nents (grains, livestock and softs) are related to macroeconomic factors of the energy sector in

general. Our results follow the same line of previous studies where unobservable components in

financial and commodity prices can be successfully mapped to the U.S. macroeconomy (Fama

and French, 1987, 1988; Ang and Piazessi, 2003; Diebold et al., 2006). Our study takes the

mapping one step forward as we show that the world, market and sector commonalities of fi-

nancial and commodity term structures have information that can ‘replicate’ movements in the

macroeconomy in an international context.

Overall, we find that the dynamic specifications proposed for the components fit the data

well in-sample. We also find some degree of heterogeneity amongst model specifications within

alternative components which hints at the possibility that ‘hybrid’ specifications obtained trough

combined models (e.g. through Bayesian averaging) might reduce forecasting uncertainty. More-

over, we uncover an economic and statistical significant relationship of the unobservable com-

ponents to the macroeconomy which suggests that these variables might serve as complement

or alternative to macroeconomic data at lower (e.g. monthly) or higher (e.g. daily) frequen-

cies. What degree of predictability power these components have at the daily frequency will be

22

uncovered in the next section.

6.2 Out-of-sample results

In this section we discuss the out-of-sample results. We start by discussing the results of the

statistical performance measures and subsequently the economic performance measures.

As discussed in Section 4.1, we perform evaluations with alternative tests that aim to ana-

lyze different features of the out-of-sample forecasts generated by our model. Given the large

set of forecasts generated by our model (cross-section, time and forward dimension) we have

summarized the results of our various tests in boxplots which are displayed in figures 9 and

10. The boxplots show the tests aggregated over the cross-section dimension for multi-step

ahead horizons h = 1, 7, 30 (daily, weekly, monthly) and for the forward dimensions τ = 1, 3, 6

(month-ahead, quarter-ahead and semester-ahead).

In terms of forecasting accuracy by means of the Clark and West test, we find that many

products generate forecasts that are statistically better than those of a random walk type model.

However, we find that at the aggregate level the forecasting accuracy of the model is only slightly

better than the random walk specification when observing the median of the distribution of test

statistics. The latter result is in line with the relative MSE results which show that the model

generates more accurate forecasts than a random walk benchmark for various products but at

the aggregate level, relative MSE are only slightly better at higher horizons as shown by the

median of the distribution.

Interestingly, however, we find that the interval forecasts generated comply with the un-

conditional, conditional and independence hypothesis at alternative forecasting horizons and

forward dimensions. The latter result holds at the aggregate level as well as at the product

specific level. While the accuracy of point forecasts vis-a-vis a naive benchmark appears to

be product specific, interval (i.e. density) forecasts generated from the model provide accurate

information with respect to the out-of-sample distribution of term structures.

As introduced in Section 4.3, we also test the economic significance of the forecasts of our

proposed model by testing the performance of a hedging strategy resulting from the variance

minimization problem of a portfolio constructed of a spot and a futures position. Results over

23

all 42 products are displayed in Figure 10. The results indicate that the model generates on

average a reduction of about 20% in portfolio variance in relation to a naive strategy. Results on

the statistical significance of the variance reduction by means of the Clark and West test show

that, as in the case of point forecasts, the reduction in variance significantly improves upon a

naive strategy at the product specific level but is only slightly better than a naive strategy at

the aggregate level.

Our results confirm findings in the rising literature on hedging in commodity markets which

show that fractional cointegrated and regime switching models have good forecasting capabilities

vis-a-vis naive benchmarks from both a statistical an economic perspective. However the results

depend many times on the product, horizon and forward dimension analyzed (Boswijk et al.,

2015; Cavalier et al., 2015; Dolatabadi and Nielsen, 2015; Dolatabadi et al., 2015).

7 Conclusion

The present article proposed a framework for modeling and forecasting financial and commod-

ity term structures in a unified global setting. Our framework not only allows analysts to

extract interesting and useful information in-sample such as time-varying variance decomposi-

tions, dynamic correlations, mapping to the macroeconomy, but also provides a value added for

forecasting both in terms of statistical and economic performance measures. The main results of

our study show that the extracted components can account for an important amount of the total

variation in the data. The mapping to the macroeconomic fundamentals show that there is, as

expected, a statistically significant relationship of the extracted components to macroeconomic

variables.

As for the out-of-sample evaluation we find that the proposed model can generate forecasts

that outperform a naive benchmark (random walk) at the product level in terms of point-forecast

accuracy and variance reductions resulting from an artificial hedging strategy. At the product

and aggregate level we find that the proposed model generates accurate interval (i.e. density)

forecasts according to the unconditional, conditional and independence hypothesis tests.

The results of this study are not only useful for academics extending versions of the NS

24

model but also for practioners who use term structures to take decisions with respect to hedg-

ing, portfolio optimization or financial and monetary policies. An interesting extension to the

present analysis would be whether other volatility structures (e.g. stochastic volatility or dy-

namic conditional correlations) can improve upon the current results in terms of statistical and

economic performance. We leave these issues to future research.

25

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29

A State-space representation

The state-space representation of our model is given by:

z1t(1)

z1t(2)...

z1t(T )...

zNt(1)...

zNt(T )

= A

l1

s1

c1

f1

...

lN

sN

cN

fN

+B

Lglb,t

Sglb,t

Cglb,t

+ C

Lcom,t

Scom,t

Ccom,t

Lfin,t

Sfin,t

Cfin,t

+D

Lene,t

Sene,t

Cene,t

Lmet,t

Smet,t

Cmet,t...

Leqt,t

Seqt,t

Ceqt,t

+ E

Fu,t

Fv,t

Fw,t

+A

εl,1t

εs,1t

εc,1t

εf,1t...

εl,Nt

εs,Nt

εc,Nt

εf,Nt

+

ε1t(1)...

ε1t(T )...

εNt(1)...

εNt(T )

= A · J +B · Xg,t + C · Xm,t +D · Xn,t + E · Ft +A · Ut + Et

= K + [B, C, D, E, A][Xg,t; Xm,t; Xn,t; Ft; Ut] + Et, (30)

= K + ΠXt + Et. (31)

where we use semi-colon (;) to denote vertical concatenation. The matrices of coefficients are

given by:

30

A=

1Λs(λ

1,1

)Λc(λ

1,1

)Λf(κ

1,c

os(1,η

),si

n(1,η

))0

···

···

0. . .

. . .. . .

. . .. . .

. . .. . .

. . .

1Λs(λ

1,T

)Λc(λ

1,T

)Λf(κ

1,c

os(T,η

),si

n(T,η

))0

···

···

0. .

.. .

.. .

.. .

.. .

.. .

.. .

.

0···

···

01

Λs(λN,1

)Λc(λN,1

)Λf(κN,c

os(

1,η

),si

n(1,η

)). . .

. . .. . .

. . .. . .

. . .. . .

. . .

0···

···

01

Λs(λN,T

)Λc(λN,T

)Λf(κN,c

os(T,η

),si

n(T,η

))

,(32)

B=

γglbl,

1Λs(λ

1,1

)γglbs,

1Λc(λ

1,1

)γglbc,

1. . .

. . .. . .

γglbl,

1Λs(λ

1,T

)γglbs,

1Λc(λ

1,,T

)γglbc,

1. . .

. . .. . .

γglbl,N

Λs(λN,1

)γglbs,N

Λc(λN,1

)γglbc,N

. . .. . .

. . .

γglbl,N

Λs(λN,T

)γglbs,N

Λc(λN,T

)γglbc,N

,(3

3)

C=

γcom

l,1

Λs(λ

1,1

)γcom

s,1

Λc(λ

1,1

)γcom

s,1

0···

0. . .

. . .. . .

. . .. . .

0

γcom

l,1

Λs(λ

1,T

)γcom

s,1

Λc(λ

1,T

)γcom

s,1

0···

0. .

.. .

.. .

.. .

.. .

.. .

.

0···

0γfin

l,N

Λs(λ

1,1

)γfin

s,N

Λc(λN,1

)γfin

s,N

. . .. . .

. . .. . .

. . .0

0···

0γfin

l,N

Λs(λN,T

)γfin

s,N

Λc(λN,T

)γfin

s,N

(3

4)

31

D=

γene

l,1

Λs(λ

1,1

)γene

s,1

Λc(λ

1,1

)γene

s,1

0···

0. . .

. . .. . .

. . .. . .

0

γene

l,1

Λs(λ

1,T

)γene

s,1

Λc(λ

1,T

)γene

s,1

0···

0. .

.. .

.. .

.. .

.. .

.. .

.

0···

0γeqt

l,N

Λs(λ

1,1

)γeqt

s,N

Λc(λN,1

)γeqt

s,N

. . .. . .

. . .. . .

. . .. . .

0···

0γeqt

l,N

Λs(λN,T

)γeqt

s,N

Λc(λN,T

)γeqt

s,N

,(3

5)

E=

Λf(κ

1,c

os(1,η

),si

n(1,η

))γu f,1

Λf(κ

1,c

os(1,η

),si

n(1,η

))γv f,1

Λf(κ

1,c

os(

1,η

),si

n(1,η

))γw f,1

. . .. . .

. . .

Λf(κ

1,c

os(T,η

),si

n(T,η

))γu f,1

Λf(κ

1,c

os(T,η

),si

n(T,η

))γv f,1

Λf(κ

1,c

os(T,η

),si

n(T,η

))γw f,1

. . .. . .

. . .

Λf(κN,c

os(

1,η

),si

n(1,η

))γu f,N

Λf(κN,c

os(1,η

),si

n(1,η

))γv f,N

Λf(κN,c

os(1,η

),si

n(1,η

))γw f,N

. . .. . .

. . .

Λf(κN,c

os(T,η

),si

n(T,η

))γu f,N

Λf(κN,c

os(T,η

),si

n(T,η

))γv f,N

Λf(κN,c

os(T,η

),si

n(T,η

))γw f,N

.(3

6)

32

Term structure conditional (co)variances and correlations are given by:

Σt = ΠQtΠ′ + Vt, (37)

ρt = diag (Σt)−1 Σtdiag (Σt)

−1 . (38)

with

Qt =

Hglb,t 03 · · · · · · · · · 03 · · · · · · · · · 03

03 Hcom,t 03 · · · · · ·...

. . ....

... 03 Hfin,t 03 · · ·...

. . ....

......

.... . .

. . ....

. . ....

03 · · · · · · 03 Hf,t 03 · · · · · · · · · 03

0 · · · · · · · · · 0 ωl,1t 0 · · · · · · 0...

. . .... 0 ωs,1t 0 · · · 0

.... . .

......

. . .. . .

. . . 0...

. . ....

.... . . 0 ωc,Nt 0

0 · · · · · · · · · 0 0 · · · · · · 0 ωf,Nt

,

=

ιglb ⊗Hglb,t 03 · · · 03

ιcom ⊗Hcom,t 03 · · · 03

......

. . ....

ιw ⊗Hf,t 03 · · · 03

01×36 ωl,1t · · · 0...

.... . .

...

01×36 0 · · · ωf,Nt

=

[Ht 036×N

0N×36 Dt

], (39)

Vt =

υ1t(1) 0 · · · · · · · · · 0

0 υ1t(2) 0 · · · 0

0 0 υ1t(3) 0 · · · 0...

.... . .

......

...

0 · · · υNt(T − 2) 0 0

0 · · · · · · 0 υNt(T − 1) 0

0 · · · · · · 0 υNt(T )

= diag([υ1t(1), υ1t(2), ..., υNt(T − 1), υNt(T )]), (40)

where 03 is a 3 × 3 matrix of zeros, Ht = [ιglb ⊗Hglb,t; ιcom ⊗Hcom,t; ...; ιf ⊗Hf,t], with ιglb =

(1, 0, ..., 0)′, ιcom = (0, 1, ..., 0)′,..., ιf = (0, 0, ..., 1)′, ⊗ is the Kronecker product and Dt =

diag([ωl,1t, ωs,1t, ..., ωc,Nt, ωf,Nt]).

33

B Estimation

In what follows we describe in more detail the estimation approach.

1. We first estimate the product’s level, slope, curvature and seasonality factors lit, sit, cit, fit

as well as the maturity and amplitude parameters λit and κit in equation (1) by means of

Nonlinear Least Squares (NLS) at each point in time. In addition, we obtain Least Squares

and Generalized Least Squares estimates of the factors lit, sit, cit and fit at each point

in time by (i) using the NLS time-varying estimates λit and κit (OLSPTV/GLSPTV),

(ii) using the mean NLS estimates per product λi and κi (OLSPRD/GLSPRD), (iii)

using ‘Mean Group’ NLS time-varying estimates averaged over sectors λnt and κnt

(OLSSTV/GLSSTV), (iv) using ‘Mean Group’ NLS mean estimates averaged over sec-

tors λn and κn (OLSSEC/GLSSEC).17 Given the candidate estimates of the factors ob-

tained via NLS, OLSPTV, GLSPTV, OLSPRD, GLSPRD, OLSSTV, GLSSTV, OLSSEC,

GLSSEC we compute Bayesian average (BAV) estimates of the factors lit, sit, cit and fit

as well as maturity λit and amplitude κit as:

lBAVit = w′itlit, (41)

sBAVit = w′itsit, (42)

cBAVit = w′itcit, (43)

fBAVit = w′itfit, (44)

λBAVit = w′itmit, (45)

where wit =exp(−0.5BICj,it)∑Jj=1 exp(−0.5BICj,it)

with BICjt the Bayesian information criterion of (1) ob-

tained from estimation type j at time t for each i. Moreover, lit, sit, cit, fit are vectors

containing the factor estimates and mit, ait are vectors containing the estimated matu-

rity and amplitude parameters obtained with the candidate estimation procedures. The

weights hold given diffuse priors and equal model prior probabilities which is assumed

here for simplicity (Hoeting et al., 1999).

2. Given the BAV estimates for the level, slope, curvature and seasonality parameters,

we demean and standardize the BAV factor estimates denoted˜lBAVit , ˜sBAVit , ˜cBAVit for

i = 1, ..., N . We extract the first principal component from each of the BAV factor

estimates denoted Lg,t, Sg,t, Cg,t, with corresponding factor loading estimates γgj,i for

j = l, s, c, g = global and i = 1, ..., N by means of Principal Component Analysis

(PCA). Let agl,it =˜lBAVit − γgl,iLg,t, a

gs,it = ˜sBAVit − γgs,iSg,t, a

gc,it = ˜cBAVit − γgc,iCg,t, for

i = 1, ..., N be the resulting residuals. We break agj,it for j = l, s, c and i = 1, ..., N

into two market groups, i.e., commodity markets (energy, metals, softs, grains, livestock)

17We experimented with median as opposed to mean estimates of the NLS λit and κit but results turn out tobe very similar.

34

and financial markets (forex, bonds, equity) and extract the first principal components

from each group denoted Lm,t, Sm,t, Cm,t with corresponding factor loading estimates

γmj,i for j = g, s, c, m = commodities, financials and i = 1, ..., N by means of PCA. Let

aml,it = agl,it− γml,iLm,t, a

ms,it = ags,it− γms,iSm,t, amc,it = agc,it− γmc,iCg,t for i = 1, ..., N . We break

amj,it for j = l, s, c and i = 1, ..., N into eight sector groups, i.e., energy, metals, softs, grains,

livestock, forex, bonds, equity and extract the first principal components from each group

denoted Ln,t, Sn,t, Cn,t with corresponding factor loading estimates γnj,i for j = g, s, c,

n = energy,metals, softs, grains, livestock, forex,bonds, equity and i = 1, ..., N by means

of PCA. In the case of the stochastic seasonality factors fit, we demean and standardize

the BAV estimates denoted˜fBAVit for i = 1, ..., N and obtain the first three principal com-

ponents of the data denoted Fu,t, Fv,t, Fw,t, with corresponding factor loading estimates

γ•f,i for • = u, v, w. Given the candidate component estimates Lg,t, Sg,t, Cg,t, Lm,t, Sm,t,

Cm,t, Ln,t, Sn,t, Cn,t, Fu,t, Fv,t, Fw,t and the BAV factor estimates obtained in the previous

step, we (re)estimate the parameters of the system specification (5)-(8) for all i = 1, ..., N

by means of system GMM. We employ one lag of the components as instruments and the

inverse of a Newey-West HAC covariance (obtained from the OLS residuals in a first step)

as weighting matrix. The idiosyncratic components ε•,it for • = l, s, c, f are the residuals

of the system GMM regressions for all i.

3. Let Xj,it = [Lj,t, Sj,t, Cj,t]′, for j = g,m, n or Xj,t = [Fu,t, Fv,t, Fw,t]

′ for j = f . The

state variable rt in the conditional mean of (9) which drives the time-variability of the

parameters d, α, β,Γ is assumed to evolve with respect to a first-order Markov chain, with

transition probability given by:

P(rt = y|rt = x) = πxy. (46)

The expression above describes the probability of switching from regime x at time t − 1

to regime y at time t. In this article we consider two regimes, that is rt = 1, 2, so that the

uncoditional (ergodic) probabilities of being in state rt = 1 or state rt = 2 are given by

π1 = (1−πxx)/(2−πxx−πyy). In what follows let It denote the information set available

to the econometrician at time t. To save on notation, we write (9) compactly as,

Yj,t = X(r)j,t = V

(r)j,t + ξj,t, (47)

where V(r)j,t = Yj,t − ξj,t = E [Yj,t|It−1] = Υj,dj,rX

(r)j,t + αj,rβ

′j,r∆

dj,r−bj,rΥj,bj,rX(r)j,t +∑P

p=1 Γpj,r∆dj,rΥp

j,bj,rX

(r)j,t for every j = g,m, n, f . We write:

Yj,t|It−1 ∼

N(

Ξ(1)t

)for π

(1)jt

N(

Ξ(2)t

)for 1− π(1)

jt

,Ξ(r)t = (dr, br, β

′r, α′r, vec(Γr)

′,diag(θ)′, diag(δ)′, ζ ′)′, (48)

35

where N (•) denotes the conditional normal distribution, Ξ(r)t is the vector of parameters

for the r-th regime and π(1)jt = P(rt = 1|It−1) is the probability of regime 1 conditional on

the information set at period t− 1. The parameters β′r, α′r, vec(Γr)

′ are concentrated out

of the Likelihood estimation and estimated via canonical correlation analysis and OLS

(Johansen and Nielsen, 2012). The conditional probability π(1)j.t is given by:

π(1)j,t = P(rt = 1|It−1) = (1− π22)

[N (Yj,t−1|rt−1 = 2)(1− π(1)

jt−1)

N (Yj,t−1|rt−1 = 1)π(1)j,t−1 +N (Yj,t−1|rt−1 = 2)(1− π(1)

j,t−1)

]

+ π11

[N (Yj,t−1|rt−1 = 1)π

(1)j,t−1

N (Yj,t−1|rt−1 = 1)π(1)j,t−1 +N (Yj,t−1|rt−1 = 2)(1− π(1)

j,t−1)

]. (49)

The likelihood function is then given by (18) and the conditional Normal distribution

given that regime r occurs at time t is given by

N (Yj,t|rt = r) =1

2|Hj,t|−1/2 exp

−1

2

(Yj,t − V(r)

j,t

)H−1j,t

(Yj,t − V(r)

j,t

). (50)

4. Given the estimated idiosyncratic components ε•,it obtained from the system GMM re-

gression in step two and the measurement errors εit(τ) obtained from Et = Zt − ΠXtusing the estimated matrix of parameters Π in (32)-(36), we employ ML to estimate the

autoregressive and GARCH(1,1) parameters in (13) and (14) for each i and τ .

C Forecasting

Following Dolatabadi et al. (2015), the multi-step ahead forecasts of the FCVAR for each j =

l, s, c, f can be obtained in the case of no regime-switching and no volatility-in-mean as

Xj,t+h|t = Υj,dXj,t+h|t + αj β′j∆

d−bΥj,bXj,t+h|t +k∑p=1

Γj,p∆dΥp

j,bXj,t+h|t (51)

Note that since Υj,• is a lag operator, the right hand side of (53) is conditional on past infor-

mation. In the case of regime switching parameters and volatility-in-mean we have

X(r)j,t+h|t = Υj,dr

X(r)j,t+h|t + αj,rβ

′j,r∆

dr−brΥj,brX

(r)j,t+h|t +

k∑p=1

Γpj,r∆drΥp

j,brX

(r)j,t+h|t (52)

+ ζj diag(Hj,t+h|t

).

with r = 1, 2 and the weigthed forecasts are then given by (23). Moreover, in order to compute

forecasts of the conditional (co)variances of the term structures we start by noting that the

36

OGARCH volatilities for each j = g,m, n, f can be computed from (12) recursively as:

Ωjt+h|t = IK + diag(θj + δj)h−1

(Ωjt+1|t − IK

), (53)

Hjt+h|t = BjΩjt+h|tB′j , (54)

where IK is an identity matrix of order K = 3. Given Hjt+h|t for j = glb, fin, com, ... in (54)

and estimates for the GARCH(1,1) volatilities of the idiosyncratic components ω•,it+h|t and of

the measurement errors υit+h|t(τ) we can compute Dt+h|t and Vt+h|t and subsequently (39), (40)

and (22) with the estimated Π.

37

Term

Str

uctu

res

Com

mod

itie

sF

inan

cia

lsE

nergy

Meta

lsG

rain

sS

oft

sL

ivest

ock

Forex

Bon

ds

Equ

ity

BR

EN

T(B

)G

old

(B)

Coco

a(B

)C

ott

on

(B)

Liv

eca

ttle

(B)

US

D/E

UR

(B)

US

Rate

s(D

)S

P500

(B)

WT

I(B

)S

ilver

(B)

Coff

ee(B

)W

hea

t(B

)L

ean

hogs

(B)

US

D/Z

AR

(B)

EU

Rate

s(E

)F

TS

E100

(B)

Gaso

lin

e(B

)C

op

per

(B)

Su

gar

(B)

Soyb

ean

s(B

)F

eed

erca

ttle

(B)

GB

P/E

UR

(B)

UK

Rate

s(D

)H

eati

ng

Oil

(B)

Alu

min

ium

(B)

Ora

nge

Soyb

ean

oil

(B)

US

D/R

MB

(B)

JP

Rate

s(D

)N

atu

ral

Gas

(B)

Lea

d(B

)L

um

ber

(B)

Rou

gh

rice

(B)

YE

N/U

SD

(B)

SA

Rate

s(D

)G

aso

il(B

)N

ickel

(B)

Ru

bb

er(B

)C

an

ola

(B)

Coal

(B)

Zin

c(B

)E

than

ol

(B)

Corn

(B)

CO

2(B

)S

teel

(B)

Macroecon

om

y

Real

Econ

om

yF

inan

cia

lan

dH

ou

sin

gM

arkets

Con

fid

en

ce

Ind

icato

rs

CP

I:C

on

sum

erP

rice

s(W

)X

RS

:E

xch

an

ge

Rate

s(D

)L

EI:

Lea

din

gE

con

om

ic(D

)IN

P:

Ind

Pro

du

ctio

n(W

)T

ST

:T

erm

Sp

read

s(D

)B

CI:

Bu

sin

ess

Con

fid

ence

(D)

MS

P:

Mon

eyS

up

ply

(W)

SV

O:

Sto

ckM

ark

etV

ola

(D)

TO

T:

Ter

ms

of

Tra

de

(W)

HP

R:

Hou

seP

rice

s(H

)W

AG

:W

ages

(HA

R)

XV

O:

FX

Vola

tility

(D)

CO

N:

Con

sum

pti

on

(D)

PE

R:

PE

Rati

os

(D)

INV

:In

ves

tmen

t(H

)D

YD

:D

ivid

end

Yie

lds

(D)

INT

:In

ven

tori

es(D

)U

NE

:U

nem

plo

ym

ent

(W)

EM

P:

Em

plo

ym

ent

(D)

Tab

le1:

Dat

ad

escr

ipti

onfo

rte

rmst

ruct

ure

san

dm

acro

econ

omic

vari

able

sco

nsi

der

ed.

Data

are

obta

ined

from

Bloomberg

(B),

Datast

ream

(D),

Haver

(H),

Eu

rop

ean

Cen

tral

Ban

k(E

),W

orld

Ban

kD

atab

ase

(W).

Ter

mst

ruct

ure

data

are

roll

ing

month

lyfu

ture

sco

ntr

act

sat

the

dail

yfr

equ

ency

from

the

per

iod

01-0

1-20

10to

10-0

1-20

15.

Mac

roec

on

om

icd

ata

are

at

the

month

lyfr

equ

ency

an

dfo

rth

esa

me

tim

ep

erio

d01

-2010

to09

-2015

.C

ountr

ies

con

sid

ered

for

the

mac

roec

on

om

icd

ata

are:

Au

stra

lia,

Bra

zil,

Can

ad

a,

Ch

ina,

EU

,F

ran

ce,

Ger

man

y,

Ind

ia,

Italy

,Jap

an,

Kor

ea,

Mal

aysi

a,R

uss

ia,

Sou

thA

fric

a,S

wed

en,

Sw

itze

rlan

d,

Th

ail

an

d,

UK

an

dU

S.

38

ESTIMATION APPROACHESType Advantages Disadvantages

NLSRelatively simple Numerical optimization for factors estimation

Product specific time-varying parameters Static factor estimationFlexible for dynamic specification No factor uncertainty correction

OLSPRD

Very simple Constant parametersNo numerical optimization for factor estimation Static factor estimation

Flexible for dynamic specification No factor uncertainty correctionProduct specific constant maturity and amplitude

Not robust under heteroskedasticity

GLSPRD

Very simple Constant parametersNo numerical optimization for factor estimation Static factor estimation

Flexible for dynamic specification No factor uncertainty correctionRobust under heteroskedasticity Product specific constant maturity and amplitude

OLSSEC

Very simple Constant parametersNo numerical optimization for factor estimation Static factor estimation

Flexible for dynamic specification No factor uncertainty correctionSector specific constant maturity and amplitude

Not robust under heteroskedasticity

GLSSEC

Very simple Constant parametersNo numerical optimization for factor estimation Static factor estimation

Flexible for dynamic specification No factor uncertainty correctionRobust under heteroskedasticity Sector specific constant maturity and amplitude

OLSPTV

Very simple Static factor estimationNo numerical optimization for factor estimation No factor uncertainty correction

Flexible for dynamic specification Not robust under heteroskedasticityProduct specific time maturity and amplitude

GLSPTV

Very simple Static factor estimationNo numerical optimization for factor estimation No factor uncertainty correction

Flexible for dynamic specificationProduct specific time-varying maturity and amplitude

Robust under heteroskedasticity

OLSSTV

Very simple Static factor estimationNo numerical optimization for factor estimation No factor uncertainty correction

Flexible for dynamic specification Not robust under heteroskedasticitySector specific time maturity and amplitude

GLSSTV

Very simple Static factor estimationNo numerical optimization for factor estimation No factor uncertainty correction

Flexible for dynamic specificationSector specific time maturity and amplitude Robust under heteroskedasticity

BAV

Weighted estimation ‘Hybrid’ estimationWeighted factor uncertainty correction Factor uncertainty reduced

Flexible for dynamic specification‘Hybrid’ dynamic specification

Weighted time-varying/constant maturity and amplitudeWeighted OLS, GLS, NLS

Table 2: Description of factor estimation approaches considered

39

Alt

ern

ati

ve

Dyn

am

icS

pecifi

cati

on

sof

Com

mon

Com

pon

ents

Mod

el

Param

ete

rS

pace

Ξj,r

tE

stim

ati

on

RW

dj,r

t=b j,r

t=

1,α′ j,rt

=0

,β′ j,rt

=0

,ς′ j

=0

,d

iag(δj)′

=0

,d

iag(Θ

j)′

=0′ ,

Γj

=0

—

VA

Rdj,r

t=b j,r

t=

1,α′ j,rt

=0

,β′ j,rt

=0

,ς′ j

=0

,d

iag(δj)′

=0

,d

iag(Θ

j)′

=0′

OL

S

CV

AR

dj,r

t=b j,r

t=

1,α′ j,rt6=

0,β′ j,rt6=

0,ς′ j

=0

,d

iag(δj)′

=0

,d

iag(Θ

j)′

=0′

Con

centr

ate

dM

L

FC

VA

Rdj,r

t=b j,r

t=dj

=b j6=

0,α′ j,rt6=

0,β′ j,rt6=

0,ς′ j

=0

,d

iag(δj)′

=0

,d

iag(Θ

j)′

=0′

Con

centr

ate

dM

L

FC

VA

R-O

GA

RC

Hdj,r

t=b j,r

t=dj

=b j6=

0,α′ j,rt

=α′ j6=

0,β′ j,rt

=β′ j6=

0,ς′ j

=0

,d

iag(δj)′>

0,

dia

g(Θ

j)′>

0′

Con

centr

ate

dM

L

RS

FC

VA

R-O

GA

RC

Hdj,r

t=b j,r

t6=

0,α′ j,rt6=

0,β′ j,rt6=

0,ς′ j

=0

,d

iag(δj)′>

0,

dia

g(Θ

j)′>

0′

Con

centr

ate

dM

Lan

dH

F

RS

FC

VA

R-O

GA

RC

H-M

dj,r

t=b j,r

t6=

0,α′ j,rt6=

0,β′ j,rt6=

0,ς′ j6=

0,

dia

g(δj)′>

0,

dia

g(Θ

j)′>

0′

Con

centr

ate

dM

Lan

dH

F

Tab

le3:

Par

am

eter

rest

rict

ion

sfo

rnes

ted

vers

ion

sof

the

dyn

amic

com

pon

ent

spec

ifica

tion.

40

Level Slope Curvature Seasonality

Product GLO MAR SEC GLO MAR SEC GLO MAR SEC SF1 SF2 SF3

BRENT0.01 -0.04 0.00 -0.09 0.01 -0.04 0.02 -0.03 0.02 0.00 0.00 0.00(0.00) (0.01) (0.03) (0.01) (0.01) (0.03) (0.01) (0.01) (0.02) (0.00) (0.00) (0.00)

WTI0.02 -0.01 0.00 -0.05 0.01 -0.04 0.03 -0.04 0.05 0.00 0.00 0.00(0.00) (0.01) (0.01) (0.01) (0.02) (0.03) (0.01) (0.01) (0.02) (0.00) (0.00) (0.00)

GASOLINE0.03 -0.00 -0.02 0.03 0.02 -0.11 0.09 -0.13 -0.21 -0.55 0.57 0.11(0.00) (0.01) (0.03) (0.05) (0.04) (0.08) (0.06) (0.09) (0.13) (0.41) (0.53) (0.35)

HEATINGOIL0.03 -0.04 -0.11 -0.03 0.06 -0.10 0.02 -0.09 -0.04 0.00 0.00 0.00(0.02) (0.05) (0.14) (0.04) (0.08) (0.16) (0.04) (0.09) (0.12) (0.00) (0.00) (0.00)

GASOIL0.03 0.00 -0.02 -0.03 0.01 -0.02 0.03 -0.03 -0.01 0.00 0.00 0.00(0.00) (0.02) (0.06) (0.00) (0.01) (0.02) (0.01) (0.01) (0.02) (0.00) (0.00) (0.00)

NATURALGAS0.03 -0.02 0.04 0.02 0.02 0.05 0.03 0.01 0.21 0.45 -0.07 -0.04(0.00) (0.01) (0.03) (0.02) (0.03) (0.06) (0.04) (0.05) (0.13) (0.38) (0.51) (0.60)

COAL0.06 0.01 -0.01 0.03 -0.02 -0.03 -0.00 0.00 -0.02 0.00 0.00 0.00(0.00) (0.01) (0.02) (0.01) (0.01) (0.02) (0.01) (0.02) (0.02) (0.00) (0.00) (0.00)

CO20.06 -0.08 0.23 0.03 -0.02 0.13 -0.01 0.10 0.04 0.00 0.00 0.00(0.04) (0.10) (0.34) (0.04) (0.10) (0.13) (0.08) (0.17) (0.23) (0.00) (0.00) (0.00)

GOLD0.03 0.02 0.03 -0.00 0.00 -0.00 0.00 -0.00 -0.01 0.00 0.00 0.00(0.01) (0.02) (0.02) (0.01) (0.01) (0.01) (0.01) (0.02) (0.03) (0.00) (0.00) (0.00)

SILVER0.05 0.05 0.12 -0.01 -0.06 -0.16 0.04 0.06 -0.20 0.00 0.00 0.00(0.02) (0.16) (0.24) (0.03) (0.12) (0.32) (0.06) (0.18) (0.41) (0.00) (0.00) (0.00)

COPPER0.06 0.00 -0.04 0.05 0.01 0.04 -0.04 0.03 -0.02 0.00 0.00 0.00(0.02) (0.02) (0.04) (0.02) (0.03) (0.07) (0.03) (0.03) (0.08) (0.00) (0.00) (0.00)

ALUMINUM0.03 -0.00 0.02 0.02 -0.02 0.01 0.00 0.01 0.02 0.00 0.00 0.00(0.00) (0.01) (0.01) (0.00) (0.01) (0.01) (0.01) (0.01) (0.02) (0.00) (0.00) (0.00)

LEAD0.02 -0.01 0.01 0.00 -0.02 0.02 0.01 0.01 0.01 0.00 0.00 0.00(0.00) (0.01) (0.01) (0.00) (0.01) (0.01) (0.00) (0.01) (0.01) (0.00) (0.00) (0.00)

NICKEL0.03 -0.02 0.03 0.01 -0.04 0.02 0.02 0.03 0.03 0.00 0.00 0.00(0.00) (0.01) (0.01) (0.00) (0.01) (0.01) (0.00) (0.01) (0.01) (0.00) (0.00) (0.00)

ZINC0.01 -0.01 0.03 0.01 -0.01 0.01 0.00 0.02 0.02 0.00 0.00 0.00(0.00) (0.01) (0.01) (0.00) (0.01) (0.01) (0.01) (0.01) (0.01) (0.00) (0.00) (0.00)

COTTON0.02 0.02 0.07 0.01 -0.04 0.04 0.12 0.10 0.34 0.35 0.27 1.35(0.00) (0.02) (0.05) (0.02) (0.10) (0.16) (0.10) (0.14) (0.19) (0.44) (0.35) (0.96)

WHEAT0.02 0.00 -0.01 -0.01 0.01 0.07 0.07 0.03 -0.16 0.00 0.00 0.00(0.00) (0.01) (0.02) (0.01) (0.01) (0.03) (0.03) (0.04) (0.07) (0.00) (0.00) (0.00)

CORN0.01 0.02 0.02 -0.01 -0.04 0.17 0.11 0.03 -0.37 -0.04 0.58 -0.58(0.01) (0.01) (0.02) (0.02) (0.04) (0.08) (0.06) (0.06) (0.16) (0.43) (0.62) (0.57)

SOYBEAN0.02 0.04 0.09 -0.01 -0.02 0.09 0.08 0.06 0.12 0.21 0.09 -0.12(0.00) (0.02) (0.06) (0.02) (0.05) (0.05) (0.03) (0.05) (0.10) (0.25) (0.21) (0.31)

SUGAR0.02 0.04 0.07 -0.00 -0.12 0.13 0.06 0.05 -0.34 -0.17 0.48 -0.08(0.02) (0.03) (0.20) (0.03) (0.05) (0.13) (0.04) (0.06) (0.43) (0.51) (1.09) (0.53)

ORANJE0.02 -0.09 -0.16 -0.02 0.06 0.20 -0.04 -0.07 -0.04 0.00 0.00 0.00(0.02) (0.17) (0.54) (0.03) (0.16) (0.53) (0.08) (0.24) (0.27) (0.00) (0.00) (0.00)

COCOA-0.01 -0.05 0.05 0.00 0.00 -0.01 -0.01 -0.00 -0.03 0.00 0.00 0.00(0.00) (0.01) (0.03) (0.01) (0.01) (0.03) (0.02) (0.02) (0.04) (0.00) (0.00) (0.00)

COFFEE0.02 0.01 0.05 0.01 -0.08 0.04 0.05 0.08 -0.09 0.00 0.00 0.00(0.01) (0.02) (0.05) (0.01) (0.01) (0.02) (0.02) (0.03) (0.06) (0.00) (0.00) (0.00)

ETHANOL0.03 0.01 -0.00 -0.03 -0.01 0.07 0.01 0.04 -0.13 -0.10 -0.35 -0.23(0.00) (0.01) (0.02) (0.02) (0.03) (0.10) (0.03) (0.04) (0.14) (0.21) (0.21) (0.39)

LUMBER-0.01 0.01 0.05 -0.00 0.02 0.08 0.09 0.04 0.17 0.00 0.00 0.00(0.02) (0.03) (0.08) (0.02) (0.05) (0.13) (0.04) (0.08) (0.21) (0.00) (0.00) (0.00)

LIVECATTLE-0.00 0.00 0.01 0.05 0.03 0.52 -0.04 -0.01 0.67 0.56 0.81 -0.43(0.02) (0.04) (0.07) (0.06) (0.05) (0.16) (0.17) (0.14) (0.54) (2.41) (1.71) (1.83)

LEANHOGS0.01 0.09 0.31 0.00 -0.04 0.21 0.22 0.31 1.43 2.99 -0.04 -0.90(0.03) (0.04) (0.10) (0.12) (0.14) (0.19) (0.26) (0.20) (0.55) (1.80) (2.22) (3.44)

FEEDERCATTLE-0.03 0.02 0.08 0.05 0.01 0.17 0.03 0.02 0.26 0.34 0.24 0.01(0.01) (0.01) (0.02) (0.02) (0.02) (0.05) (0.05) (0.03) (0.11) (0.43) (0.40) (0.44)

USDEUR-0.00 0.01 0.01 -0.02 -0.02 0.00 0.03 -0.01 -0.01 0.00 0.00 0.00(0.00) (0.01) (0.01) (0.00) (0.00) (0.01) (0.00) (0.00) (0.01) (0.00) (0.00) (0.00)

ZARGBP-0.05 0.04 0.08 -0.01 -0.00 0.02 0.01 -0.03 0.03 0.00 0.00 0.00(0.01) (0.03) (0.03) (0.02) (0.02) (0.05) (0.04) (0.07) (0.20) (0.00) (0.00) (0.00)

GBPEUR0.01 0.01 -0.00 -0.00 -0.01 -0.00 0.00 0.00 -0.01 0.00 0.00 0.00(0.00) (0.00) (0.01) (0.00) (0.00) (0.00) (0.00) (0.00) (0.01) (0.00) (0.00) (0.00)

ZARUSD-0.05 0.04 0.04 -0.02 0.01 0.03 0.01 -0.06 0.03 0.00 0.00 0.00(0.01) (0.01) (0.01) (0.00) (0.01) (0.01) (0.01) (0.02) (0.04) (0.00) (0.00) (0.00)

JPYUSD-0.03 0.02 -0.01 0.00 0.00 0.00 -0.00 0.00 0.00 0.00 0.00 0.00(0.00) (0.01) (0.01) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

RENUSD-0.01 -0.01 0.01 -0.01 -0.02 0.02 0.01 0.01 0.01 0.00 0.00 0.00(0.00) (0.00) (0.01) (0.00) (0.00) (0.00) (0.00) (0.01) (0.01) (0.00) (0.00) (0.00)

USINTR0.01 -0.03 0.08 0.02 -0.01 0.01 -0.02 0.11 0.00 0.00 0.00 0.00(0.02) (0.04) (0.06) (0.03) (0.05) (0.06) (0.03) (0.07) (0.06) (0.00) (0.00) (0.00)

41

EUINTR-0.48 -0.19 0.58 -0.68 -0.54 0.39 0.83 -0.34 0.41 0.00 0.00 0.00(0.06) (0.12) (0.14) (0.05) (0.07) (0.10) (0.06) (0.15) (0.16) (0.00) (0.00) (0.00)

SAINTR-0.20 -0.09 0.40 -0.31 -0.23 0.35 0.27 -0.05 0.41 0.00 0.00 0.00(0.04) (0.09) (0.12) (0.04) (0.07) (0.09) (0.05) (0.09) (0.10) (0.00) (0.00) (0.00)

UKINTR-0.01 -0.04 0.03 0.11 -0.25 -0.39 0.07 -0.53 0.40 0.00 0.00 0.00(0.00) (0.01) (0.01) (0.09) (0.10) (0.13) (0.09) (0.21) (0.17) (0.00) (0.00) (0.00)

JPINTR0.00 0.01 0.01 0.01 0.00 0.02 -0.01 0.01 0.03 0.00 0.00 0.00(0.00) (0.01) (0.01) (0.01) (0.00) (0.01) (0.01) (0.01) (0.01) (0.00) (0.00) (0.00)

CHINTR-0.06 -0.02 0.05 -0.04 -0.11 0.02 0.09 0.10 0.03 0.00 0.00 0.00(0.02) (0.04) (0.06) (0.02) (0.04) (0.05) (0.02) (0.04) (0.04) (0.00) (0.00) (0.00)

SP500TSX60-0.01 0.02 0.04 -0.00 0.00 -0.00 -0.00 -0.00 -0.01 0.00 0.00 0.00(0.00) (0.01) (0.02) (0.00) (0.00) (0.01) (0.00) (0.00) (0.02) (0.00) (0.00) (0.00)

FTSE100-0.02 0.03 0.05 -0.01 0.00 0.04 0.01 -0.01 0.11 0.00 0.00 0.00(0.00) (0.01) (0.03) (0.01) (0.01) (0.05) (0.02) (0.02) (0.21) (0.00) (0.00) (0.00)

Table 4: Component loadings for level, slope, curvature and seasonality factors

42

Glo

bal

Mark

etS

ecto

rS

easo

nality

ΞG

LO

CO

MF

INO

AG

ME

TG

RA

SO

FL

IVF

EX

BO

NE

QT

SE

A

β1,1

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)

β2,1

-0.2

12

-0.3

71

-4.7

70

-0.2

94

-3.7

05

1.6

61

2.2

83

-4.5

80

-2.4

00

-0.5

22

3.6

30

0.1

90

(0.1

29)

(0.1

50)

(0.3

19)

(0.1

25)

(0.1

74)

(0.1

49)

(0.1

01)

(0.7

65)

(1.5

84)

(0.1

44)

(3.2

04)

(0.1

02)

β3,1

-2.3

08

0.4

44

-9.4

74

-0.0

87

-4.3

89

1.5

37

-0.8

30

-5.8

32

4.4

14

1.2

22

-12.6

68

-1.1

24

(0.2

20)

(0.1

60)

(0.4

40)

(0.1

07)

(0.1

55)

(0.1

37)

(0.0

81)

(0.8

79)

(2.4

17)

(0.1

80)

(3.6

35)

(0.1

67)

β1,2

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

1.0

00

(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)(NaN

)

β2,2

-0.1

25

-0.4

64

0.5

15

-0.2

87

-2.8

66

1.2

24

2.2

10

-1.4

20

-5.2

65

-3.5

99

-5.1

12

-0.0

14

(0.1

20)

(0.1

34)

(0.1

24)

(0.1

28)

(0.1

25)

(0.1

01)

(0.1

02)

(0.5

35)

(0.7

55)

(0.4

69)

(3.7

21)

(0.0

71)

β3,2

-2.1

38

0.2

26

0.5

68

-0.0

85

-0.9

22

0.7

04

-0.7

50

-4.0

43

9.0

21

5.4

12

6.8

97

-0.5

58

(0.2

03)

(0.1

52)

(0.1

59)

(0.1

10)

(0.1

19)

(0.0

82)

(0.0

82)

(0.5

33)

(1.2

90)

(0.5

69)

(4.2

10)

(0.1

07)

d1

0.9

62

0.7

59

0.8

36

0.9

78

0.9

88

0.8

76

0.9

19

0.9

01

0.8

52

0.5

51

0.3

35

0.8

43

(0.4

51)

(0.1

37)

(0.2

80)

(0.6

97)

(0.4

55)

(0.3

98)

(0.1

00)

(0.3

00)

(0.6

49)

(0.0

98)

(0.0

12)

0.2

15

d2

0.9

96

0.6

29

0.5

43

0.9

89

0.8

49

0.6

47

0.9

65

0.5

25

0.9

85

0.9

90

1.0

00

0.5

87

(0.4

72)

(0.2

40)

(0.1

49)

(2.6

76)

(0.3

78)

(0.0

47)

(0.3

00)

(0.0

80)

(0.8

30)

(3.7

33)

(1.4

34)

(0.5

41)

p11

0.9

15

0.7

25

0.7

73

0.5

68

0.9

60

0.8

99

0.9

89

0.8

01

0.1

77

0.0

34

0.5

11

0.9

90

(0.5

05)

(0.3

59)

(0.3

58)

(0.5

79)

(0.4

47)

(0.1

84)

(0.4

90)

(0.3

90)

(0.9

90)

(0.9

90)

(0.2

09)

(0.9

90)

p22

0.9

33

0.6

18

0.2

33

0.9

88

0.9

83

0.8

90

0.8

56

0.9

66

1.0

00

0.9

80

0.5

48

0.5

01

(0.5

20)

(0.3

04)

(0.9

90)

(0.3

27)

(0.4

90)

(0.3

16)

(0.9

90)

(0.9

90)

(0.9

90)

(0.9

90)

(0.1

99)

(0.1

90)

Tab

le5:

Est

imat

ion

resu

lts

of

lon

g-r

un

,fr

act

ion

alco

effici

ents

and

tran

siti

onp

rob

abil

itie

sfo

rth

efu

llR

SF

CV

AR

-OG

AR

CH

-Msp

ecifi

-ca

tion

.

43

Non-Regime Switching Regime Switching

Component Diag VAR CVAR FCVAR FCVAR-OG RSFCVAR-OG RSFCVAR-OGM

GLO

L 4018.638 4010.820 4012.203 4274.291 4314.816 4431.828

LRp 0.000 1.000 0.096 0.000 0.000 0.000

COM

L 3057.168 3068.470 3040.458 3285.717 3282.048 3216.401

LRp 0.000 0.000 1.000 0.000 1.000 1.000

FIN

L 2167.301 2171.892 2190.021 2725.699 2866.797 2693.152

LRp 0.000 0.002 0.000 0.000 0.000 1.000

ENE

L 3076.458 3082.519 3085.464 2848.062 2543.828 2879.110

LRp 0.000 0.000 0.015 1.000 1.000 0.000

MET

L 2225.004 2213.219 2228.412 1994.807 2095.854 2198.392

LRp 0.000 1.000 0.000 1.000 0.000 0.000

GRA

L 2160.538 2143.805 2145.606 3513.031 3673.814 3602.652

LRp 0.000 1.000 0.058 0.000 0.000 1.000

SOF

L 2655.370 2675.327 2679.610 2717.519 2932.087 2938.100

LRp 0.000 0.000 0.003 0.000 0.000 1.000

LIV

L 3029.695 3024.042 3031.817 3666.757 3748.355 3694.890

LRp 0.000 1.000 0.000 0.000 0.000 1.000

FEX

L 2757.695 2755.534 2767.369 3163.390 2424.060 2830.182

LRp 0.000 1.000 0.000 0.000 1.000 0.000

BON

L 3517.987 3521.230 3536.950 3937.453 3996.207 3944.632

LRp 0.000 0.011 0.000 0.000 0.000 1.000

EQT

L 1089.600 1064.085 1036.178 1143.612 1144.499 1160.232

LRp 0.000 0.000 0.000 0.000 1.000 1.000

SEA

L 1574.418 1584.657 1590.112 2183.107 2252.260 2354.177

LRp 0.000 0.000 0.001 0.000 0.000 0.000

Table 6: Diagnostics for alternative dynamic component specifications

44

Glo

bal

Com

modit

ies

Fin

ancia

lsE

nerg

yM

eta

lsG

rain

s

ΞL

EV

SL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

αj,1

-0.0

03

-0.0

02

0.0

03

0.0

04

-0.0

05

0.0

09

-0.0

02

0.0

01

-0.0

04

-0.2

57

-0.0

01

0.0

13

-0.0

01

0.0

02

-0.0

06

0.0

01

0.0

02

-0.0

02

(0.0

01)

(0.0

01)

(0.0

02)

(0.0

02)

(0.0

02)

(0.0

02)

(0.0

01)

(0.0

01)

(0.0

02)

(0.0

19)

(0.0

02)

(0.0

02)

(0.0

02)

(0.0

02)

(0.0

03)

(0.0

03)

(0.0

03)

(0.0

03)

αj,2

0.0

15

0.0

08

-0.0

08

0.0

01

0.0

04

-0.0

10

0.0

17

-0.0

13

0.0

35

0.0

01

0.0

08

-0.0

04

-0.0

06

-0.0

02

-0.0

01

-0.0

78

-0.1

62

-0.0

29

(0.0

03)

(0.0

01)

(0.0

02)

(0.0

03)

(0.0

02)

(0.0

03)

(0.0

02)

(0.0

02)

(0.0

06)

(0.0

02)

(0.0

02)

(0.0

02)

(0.0

03)

(0.0

02)

(0.0

03)

(0.0

10)

(0.0

23)

(0.0

05)

Γj1,1

0.1

91

-0.2

12

-0.1

88

0.6

39

-0.1

36

0.0

62

0.5

72

-0.0

52

-0.1

11

-0.0

77

-0.1

64

-0.1

41

0.2

24

-0.0

85

-0.0

76

0.4

62

0.0

23

-0.0

25

(0.0

37)

(0.0

48)

(0.0

31)

(0.0

30)

(0.0

97)

(0.0

75)

(0.0

38)

(0.0

48)

(0.0

29)

(0.1

20)

(0.0

69)

(0.0

88)

(0.0

33)

(0.0

47)

(0.0

33)

(0.0

71)

(0.0

28)

(0.0

37)

Γj1,2

-0.0

21

0.3

86

-0.0

04

-0.0

13

0.4

85

0.1

34

-0.0

40

0.3

73

0.0

68

0.1

17

0.3

00

-0.0

43

-0.0

82

0.3

25

-0.0

44

0.0

49

0.4

80

0.0

18

(0.0

28)

(0.0

33)

(0.0

25)

(0.0

31)

(0.0

41)

(0.0

37)

(0.0

44)

(0.0

79)

(0.0

38)

(0.0

68)

(0.0

76)

(0.0

42)

(0.0

25)

(0.0

34)

(0.0

24)

(0.0

43)

(0.0

35)

(0.0

37)

Γj2,1

-0.0

98

-0.1

89

0.2

77

-0.0

22

0.2

45

0.5

33

0.0

25

-0.0

04

0.2

75

0.0

90

-0.0

39

0.3

15

-0.0

73

-0.0

69

0.2

22

-0.0

07

-0.0

55

0.5

10

(0.0

47)

(0.0

54)

(0.0

34)

(0.0

29)

(0.0

82)

(0.0

63)

(0.0

55)

(0.0

47)

(0.0

40)

(0.0

41)

(0.0

37)

(0.0

30)

(0.0

36)

(0.0

48)

(0.0

35)

(0.0

31)

(0.0

27)

(0.0

45)

Γj2,2

0.1

64

-0.2

10

-0.1

61

0.8

59

-0.1

01

0.0

42

0.9

59

-0.0

44

-0.1

16

-0.0

59

0.1

99

0.0

17

0.4

15

-0.0

81

-0.0

95

1.0

27

0.2

51

-0.0

07

(0.0

35)

(0.0

41)

(0.0

27)

(0.0

29)

(0.0

56)

(0.0

48)

(0.0

44)

(0.0

47)

(0.0

45)

(0.0

94)

(0.0

31)

(0.0

39)

(0.0

35)

(0.0

49)

(0.0

36)

(0.0

95)

(0.0

83)

(0.0

59)

Γj3,1

-0.0

15

0.3

48

0.0

11

-0.0

17

0.7

55

0.1

05

0.0

27

0.9

48

0.0

75

0.1

15

0.2

99

-0.0

38

-0.0

74

0.5

13

-0.0

55

0.4

73

1.3

16

0.0

15

(0.0

30)

(0.0

38)

(0.0

28)

(0.0

24)

(0.0

27)

(0.0

24)

(0.0

37)

(0.0

54)

(0.0

44)

(0.0

63)

(0.0

72)

(0.0

41)

(0.0

26)

(0.0

34)

(0.0

27)

(0.1

34)

(0.0

84)

(0.0

80)

Γj3,2

-0.1

02

-0.1

89

0.2

22

0.0

20

0.1

58

0.7

70

-0.0

97

-0.0

74

0.7

09

0.0

87

-0.0

61

0.2

92

-0.0

74

-0.0

07

0.3

78

0.0

91

0.0

20

0.8

48

(0.0

50)

(0.0

63)

(0.0

39)

(0.0

26)

(0.0

54)

(0.0

45)

(0.0

98)

(0.1

01)

(0.0

56)

(0.0

30)

(0.0

32)

(0.0

29)

(0.0

35)

(0.0

48)

(0.0

39)

(0.0

39)

(0.0

31)

(0.0

35)

ζj

0.0

18

-0.0

15

-0.0

08

0.0

20

0.0

17

0.0

01

0.0

08

0.0

51

0.0

17

-0.0

40

0.0

70

0.0

04

0.0

05

-0.0

03

0.0

20

0.0

50

0.0

11

-0.0

01

(0.0

07)

(0.2

53)

(0.1

72)

(0.0

23)

(0.1

74)

(0.1

51)

(0.0

04)

(0.1

53)

(0.0

95)

(0.1

12)

(0.1

86)

(0.1

91)

(0.1

60)

(0.2

05)

(0.1

46)

(0.0

88)

(0.1

29)

(0.1

27)

Θj

0.8

90

0.7

18

0.7

39

0.3

57

0.3

11

0.3

42

0.4

59

0.5

70

0.4

22

0.2

65

0.5

10

0.4

83

0.5

50

0.9

42

0.8

67

0.3

64

0.4

29

0.3

65

(0.2

07)

(0.2

26)

(0.1

32)

(0.0

70)

(0.0

64)

(0.0

82)

(0.1

74)

(0.0

37)

(0.1

20)

(0.1

57)

(0.1

03)

(0.8

20)

(0.2

39)

(0.1

41)

(0.1

21)

(0.0

58)

(0.0

85)

(0.0

47)

δj

0.0

40

0.0

74

0.0

96

0.3

40

0.4

12

0.3

85

0.1

03

0.1

57

0.1

70

0.7

17

0.1

31

0.0

17

0.0

56

0.0

39

0.0

84

0.4

78

0.2

62

0.2

60

(0.0

04)

(0.0

31)

(0.0

09)

(0.1

08)

(0.1

46)

(0.0

51)

(0.0

59)

(0.2

60)

(0.0

64)

(0.2

86)

(0.0

20)

(0.9

00)

(0.0

33)

(0.1

87)

(0.1

67)

(0.1

17)

(0.1

39)

(0.1

36)

Tab

le7:

Res

ult

sof

shor

t-ru

n,

GA

RC

Han

dvo

lati

lity

in-m

ean

coeffi

cien

tsfo

rth

efu

llR

SF

CV

AR

-OG

AR

CH

-Msp

ecifi

cati

on

(Part

I).

45

Soft

sL

ivest

ock

Fore

ign

Exchange

Bonds

Equit

ySeaso

naliti

es

ΞL

EV

SL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

SF

1SF

2SF

3

αj,1

0.0

05

-0.0

07

0.0

06

0.0

05

0.0

02

-0.0

06

-0.0

06

0.0

09

0.0

50

-0.0

08

0.0

02

0.0

12

0.0

21

0.0

19

-0.0

48

0.0

02

-0.0

06

0.0

04

(0.0

04)

(0.0

04)

(0.0

04)

(0.0

01)

(0.0

01)

(0.0

01)

(0.0

01)

(0.0

01)

(0.0

01)

(0.0

03)

(0.0

03)

(0.0

04)

(0.0

06)

(0.0

10)

(0.0

08)

(0.0

03)

(0.0

02)

(0.0

03)

αj,2

-0.0

58

0.3

94

0.3

94

0.0

30

-0.0

08

-0.0

10

-0.0

02

0.0

02

0.0

04

0.0

01

-0.0

04

0.0

02

-0.0

39

-0.0

04

0.0

61

0.2

75

0.0

43

-0.8

73

(0.0

14)

(0.0

79)

(0.0

77)

(0.0

05)

(0.0

02)

(0.0

04)

(0.0

01)

(0.0

01)

(0.0

01)

(0.0

01)

(0.0

01)

(0.0

01)

(0.0

28)

(0.0

14)

(0.0

59)

(0.0

38)

(0.0

07)

(0.0

88)

Γj1,1

0.3

43

-0.1

47

0.1

15

0.5

02

0.0

60

0.0

97

0.4

59

0.0

59

0.0

43

1.0

28

-0.0

79

0.0

56

1.2

16

-0.0

90

0.3

33

0.5

33

-0.0

73

-0.0

24

(0.0

55)

(0.0

28)

(0.0

23)

(0.0

91)

(0.0

52)

(0.0

79)

(0.0

51)

(0.0

69)

(0.0

54)

(0.0

46)

(0.0

35)

(0.0

41)

(0.0

83)

(0.1

22)

(0.1

26)

(0.0

27)

(0.0

31)

(0.0

27)

Γj1,2

-0.1

33

0.2

72

0.0

13

-0.0

66

0.4

07

-0.0

41

0.1

43

0.5

22

-0.0

56

-0.0

15

0.8

95

0.1

04

-0.0

83

1.1

68

0.4

13

0.0

24

0.4

82

-0.0

68

(0.0

42)

(0.0

39)

(0.0

25)

(0.0

44)

(0.0

43)

(0.0

47)

(0.0

47)

(0.0

86)

(0.0

49)

(0.0

37)

(0.0

45)

(0.0

37)

(0.1

41)

(0.1

88)

(0.2

91)

(0.0

30)

(0.0

34)

(0.0

41)

Γj2,1

0.0

74

0.0

10

0.4

25

-0.0

49

-0.0

38

0.3

39

0.1

25

0.2

18

0.1

56

-0.0

40

0.0

57

0.9

07

0.1

62

0.1

82

0.4

38

-0.0

22

0.0

26

0.5

16

(0.0

35)

(0.0

40)

(0.0

27)

(0.0

37)

(0.0

46)

(0.0

42)

(0.0

64)

(0.1

04)

(0.0

71)

(0.0

47)

(0.0

39)

(0.0

34)

(0.0

97)

(0.1

95)

(0.2

67)

(0.0

23)

(0.0

36)

(0.0

45)

Γj2,2

0.2

89

-0.0

47

0.0

78

0.9

58

0.3

55

0.4

64

0.2

64

0.0

72

0.0

24

0.3

98

-0.0

55

0.0

40

0.2

47

-0.1

31

0.1

28

0.0

08

0.3

00

0.0

30

(0.0

71)

(0.0

85)

(0.0

84)

(0.0

61)

(0.0

94)

(0.0

87)

(0.0

51)

(0.0

61)

(0.0

48)

(0.0

65)

(0.0

25)

(0.0

44)

(0.1

39)

(0.1

02)

(0.1

08)

(0.2

64)

(0.2

39)

(0.4

93)

Γj3,1

-0.2

51

-0.3

68

0.2

07

-0.0

10

0.9

48

-0.0

88

0.1

66

0.3

48

-0.0

28

0.0

33

0.2

38

0.1

13

-0.2

23

0.1

26

-0.0

24

-0.1

40

0.9

34

-0.0

52

(0.2

36)

(0.1

62)

(0.1

46)

(0.0

67)

(0.0

54)

(0.0

70)

(0.0

41)

(0.0

71)

(0.0

43)

(0.0

32)

(0.0

55)

(0.0

31)

(0.2

12)

(0.1

82)

(0.1

40)

(0.0

48)

(0.0

44)

(0.0

67)

Γj3,2

-0.0

44

-0.5

44

0.5

51

0.0

03

-0.1

59

0.7

99

0.2

82

0.2

42

0.1

46

0.0

11

0.0

89

0.2

57

-0.0

53

0.0

96

0.0

72

2.9

60

-1.0

66

0.6

90

(0.2

60)

(0.1

62)

(0.1

48)

(0.1

43)

(0.0

88)

(0.1

24)

(0.0

45)

(0.0

74)

(0.0

47)

(0.0

30)

(0.0

27)

(0.0

23)

(0.1

09)

(0.1

83)

(0.2

29)

(0.2

72)

(0.3

90)

(0.4

42)

ζj

-0.0

00

-0.0

28

0.0

11

-0.0

05

-0.0

15

0.0

15

0.0

17

-0.0

09

-0.0

21

0.0

07

0.0

29

0.0

03

-0.0

08

-0.0

45

0.0

15

-0.0

31

-0.0

17

-0.0

19

(1.0

00)

(1.0

00)

(1.0

00)

(0.1

85)

(0.1

30)

(0.1

13)

(0.1

26)

(0.1

34)

(0.1

25)

(0.2

11)

(0.2

16)

(0.2

01)

(0.0

91)

(0.0

27)

(0.0

14)

(1.0

00)

(1.0

00)

(1.0

00)

Θj

0.9

69

0.9

07

0.9

47

0.1

48

0.8

88

0.3

60

0.9

71

0.7

15

0.9

14

0.5

14

0.4

99

0.5

03

0.4

72

0.6

88

0.0

01

0.4

98

0.4

15

0.9

18

(0.4

00)

(0.3

30)

(0.5

10)

(0.2

10)

(0.0

90)

(0.1

53)

(0.2

63)

(0.1

21)

(0.3

73)

(0.1

96)

(0.1

34)

(0.5

00)

(0.1

03)

(0.0

29)

(0.0

19)

(0.2

80)

(0.1

50)

(0.6

40)

δj

0.0

26

0.0

71

0.0

47

0.0

03

0.1

01

0.0

75

0.0

20

0.1

14

0.0

78

0.1

05

0.0

92

0.0

12

0.2

25

0.2

30

0.9

20

0.0

03

0.1

64

0.0

66

(0.0

04)

(0.0

01)

(0.0

10)

(0.2

10)

(0.0

41)

(0.0

72)

(0.0

01)

(0.0

20)

(0.0

38)

(0.0

34)

(0.0

94)

(0.0

09)

(0.1

06)

(0.0

90)

(0.5

00)

(0.0

50)

(0.9

00)

(0.0

01)

Tab

le8:

Res

ult

sof

short

-ru

n,

GA

RC

Han

dvo

lati

lity

in-m

ean

coeffi

cien

tsfo

rth

efu

llR

SF

CV

AR

-OG

AR

CH

-Msp

ecifi

cati

on

(Part

II).

46

Glo

bal

Com

modit

ies

Fin

ancia

lsE

nerg

yM

eta

lsG

rain

s

Macro

Com

p.

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

CO

NS

-12.0

12.5

7-4

.50

0.2

177.4

4-2

.21

-4.6

412.4

9-4

.20

-1.4

71.2

12.1

0-1

.59

-4.7

3-3

.61

-0.1

0-0

.34

0.4

6

XR

S1

-1.3

00.7

1-1

.26

1.7

82.1

0-1

.18

-1.9

12.2

10.6

71.2

7-1

.67

-0.9

52.0

11.9

20.9

80.7

20.6

91.4

3

XR

S2

0.6

7-1

.92

2.0

71.1

6-6

.35

-1.7

0-0

.62

-1.7

00.0

50.1

2-5

.00

1.5

2-8

.06

0.2

10.9

7-1

.30

8.1

9-6

.42

CP

I11.4

4-1

.37

1.1

6-1

.23

0.1

10.5

11.8

9-0

.94

-1.5

9-1

.22

-1.2

91.1

01.0

3-1

.01

-1.3

4-1

.05

-1.0

6-3

.28

CP

I21.1

3-1

.22

1.2

0-1

.17

-0.9

31.5

51.1

9-1

.16

-1.1

7-1

.24

-1.4

21.1

7-0

.30

-0.9

5-1

.48

-1.4

31.1

6-0

.94

INP

1-0

.98

0.9

4-1

.24

1.3

63.2

4-2

.10

-1.2

81.2

90.5

61.1

81.2

6-0

.96

1.5

70.6

31.4

50.8

5-2

.38

0.5

8

INP

2-1

.10

1.1

4-1

.35

1.3

1-0

.22

-1.2

0-1

.52

1.4

51.1

61.2

81.0

3-1

.51

3.7

30.8

31.3

71.7

5-1

.61

0.4

5

TST

1-1

.29

1.6

0-1

.55

1.0

20.1

7-0

.33

-0.2

71.0

01.5

31.2

91.6

1-1

.65

3.8

71.2

50.6

81.4

7-1

.72

2.5

0

TST

21.2

7-1

.24

1.4

6-1

.31

-3.4

01.2

00.9

9-1

.42

-1.1

0-1

.19

-0.3

81.0

9-3

.57

-1.2

5-0

.97

-0.5

90.8

7-1

.48

BC

I1-1

.21

1.4

3-1

.47

1.1

81.7

7-0

.50

-0.8

10.9

61.3

91.3

01.6

6-1

.43

2.9

70.6

00.3

31.1

7-1

.13

1.4

2

BC

I2-1

.20

1.1

9-1

.22

1.2

31.2

0-1

.47

-0.9

81.1

61.1

91.2

21.1

4-1

.13

0.6

41.0

01.3

20.9

9-1

.06

2.0

3

LE

I11.1

9-1

.31

1.3

4-1

.21

-1.1

51.0

40.9

6-1

.05

-1.4

1-1

.27

-1.2

91.2

7-1

.45

-0.8

0-1

.02

-1.0

70.8

4-1

.72

LE

I21.3

9-1

.95

2.0

1-0

.83

-0.7

3-0

.66

0.6

9-1

.02

-1.5

7-1

.29

-2.2

91.6

4-6

.90

-0.6

3-0

.25

-0.9

21.2

4-4

.10

XV

O1

1.1

0-1

.18

1.3

9-1

.22

-1.3

61.8

50.3

3-0

.99

-1.0

7-1

.24

-1.4

91.1

70.3

7-0

.02

-2.0

4-0

.98

2.4

3-3

.95

XV

O2

-1.2

01.2

5-1

.26

1.2

20.9

0-1

.33

-1.0

71.1

31.3

21.2

71.4

0-1

.21

0.5

70.9

71.5

41.1

4-1

.00

1.7

7

PE

R1

-1.2

51.3

7-1

.55

1.3

55.3

70.2

1-0

.92

1.0

11.4

81.3

00.1

4-1

.27

3.3

72.0

50.8

50.6

30.4

92.4

2

PE

R2

1.4

5-1

.61

1.7

8-1

.09

-1.2

8-0

.19

1.0

1-1

.19

-1.3

5-1

.27

-1.4

01.5

4-3

.63

-0.9

8-0

.21

-0.6

81.4

4-3

.27

DY

D1

-1.1

31.0

0-1

.23

1.5

76.6

30.7

8-2

.38

0.9

21.0

41.1

5-1

.83

-0.8

42.3

23.3

51.1

10.4

12.9

72.2

6

DY

D2

1.2

3-1

.13

1.3

1-1

.30

-0.4

00.9

31.7

0-1

.14

-1.1

1-1

.19

-0.9

71.0

00.9

4-1

.11

-2.0

3-0

.75

1.1

6-3

.23

MSP

1-1

.34

1.4

1-1

.37

1.2

12.7

3-0

.82

-1.2

11.2

21.1

81.2

61.3

8-1

.25

1.2

41.0

90.8

81.1

1-0

.92

1.6

7

MSP

21.1

5-1

.19

1.2

1-1

.18

-4.0

81.5

31.1

4-1

.12

-0.9

4-1

.16

-1.2

81.0

7-0

.89

-0.9

3-1

.38

-0.8

81.6

0-1

.03

SV

O1

-1.0

80.4

6-0

.77

1.7

23.7

2-1

.54

-2.7

91.8

40.4

41.0

7-2

.53

-0.5

41.9

62.5

01.3

00.4

01.6

80.2

4

SV

O2

1.1

5-1

.27

1.3

4-1

.21

1.2

90.8

21.0

1-1

.17

-1.3

6-1

.27

-1.2

81.4

6-2

.50

-0.8

1-1

.08

-1.2

11.1

2-1

.46

CO

N1

1.2

6-1

.10

0.9

2-1

.33

-0.2

91.8

71.3

8-1

.05

-1.2

2-1

.17

-0.6

61.0

11.2

0-1

.44

-1.4

9-1

.22

0.4

80.4

3

CO

N2

-1.1

61.2

2-1

.01

1.1

0-0

.44

-2.0

2-0

.86

0.7

51.5

71.1

91.0

7-0

.94

-2.5

31.0

82.1

81.3

8-0

.71

2.1

0

TO

T1

-1.0

60.9

9-1

.58

1.3

65.0

2-1

.36

0.1

71.0

31.0

11.1

60.8

0-0

.97

5.7

5-0

.45

-0.3

5-0

.65

-3.1

82.2

3

TO

T2

-0.0

81.5

0-1

.04

-0.2

0-2

4.7

81.6

92.0

9-2

.13

3.4

61.0

9-2

.36

-0.3

4-4

.06

0.2

23.9

64.2

18.7

612.5

2

HP

R1

1.1

5-1

.79

1.9

7-0

.94

4.1

7-1

.49

-1.1

8-0

.71

-2.5

7-1

.51

-3.4

12.2

9-6

.28

0.6

9-0

.89

-1.8

21.6

3-5

.14

HP

R2

-2.1

21.2

0-0

.86

1.1

5-4

.17

7.8

5-4

.09

1.2

6-0

.23

0.9

4-0

.37

-2.6

37.7

56.3

5-1

.92

0.2

21.9

3-8

.81

WA

G1

-1.3

33.3

8-2

.82

-0.2

6-5

.10

4.9

51.9

1-0

.19

2.9

71.3

45.1

6-2

.58

10.0

7-0

.09

-0.2

01.8

3-0

.95

6.6

2

WA

G2

-1.2

71.9

8-1

.39

0.5

4-6

.73

-1.0

20.3

80.2

72.8

51.2

63.8

9-1

.03

-3.5

6-0

.40

3.5

31.8

30.0

75.9

6

INS1

1.1

0-1

.33

1.4

2-1

.15

-1.6

71.0

60.2

9-0

.91

-1.2

5-1

.27

-2.0

21.5

3-3

.41

-0.3

3-0

.92

-1.4

62.3

3-1

.14

INS2

1.1

1-1

.29

1.2

7-1

.16

-0.0

50.9

80.9

7-0

.98

-1.3

9-1

.29

-1.6

21.2

9-1

.33

-0.7

5-1

.20

-1.4

30.8

3-1

.29

INT

1-1

.26

1.4

5-1

.50

1.1

2-1

.49

-0.0

8-1

.11

1.2

81.6

11.3

31.5

9-1

.53

2.3

90.9

91.0

71.2

9-0

.69

3.2

3

INT

2-1

.10

1.1

2-1

.26

1.3

31.1

8-0

.62

-1.5

11.3

51.1

01.3

00.9

3-1

.21

1.8

61.1

20.8

61.1

1-0

.22

0.9

7

UN

E1

-1.5

21.2

1-1

.00

1.3

72.1

5-1

.05

-2.9

51.3

80.8

91.1

0-0

.50

-0.9

61.2

72.8

80.4

91.1

70.6

41.7

8

UN

E2

-1.1

51.1

2-1

.56

1.4

911.8

9-1

.37

-2.0

02.2

80.3

61.1

21.1

5-0

.84

3.5

31.4

00.5

10.3

5-0

.30

-2.1

9

EM

P1

-1.2

61.8

4-1

.56

0.8

41.9

7-0

.76

0.1

00.4

51.2

61.3

02.8

8-1

.44

-0.1

90.8

63.2

41.2

4-3

.12

1.5

5

EM

P2

1.1

5-1

.23

1.2

4-1

.18

0.4

71.3

70.4

9-0

.98

-1.4

6-1

.26

-1.6

41.3

0-1

.44

-0.5

1-1

.24

-1.3

40.9

9-1

.58

Tab

le9:

Mac

roec

on

om

icm

app

ing

resu

lts

(Par

tI)

.N

ote:

Th

eta

ble

show

st-

rati

osro

bu

stu

nd

erh

eter

oske

dsa

tici

tyan

dau

toco

rrel

ati

on

.A

cronym

des

crip

tion

sca

nb

efo

un

din

Tab

le1.

An

ind

ex1,

2re

fers

toth

efi

rst

and

seco

nd

pri

nci

pal

com

pon

ent

ob

tain

edfr

om

the

19

cou

ntr

ies

un

der

con

sid

erat

ion

wit

hin

ap

arti

cula

rca

tego

ry(c

onsu

mp

tion

,ex

chan

gera

tes,

etc)

.

47

Soft

sL

ivest

ock

Fore

ign

Exchange

Bonds

Equit

y

Macro

Com

p.

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

LE

VSL

OC

UR

CO

NS

-1.6

11.2

22.1

61.5

2-1

.37

-1.0

0-1

.44

-2.3

91.4

60.4

10.7

91.1

41.4

00.7

3-1

.59

XR

S1

-0.0

4-1

.05

1.3

3-1

.41

-6.3

11.6

9-1

.51

0.3

5-0

.89

0.7

8-1

.42

-1.1

6-1

.04

-0.8

61.2

7

XR

S2

-9.2

70.5

0-3

.68

-0.2

49.3

6-0

.83

0.5

7-9

.15

-1.5

2-2

.17

1.2

60.4

9-0

.28

0.9

20.5

8

CP

I1-0

.18

1.2

30.5

11.2

3-3

.03

-0.9

3-1

.20

1.4

21.2

7-1

.34

1.3

41.3

11.1

81.0

3-1

.13

CP

I2-1

.38

1.2

3-1

.27

1.2

10.4

3-1

.22

-1.0

60.8

71.1

7-1

.36

1.2

91.2

61.2

31.0

6-1

.26

INP

10.0

8-1

.32

1.5

4-1

.35

-2.5

71.5

3-0

.24

-0.4

5-1

.28

1.4

2-1

.55

-1.2

8-1

.15

-1.0

71.1

6

INP

2-0

.40

-1.1

2-0

.01

-1.4

3-1

.54

1.6

52.5

0-2

.62

-1.5

01.2

6-1

.43

-1.1

8-1

.37

-1.4

81.3

7

TST

10.4

6-1

.27

1.0

2-1

.31

-0.8

31.4

12.3

1-1

.02

-1.2

81.2

6-1

.11

-1.1

7-1

.38

-1.2

31.3

3

TST

2-1

.88

1.3

3-1

.75

1.3

32.7

4-1

.54

-0.5

90.1

21.1

4-1

.18

1.2

81.2

51.2

61.0

8-1

.29

BC

I10.7

5-1

.31

1.6

3-1

.29

-1.9

41.4

22.3

4-1

.89

-1.2

91.3

2-1

.21

-1.1

9-1

.34

-0.7

41.4

2

BC

I21.4

2-1

.27

1.0

6-1

.25

-0.8

91.3

10.7

6-0

.67

-1.1

41.2

9-1

.24

-1.2

2-1

.21

-1.1

11.2

2

LE

I1-0

.75

1.2

8-1

.58

1.2

71.0

2-1

.34

-1.9

71.5

81.2

2-1

.31

1.1

91.1

71.3

00.8

8-1

.35

LE

I2-1

.49

1.6

1-2

.24

1.3

51.6

3-1

.67

-4.2

21.3

31.1

6-1

.16

0.8

81.1

21.5

90.4

6-1

.70

XV

O1

-0.4

81.3

2-0

.90

1.1

81.5

0-1

.31

0.1

0-0

.10

1.1

7-1

.58

1.6

11.2

71.2

30.8

0-1

.13

XV

O2

0.9

7-1

.31

1.4

1-1

.28

-0.2

21.2

91.1

0-0

.89

-1.1

71.2

6-1

.30

-1.2

5-1

.24

-1.0

91.2

7

PE

R1

1.8

0-1

.22

2.6

9-1

.30

-0.4

11.3

22.0

2-0

.57

-0.9

90.9

5-0

.83

-0.9

4-1

.29

-0.5

01.4

2

PE

R2

-1.9

21.3

1-2

.04

1.3

3-0

.14

-1.4

7-2

.93

0.9

71.2

6-1

.25

1.2

41.2

11.4

00.9

4-1

.51

DY

D1

1.7

8-0

.95

1.7

6-1

.20

1.6

80.9

30.7

00.1

1-0

.86

0.6

7-0

.46

-0.7

0-1

.11

-0.1

01.3

7

DY

D2

-0.2

11.2

0-1

.47

1.3

5-0

.06

-1.3

7-1

.03

0.6

61.2

5-1

.44

1.5

51.2

71.2

20.8

9-1

.31

MSP

11.8

3-1

.33

1.3

2-1

.26

-0.8

51.2

41.3

5-0

.43

-1.0

61.3

7-1

.20

-1.2

6-1

.24

-0.9

11.2

6

MSP

2-1

.40

1.2

7-1

.60

1.2

01.6

8-1

.23

-0.3

50.4

41.1

7-1

.34

1.1

71.1

81.1

61.0

3-1

.14

SV

O1

0.2

4-0

.94

-0.5

3-1

.27

-0.3

61.1

1-1

.30

-0.4

4-1

.07

0.5

9-1

.12

-1.1

5-0

.82

-0.6

71.2

9

SV

O2

0.6

61.2

3-0

.86

1.3

30.7

0-1

.39

-1.5

61.8

01.4

3-1

.14

1.3

81.2

61.2

91.3

1-1

.34

CO

N1

-1.8

91.3

2-0

.59

1.2

30.9

3-1

.11

0.5

81.0

41.1

4-1

.22

1.4

21.3

91.1

61.5

3-1

.08

CO

N2

1.7

5-1

.34

1.0

4-1

.11

0.4

81.0

70.9

8-1

.27

-1.0

71.5

0-1

.34

-1.2

7-1

.16

-1.0

71.0

9

TO

T1

0.5

0-1

.31

2.8

7-1

.43

-4.6

21.9

70.5

0-0

.54

-1.3

00.9

6-1

.26

-1.0

4-1

.11

-1.1

51.3

2

TO

T2

2.4

0-0

.72

2.8

8-0

.01

-6.9

20.5

53.7

71.1

30.3

12.7

0-1

.89

-0.8

8-0

.51

2.6

91.4

5

HP

R1

2.7

31.3

20.0

31.4

01.9

1-1

.81

-4.5

84.1

61.6

7-1

.26

1.7

71.2

21.4

51.2

0-1

.50

HP

R2

-4.6

5-0

.34

-3.3

4-1

.36

6.2

60.0

0-1

.08

-6.0

4-1

.62

-0.2

3-1

.90

-1.5

9-1

.29

-1.8

71.5

3

WA

G1

2.9

0-1

.87

3.5

8-0

.87

6.5

40.9

07.6

31.1

8-0

.52

1.2

4-0

.28

-0.8

0-1

.68

-0.5

61.5

4

WA

G2

2.6

9-1

.83

2.3

8-0

.95

1.0

01.0

12.3

7-0

.30

-0.7

92.0

6-1

.98

-1.7

1-1

.37

-1.0

51.0

7

INS1

1.1

31.3

6-0

.20

1.3

50.7

8-1

.39

-1.1

82.2

51.5

3-1

.28

1.3

91.1

61.2

11.5

4-1

.15

INS2

-0.8

41.2

5-1

.42

1.2

31.3

7-1

.31

-1.7

11.6

41.2

6-1

.38

1.2

91.2

41.2

40.8

9-1

.33

INT

10.7

2-1

.17

1.5

6-1

.31

-0.7

31.4

63.0

0-1

.34

-1.1

61.3

1-1

.26

-1.2

1-1

.39

-0.7

41.5

5

INT

20.9

2-1

.09

1.6

1-1

.27

-1.9

11.3

51.1

1-0

.79

-1.1

61.2

8-1

.31

-1.2

5-1

.22

-0.6

61.4

3

UN

E1

2.2

2-1

.19

-0.0

7-1

.24

1.8

20.9

30.7

1-1

.03

-1.1

61.1

7-1

.19

-1.3

8-1

.21

-0.9

91.3

5

UN

E2

6.3

7-1

.27

2.8

6-1

.23

-3.1

81.4

40.4

03.8

7-0

.50

1.1

9-0

.97

-1.1

8-1

.14

-0.1

81.2

5

EM

P1

0.5

0-1

.58

1.4

8-1

.08

2.8

90.8

41.8

10.6

7-1

.07

1.6

3-1

.09

-1.1

8-1

.25

-0.9

11.0

2

EM

P2

-0.8

81.2

7-1

.37

1.2

92.7

7-1

.49

-1.4

82.1

31.3

2-1

.24

1.2

61.2

21.3

01.1

7-1

.32

Tab

le10:

Macr

oec

onom

icm

ap

pin

gre

sult

s(P

art

II).

Not

e:T

he

tab

lesh

ows

t-ra

tios

rob

ust

un

der

het

eros

ked

ast

icit

yan

dau

toco

rrel

ati

on

.A

cronym

des

crip

tion

sca

nb

efo

un

din

Tab

le1.

An

ind

ex1,

2re

fers

toth

efi

rst

and

seco

nd

pri

nci

pal

com

pon

ents

,re

spec

tive

ly,

obta

ined

from

the

19

cou

ntr

ies

un

der

con

sid

erat

ion

wit

hin

apar

ticu

lar

cate

gory

(con

sum

pti

on,

exch

ange

rate

s,et

c).

48

Level

Slo

pe

Curv

atu

reSeaso

nality

Pro

duct

GL

OM

AR

SE

CID

IG

LO

MA

RSE

CID

IG

LO

MA

RSE

CID

ISF

1SF

2SF

3SID

BR

EN

T0.2

812.7

75.6

52.7

320.9

70.8

725.5

84.2

95.0

510.8

96.6

54.2

90.0

00.0

00.0

00.0

0

WT

I1.9

72.1

29.9

40.1

07.8

00.4

923.7

71.0

78.8

212.2

330.6

31.0

70.0

00.0

00.0

00.0

0

GA

SO

LIN

E0.0

20.0

02.9

00.0

00.0

20.0

21.4

71.9

80.5

81.5

62.6

81.9

857.7

227.5

31.5

30.0

0

HE

AT

ING

OIL

0.2

41.0

665.6

22.1

60.2

01.3

915.5

93.1

70.2

65.5

91.5

53.1

70.0

00.0

00.0

00.0

0

GA

SO

IL3.8

90.0

958.2

40.2

63.2

60.5

29.9

10.3

87.9

513.5

81.5

50.3

80.0

00.0

00.0

00.0

0

NA

TU

RA

LG

AS

0.0

10.0

33.2

00.0

00.0

00.0

10.1

70.5

40.0

30.0

01.2

50.5

417.9

10.2

90.2

275.7

9

CO

AL

17.7

93.1

940.6

50.2

44.8

52.1

021.4

50.4

50.1

00.0

38.7

00.4

50.0

00.0

00.0

00.0

0

CO

20.1

61.2

458.9

814.7

40.0

40.0

54.7

48.9

50.0

11.9

40.2

28.9

50.0

00.0

00.0

00.0

0

GO

LD

5.2

59.1

843.8

46.3

10.0

20.1

60.5

08.9

20.0

20.2

116.6

78.9

20.0

00.0

00.0

00.0

0

SIL

VE

R0.2

91.0

613.1

410.9

50.0

10.6

312.2

05.5

00.4

71.3

348.9

35.5

00.0

00.0

00.0

00.0

0

CO

PP

ER

7.8

70.2

231.5

75.0

84.5

30.5

517.0

11.9

212.7

97.1

99.3

51.9

20.0

00.0

00.0

00.0

0

AL

UM

INU

M6.9

00.0

824.5

11.0

82.4

64.9

68.1

90.9

30.0

91.4

248.4

50.9

30.0

00.0

00.0

00.0

0

LE

AD

4.1

02.7

224.3

31.9

90.2

06.3

923.6

80.1

22.9

37.4

725.9

50.1

20.0

00.0

00.0

00.0

0

NIC

KE

L4.8

33.7

926.2

10.7

50.1

87.1

64.4

10.2

12.6

711.3

738.1

90.2

10.0

00.0

00.0

00.0

0

ZIN

C0.3

14.8

736.8

60.3

51.0

91.6

36.0

70.1

70.1

39.3

139.0

50.1

70.0

00.0

00.0

00.0

0

CO

TT

ON

0.0

00.0

10.1

30.0

10.0

00.0

50.0

41.0

90.4

90.5

22.6

91.0

910.6

22.8

518.2

962.1

2

WH

EA

T0.6

70.1

10.4

10.6

30.0

30.4

315.3

80.8

613.2

33.0

364.3

50.8

60.0

00.0

00.0

00.0

0

CO

RN

0.0

10.0

80.0

80.0

80.0

10.1

83.4

53.2

21.6

10.2

513.4

53.2

20.6

451.4

822.2

50.0

0

SO

YB

EA

N0.0

10.3

71.2

00.7

00.0

00.0

91.5

91.3

71.1

10.7

92.1

61.3

719.3

51.9

83.1

164.7

8

SU

GA

R0.0

20.3

50.7

40.0

60.0

01.1

85.3

64.7

10.5

60.6

520.4

64.7

114.1

645.8

41.1

80.0

0

OR

AN

JE

0.1

17.8

812.3

62.2

70.0

91.4

349.8

98.1

71.5

55.6

22.4

68.1

70.0

00.0

00.0

00.0

0

CO

CO

A0.4

036.1

324.2

60.9

10.0

00.1

32.5

44.1

00.8

50.0

726.4

94.1

00.0

00.0

00.0

00.0

0

CO

FF

EE

0.4

21.0

86.9

81.3

60.1

38.9

511.8

22.8

77.5

722.7

033.2

52.8

70.0

00.0

00.0

00.0

0

ET

HA

NO

L0.1

10.0

40.0

00.0

20.1

00.0

13.9

29.0

30.0

30.7

37.5

79.0

39.6

048.2

611.5

60.0

0

LU

MB

ER

0.0

40.1

02.4

610.1

10.0

00.2

913.4

511.7

09.4

92.9

537.6

911.7

00.0

00.0

00.0

00.0

0

LIV

EC

AT

TL

E0.0

00.0

00.0

00.0

00.0

00.0

00.6

80.2

70.0

00.0

02.8

00.2

71.1

71.1

20.4

893.2

1

LE

AN

HO

GS

0.0

00.0

00.0

00.0

00.0

00.0

00.0

00.4

10.0

00.0

00.2

30.4

10.4

70.0

00.0

498.4

4

FE

ED

ER

CA

TT

LE

0.0

10.0

10.1

10.0

00.0

20.0

10.7

50.1

90.0

30.0

45.1

00.1

95.2

41.3

60.0

086.9

6

USD

EU

R0.1

01.6

53.5

20.0

84.3

210.5

20.9

80.1

417.8

17.7

353.0

40.1

40.0

00.0

00.0

00.0

0

ZA

RG

BP

1.6

72.7

525.9

08.2

80.1

00.0

11.7

512.8

80.1

511.4

722.1

412.8

80.0

00.0

00.0

00.0

0

GB

PE

UR

3.7

08.0

23.0

90.0

70.3

17.0

82.1

50.0

22.1

84.4

668.9

00.0

20.0

00.0

00.0

00.0

0

ZA

RU

SD

2.2

63.2

49.7

40.1

50.1

80.6

23.9

50.0

90.1

652.0

227.5

10.0

90.0

00.0

00.0

00.0

0

JP

YU

SD

33.1

548.2

76.1

20.5

70.0

10.0

20.9

40.0

00.0

10.0

410.8

60.0

00.0

00.0

00.0

00.0

0

RE

NU

SD

0.7

64.0

86.8

90.4

90.4

814.2

816.8

90.3

14.7

323.8

326.9

60.3

10.0

00.0

00.0

00.0

0

USIN

TR

0.0

40.3

81.6

917.8

40.0

40.1

90.0

521.6

10.3

136.2

40.0

021.6

10.0

00.0

00.0

00.0

0

EU

INT

R2.5

41.0

75.7

110.3

53.9

48.7

82.0

211.2

718.1

121.1

33.8

011.2

70.0

00.0

00.0

00.0

0

SA

INT

R1.7

80.9

910.9

518.3

43.3

86.9

36.6

912.1

28.5

01.9

416.2

612.1

20.0

00.0

00.0

00.0

0

UK

INT

R0.0

00.0

70.0

20.0

00.1

82.9

23.0

56.9

90.1

973.9

95.6

06.9

90.0

00.0

00.0

00.0

0

JP

INT

R0.2

75.0

95.2

00.0

20.9

00.0

08.6

60.0

26.0

643.2

130.5

50.0

20.0

00.0

00.0

00.0

0

CH

INT

R0.1

50.0

80.2

029.8

20.0

62.2

40.0

229.0

10.9

78.3

20.1

229.0

10.0

00.0

00.0

00.0

0

SP

500T

SX

60

0.7

14.7

081.0

00.6

00.0

00.0

93.2

10.0

20.0

20.4

59.1

90.0

20.0

00.0

00.0

00.0

0

FT

SE

100

0.0

90.6

211.5

10.7

70.0

10.0

020.8

70.3

70.0

30.8

064.5

50.3

70.0

00.0

00.0

00.0

0

Tab

le11:

Per

centa

ge

vari

ance

exp

lain

edby

glob

al,

mar

ket,

sect

oran

did

iosy

ncr

ati

cco

mp

on

ents

.

49

1020

30

0123

τλ,κ

BR

EN

T

1020

30

0123

τ

λ,κ

WT

I

1020

30

0123

τ

λ,κ

GA

SO

LIN

E

1020

30

0123

τ

λ,κ

HE

AT

ING

OIL

1020

30

0123

τ

λ,κ

GA

SO

IL

1020

30

0123

τ

λ,κ

NA

TU

RA

LGA

S

1020

30

0123

τ

λ,κ

CO

AL

1020

30

0123

τ

λ,κ

CO

2

1020

30

0123

τ

λ,κG

OLD

1020

30

0123

τλ,κ

SIL

VE

R

1020

30

0123

τ

λ,κ

CO

PP

ER

1020

30

0123

τ

λ,κ

ALU

MIN

UM

1020

30

0123

τ

λ,κ

LEA

D

1020

30

0123

τ

λ,κ

NIC

KE

L

1020

30

0123

τ

λ,κ

ZIN

C

1020

30

0123

τ

λ,κ

CO

TT

ON

1020

30

0123

τ

λ,κ

WH

EA

T

1020

30

0123

τ

λ,κC

OR

N

1020

30

0123

τλ,κ

SO

YB

EA

N

1020

30

0123

τ

λ,κ

SU

GA

R

1020

30

0123

τ

λ,κ

OR

AN

JE

1020

30

0123

τ

λ,κ

CO

CO

A

1020

30

0123

τ

λ,κ

CO

FF

EE

1020

30

0123

τ

λ,κ

ET

HA

NO

L

1020

30

0123

τ

λ,κ

LUM

BE

R

1020

30

0123

τ

λ,κ

LIV

EC

AT

TLE

1020

30

0123

τ

λ,κLE

AN

HO

GS

1020

30

0123

τλ,κ

FE

ED

ER

CA

TT

LE

1020

30

0123

τ

λ,κ

US

DE

UR

1020

30

0123

τ

λ,κ

ZA

RG

BP

1020

30

0123

τ

λ,κ

GB

PE

UR

1020

30

0123

τ

λ,κ

ZA

RU

SD

1020

30

0123

τ

λ,κ

JPY

US

D

1020

30

0123

τ

λ,κ

RE

NU

SD

1020

30

0123

τ

λ,κ

US

INT

R

1020

30

0123

τ

λ,κE

UIN

TR

1020

30

0123

τλ,κ

SA

INT

R

1020

30

0123

τ

λ,κ

UK

INT

R

1020

30

0123

τ

λ,κ

JPIN

TR

1020

30

0123

τ

λ,κ

CH

INT

R

1020

30

0123

τ

λ,κ

SP

500T

SX

60

1020

30

0123

τ

λ,κ

FT

SE

100

Le

vel

Slo

peC

urva

ture

Sea

sona

lity

Sum

Fig

ure

1:E

stim

ated

bas

elo

adfu

nct

ion

sp

erp

rod

uct

.

50

Fig

ure

2:M

od

el-b

ased

term

stru

ctu

res

over

tim

e.

1215

3.9

4.5

4.8

zit(τ)

BR

EN

T

1215

3.8

4.4

4.7

zit(τ)

WT

I

1215

4.8

5.5

5.8

zit(τ)

GA

SO

LIN

E

1215

5.1

5.6

5.8

zit(τ)

HE

AT

ING

OIL

1215

6.1

6.7

7.0

zit(τ)

GA

SO

IL

1215

0.7

1.4

1.9

zit(τ)

NA

TU

RA

LGA

S

1215

4.0

4.5

4.9

zit(τ)

CO

AL

1215

0.8

2.2

5.4

zit(τ)

CO

2

1215

7.0

7.3

7.6

zit(τ)

GO

LD

1215

2.7

3.2

3.9

zit(τ)

SIL

VE

R

1215

5.4

5.8

6.1

zit(τ)

CO

PP

ER

1215

7.4

7.7

8.0

zit(τ)

ALU

MIN

UM

1215

7.4

7.7

8.0

zit(τ)

LEA

D

1215

9.3

9.8

10.3

zit(τ)

NIC

KE

L

1215

7.5

7.7

7.9

zit(τ)

ZIN

C

1215

4.0

4.4

5.4

zit(τ)

CO

TT

ON

1215

6.1

6.6

6.9

zit(τ)

WH

EA

T

1215

5.8

6.2

6.8

zit(τ)

CO

RN

1215

6.8

7.1

7.5

zit(τ)

SO

YB

EA

N

1215

2.4

3.0

3.6

zit(τ)

SU

GA

R

1215

4.6

5.0

5.4

zit(τ)

OR

AN

JE

1215

7.6

7.9

8.2

zit(τ)

CO

CO

A

1215

4.6

5.3

5.8

zit(τ)

CO

FF

EE

1215

0.3

0.6

1.3

zit(τ)E

TH

AN

OL

1215

5.4

5.8

6.1

zit(τ)

LUM

BE

R

1215

4.2

4.9

5.5

zit(τ)

LUM

BE

R

1215

3.6

4.4

5.3

zit(τ)

LUM

BE

R

1215

4.7

5.1

5.7

zit(τ)

LUM

BE

R

1215

0.0

0.3

0.4

zit(τ)

US

DE

UR

1215

2.3

2.7

3.1

zit(τ)

ZA

RG

BP

1215

−0.

4

−0.

2

−0.

1

zit(τ)

GB

PE

UR

1215

1.8

2.3

2.8

zit(τ)Z

AR

US

D

1215

4.3

4.5

4.8

zit(τ)

JPY

US

D

1215

1.8

1.9

1.9

zit(τ)

RE

NU

SD

1215

4.3

4.5

4.6

zit(τ)

US

INT

R

1215

3.4

4.5

4.7

zit(τ)

EU

INT

R

1215

1.7

3.3

4.5

zit(τ)

SA

INT

R

1215

4.2

4.5

4.8

zit(τ)

UK

INT

R

1215

4.5

4.6

4.6

zit(τ)

JPIN

TR

1215

3.1

4.0

4.6

zit(τ)C

HIN

TR

1215

6.4

6.6

6.8

zit(τ)

SP

500T

SX

60

1215

8.4

8.7

9.0

zit(τ)

FT

SE

100

51

Figure 3: Selected model-based term structures over time and forward month.

52

Figure 4: Selected model-based volatility term structures over time and forward month.

53

Fig

ure

5:E

stim

ated

λit

’sp

erp

rod

uct

.

1215

0.0

0.5

1.0

λ(t)

BR

EN

T

1215

0.0

0.5

1.0

λ(t)

WT

I

1215

0.5

1.8

λ(t)

GA

SO

LIN

E

1215

0.5

1.6

λ(t)

HE

AT

ING

OIL

1215

0.0

1.0

2.0

λ(t)

GA

SO

IL

1215

0.4

1.9

λ(t)

NA

TU

RA

LGA

S

1215

0.2

1.1

λ(t)

CO

AL

1215

0.0

1.0

2.0

λ(t)

CO

2

1215

0.1

0.6

λ(t)

GO

LD

1215

0.3

1.3

λ(t)

SIL

VE

R

1215

0.8

1.4

λ(t)

CO

PP

ER

1215

1.5

2.0

2.5

λ(t)

ALU

MIN

UM

1215

0.0

0.5

1.0

λ(t)

LEA

D

1215

0.3

1.1

λ(t)

NIC

KE

L

1215

0.0

1.0

2.0

λ(t)

ZIN

C

1215

0.8

1.9

λ(t)

CO

TT

ON

1215

0.8

1.7

λ(t)

WH

EA

T

1215

0.0

1.0

2.0

λ(t)

CO

RN

1215

0.7

1.9

λ(t)

SO

YB

EA

N

1215

1.0

2.0

λ(t)

SU

GA

R

1215

1.2

2.0

λ(t)

OR

AN

JE

1215

0.6

1.8

λ(t)

CO

CO

A

1215

0.0

1.0

2.0

λ(t)

CO

FF

EE

1215

0.7

1.5

λ(t)

ET

HA

NO

L

1215

0.0

1.0

2.0

λ(t)

LUM

BE

R

1215

0.8

1.7

λ(t)

LIV

EC

AT

TLE

1215

0.5

1.4

λ(t)

LEA

NH

OG

S

1215

0.0

1.0

2.0

λ(t)

FE

ED

ER

CA

TT

LE

1215

0.0

0.5

1.0

λ(t)

US

DE

UR

1215

0.0

1.0

2.0

λ(t)

ZA

RG

BP

1215

−2.

0

0.0

2.0

λ(t)

GB

PE

UR

1215

−2.

0

0.0

2.0

λ(t)

ZA

RU

SD

1215

−2.

0

0.0

2.0

λ(t)

JPY

US

D

1215

0.0

0.5

1.0

λ(t)

RE

NU

SD

1215

0.0

0.5

1.0

λ(t)

US

INT

R

1215

0.1

1.0

λ(t)

EU

INT

R

1215

0.4

0.9

λ(t)

SA

INT

R

1215

0.0

1.0

2.0

λ(t)

UK

INT

R

1215

−2.

0

0.0

2.0

λ(t)

JPIN

TR

1215

1.5

2.0

λ(t)C

HIN

TR

1215

0.7

1.1

λ(t)

SP

500T

SX

60

1215

0.5

1.4

λ(t)

FT

SE

100

54

Fig

ure

6:S

elec

ted

esti

mat

edκit

’sp

erp

rod

uct

.

1012

1315

2.0

2.7

3.3

κ(t)

GA

SO

LIN

E

1012

1315

2.0

2.6

3.1

κ(t)

NA

TU

RA

LGA

S

1012

1315

1.1

2.3

3.1

κ(t)

CO

TT

ON

1012

1315

1.0

2.5

3.0

κ(t)

CO

RN

1012

1315

1.9

2.6

3.1

κ(t)

SO

YB

EA

N

1012

1315

1.2

2.3

3.0

κ(t)

SU

GA

R

1012

1315

0.7

2.4

3.4

κ(t)

LIV

EC

AT

TLE

1012

1315

0.7

2.4

3.4

κ(t)

LEA

NH

OG

S

1012

1315

1.5

2.6

3.5

κ(t)

FE

ED

ER

CA

TT

LE

55

Fig

ure

7:

Sel

ecte

dex

amp

les

ofB

AV

fact

ores

tim

ate

s

1012

1315

4.1

4.4

4.7

Levels

WT

I

1012

1315

7.5

7.7

7.9

Levels

LEA

D

1012

1315

2.5

3.0

3.2

Levels

SU

GA

R

1012

1315

−0.

3

−0.

2

−0.

1

Levels

GB

PE

UR

1012

1315

−0.

5

0.0

0.4

Slopes

WT

I

1012

1315

−0.

1

−0.

0

0.1

Slopes

LEA

D

1012

1315

−1.

1

0.1

1.5

Slopes

SU

GA

R

1012

1315

−0.

1

−0.

0

0.0

Slopes

GB

PE

UR

OLS

PR

DG

LSP

TV

OLS

SE

CB

AV

56

Fig

ure

8:

Est

imat

edG

lob

al,

Mar

ket

and

Sec

tor

com

pon

ents

.

0512

0215

−9.

7

0.1

7.2

Glo

bal

0512

0215

−6.

8

0.0

7.1

Com

mod

ities

0512

0215

−5.

8

0.0

5.1

Fin

anci

als

0512

0215

−3.

0

−0.

0

4.0

Ene

rgy

0512

0215

−3.

0

0.1

4.7

Met

als

0512

0215

−8.

3

0.0

3.7

Gra

ins

0512

0215

−3.

0

0.1

3.1

Sof

ts

0512

0215

−3.

9

0.0

4.5

Live

stoc

k

0512

0215

−5.

1

0.0

5.2

For

eign

Exc

hang

e

0512

0215

−4.

2

−0.

0

3.0

Bon

ds

0512

0215

−10

.7

0.0

7.3

Equ

ity

0512

0215

−3.

9

0.0

4.8

Sea

sona

litie

s

S

lope

Leve

lC

urva

ture

57

Fig

ure

9:V

aria

nce

dec

omp

osit

ion

sp

erp

rod

uct

over

month

.

1215

0.0

0.5

1.0

BR

EN

T

1215

0.0

0.5

1.0

WT

I

1215

0.0

0.5

1.0

GA

SO

LIN

E

1215

0.0

0.5

1.0

HE

AT

ING

OIL

1215

0.0

0.5

1.0

GA

SO

IL

1215

0.0

0.5

1.0

NA

TU

RA

LGA

S

1215

0.0

0.5

1.0

CO

AL

1215

0.0

0.5

1.0

CO

2

1215

0.0

0.5

1.0

GO

LD

1215

0.0

0.5

1.0

SIL

VE

R

1215

0.0

0.5

1.0

CO

PP

ER

1215

0.0

0.5

1.0

ALU

MIN

UM

1215

0.0

0.5

1.0

LEA

D

1215

0.0

0.5

1.0

NIC

KE

L

1215

0.0

0.5

1.0

ZIN

C

1215

0.0

0.5

1.0

CO

TT

ON

1215

0.0

0.5

1.0

WH

EA

T

1215

0.0

0.5

1.0

CO

RN

1215

0.0

0.5

1.0

SO

YB

EA

N

1215

0.0

0.5

1.0

SU

GA

R

1215

0.0

0.5

1.0

OR

AN

JE

1215

0.0

0.5

1.0

CO

CO

A

1215

0.0

0.5

1.0

CO

FF

EE

1215

0.0

0.5

1.0

ET

HA

NO

L

1215

0.0

0.5

1.0

LUM

BE

R

1215

0.0

0.5

1.0

LIV

EC

AT

TLE

1215

0.0

0.5

1.0

LEA

NH

OG

S

1215

0.0

0.5

1.0

FE

ED

ER

CA

TT

LE

1215

0.0

0.5

1.0

US

DE

UR

1215

0.0

0.5

1.0

ZA

RG

BP

1215

0.0

0.5

1.0

GB

PE

UR

1215

0.0

0.5

1.0

ZA

RU

SD

1215

0.0

0.5

1.0

JPY

US

D

1215

0.0

0.5

1.0

RE

NU

SD

1215

0.0

0.5

1.0

US

INT

R

1215

0.0

0.5

1.0

EU

INT

R

1215

0.0

0.5

1.0

SA

INT

R

1215

0.0

0.5

1.0

UK

INT

R

1215

0.0

0.5

1.0

JPIN

TR

1215

0.0

0.5

1.0

CH

INT

R

1215

0.0

0.5

1.0

SP

500T

SX

60

1215

0.0

0.5

1.0

FT

SE

100

Le

vl−

GLO

Levl

−M

KT

Levl

−S

EC

Levl

−ID

IS

lop−

GLO

Slo

p−M

KT

Slo

p−S

EC

Slo

p−ID

IC

urv−

GLO

Cur

v−M

KT

Cur

v−S

EC

Cur

v−ID

IS

eas−

GLO

Sea

s−M

KT

Sea

s−S

EC

Sea

s−ID

I

58

Fig

ure

10:

Ou

t-of

-sam

ple

fore

cast

ing

resu

lts

(sta

tist

ical

mea

sure

s).

17

300.

0

1.0

3.0

h

MSE

τ =

1

17

300.

0

1.0

3.0

h

MSE

τ =

3

17

300.

0

1.0

3.0

h

MSE

τ =

6

17

30−

20−

100.01020

h

Clark West

17

30−

20−

100.01020

h

Clark West

17

30−

20−

100.01020

h

Clark West

17

300.

000.4

0.81

h

Cond. Coverage

17

300.

000.4

0.81

h

Cond. Coverage1

730

0.000.

4

0.81

h

Cond. Coverage

17

300.

000.4

0.81

h

Uncond. Coverage

17

300.

000.4

0.81

h

Uncond. Coverage

17

300.

000.4

0.81

h

Uncond. Coverage1

730

0.000.

4

0.81

h

Indep. Hypothesis

17

300.

000.4

0.81

h

Indep. Hypothesis

17

300.

000.4

0.81

hIndep. Hypothesis

59

Fig

ure

11:

Ou

t-of

-sam

ple

fore

cast

ing

resu

lts

(eco

nom

icm

easu

res)

.

17

30−

300.010

h

VARED−%

FC

VA

R−

OG

AR

CH

17

30−

300.010

hVARED−%

RS

FC

VA

R−

OG

AR

CH

17

30−

300.010

h

VARED−%

RS

FC

VA

R−

OG

AR

CH

−M

17

30−

300.010

h

VARED−%

BA

V

17

30

0.0

5.0

h

VARED−CW

17

30

0.0

5.0

h

VARED−CW

17

30

−9.

0

−5.

0

0.0

5.0

9.0

h

VARED−CW

17

30

−5.

0

0.0

5.0

9.0

h

VARED−CW

60