DP2018/04

Real-time forecasting with macro-finance

models in the presence of a zero lower bound

Leo Krippner and Michelle Lewis

March 2018

JEL classification: C43, E43

www.rbnz.govt.nz

Discussion Paper Series

ISSN 1177-7567

DP2018/04

Real-time forecasting with macro-finance models in

the presence of a zero lower bound∗

Leo Krippner and Michelle Lewis†

Abstract

We investigate the forecasting performance of a joint model of macroeconomicand yield curve components for the United States, using data as wouldhave been available in real time. Relative to a standard macroeconomicmodel benchmark, our results show a clear benefit from including yieldcurve information when forecasting inflation and the Federal Funds Rate, forhorizons up to the four years that we tested. We find some real-time forecastimprovement for capacity utilization, our variable representing real economicactivity, but only for longer horizons, and similarly when using macroeconomicdata to help forecast yield curve components. Using a shadow/lower-boundterm structure model allows the ready extension of our forecasting frameworkto include the unconventional period of monetary policy, and we obtain verysimilar results to those already mentioned for the conventional period.

∗ The Reserve Bank of New Zealand’s discussion paper series is externally refereed. Theviews expressed in this paper are those of the author(s) and do not necessarily reflectthe views of the Reserve Bank of New Zealand. We would like to thank participants atthe 2015 New Zealand Econometrics Study Group, the 2015 New Zealand Associationof Economists conference, Glen Rudebusch, Ken West, and colleagues at the ReserveBank of New Zealand for helpful discussion and comments.† Leo Krippner: Economics Department, Reserve Bank of New Zealand, 2 The Terrace,PO Box 2498, Wellington, New Zealand. Email address: leo.krippner@rbnz.govt.nz.Michelle Lewis: Economics Department, Reserve Bank of New Zealand, 2 The Terrace,PO Box 2498, Wellington, New Zealand. Email address: michelle.lewis@rbnz.govt.nzISSN 1177-7567 c©Reserve Bank of New Zealand

Non-technical summary

We investigate the real-time forecasting performance of macro-finance vectorautoregression models, which incorporate macroeconomic data and yield curvecomponent estimates as would have been available at the time of each forecast,for the United States.

Our results show a clear benefit from using yield curve information whenforecasting macroeconomic variables, both prior to the Global FinancialCrisis and continuing into the period where the lower-bound constrainedshorter-maturity interest rates. The forecasting gains, relative to traditionalmacroeconomic models, for inflation and the Federal Funds Rate are generallystatistically significant and economically material for the horizons up to thefour years that we tested. However, macro-finance models do not improvethe real-time forecasts over shorter horizons for capacity utilization, ourvariable representing real economic activity. This is in contrast to the relatedrecent macro-finance literature, which establishes such results (as do we) withpseudo real-time, i.e. truncated final-vintage, data. Nevertheless, for longerhorizons that are more relevant for central bankers, yield curve informationdoes improve activity forecasts.

Overall, our results suggest that the yield curve contains fundamental informa-tion about the likely evolution of the macroeconomy. We find less convincingevidence for the reverse direction, which is likely because expectations ofmacroeconomic variables are already reflected in the yield curve. However, forlonger horizons, we find there are still some gains from using macroeconomicvariables to forecast the yield curve.

1 Introduction

In this paper, we investigate the forecasting performance of a joint modelof macroeconomic and yield curve data for the United States (US), usingreal-time data and including the lower-bound period. We are motivatedby previous related literature, to be outlined shortly below, that has foundrelationships between macroeconomic and yield curve data that appear to beempirically useful to forecasters. However, these studies have generally usedin-sample or pseudo-real-time data (i.e. truncated final-vintage data), ratherthan the genuine real-time data that would actually have been available atthe time. Furthermore, for reasons we outline further below, the methodsemployed in previous studies are not strictly applicable to the lower-boundenvironments experienced by many developed economies since the GlobalFinancial Crisis. We show how the lower-bound constraint may readily beallowed for in such analysis.

The initial work linking macroeconomic and yield curve data began in thelate 1980s with the observation that flatter yield curve slopes (i.e. the spreadbetween long-term and short-term rates, such as the 10-year governmentbond and three-month Treasury bill) provided a leading indicator for slowerfuture output growth or recessions. For example, Estrella and Hardouvelis(1991) provided the first comprehensive statistical study into the relationshipfor the US, finding that the bond spread is useful for forecasting economicactivity, particularly 4-6 quarters ahead. Subsequent studies have generallyconfirmed such predictive power, albeit with variation across countries and/oramong different sample periods. The related literature is far too vast to citehere, and we refer readers to Wheelock and Wohar (2009) for a comprehensivesurvey.1 Regarding inflation, a parallel literature tests the yield curve as apredictor of inflation; see Stock and Watson (2003) for a survey.

The more recent literature has employed term structure models to investigatethe joint dynamics of macroeconomic and yield curve data. Term structuremodels offer the advantage of summarizing all yield curve data with justseveral factors, rather than selecting a given spread. Furthermore, they do soin a theoretically consistent manner if the arbitrage-free condition is imposed.

The seminal article of Ang and Piazzesi (2003) investigates the relationshipfrom macroeconomic variables to arbitrage-free latent factors of the yieldcurve within a structural VAR. It finds that a large proportion of the variation

1 Also see www.newyorkfed.org/research/capital_markets/ycfaq.html for an exten-sive bibliography. Rudebusch and Williams (2009) is the latest confirmation, for theUS, on the predictive power of the yield curve slope that we are aware of.

1

in yields can be explained by output and inflation data. Extending that to aforecasting perspective using pseudo real-time data, the authors find significantimprovements when forecasting yields. Similarly, Moench (2008) uses anarbitrage-free term structure model with a large number of macroeconomicvariables in a factor augmented VAR (FAVAR) model to forecast US bondyields. Up to 12 quarters ahead, the pseudo real-time forecasts from thatmacro-finance FAVAR outperform those from yield curve models, such asthe three-factor Duffee (2002) model and the Diebold and Li (2006) dynamicNelson and Siegel (1987) model.

In the reverse direction, Ang, Piazzesi and Wei (2006) finds that using anarbitrage-free term structure model, rather than just the bond spread, inconjunction with pseudo real-time GDP growth data improves US GDPforecasts out to 12 quarters. Ang, Bakaert, and Wei (2006) obtain similarresults. Related literature has used term structure models to investigatethe joint dynamics of macroeconomic and yield curve data, but not in aforecasting context.2

However, Ghysels, Horan, and Moench (2012) warns that using in-sample orpseudo-real-time data may overstate Treasury yield forecast improvements.Specifically, much of the predictive power of macroeconomic data for bondyields disappears when using real-time data. Our exercise therefore tests thepotential for forecasting improvements from a joint model of macroeconomicand yield curve data (our macro-finance model) using real-time data. Giventhe literature discussed earlier, we also test for potential forecast improvementsfrom yields to the macroeconomy.

The framework we use is analogous to that in Diebold, Rudebusch, andAruoba (2006), which considers the bidirectional relationship between USmacroeconomic data (capacity utilization, inflation, and the Federal FundsRate) and the yield curve, where the latter is summarized using estimatedNelson Siegel (1987) Level, Slope, and Bow factors.3 Our application hasthree main differences. First, our focus is on the forecasting performancefrom the model, rather than establishing the in-sample relationships betweenthe macroeconomic variables and yield curve components as in Diebold etal. (2006). Second, in the period prior to the lower-bound constraint on

2 Examples include Kozicki and Tinsley (2001), Piazzesi (2005), Ang and Piazzesi (2003),Dewachter and Lyrio (2006), Balfoussia and Wickens (2007), Ludvigson and Ng (2009),Joslin, Priebsch, and Singleton (2014), Bikbov and Chernov (2010), and Wright (2011).

3 Those authors find yield curve factors explain a significant proportion of variation inmacroeconomic variables, and the reverse relationship is also important but to a lesserdegree. The authors also link the Level component to inflation, the Slope to economicactivity, while the Bow appears unrelated to the key macroeconomic variables.

2

nominal interest rates, we use an arbitrage-free version of the Nelson-Siegelmodel (ANSM), as detailed in Krippner (2006) and Christensen, Diebold, andRudebusch (2011). Third, to accommodate the period where shorter-maturityinterest rates were constrained by the near-zero policy rate setting in the USfollowing the Global Financial Crisis but additional monetary accommodationwas delivered via unconventional means, we incorporate the ANSM within ashadow/lower-bound framework, as detailed in Krippner (2011, 2015) andChristensen and Rudebusch (2015).

The reason for using the shadow/lower-bound model in our full-sample exerciseis to avoid the distortion that would otherwise be present in the yield curvecomponents between the non-lower-bound and lower-bound periods. That is,whether using a 10-year less 3-month spread or estimated latent slope factor,the constraint on shorter-maturity interest rates would lead usual measures ofthe yield curve slope to understate the degree of monetary accommodation inthe lower-bound period. That in turn would distort associated macroeconomicoutcomes relative to the unconstrained period. Using yield curve componentsestimated from a shadow/lower-bound term structure model overcomes thatpotential distortion, because the components of the shadow yield curve moveexactly like the arbitrage-free version of the Nelson-Siegel model in the pre-lower-bound period, and continue to move freely in the lower-bound period.

At each point in time, we estimate the model to obtain Level, Slope, andBow state variables. We then use those in a vector autoregression (VAR)model with the capacity utilization, inflation, and Federal Funds Rate datathat was also available at the time (allowing for publication lags) to producejoint forecasts of macroeconomic and yield curve data.

Our results first confirm the pseudo-realtime forecasting results from theliterature. That is, the forecasts from our joint macro-finance model generallyoutperform the forecasts of macroeconomic variables from the models esti-mated using only macroeconomic data and, to a lesser extent, the forecasts ofyield curve variables from the models estimated using only yield curve data.However, the results are weaker when the stricter real-time considerationsare imposed. Specifically, the forecast improvement from including the termstructure almost disappears for capacity utilization, with gains remainingonly for longer forecast horizons. Nevertheless, the information gain at thelonger-horizons should still be of use to policy makers given this horizon isconsistent with medium-term objectives. The forecast improvement is morerobust for inflation, even when incorporating real-time economic activity inthe forecasting model.

The remainder of the paper proceeds as follows. Section 2 details the macro-

3

finance VAR models that we use for our forecasting comparison exercises.In section 3, we outline the data used in the models, including the modelswe use to estimate the state variables that summarize the yield curve data.Section 4 presents and discusses the results, and section 5 concludes.

2 Forecasting models

In this section we detail the models to be used in our forecasting exercises,including the yields-only and macro-only subset models to be used as bench-marks. All of the models are VARs estimated using ordinary least squares,and the appropriate lag length for the VARs are selected using the BayesianInformation Criterion (BIC).4 In all cases, for the repeated estimations forthe pseudo-real-time and real-time forecasting, we find the optimal lag lengthof 1, which conveniently allows us to present the models in their full formbelow.

2.1 Macro-only model

The macro-only subset model using the Diebold et al. (2006) data takes theform of a small traditional VAR containing economic activity, inflation, andthe policy rate, i.e.:ytπt

rt

=

a10a20a30

+

a11 a12 a13a21 a22 a23a31 a32 a33

yt−1

πt−1

rt−1

+

ea1tea2tea3t

(1)

where yt, πt, and rt are respectively capacity utilization, core CPI inflation,and the Federal Funds Rate rate, which are discussed in section 3.1. Themacro-only model serves as the benchmark for forecasts of macroeconomicvariables from the macro-finance models below.5

4 The BIC results are not reported here but are available upon request.5 Obviously, many alternative benchmark models could be used. We use VAR forecaststo be consistent with the yields-only VAR model and the macro-finance models.

4

2.2 Yields-only model

The yields-only subset model is an unconstrained VAR as follows:LtStBt

=

b10b20b30

+

b11 b12 b13b21 b22 b23b31 b32 b33

Lt−1

St−1

Bt−1

+

eb1teb2teb3t

(2)

where Lt, St, and Bt are respectively the Level, Slope, and Bow state variablesestimated from the arbitrage-free Nelson-Siegel model or the lower-boundaugmented version of that model. We provide the details of those models insections 3.1 and 3.2.6

The yields-only model serves as a benchmark for forecasting the yield curvewith only yield curve information, which will be compared against the macro-finance models below.

2.3 Unrestricted macro-finance model

The simplest macro-finance model is obtained by using the macroeconomicvariables and yield curve state variable within a single unconstrained VAR,i.e.:7

ytπtrtLtStBt

=

c10c20c30c40c50c60

+

c11 c12 c13 c14 c15 c16c21 c22 c23 c24 c25 c26c31 c32 c33 c34 c35 c36c41 c42 c43 c44 c45 c46c51 c52 c53 c54 c55 c56c61 c62 c63 c64 c65 c66

yt−1

πt−1

rt−1

Lt−1

St−1

Bt−1

+

ec1tec2tec3tec4tec5tec6t

(3)

The bolded parameters in the top-left quadrant of equation 3 relate to themacro-only VAR, while the bolded parameters in the bottom-right quadrantrelate to yields-only VAR. The parameters in the top-right quadrant allowthe yield curve components to help forecast the macroeconomic variables,and vice-versa for the parameters in the bottom-left quadrant.

6 Yield curve forecasts could be obtained directly from the estimated term structuremodel, but we use VAR forecasts to be consistent with the macro-only VAR model andthe macro-finance models.

7 Our macro-finance VAR models are a therefore the result of a two-step estimation,first to obtain the yield curve state variables and then to estimate the VAR modelitself. A one-step estimation would be possible within a full state space formulation,but the repeated estimations required for our real-time forecasting application wouldbe too computationally burdensome. Ang, Piazzesi, and Wei (2006) also use a two-stepestimation for the same reason.

5

2.4 Restricted macro-finance model

Restrictions may be applied to the macro-finance model to obtain parsimonythat may improve forecast performance. The first set of restrictions weimpose is so the policy rate does not affect other variables (but it can affectitself). The motivation is that the Level, Slope, and Bow variables alreadysummarize the information from the full yield curve, including an impliedpolicy rate estimate, so estimating an additional influence from the policy rateis redundant. The second set of restrictions we impose is so the Bow statevariable does not affect any of the macroeconomic variables. This restrictionreflects the lack of theoretical and empirical evidence, as discussed in Dieboldet al. (2006), about a relationship between the Bow and macroeconomicvariables. The final restriction we impose is a zero steady-state value for Lt,given that that its persistence is consistent with a near-unit-root process.

The final restricted model we use is therefore as follows:ytπtrtLtStBt

=

d10d20d300d50d60

+

d11 d12 0 d14 d15 0d21 d22 0 d25 d26 0d31 d32 d33 d34 d35 0d41 d42 0 d44 d45 d46d51 d52 0 d54 d55 d56d61 d62 0 d64 d65 d66

yt−1

πt−1

rt−1

Lt−1

St−1

Bt−1

+

ed1ted2ted3ted4ted5ted6t

(4)

3 Data and forecast production

In this section we detail the data used in the macrofinance models. Section 3.1details the source of the real-time macroeconomic data. Section 3.2 discussesthe real-time estimation of the Level, Slope, and Bow state variables for ourpre-LB exercise, and section 3.3 discusses the estimation for the full sampleincluding the LB period. In section 3.4 we detail how the data are used inour forecasting exercises, and the evaluation of the forecasts is discussed insection 3.5.

3.1 Macroeconomic data

As noted in section 2.1, the macroeconomic variables we use in our modelsare the following monthly series:

6

• capacity utilization, which we demean to create a similar concept tothe output gap often used by central banks;8

• core CPI inflation, which is calculated as annualized monthly log differ-ences of the price level;9 and

• the effective Fed Funds rate.

For the pseudo real-time forecasting exercise, we use the final vintages ofthe series above. These were obtained from the Federal Reserve Bank of St.Louis FRED website (www.research.stlouisfed.org/fred2/) at the endof 2015, when we started our project.

For the real-time macroeconomic data, i.e. the data that actually wouldhave been available to the forecaster at each point in time, we obtain thevintages from the Federal Reserve Bank of St. Louis ALFRED website(www.alfred.stlouisfed.org/). Capacity utilization and core CPI inflationare subject to minor changes between vintages, which mostly reflects seasonaladjustment, but it may also be due to a fuller set of information, changesin methodology, or technical reasons. Our real-time demeaning of capacityutilization, to reflect the amount of capacity pressure as would have beengauged in real time, also results in changes to the series over time. Real-timemonthly data vintages are used from December 1996 and the final vintage isDecember 2015. Figure 1 plots the capacity utilization and inflation data,and also serves to illustrate the extent of revisions as time evolves. Note thatthe Federal Fund Rate is observed at the end of the month and that serieshas no revisions relative to the historical vintages.

8 In principle, it would be more ideal to have used the actual output gap. However, to ourknowledge, no real-time output gap vintages series are available, and trying to createsuch a series would be open to subjectivity on how to do so (e.g. on model specificationand parameter choices). However, we have undertaken in-sample and pseudo-realtimeforecast exercises using the official Congressional Budget Office (CBO) 2014 measureof potential output and 2014Q1 GDP data. These obtain similar results to those wereport for capacity utilization.

9 For robustness, we also tested core PCE, headline PCE, and headline CPI inflation.All results were similar, but strongest overall for headline CPI inflation. We use coreCPI inflation because it is used in much of the literature, and it is more conceptuallyconsistent to expect financial market prices to reflect the trend component of inflationrather than the idiosyncratic components that are sometimes present in headline inflationmeasures.

7

Figure 1: Real-time macroeconomic dataDemeaned US capacity utilization

1985 1990 1995 2000 2005 2010 2015−14

−12

−10

−8

−6

−4

−2

0

2

4

6US annualized core inflation

1985 1990 1995 2000 2005 2010 2015−2

−1

0

1

2

3

4

5

6

7

8

Note:

Vintages from St. Louis ALFRED, with means subtracted for capacity utilization.

3.2 Arbitrage-free Nelson-Siegel model

Prior to the Global Financial Crisis (GFC), nominal yields were not con-strained by the lower bound. Hence it is valid to apply the standard arbitrage-free Nelson-Siegel model (hereafter ANSM) developed in Krippner (2006) andChristensen, Diebold, and Rudebusch (2011).

The heart of the ANSM is the following expression for forward rates:

f(t, u) = Lt + St · exp (−φu) +Bt · φτ exp (−φu) + V Ef (u) (5)

where f(t, u) is the instantaneous forward rate at time t as a function offorward horizon u, Lt, St, and Bt are the state variables, 1 and the functionsof φ are the forward rate factor loadings associated with each state variable,and V Ef (u) represents the volatility effect for forward rates required to makethe model arbitrage free (see Krippner (2015) for the full expression).

The interest rates, R (t, τ), at observation data t and as a function of time tomaturity τ , are then given by:

R(t, τ) =1

τ

∫ τ

0

f(t, u) du

= Lt + St

(1− exp (−φτ)

φτ

)+Bt

(1− exp (−φτ)

φτ− exp (−φτ)

)+ V ER(τ) (6)

where the state variables remain as for the forward rate expression, 1 andthe functions of φ are now the interest rate factor loadings associated with

8

Figure 2: Nelson-Siegel interest rate factor loadings

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time to maturity (τ)

LevelSlopeBow

Note: Example of arbitrage-free Nelson-Siegel factor loadings with φ = 0.79.

each state variable, and V ER(τ) represents the volatility effect for interestrates required to make the model arbitrage free (see Christensen, Diebold,and Rudebusch (2011) or Krippner (2015) for the full expression).10

The intuition underlying the names Level, Slope, and Bow comes from theshape of the factor loadings, which are plotted in figure 2. The first loadingis constant by maturity and in practice reflects the long-horizon Level of theyield curve. The second loading represents long-maturity yields relative toshort-maturity yields, so it reflects the Slope of the yield curve. The thirdloading represents mid-maturity yields relative to short- and long-maturityyields, so it reflects the Bow (or Curvature) of the yield curve. Note that φ isan estimated parameter, and it determines the decays of the Slope and Bowfactor loadings and the position of the local maximum for the Bow factorloading.

We estimate the ANSM using the Kalman filter, as detailed in Christensen,Diebold, and Rudebusch (2011) and Krippner (2015), with end-of-monthzero coupon bond data, from the end of 1985 until the end of 2007, which isprior to the events associated with the GFC and the subsequent lower-boundperiod. The start of the sample period is chosen to capture the period ofthe great moderation and inflation stability in the US, and it also avoids the

10 Ignoring the volatility effect gives the original Nelson-Siegel model, in forward rate orinterest rate terms, which is not arbitrage-free.

9

structural break in the early 1980s.11 The maturities we use are 0.25, 0.5, 1,2, 3, 5, 7, and 10 years, which follows Diebold et al. (2006) and is consistentwith much of the literature.

To be consistent with the real-time vintages of macroeconomic data, werecursively estimate the ANSM with the sample period expanding month-by-month. Hence, the first vintage is the estimation using the sample up toDecember 1996, and the final vintage uses the sample up to December 2007.

Figure 3 contains the real-time estimates of the ANSM Level, Slope, andBow. It turns out that the real-time updates show little variation relative tohistorical estimates, so plotting all of the vintages together looks like a singleseries. For the Level state component, the largest difference relative to the lastvintage is 24 basis points and the absolute mean difference is just three basispoints. The Slope and Bow components also show little real-time variation,with absolute mean differences of four and 12 basis points respectively.12

3.3 Shadow/lower-bound arbitrage-free Nelson-Siegelmodel

After the GFC, the US Federal Reserve cut the policy rate to near-zero levels,and so the lower bound for nominal yields became a material constraint. Asdetailed in Krippner (2015), it is no longer valid to apply the ANSM in such anenvironment, essentially because the ANSM would be mis-specified relative tothe properties of the observable data.13 However, it is valid to use the ANSMas the shadow term structure representation within the shadow/lower-boundframework developed by Krippner (2011, 2015), and we will call this model

11 Estrella, Rodrigues, and Schich (2003) and Joslin, Priebsch, and Singleton (2014) finda structural change around that period.

12 The realtime estimation exercise was also done for the yield curve out to 30 years.These estimates showed more realtime variation, with the largest difference peaking at111 basis points.

13 The ANSM implies that short-maturity rates are free to move below the lower boundand they maintain a constant volatility over the sample period, whereas the short-maturity interest rate data is constrained by the lower bound and has markedly lowervolatility relative to the pre-LB period. Also see Christensen and Rudebusch (2015),Bauer and Rudebusch (2015), and Wu and Xia (2016) for further discussion on theinconsistencies of non-lower-bound models when applied near the lower bound.

10

Figure 3: ANSM real-time estimatesANSM Level

1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 20074

5

6

7

8

9

10

11

Years

Per

cent

Realtime estimates

ANSM Slope

1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007

−6

−4

−2

0

Years

Per

cent

ANSM Bow

1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 2007−8

−6

−4

−2

0

2

4

Years

Per

cent

ANSM Phi

1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 20070.5

0.55

0.6

0.65

Years

Per

cent

Note: Real-time estimates of ANSM state variables. The sample starts December 1985

and the realtime vintages start in December 1996. The sample ends in December 2007,

which is prior to the onset of the GFC and the lower-bound environment.

11

the KANSM.14

The intuition underlying the KANSM is that the observed short rate r (t) =max {r (t) , rLB}, where r (t) is a Gaussian diffusion for the shadow short rate,and rLB is the lower-bound parameter. With a lower bound value of zeroimposed,15 the lower bound forward rate curve is then as follows:

f (t, u) = f (t, u) · Φ[f (t, u)

ω (u)

]+ ω (u) · 1√

2πexp

(−1

2

[f (t, u)

ω (u)

]2)(7)

where f (t, u) is the lower bounded forward rate, ω (u) is the shadow shortrate volatility function, and Φ [·] is the cumulative normal density function,and f (t, u) is the ANSM forward rate expression from equation 5. The LBinterest rate curve is obtained by the straightforward univariate numericalintegration of the following expression:

R(t, τ) =1

τ

∫ τ

0

f(t, u) du (8)

The first plot in figure 4 provides an example of applying the KANSM in alower-bound period, i.e. where the lower-bound constrains shorter-maturityinterest rates. The second plot is an example for a non-LB period, where allinterest rates are sufficiently high for the lower bound to be an immaterialconstraint.

Across both non-LB and LB periods, the KANSM still obtains Level, Slope,and Bow state variables, and these are plotted in figure 4. The associatedforward rate and interest rates factor loadings, and their interpretation,remain as for the ANSM, except they now represent the shadow yield curve.Importantly, the shadow term structure estimated from the KANSM isessentially coincident with the ANSM estimates in the pre-LB period, asin the second plot of figure 2. Correspondingly, the Level, Slope, and Bowestimates in figures 3 and 5 are almost identical over the 1996-2007 period.16

14 The Krippner (2011, 2015) framework is developed in continuous time, which ac-commodates the continuous-time ANSM. Wu and Xia (2015) develop a discrete-timeequivalent to the Krippner (2011, 2015) framework. Both frameworks are very tractableapproximations to the shadow/lower-bound framework suggested in Black (1995).

15 It is possible to estimate a lower bound parameter, but we obtained implausibly highestimates when the recursive samples span only a short time in the lower bound period(because the models effectively uses the free parameter as an extra degree of freedom).Imposing a LB value of zero for all recursive samples avoids this issue.

16 When the lower bound is not a mateial constraint on r (t) or its expectations, thenr (t) = max {r (t) , rLB} may be treated as r (t) = r (t), and so the KANSM becomesthe standard ANSM.

12

Figure 4: Examples of applying the KANSM

0 2 4 6 8 10

Time to maturity (years)

-5

0

5

Per

cent

age

poin

ts

Lower-bound period

0 2 4 6 8 10

Time to maturity (years)

-5

0

5 Non-LB period

yield curve datamodel yield curveshadow yield curveoption effect

But after the onset of the LB period, the shadow term structure can adoptnegative values for shorter maturities, and so the Slope estimate in particularcan continue to vary freely. Hence, the KANSM allows us to extend thesample period from December 2007 to December 2014, without the yieldcurve components being subject to the distortion that the ANSM would incurbetween the non-LB and LB periods.

As an interesting aside, figure 6 plots the final vintage of annualized core CPIinflation with the KANSM Level estimate, and the final vintage of capacityutilization (and the CBO output gap) with the KANSM Slope estimate. Thecorrelations in both plots are highly significant, at 0.62 and 0.59 respectively,which is consistent with the in-sample results from Diebold et al. (2006). Therespective correlations for the ANSM model to 2007 are very similar. It isthese sort of inter-relationships that the forecasting exercise tries to exploit,but in real time.

3.4 Producing the forecasts

We undertake four sets of model forecasts, i.e. pseudo real-time forecasts andgenuine real-time forecasts for the pre-GFC sample, and then we repeat thoseexercises for the full sample.

The pseudo real-time forecasts are straightforward to produce. We simply

13

Figure 5: KANSM real-time estimatesKANSM Level

1990 1995 2000 2005 2010 20151

2

3

4

5

6

7

8

9

10

11

Years

Per

cent

Realtime estimates

KANSM Slope

1990 1995 2000 2005 2010 2015

−6

−4

−2

0

Years

Per

cent

KANSM Bow

1990 1995 2000 2005 2010 2015−12

−10

−8

−6

−4

−2

0

2

4

Years

Per

cent

KANSM Phi

1998 2000 2002 2004 2006 2008 2010 2012 20140.5

0.55

0.6

0.65

Years

Per

cent

Note: Realtime estimates of KANSM state variables. The sample starts in December 1985

and the realtime vintages start in December 1996. The sample ends in December 2015.

Figure 6: Macro-finance linkagesUS Level and inflation

1985 1990 1995 2000 2005 2010 2015−2

0

2

4

6

8

10

12

LevelCore CPI (annualised)

US Slope and real activity

1985 1990 1995 2000 2005 2010 2015

−10

−5

0

5

SlopeCapacity utilisationOutput gap

Note: The correlation between the Level factor and core CPI inflation is 0.62. The

correlation between the Slope factor and capacity utilisation is 0.59.

14

Figure 7: Accommodating data unavailable in real time

Time Variable r L S B π y ⁞ t-3 t-2 t-1 t O O t+1 x x x x x x t+2 x x x x x x ⁞ x x x x x x

Note: Capacity utilization and inflation data are unavailable at time t, and ’O’ denotes

the “nowcast” for those variables using t− 1 data. “x” denotes the forecasts from the

resultant balanced panel.

use the pseudo real-time series (i.e the final-vintage data truncated at theend of each month) as a complete balanced panel of data to estimate theVAR models outlined in section 2, and then use those estimated models toobtain forecasts for each variable from t+ 1 until t+ 48 (fours years ahead).

Producing genuine real-time forecasts is a little more involved, partly becausehistorical vintages need to be used, but also because the one-month publicationlag for capacity utilization and inflation data needs to taken into account.That is, both of those variables for a given month are only released duringthe following month. Hence, those data need to treated as missing at the endof month, and so we therefore have an unbalanced panel of data when thereal-time forecasts are undertaken.

We accommodate the missing capacity utilization and inflation observationsas illustrated in figure 7. That is, at each point in time t, any VAR modelwith missing capacity utilization and inflation data is estimated up to t− 1,and that t− 1 VAR is used to “now-cast” the missing observations at time t.The VAR model is then re-estimated with the balanced panel, including thenow-cast “observations”, up to time t, and that time t VAR model is used toproduce forecasts, again from t+ 1 until t+ 48.17

17 A one-step estimation using a state space formulation would provide a more formalresolution of the unbalanced panel. However, as mentioned in footnote 7, the repeatedestimations required for the real-time forecasts would be too computationally onerous.The key point for our real-time forecast exercises is that any missing data are treatedin the same way in the benchmark and macro-finance models.

15

Both the pseudo real-time and real-time forecasts use an expanding sample.This follows the literature, and reflects the relatively short sample sizesinvolved.

3.5 Evaluating forecast performance

For the macroeconomic variables, we use forecasts relative to the final datavintages to evaluate the performance of the model estimates. Given thestability in the real-time estimates, we have also chosen to evaluate theforecasts of yield curve components against their final vintage. An alternativewould be to calculate yield forecasts for some specific maturities and comparethose to realized yield series, but evaluating the yield curve componentsdirectly provides a general test across the entire yield curve.

We calculate the root mean squared forecast error (RMSFE) for our forecastsand evaluate those against the RMSFEs from appropriate benchmark models.For macroeconomic variables, the benchmark model is the macro-only model.For yield curve factors, the benchmark model is the yields-only model. Wehave also calculated single equation AR(1) forecasts for all variables, whichis a standard benchmark for forecasts from VAR models. Note that we donot subject forecasts of the Federal Funds Rate to be bounded by zero in anyof our models. It would be possible to impose values of zero on any negativeforecasts, but the key point in our relative forecasting exercise is that allmodels have identical treatment.

We test the statistical significance of the RMSFE differences relative tobenchmark models using the Diebold and Mariano (1995) test for non-nestedmodels, with the Clark and West (2007) correction for nested models. Forexample, the AR(1), macro-only, and yields-only models are nested withinthe unrestricted macro-finance VAR model. We use one-sided tests, givenwe are interested in model outperformance over the benchmark. Note thatwhenever forecasts of monthly data are made for horizons, h, greater than onemonth, the time series of forecast errors will overlap, which produces serialcorrelation. We therefore use a Newey-West estimator with a window lengthof h− 1 to correct for that autocorrelation when calculating the statisticalsignificance.

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4 Results

In this section we present the results of our forecasting exercises. Section 4.1contains the pre-GFC results from December 1986 to December 2007, andsection 4.2 contains the full sample results from December 1986 to December2014. We report the results grouped by macroeconomic variables, wherethe benchmark is the macro-only VAR model, and yield curve components,where the benchmark is the yields-only model. The outright RMSFEs fromthe benchmark model are reported for each variable, with the units as thevariables were used in the VARs, i.e. index points for capacity utilization,and annualized percentage points for CPI inflation, the Federal Funds Rate,and the yield curve components. The remaining entries in each line of thetable are the relative RMSFEs for the forecasts from the alternative models.A ratio smaller than one, in bold type, means the alternative model providesa better forecast than the benchmark model. We will typically refer to thepercentage improvement, e.g. if the relative RMSFE is 0.80, then the forecastimprovement is 20 percent relative to the benchmark model. The indicators“*”, “**”, and “***” respectively denote the 10, 5, and 1 percent levels ofstatistical significance.

Note that entries of “n/a” in the table indicate that the forecast errors arezero by definition. This occurs for all of the pseudo real-time forecasts forthe h = 0 horizon, where the data is implicitly assumed to be known at theend of the month. The only occurrence in the real-time forecasts is for theFederal Funds Rate rate at the h = 0 horizon, because that data is observedat the end of the month and the final vintage contains no revisions relativeto the historical vintages. Conversely, the nowcasts of demeaned capacityutilization and inflation, and the historical vintage estimates of the yieldcurve components are compared to their final (revised) vintages, and so therewill be some forecast error at the h = 0 horizon.

4.1 Pre-GFC forecast results

Table 1 contains the forecast performance results for the macroeconomicvariables over the pre-GFC sample. The first point of note is that thepseudo real-time results are consistent with those already reported in theliterature. That is, including yield curve information improves macroeconomicforecasts relative to the macro-only model. The improvements are moreapparent for longer horizons, and are generally better for the restrictedversus the unrestricted macro-finance model. Inflation and the FFR are

17

the variables with the largest and most consistent forecast improvements,with respective gains of 35 and 43 percent at the four-year horizon (andthe nowcast for inflation beats the macro-only benchmark by 10 percent).Those improvements are highly statistically significant, and very material ineconomic significance. On the latter, for example, using the table entries tocalculate 4.25× (1− 0.57) gives a 1.83 percentage point forecast improvementfor the FFR.

The second point of note is that, for capacity utilization, the real-time forecastimprovements are markedly smaller than those for pseudo real-time forecasts.Indeed, the macro-finance models underperform the benchmark out to twoor three years, respectively for the restricted and unrestricted macro-financemodels. The maximum outperformance is at the four year horizon, but onlyto the extent of a (statistically significant) 11 percent improvement for therestricted model, compared to 18 percent in the same model in pseudo realtime.

The third point of note is that, while there is some deterioration, the real-timeforecasts of inflation and the FFR largely maintain the profile, magnitude,and of the outperformances in the pseudo real-time results. Finally, themacro-finance models outperform the AR benchmark, both in pseudo realtime and in real time.

Table 2 contains the forecast performance results for the yield curve com-ponents over the pre-GFC sample. The pseudo real-time results are againconsistent with those already reported in the literature, in that includingmacroeconomic information improves yield curve forecasts relative to theyields-only model. However, unlike the macroeconomic results, the forecastimprovements are not consistent across all horizons, with the Level and Slopeforecasts underperforming the yields-only benchmark for shorter horizons.Also, there is no obvious advantage from the restricted versus the unrestrictedmacro-finance model. In addition, when there are forecast improvements, theyare typically less impressive than for the macroeconomic variables. Neverthe-less, the forecast improvements for long horizons are statistically significantand economically material, particularly for the Slope variable where the 33percent improvement translates to a 1.14 percentage point improvement inthe RMSFE relative to the yields-only model.

The real-time forecasts of the yield curve components largely maintain themixed results noted for the pseudo real-time results, and there is somedeterioration in the Slope and Bow components. Surprisingly, the real-timeLevel results are better than their pseudo real-time counterparts.

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Table 1: Pre-GFC forecast results for macroeconomic variablesCapacity utilization, pseudo real-time Capacity utilization, actual real-time

Macro MF MF AR Macro MF MF ARh BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a 2.17 1.03 1.04 1.031 0.42 0.98 *** 1.01 1.02 2.16 1.06 1.08 1.052 0.59 0.94 *** 0.98 1.03 2.19 1.08 1.11 1.063 0.80 0.91 *** 0.96 1.04 2.21 1.09 1.14 1.086 1.38 0.91 ** 0.93 1.05 2.36 1.11 1.19 1.1312 2.58 0.93 ** 0.90 1.06 2.63 1.10 1.20 1.2324 5.14 0.94 * 0.86 ** 1.06 3.05 1.12 1.09 1.4136 6.11 0.90 * 0.81 ** 1.09 3.32 1.00 0.89 1.6248 6.82 0.87 0.82 ** 1.07 3.42 0.99 * 0.89 ** 1.80

Inflation, pseudo real-time Inflation, actual real-timeMacro MF MF AR Macro MF MF AR

h BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a 1.17 0.91 *** 0.90 ** 1.081 1.15 0.89 *** 0.88 *** 1.10 1.18 0.86 *** 0.86 *** 1.152 1.18 0.89 *** 0.88 ** 1.16 1.19 0.88 *** 0.88 *** 1.213 1.18 0.88 *** 0.88 *** 1.21 1.19 0.88 *** 0.88 *** 1.246 1.20 0.88 *** 0.87 ** 1.24 1.21 0.88 *** 0.87 ** 1.2412 1.23 0.93 *** 0.91 1.22 1.22 0.94 ** 0.91 1.2324 1.44 0.87 *** 0.83 *** 1.07 1.48 0.88 *** 0.82 *** 1.0536 1.68 0.79 *** 0.75 *** 0.97 1.64 0.83 *** 0.75 *** 0.9448 1.69 0.72 *** 0.65 *** 0.97 1.66 0.78 *** 0.69 *** 0.98

Fed. Funds Rate, pseudo real-time Fed. Funds Rate, actual real-timeMacro MF MF AR Macro MF MF AR

h BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a n/a n/a n/a n/a1 0.16 0.74 *** 0.88 *** 1.09 0.17 0.73 *** 0.87 *** 1.022 0.32 0.75 *** 0.90 ** 1.08 0.32 0.74 *** 0.89 ** 1.013 0.47 0.76 *** 0.91 ** 1.08 0.47 0.76 *** 0.91 ** 1.006 0.91 0.82 *** 0.95 * 1.05 0.90 0.84 *** 0.96 0.96 *12 1.73 0.87 ** 0.97 1.01 1.70 0.90 ** 1.00 0.92 **24 2.81 0.88 * 0.90 1.03 2.72 0.99 * 0.98 0.96 *36 3.58 0.77 *** 0.67 *** 0.99 * 3.44 0.91 *** 0.79 *** 1.0048 4.25 0.72 *** 0.57 *** 0.87 *** 3.88 0.87 *** 0.65 *** 0.97Notes: The Macro Benchmark (BM) results are RMSFEs for each horizon h. The remaining

results are RMSFEs relative to the BM results. “*”, “**”, and “***” are respectively 10,

5, and 1 percent levels of statistical significance based on the one-sided Diebold-Mariano

-West test, with the Clark-West correction for the nested models (MF Unres. and AR).

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Table 2: Pre-GFC forecast results for yield curve componentsLevel, pseudo real-time Level, actual real-time

Yields MF MF AR Macro MF MF ARh BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a 0.04 1.00 1.00 1.001 0.27 1.02 1.00 0.99 * 0.27 0.98 ** 0.98 ** 0.99 **2 0.36 1.02 0.97 * 0.98 ** 0.37 0.97 *** 0.96 ** 0.98 **3 0.41 1.02 0.96 * 0.97 ** 0.42 0.95 *** 0.95 ** 0.97 **6 0.61 1.01 0.94 0.97 ** 0.62 0.93 ** 0.92 * 0.97 **12 0.94 0.99 0.96 0.98 0.92 0.91 ** 0.91 0.9824 1.08 0.97 1.03 0.99 1.03 0.92 0.96 1.0236 1.28 0.91 * 1.01 0.94 *** 1.28 0.87 ** 0.90 * 0.9648 1.47 0.88 ** 0.96 0.95 *** 1.46 0.86 *** 0.84 *** 0.94 ***

Slope, pseudo real-time Slope, actual real-timeYields MF MF AR Macro MF MF AR

h BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a 0.06 1.00 1.00 1.001 0.37 1.01 1.04 1.05 0.37 1.00 1.03 1.042 0.54 1.01 1.06 1.09 0.53 1.01 1.07 1.073 0.67 1.01 1.08 1.11 0.66 1.03 1.12 1.096 1.09 0.99 1.09 1.10 1.08 1.06 1.18 1.0812 1.89 0.94 1.05 1.05 1.87 1.05 1.18 1.0224 2.96 0.82 *** 0.89 * 0.94 2.97 0.97 1.04 0.87 *36 3.56 0.70 *** 0.74 *** 0.92 3.73 0.81 *** 0.84 *** 0.8648 3.46 0.67 *** 0.67 *** 0.92 3.73 0.71 ** 0.69 *** 0.86

Bow, pseudo real-time Bow, actual real-timeYields MF MF AR Macro MF MF AR

h BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a 0.09 1.00 1.00 1.001 0.86 1.00 0.99 0.99 * 0.85 1.01 1.01 1.002 1.13 1.01 0.98 0.98 ** 1.11 1.02 1.01 0.98 *3 1.37 1.00 0.97 0.97 ** 1.33 1.01 1.00 0.98 **6 1.87 0.97 * 0.91 ** 0.96 ** 1.76 1.00 0.96 0.96 **12 2.25 0.90 *** 0.81 *** 0.94 ** 2.19 0.96 0.90 0.94 **24 2.72 0.87 *** 0.77 *** 0.90 ** 2.71 0.96 * 0.87 ** 0.89 **36 3.18 0.91 *** 0.84 *** 0.92 *** 2.94 0.94 *** 0.83 *** 0.87 ***48 3.67 0.94 0.88 ** 0.97 *** 2.90 0.99 0.85 ** 0.92 ***Notes: The Yields Benchmark (BM) results are RMSFEs by horizon h. The remaining

results are RMSFEs relative to the BM results. “*”, “**”, and “***” are respectively 10,

5, and 1 percent levels of statistical significance based on the one-sided Diebold-Mariano

-West test, with the Clark-West correction for the nested models (MF Unres. and AR).

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Table 3: Full-sample forecast results for macroeconomic variablesCapacity utilization, pseudo real-time Capacity utilization, actual real-time

Macro MF MF AR Macro MF MF ARh BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a 1.85 1.02 1.02 1.021 0.53 0.99 *** 0.99 1.00 1.92 1.04 1.05 1.032 0.85 0.97 *** 0.97 * 0.99 2.04 1.04 1.06 1.033 1.17 0.96 *** 0.96 * 0.99 * 2.19 1.04 1.07 1.036 2.19 0.95 *** 0.96 * 0.99 2.81 1.01 1.06 1.0312 3.87 0.94 *** 0.95 ** 0.99 3.90 0.98 ** 1.03 1.0324 5.70 0.90 ** 0.88 ** 1.03 4.24 0.94 ** 0.98 1.1536 6.87 0.81 ** 0.79 *** 1.07 4.63 0.82 * 0.85 ** 1.3248 8.11 0.74 * 0.74 ** 1.09 5.28 0.73 * 0.78 * 1.43

Inflation, pseudo real-time Inflation, actual real-timeMacro MF MF AR Macro MF MF AR

h BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a 1.05 0.99 *** 1.00 1.111 1.05 0.97 *** 0.98 1.12 1.08 0.95 *** 0.96 1.212 1.08 0.97 *** 0.98 1.22 1.09 0.97 *** 0.98 1.283 1.09 0.96 *** 0.96 1.28 1.09 0.97 *** 0.97 1.316 1.09 0.97 *** 0.95 1.34 1.09 0.98 *** 0.95 1.3512 1.11 1.03 0.95 1.33 1.09 1.04 0.96 1.3524 1.48 0.84 *** 0.80 *** 1.02 1.47 0.87 *** 0.81 *** 1.0436 1.73 0.75 *** 0.72 *** 0.88 1.58 0.80 *** 0.74 *** 0.90 *48 1.89 0.65 *** 0.61 *** 0.83 * 1.71 0.71 *** 0.66 *** 0.89

Fed. Funds Rate, pseudo real-time Fed. Funds Rate, actual real-timeMacro MF MF AR Macro MF MF AR

h BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a n/a n/a n/a n/a1 0.18 0.80 *** 0.86 *** 0.99 *** 0.19 0.77 *** 0.85 *** 0.93 ***2 0.33 0.81 *** 0.88 *** 0.97 *** 0.34 0.78 *** 0.86 *** 0.91 ***3 0.48 0.82 *** 0.89 *** 0.96 *** 0.49 0.80 *** 0.88 *** 0.89 ***6 0.90 0.86 *** 0.92 ** 0.92 ** 0.89 0.85 *** 0.92 ** 0.85 ***12 1.71 0.85 *** 0.94 0.87 ** 1.67 0.86 *** 0.95 0.80 ***24 2.91 0.83 *** 0.88 * 0.84 ** 2.73 0.89 *** 0.94 0.79 ***36 3.98 0.71 *** 0.73 *** 0.78 ** 3.61 0.82 *** 0.80 *** 0.80 **48 5.04 0.64 *** 0.64 *** 0.66 ** 4.36 0.75 *** 0.70 *** 0.74 **Notes: The Macro Benchmark (BM) results are RMSFEs for each horizon h. The remaining

results are RMSFEs relative to the BM results. “*”, “**”, and “***” are respectively 10,

5, and 1 percent levels of statistical significance based on the one-sided Diebold-Mariano

-West test, with the Clark-West correction for the nested models (MF Unres. and AR).

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Table 4: Full-sample forecast results for yield curve componentsLevel, pseudo real-time Level, actual real-time

Yields MF MF AR Macro MF MF ARh BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a 0.04 1.00 1.00 1.001 0.35 1.01 1.01 1.00 0.35 1.01 1.00 1.002 0.50 1.01 1.00 1.00 0.51 1.01 1.00 1.003 0.56 1.01 1.00 1.00 0.57 1.01 1.00 1.006 0.80 1.00 * 1.01 1.02 0.81 0.99 1.00 1.0212 1.09 0.99 * 1.03 1.05 1.08 0.99 1.01 1.0624 1.17 0.98 1.10 1.10 1.16 0.95 ** 1.07 1.1336 1.32 0.93 ** 1.11 1.12 1.33 0.86 ** 1.07 1.1748 1.55 0.88 *** 1.04 1.07 1.58 0.87 ** 1.00 1.11

Slope, pseudo real-time Slope, actual real-timeYields MF MF AR Macro MF MF AR

h BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a 0.05 1.00 1.00 1.001 0.44 0.99 1.00 1.00 0.43 0.99 1.00 1.002 0.63 0.99 1.00 1.01 0.62 0.99 1.01 1.003 0.75 0.98 1.01 1.01 0.74 0.99 1.02 0.99 **6 1.19 0.96 1.01 0.99 * 1.18 1.00 1.06 0.97 **12 1.90 0.97 1.04 0.96 * 1.88 1.04 1.12 0.94 **24 2.70 0.92 1.04 0.92 * 2.66 1.03 1.13 0.86 **36 3.21 0.91 1.07 0.91 3.23 0.96 1.09 0.85 **48 3.11 0.83 1.04 0.91 3.18 0.79 * 0.96 0.86 *

Bow, pseudo real-time Bow, actual real-timeYields MF MF AR Macro MF MF AR

h BM Unres. Res. BM Unres. Res.0 n/a n/a n/a n/a 0.08 1.00 1.00 1.001 0.92 1.00 1.00 1.00 0.91 1.00 1.00 1.002 1.23 1.01 0.99 1.00 1.22 1.01 1.01 1.003 1.46 1.01 0.99 1.00 1.44 1.01 1.01 1.016 2.00 0.99 0.97 1.03 1.94 1.01 1.01 1.0412 2.46 0.93 * 0.92 1.09 2.44 0.97 0.98 1.1024 3.13 0.91 * 0.86 * 1.10 3.20 0.98 0.94 1.1236 3.63 0.92 *** 0.87 *** 1.07 3.63 0.97 * 0.90 *** 1.0948 3.95 0.91 * 0.86 *** 1.05 3.62 0.96 0.87 *** 1.07Notes: The Yields Benchmark (BM) results are RMSFEs by horizon h. The remaining

results are RMSFEs relative to the BM results. “*”, “**”, and “***” are respectively 10,

5, and 1 percent levels of statistical significance based on the one-sided Diebold-Mariano

-West test, with the Clark-West correction for the nested models (MF Unres. and AR).

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4.2 Full-sample forecast results

Tables 3 and 4 respectively contain the forecast performance for the macroe-conomic variables and the yield curve components for the full sample. Themain points of note parallel those already outlined for the pre-GFC results.Specifically, for the macroeconomic variables, both the pseudo real-time andreal-time forecast improvements remain very similar to the results for thepre-GFC sample. Indeed, the real-time forecasts of capacity utilization rel-ative to the pseudo real-time forecasts no longer deteriorate as much, withthe respective improvements of 27 and 22 percent from the unrestricted andrestricted macro-finance models at the four-year horizon comparable to theimprovements for inflation and the FFR.

Regarding the yield curve components, both the pseudo real-time and real-time results forecast performances remain mixed by horizon and model. Ingeneral, there appears to be a larger deterioration for the real-time forecastsrelative to the pseudo real-time forecasts in the full sample compared to thepre-GFC sample.

4.3 Discussion of results

The results in the previous two sections provide the basis for several implica-tions and related discussion.

First, there is a clear benefit from using yield curve information when under-taking forecasts of macroeconomic variables in real time. This is an intuitiveresult for the FFR, because the yield curve should contain information aboutthe expected path of the policy rate. Similarly, the improvement in theinflation forecasts is consistent with the empirical and theoretical link be-tween inflation forecasts and the inflation component in nominal interestrates. While the capacity utilization results are consistent with literatureshowing that yield curve information helps to forecast economic downturns,the caveat from our analysis is that the short- and medium-horizon forecastimprovements that appear possible with pseudo real-time analysis are notobtained in the real-time setting.

Second, the real-time forecast improvements are greatest at longer horizons,even including material gains for capacity utilization. These longer-horizonresults are particularly useful for central banks. That is, central bankstypically set monetary policy to target macroeconomic outcomes/objectivesover longer horizons. So long as the level and shape of the prevailing yield

23

curve is consistent with those desired outcomes/objectives, then policy makerscan be more tolerant of any deviations over shorter horizons. Or if the yieldcurve is inconsistent with the desired outcomes/objectives, then policy makerscan influence its level and shape by adjusting the settings and indications ofmonetary policy.

Third, yield curve information, at least in the way we have obtained andused it, continues to improve the forecasts of macroeconomic variables in realtime even when the lower-bound period is included. That is, even while theconstraint on short-maturity interest rates distorts the shape of yield curveand its potential information content, estimating the yield curve componentsusing a shadow/lower-bound framework obtains components that vary asfreely as in the pre-LB period. In turn, those estimated components continueto improve the forecasts of macroeconomic variables relative to the macro-onlymodel.

Fourth, there is some benefit from using macroeconomic information whenundertaking forecasts of the yield curve in real time, but the gains are ingeneral not as consistent, statistically significant, and economically materialas in the reverse direction. The likely reason is that financial market vari-ables should already reflect expectations of macroeconomic variables. Hence,there should be little systematic benefit from including the macroeconomicinformation when forecasting the yield curve. Nevertheless, there appears tobe some benefit at longer horizons.

5 Conclusion

We investigate the real-time forecasting performance of macro-finance vectorautoregression models, incorporating macroeconomic data and yield curvecomponent estimates, for the United States.

Our results show a clear benefit from using yield curve information whenforecasting macroeconomic variables, both prior to the Global FinancialCrisis and continuing into the period where the lower-bound constrainedshorter-maturity interest rates. The forecasting gains, relative to traditionalmacroeconomic models, for inflation and the Federal Funds Rate are generallystatistically significant and economically material for the horizons up to thefour years that we tested. However, macro-finance models do not improvethe real-time forecasts over shorter horizons for capacity utilization, ourvariable representing real economic activity. This is in contrast to the related

24

recent macro-finance literature, which establishes such results (as do we) withpseudo real-time, i.e. truncated final-vintage, data. Nevertheless, for longerhorizons that are more relevant for central bankers, yield curve informationdoes improve activity forecasts.

Overall, our results suggest that the yield curve contains fundamental informa-tion about the likely evolution of the macroeconomy. We find less convincingevidence for the reverse direction, which is likely because expectations ofmacroeconomic variables are already reflected in the yield curve. However, forlonger horizons, we find there are still some gains from using macroeconomicvariables to forecast the yield curve.

There are several avenues in which our analysis could be refined. One is toundertake the forecasts in state space models, rather than the VARs we haveused. The estimation would then be more efficient and it would also providea more formal resolution of accommodating unbalanced data panels whenforecasting in real time. Another avenue is to impose further parsimony in themacro-finance models using restrictions suggested the policy expectation andterm-premia components of the yield curve models. While these may lead tofurther gains in real-time forecast performance, we expect any improvementsto be modest relative to the straightforward and readily applicable approachwe have presented in this article.

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References

Ang, A. and Piazzesi, M. 2003. A no-arbitrage vector autoregressionof term structure dynamics with macroeconomic and latent variables.Journal of Monetary Economics, 50, 745-787.

Ang, A., Piazzesi, M., and Wei, M. 2006. What does the yield curvetell us about GDP growth?, Journal of Econometrics. 131, 359-403.

Ang, A., Bekaert, G., and Wei, M. 2007. Do macro variables, assetmarkets, or surveys forecast inflation better? Journal of MonetaryEconomics, 54, 1163-1212.

Balfoussia, H., and Wickens, M. 2007. Macroeconomic sources of riskin the term structure. Journal of Money, Credit and Banking, 39,205-236.

Bauer and Rudebusch 2015. Monetary Policy Expectations at the ZeroLower Bound. Journal of Money, Credit and Banking, 48, 1439-1465.

Black, F. 1995. Interest rates as options. Journal of Finance 50, 1371-1376.

Bikbov, R., and Chernov, M. 2010. No-arbitrage macroeconomic de-terminants of the yield curve. Journal of Econometrics, 159, 166-182.

Christensen, J., Diebold, F., and Rudebusch, G. 2011. The affinearbitrage-free class of Nelson-Siegel term structure models. Journal ofEconometrics, 164, 4-20.

Christensen, J. ,and Rudebusch, G. 2015. Estimating shadow-rateterm structure models with near-zero yields. Journal of FinancialEconometrics, 13, 226-259.

Clark, T., and West, K. 2007. Approximately normal tests for equalpredictive accuracy in nested models. Journal of Econometrics, 138,291-311.

Dewachter, H., and Lyrio, M. 2006. Macro factors and the termstructure of interest rates. Journal of Money, Credit and Banking, 38,119-140.

Diebold, F., and Mariano, R. 1995. Comparing predictive accuracy.Journal of Business and Economic Statistics, 13, 253-63.

26

Diebold, F., Rudebusch, G., and Aruoba, B. 2006. The macroeconomyand the yield curve: a dynamic latent factor approach. Journal ofEconometrics, 131, 309-338.

Duffee, G. 2002. Term premia and interest rate forecasts in affinemodels. Journal of Finance, 57, 405-443.

Estrella, A., and Hardouvelis, G. 1991. The term structure as a pre-dictor of real economic activity. Journal of Finance, 46, 555-76.

Estrella, A., and Rodrigues, A. and Schich, S. 2002. How stable is thepredictive power of the yield curve? Evidence from Germany and theUnited States. Review of Economics and Statistics, 85, 629-644.

Ghysels, E., Horan, C., and Moench, E. 2012. Forecasting through therear-view mirror: data revisions and bond return predictability. TheReview of Financial Studies, 31(2), 678-714.

Gurkaynak, R., Sack, B., and Wright, J. 2007. The U.S. Treasuryyield curve: 1961 to the present. Journal of Monetary Economics, 54,2291-2304.

Joslin, S., Priebsch M., and Singleton, K. 2014. Risk premiums indynamic term structure models with unspanned macro risks TheJournal of Finance, 69, 11971233.

Kozicki, S., and Tinsley, P.A. 2005. What do you expect? Imperfectpolicy credibility and tests of the expectations hypothesis. Journal ofMonetary Economics, 52, 421-447.

Krippner, L. 2006. A theoretically-consistent version of the Nelsonand Siegel class of yield curve models. Applied Mathematical Finance,13, 39-59.

Krippner, L. 2011. Modifying Gaussian term structure models wheninterest rates are near the zero lower bound. Discussion paper, Centrefor Applied Macroeconomic Analysis 36/2011.

Krippner, L. 2015. Term Structure Modeling at the Zero Lower Bound:A Practitioners Guide. Palgrave-Macmillan.

Ludvigson, S., and Ng, S. 2011. A factor analysis of bond risk premia.In A. Ulah and D. Giles (ed). Handbook of Empirical Economics andFinance, Chapman and Hall, 313-372.

27

Moench, E. 2008. Forecasting the yield curve in a data-rich environ-ment: a no-arbitrage factor-augmented VAR approach. Journal ofEconometrics, 146, 26-43.

Nelson, C. and Siegel, A. 1987. Parsimonious modelling of yield curves.Journal of Business, 473-489.

Stock, J., and Watson, W. 2003. Forecasting output and inflation: therole of asset prices. Journal of Economic Literature, 41, 788-829.

Piazzesi, M. 2005. Bond yields and the Federal Reserve. Journal ofPolitical Economy, 113, 311-344.

Rudebusch, G., and Williams, J. 2009. Forecasting recessions: thepuzzle of the enduring power of the yield curve. Journal of Businessand Economic Statistics, 27, 492-503.

Wheelock, D. and Wohar M. 2009. Can the term spread predict outputgrowth and recessions? a survey of the literature. Review, FederalReserve Bank of St. Louis, issue Sep, 419-440.

Wright, J. 2011. Term premia and inflation uncertainty: empiricalevidence from an international panel dataset. American EconomicReview, 101, 1514-1534.

Wu, C. and Xia, F. 2015. Measuring the macroeconomic impact ofmonetary policy at the zero lower bound. Journal of Money, Credit,and Banking, 48, 253-291.

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