Forecasting the term structures of Treasury and corporate yields …download.xuebalib.com/xuebalib.com.27723.pdf · affine equilibrium models (Cox, Ingersoll, & Ross, 1985 ;Vasicek,

International Journal of Forecasting 27 (2011) 579–591www.elsevier.com/locate/ijforecast

Forecasting the term structures of Treasury and corporate yieldsusing dynamic Nelson-Siegel models

Wei-Choun Yua,∗, Eric Zivotb

a Winona State University, MN, United Statesb University of Washington, Seattle, WA, United States

Abstract

We extend Diebold and Li’s dynamic Nelson-Siegel three-factor model to a broader empirical prospective by including theevaluation of the state space approach and by using nine different ratings for corporate bonds. We find that the dynamic Nelson-Siegel factor AR(1) model outperforms other competitors on the out-of-sample forecast accuracy, especially on the investment-grade bonds for the short-term forecast horizon and on the high-yield bonds for the long-term forecast horizon. The dynamicNelson-Siegel factor state space model, however, becomes appealing on the high-yield bonds in the short-term forecast horizon,where the factor dynamics are more likely time-varying and parameter instability is more probable in the model specification.c⃝ 2010 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

Keywords: Term structures; Treasury yields; Corporate yields; Nelson-Siegel model; Factor model; AR(1); VAR(1); Out-of-sample forecastingevaluations

s

1. Introduction

Forecasting the term structure of interest rates playsa crucial role in portfolio management, household fi-nance decisions, business investment planning, andpolicy formulation. Improved methods for yield curvemodeling and forecasting have recently been gainingmore and more attention. It is well known that theno-arbitrage models (Ho & Lee, 1986; Hull & White,1990) focus only on term-structure cross-sectional

∗ Corresponding author. Tel.: +1 507 457 2982; fax: +1 507 4575697.

E-mail address: [email protected] (W.-C. Yu).

0169-2070/$ - see front matter c⃝ 2010 International Institute of Forecadoi:10.1016/j.ijforecast.2010.04.002

fitting. Although they are the most common, theaffine equilibrium models (Cox, Ingersoll, & Ross,1985; Vasicek, 1977; Dai & Singleton, 2000) onlylook at instantaneous short rates, and thus result inpoor yield curve forecasts (Duffee, 2002). To pro-vide an improved forecasting model for the yieldcurve, Diebold and Li (2006, hereafter DL) extendedthe parsimonious three-factor (exponential compo-nents) yield curve model of Nelson and Siegel (1987)to the dynamic form. Compared to the no-arbitrageand affine equilibrium models, DL’s dynamic Nelson-Siegel factor autoregressive of order 1 (AR(1)) modelproduced superior out-of-sample forecasts of US Trea-sury yields, especially at the 1-year-ahead horizon.

ters. Published by Elsevier B.V. All rights reserved.

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580 W.-C. Yu, E. Zivot / International Journal of Forecasting 27 (2011) 579–591

DL used a simple two-step approach, in which theyfirst estimate three factors, and then model and fore-cast them. Diebold, Rudebusch, and Aruoba (2006,hereafter DRA) proposed a one-step approach thatuses a state space model to perform factor estima-tion, modeling, and forecasting simultaneously. Theyargued that the one-step state space approach providesa unified framework, and should improve the out-of-sample forecasts. However, they did not perform anyout-of-sample forecast evaluations in their study. Inthis paper, we bridge the empirical gap and evaluatethe out-of-sample forecasting performances of one-step and two-step dynamic Nelson-Siegel models, aswell as those of some common competing models. Wefind, surprisingly, that the state space approach doesnot improve the out-of-sample forecasts for Treasuryyields: the simple two-step AR(1) method of DL is stillthe most accurate forecasting model.

Many financial institutions and academies have de-voted substantial resources to the measurement andmanagement of credit (default) risk over the pastdecade. However, less attention has been paid to themodeling and time series forecasting of credit yieldcurves. To partially fill this gap, we use the dynamicNelson-Siegel three-factor model, with both the one-step and two-step approaches, to model and forecastAAA, AA, A+, A, A−, BB+, BB−, B, and B− ratedcorporate bond yields. The different nature of interestrate data relative to various credit rating bonds pro-vides a broader universe in which to evaluate the mod-els’ predictability. We find that Treasury yields andinvestment-grade (AAA, AA, A+) corporate yieldshave similar results: the two-step AR(1) method pro-duces superior and robust out-of-sample forecasts. Wefind, however, that the one-step state space approachbeats the AR(1) approach for the high-yield (specu-lative) bonds (A−, BB+, BB−, B, B−), where themodel parameters are more likely to be time-varying.

It is desirable to know the long-term forecast per-formances of yield curve models for long-term invest-ments and portfolio management, especially for longermaturity bonds. A successful model for short-termforecasting, however, does not necessarily producegood long-term forecasts. DL only provided short-term out-of-sample forecasts (1-, 6-, and 12-month-ahead forecasts). In addition to short-term forecasts,we also evaluate long-term (36- and 60-month-ahead)forecasts of Treasury and corporate yields. We find

that the performances of the models’ short-run andlong-run predictions are different, and that the long-term forecasts of high yield bonds from the two-stepAR(1) model are very robust.

In summary, we make four contributions to theliterature. First, we evaluate and compare the fore-casting performances of the one-step and two-stepdynamic Nelson-Siegel three-factor models. Second,we model and forecast Treasury and corporate bondyields. Third, we evaluate both short-run and long-run forecasts. Fourth, we provide evidence that thelevel factors include the additional credit risk, which isasymmetrically proportional to non-investment gradebonds with a higher leverage ratio.

The rest of the paper is organized as follows. Sec-tion 2 presents the model and the data; Section 3reports the forecasting evaluation; and Section 4 con-cludes.

2. Model and data

2.1. Nelson-Siegel model

Nelson and Siegel (1987) introduced a parsimoni-ous and influential three-factor model for zero couponbond yields,1

yt (τi ) = β1t + β2t

1 − e−λτi

λτi

+ β3t

1 − e−λτi

λτi− e−λτi

, (1)

where yt denotes the yield at time t , and τi is the ma-turity of bond i , which usually ranges from 3 monthsto 30 years. The parameter λ determines the rate ofexponential decay. The three factors are β1t , β2t , andβ3t . The factor loading on β1t is 1, and loads equallyat all maturities. A change in β1t changes all yieldsuniformly, and it is therefore called the level factor.As the value of τi increases, β1t plays a more impor-tant role in forming yields relative to the smaller fac-tor loadings on β2t and β3t . In the limit, yt (∞) = β1t ,and thus β1t is also called the long-term factor. Thefactor loading on β2t is (1 − e−λτi )/λτi , which is a

1 The original Nelson-Siegel model is slightly different from Eq.(1), which is a modified equation from DL. DL explain the reasonsfor this revision.

W.-C. Yu, E. Zivot / International Journal of Forecasting 27 (2011) 579–591 581

function which decays quickly and monotonically tozero as τι increases. It loads short rates more heavilythan long rates, and as a consequence, it changes theslope of the yield curve. Hence, β2t is a short-term fac-tor, which is also called the slope factor.2 The factorloading on β3t is (1 − e−λτi )/λτi − e−λτi , which is afunction starting at zero (not short-term) and decayingto zero (not long-term), with a humped shape in themiddle. It loads medium rates more heavily. Accord-ingly, β3t is the medium-term factor. It is also calledthe curvature factor, because an increase in β3t willincrease the yield curve curvature.

The Nelson-Siegel three factor model can capturevarious different kinds of yield curves, including up-ward sloping, downward sloping, humped and invertedhumped. It is well known that the exponential compo-nent model has competitive in-sample predictions, inspite of its parsimonious structure.

2.2. VAR(1) and AR(1) factor models—two-stepapproach

Despite its success in the cross-sectional interpo-lation of yields across maturities, the Nelson-Siegelmodel is not well suited to out-of-sample time se-ries forecasts, because the factors β1t , β2t and β3t donot stay constant over time. This motivates DL to ex-tend the Nelson-Siegel model to a dynamic form inwhich the factors are time varying. They proposed twobenchmark models for the evolution of the factors: amultivariate vector autoregressive VAR(1) model forβt = (β1t , β2t , β3t )’, and univariate AR(1) models forβi t (i = 1, 2, 3). DL followed a two-step method inwhich they first estimate the three factors β1t , β2t , andβ3t by least squares period by period, assuming a fixedvalue for λ, and then use the estimated factors β1t , β2t ,and β3t to forecast future values of the factors. For ex-ample, the forecasts from the unconstrained VAR(1)model are computed using

βt+h|t = c + Γ βt , (2)

where c and Γ are estimated by regressing βt on anintercept and βt−k . Similarly, the forecasted factorsfrom the univariate AR(1) models are computed using

β j,t+h|t = c j + γ j β j t , j = 1, 2, 3. (3)

2 Some authors define the yield curve slope as yt (∞) − yt (0),which (from Eq. (1)) equals β1t − (β1t + β1t ) = −β2t .

Given the projected factors computed in Eqs. (2) and(3), the predicted term structure of yields is formedusing

yt+h|t (τi ) = β1,t+h|t + β2,t+h|t

1 − e−λτi

λτi

+ β3,t+h|t

1 − e−λτi

λτi− e−λτi

. (4)

We follow this two-step method for forecast hori-zons of h = 1, 6, 12, 36, and 60 months. Follow-ing DRA, we fix λ = 0.077 to simplify the compu-tations.3 We note that a prudent choice of λ may beable to enhance the model’s forecasting ability. Forexample, Yu and Salyards (2009) found that there isa substantial difference in the optimal value of λ be-tween investment-grade and speculative-grade bonds.Koopman, Mallee, and der Wel (in press) allowed λ tobe time-varying by treating it as a fourth factor whichis modeled jointly with the other factors. Because themain focus of this study is on evaluating the one-stepstate space approach relative to the two-step approach,we fix λ, as was done by DL and DRA.

2.3. State space model—one-step approach

DRA argue that the two-step procedure used byDL suffers from the fact that the parameter estimationand signal extraction uncertainty associated with thefirst step are not considered in the second step. Theypropose a linear Gaussian state space approach whichuses a one-step Kalman filter, a recursive procedurefor computing the optimal estimator of the state vec-tor at time t given the information available at time t ,to perform parameter estimation and signal extractionsimultaneously in the dynamic Nelson-Siegel model.They suggest that the one-step approach is better thanthe two-step approach because the simultaneous es-timation of all parameters produces correct inferencevia standard theory.

The measurement (observation) equation of thestate space form of the dynamic Nelson-Siegel model

3 Using λ = 0.077 implies that the loading on the curvature factoris maximized at a maturity of 23.3 months. If λ = 0.061, the valueused by DL, the curvature factor loading reaches its maximum at amaturity of 30 months.


isyt (τ1)

yt (τ2)...

yt (τn)

=

1

1 − e−λτ1

λτ1

1 − e−λτ1

λτ1− e−λτ1

......

...

11 − e−λτn

λτn

1 − e−λτn

λτn− e−λτn

×

β1tβ2tβ3t

+

εt (τ1)

εt (τ2)...

εt (τn)

(5)

or

yt(13×1)

= Z(13×3)

· βt(3×1)

+ εt(13×1)

, (6)

which relates the observed yields to the latent factors(state variables) and measurement errors. The transi-tion equation describes the evolution of the state vari-ables as a first-order Markov process, and is given byβ1t

β2tβ3t

=

φ11 φ12 φ13φ21 φ22 φ23φ31 φ32 φ33

β1t−1β2t−1β3t−1

+

η1tη2tη3t

(7)

or

βt+1(3×1)

= T(3×3)

· βt(3×1)

+ ηt(3×1)

. (8)

To identify the model and simplify the computations,we restrict the transition matrix T to be diagonal, as,theoretically and empirically, the three factors shouldshow little correlation, as was argued by DRA. We as-sume that the measurement and transition disturbancesare Gaussian white noise, and are diagonal and orthog-onal to each other, as is the standard treatment of thestate space model (see Durbin & Koopman, 2001):

εtηt

∼ i id N

[00

,

diag(σ 2ε1, . . . , σ

2ε13) 0

0 diag(σ 2η1, . . . , σ

2η3)

]. (9)

The initial value of β is assumed to be N (u, P). Weuse the mean of the factors estimated from the first stepof the one-step AR(1) approach as the value for u. Wespecify P = cI3, where c is a large number (e.g., 106),to give a diffuse initialization. Because the Kalmanfilter prediction equations produce one-step-ahead

predictions of the state vector, we can compute the out-of-sample forecasts from these equations by extendingthe data set with a set of missing values. For example,a sequence of h missing values at the end of the samplewill produce a set of h-step-ahead forecasts.4

2.4. Data

We use the end-of-month zero-coupon bond ratesof US Treasury bonds, and Standard and Poor’srating AAA, AA, A+, A, A−, BB+, BB−, B, andB− corporate bonds, taken from Bloomberg over theperiod December 1994 to April 2006. The maturitiesof bonds include 1–10, 15, 20, and 30 years. For afurther description of this dataset, see Yu and Salyards(2009).

2.5. Dynamic Nelson-Siegel models—estimation re-sults

For the two-step approach we first use least squaresto estimate the level, slope, and curvature factors pe-riod by period using Eq. (1). These factors are usedas data in the VAR(1) and AR(1) models in the sec-ond step. In the one-step approach, we estimate thestate space model (Eqs. (6) and (8)) by maximum like-lihood. Figs. 1 to 4, respectively, show the time se-ries of estimated factors from the two approaches, aswell as their common empirical proxies for the Trea-sury, A+ corporate, BB+ corporate, and B− corporateyields. The empirical counterpart of the level factor isthe average of the short-term (1 year), medium-term(2 year), and long-term (10 year) yields: (y1 + y2 +

y10)/3; the empirical counterpart of the slope factoris y1 − y10; and the empirical counterpart of the cur-vature factor is 2 × y2 − y1 − y10.

Dewachter and Lyrio (2006) provide a macroeco-nomic interpretation of the latent factors of the Trea-sury yields: the level factor expresses the long-terminflation expectation; the slope factor represents busi-ness cycle conditions; and the curvature factor repre-sents a clear independent monetary policy factor. Forthe slope factor series that we obtained from the AR(1)

4 We estimate the linear Gaussian state space model usingS-PLUS 7.0 with the SsfPack functions implemented in theS+Finmetrics 2.0 module, as described by Zivot, Wang, andKoopman (2004) and Zivot and Wang (2006).


87

65

43

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Level factor via AR(1) and state space approach and empirical counterparts

Slope factor via AR(1) and state space approach and empirical counterparts

Curvature factor via AR(1) and state space approach and empirical counterparts

-10

12

-2-3

-4-5

-6

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

-18

-16

-14

-12

-10

-8-6

-4-2

02

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Fig. 1. Treasury yields three factors: level, slope, curvature.

and state space approaches, the Treasury and cor-porate bonds have similar dynamics, capturing real-time business cycle conditions. For the level factorseries, Treasury and A+ corporate also have similardynamics, which reconciles the real-time long-terminflation expectation. However, the level factor departs

87

65

43


1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006


-10

1-2

-3-4

-5

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006


-12

-10

-8-6

-4-2

0

Fig. 2. A+ corporate yields three factors: level, slope, curvature.

from the inflation expectation dynamics under the non-investment corporate bonds (BB+, and B−). Becauseof the surge of level factors between 1998 and 2003for BB+ and B− since the 1998 Russian default, weargue that the level factors include the additional credit


109

87

65


1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006


-10

1-2

-3-4

-5

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006


0-2

-4-6

-8-1

0-1

2-1

4-1

6

Fig. 3. BB+ corporate yields three factors: level, slope, curvature.

risk, which is asymmetrically proportional to non-investment grade bonds with a higher leverage ratio.The context is consistent with the findings of Yu andSalyards (2009), who argue that three factors are suf-ficient to explain the variation in all corporate yields


1618

1412

108

6

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006


-20

-4-6

-8-1

0


105

0-5

-10

-15

-20

-25

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Fig. 4. B− corporate yields three factors: level, slope, curvature.

in different ratings. Regarding the curvature factor se-ries, Treasury, A+, and BB+ have similar dynamics,responding to independent monetary policy. The dif-ferent pattern of the curvature factor of B− suggeststhat highly risky bonds are not sensitive to monetarypolicy.


3. Forecast evaluation

In this section, we assess the out-of-sample predic-tive accuracies of various models. We consider fore-casts from models that fit to yields-only, as well as onemodel that utilizes additional macroeconomic infor-mation.

3.1. Yields-only models

In addition to the one-step and two-step dynamicNelson-Siegel models, we also consider a subset ofcompeting forecasting models used by DL. The yields-only models we consider are: Model 1: VAR(1) NSfactor; Model 2: AR(1) NS factor; Model 3: statespace NS factor; Model 4: random walk; and Model 5:VAR(1) on yield levels. The random walk model usesthe current value as the forecast at all horizons. TheVAR(1) on yield levels presumes that there are neitherintegrated nor co-integrated series in the data.

3.2. The yields-macro model

In addition to the yields-only models, we also con-sider Model 6 — a yield curve model with ma-croeconomic variables. It is well known that macrovariables are related to the dynamics of yields curves,and their inclusion in yields-only models shouldimprove the forecasts. For example, Ludvigson andNg (2009) find that real and inflation macro variableshave predictive power for future government bondyields. Ang and Piazzesi (2003) find that the 1-month-ahead out-of-sample VAR forecast performance ofTreasury yields is improved when no-arbitrage rest-rictions are imposed and macro variables are inco-rporated. Cochrane and Piazzesi (2005) study thetime-varying risk premia of government bonds andfind an additional return-forecasting factor. DRAalso incorporate three macroeconomic variables(manufacturing capacity utilization, the federal fundsrate, and annual price inflation) into their state spacemodel, and analyze the impulse response functionsand variance decompositions of the yields-macromodel. Instead of analyzing the in-sample interactionsbetween yields and these macroeconomic variables, asDRA did, we focus on the out-of-sample forecast gainfrom including the macroeconomic variables.

Our Model 6 is a two-step model that integratesthree macro variables into Model 1 (VAR(1) NS fac-tor): the annual personal consumption expenditure in-flation rate (INFL), the log of manufacturing capacityutilization (CU), and the effective federal funds rate(FFR).5 The first step is the same as in Model 1. Thesecond step of Model 6 uses

β1t+h|tβ2t+h|tβ3t+h|t

CUt+h|tF F Rt+h|t

I N F L t+h|t

=

c1c2c3c4c5c6

+

γ11 γ12 γ13 γ14 γ15 γ16γ21 γ22 γ23 γ24 γ25 γ26γ31 γ32 γ33 γ34 γ35 γ36γ41 γ42 γ43 γ44 γ45 γ46γ51 γ52 γ53 γ54 γ55 γ56γ61 γ62 γ63 γ64 γ65 γ66

β1tβ2tβ3t

CUtF F Rt

I N F L t

(10)

to forecast future factors. We do not incorporate macrovariables into Model 3 (state space NS factor), becausewe assume that the transition matrix T is diagonal,and thus there will be no information gain by includ-ing macro variables. We also do not incorporate macrovariables into Model 5 (VAR(1) on yield levels), fortwo reasons. First, we believe that macro variables willnot improve the prediction accuracy of an unrestrictedVAR(1) on yield levels, as was concluded by Ang andPiazzesi (2003). Second, Model 5 already has 13 vari-ables for different maturities; therefore, adding threemore variables will further reduce the degrees of free-dom of an already high-dimensional VAR.

3.3. Out-of-sample forecasts

For the one-step-ahead out-of-sample forecasts, weuse data from 1994:12 to 2004:4 as the in-sampleperiod, and from 2004:5 to 2006:4 (24 predictions) asthe out-of-sample period. We use a rolling window toestimate the parameters of each model based on themost up-to-date information available at the time ofthe forecast. For example, we use data from 1994:12

5 Macro data were obtained online from the Federal ReserveEconomic Data (FRED) database.


to 2004:4 to forecast 2004:5, data from 1995:1 to2004:5 to forecast 2004:6, and so on. For longerforecast horizons, we have to decrease the in-samplesize in order to keep 24 out-of-sample predictions.For instance, to compute the 6-month-ahead forecast,we use data from 1994:12 to 2003:11, and our firstprediction will be for 2004:5.

To evaluate the out-of-sample forecasting perfor-mances, we use the root mean squared forecast errors(RMSE),−

(yt+h − yModelt+h|t )2/24

1/2, (11)

where yt+h is the realized yield and yModelt+h|t is the

prediction made by the models. In Table 1, we showthe evaluation results for 1-, 3-, 5-, 10-, and 30-year maturities for Treasury bonds. The bold numbersin the bottom row of each model represent theiraverage RMSEs across the 1-, 3-, 5-, 10-, and 30-year maturities. To simplify the evaluation output forthe nine corporate bonds in Tables 2 and 3, we onlypresent the average RMSEs of the 1-, 3-, 5-, 10-, and30-year maturities. For the last two columns in eachtable, we also present the average short-term RMSEs:(1-month-ahead + 6-month-ahead + 12-month-aheadRMSEs)/3; and the average long-term RMSEs: (36-month-ahead + 60-month-ahead RMSEs)/2.

3.4. Forecast results

First, consider the results for Treasury yields in Ta-ble 1. Model 2 (NS factor AR(1) model) has the low-est average short-term RMSE, 0.485, and the secondlowest average long-term RMSE, 1.954. Model 4 (ran-dom walk) has the lowest long-term RMSE, 1.663. Ta-ble 2 illustrates similar results for AAA and A+ cor-porate yields. Model 2 produces the best short-termforecasts while Model 4 produces the best long-termforecasts. For AA corporate yields, Model 2 is thebest for both short- and long-term forecasts. For theremaining corporate yields, the results are mixed, es-pecially for short-term forecasts (1-, 6-, and 12-month-ahead forecasts). For the short-term forecast horizon,Model 5 (VAR(1) on yields level) is best for the A(RMSE: 0.569) and A− (RMSE: 0.513) corporatebonds. Table 3 shows that Model 4 is the best for theBB− (RMSE: 0.638) and B yields (RMSE: 0.560).Model 3 (NS factor state space approach) is supe-rior for BB+ (RMSE: 0.556) and B− (RMSE: 0.739).

For the long-term forecast horizon (36 and 60 monthsahead), Model 2 is the best for the BB+ (RMSE:2.408), BB− (RMSE: 2.42), B (RMSE: 3.023), andB− (RMSE: 3.375) yields.6

3.5. Short-term forecasts for the one-step and two-step approaches

Our results show that a model which has a good in-sample fit may not necessarily produce the best out-of-sample forecasts. In addition, when the credit risks(spreads) increase, the NS factor AR(1) short-termforecast performance deteriorates and the state spacemodel performance improves. The average short-termRMSE increases from 0.485 for Treasury yields to1.056 for B− yields. Although Model 3 is alwayssuboptimal for forecasting all types of investment-grade bonds, its short-term forecasts are very stablefor non-investment grade bonds. Its lowest RMSEs arefor BB+ (0.556) and B− (0.739) yields, while it is thesecond lowest for the A− (0.562), BB− (0.562), and B(0.602) yields. In general, Model 3 provides the mostreliable out-of-sample forecasts for high-risk bonds.

A possible explanation for why the state spacemodel predicts better on high-risk yields while theAR(1) model predicts better on investment-gradeyields is as follows. The state space model, which wasdesigned for time-varying parameters, is most robustin the environment of parameter instability, nonlinear-ity, and non-normality for which speculative bonds aremost likely to exist. As Metz and Cantor (2006) pointout, although the distribution of bond ratings is fairlystable over time, the relationship between the ratingsand their corresponding credit factors (such as interestcoverage) is not stable over time, particularly for spec-ulative bonds. In addition, Cantor and Packer (1996)argue that rating announcements have immediate ef-fects on market pricings for speculative bonds, leadingto more unstable dynamics. In the context of the termstructure theory, Dai and Singleton (2003) summarizethat one would use the diffusion process for modeling

6 Finally, it is worth noting that Model 1 (NS factor VAR(1)) isinferior to Model 2 (NS factor AR(1)) for all kinds of bonds becausethe three factors (level, slope, curvature) constructed by exponentialcomponents are supposed to be uncorrelated. Consequently, thisresult supports our assumption regarding Model 3, in which thetransition matrix T is diagonal.


Table 1Treasury yields out-of-sample forecast evaluation: Root Mean Squared Errors (RMSEs).

Maturities 1-month-ahead

6-month-ahead

12-month-ahead

36-month-ahead

60-month-ahead

Average1, 6, 12 m

Average36, 60 m

Model 1 1 year 0.230 0.904 1.903 11.977 2.526 1.012 7.251NS factors 3 year 0.352 0.814 1.442 6.852 1.946 0.869 4.399VAR(1) 5 year 0.278 0.381 0.678 4.086 1.835 0.446 2.960

10 year 0.248 0.331 0.408 2.226 1.417 0.329 1.82230 year 0.318 0.649 0.931 2.653 1.413 0.633 2.033Average 0.285 0.616 1.072 5.559 1.827 0.658 3.693

Model 2 1 year 0.217 0.519 0.786 4.062 2.524 0.507 3.293NS factors 3 year 0.292 0.400 0.448 2.207 2.010 0.380 2.109AR(1) 5 year 0.291 0.393 0.437 1.554 1.908 0.374 1.731


Model 3 1 year 0.695 0.481 1.108 6.622 1.792 0.761 4.207NS factors 3 year 0.417 0.975 1.726 6.956 1.223 1.039 4.090State space 5 year 0.296 0.625 1.180 4.992 0.953 0.700 2.972


Model 4 1 year 0.263 0.887 1.544 2.170 2.761 0.898 2.465Random walk 3 year 0.281 0.633 1.087 1.561 2.294 0.667 1.927

5 year 0.282 0.484 0.708 1.128 2.182 0.492 1.65510 year 0.248 0.356 0.506 0.760 1.662 0.370 1.21130 year 0.207 0.477 0.755 0.900 1.211 0.480 1.055Average 0.256 0.568 0.920 1.303 2.022 0.581 1.663

Model 5 1 year 0.275 0.949 1.880 38.387 2.880 1.035 20.633Yields level 3 year 0.331 0.709 1.247 29.743 2.442 0.762 16.093VAR(1) 5 year 0.327 0.499 0.793 26.793 2.556 0.540 14.675


Model 6 1 year 0.216 0.541 0.632 3.735 3.048 0.463 3.391NS factors + 3 year 0.289 0.545 0.685 3.084 2.479 0.506 2.781Macro variables 5 year 0.313 0.628 0.888 2.683 2.198 0.610 2.440VAR(1) 10 year 0.273 0.645 0.970 1.980 1.743 0.630 1.862

30 year 0.330 0.804 1.289 1.813 1.790 0.808 1.801Average 0.284 0.632 0.893 2.659 2.251 0.603 2.455

1. The table reports the out-of-sample forecast evaluation based on the root mean squared error (RMSE).2. The in-sample period is 1994:12–2004:4, and the out-of-sample period is 2005:5–2006:4.3. RMSE = [

∑(yt+h − yModel

t+h|t )2/24]1/2, where yt+h is the realized data and yModel

t+h|t is the predicted data.4. The table evaluates the 1-, 6-, 12-, 36-, and 60-month-ahead forecasts.5. The average RMSE for 1, 6, and 12 m is the mean of the (1 m + 6 m + 12 m) RMSEs, while the average RMSE for 36, 60 m is the mean ofthe (36 m + 60 m) RMSEs.

default-free bond yields, but the jump-diffusion pro-cess for modeling defaultable bond yields. For exam-ple, Fig. 4 shows that the level factor of the B− yieldexhibits a steep structural shift (jump) from 18 to 10in early 2003. It would therefore follow to conclude

that the state space model is more capable of a jump-diffusion data generating process (DGP), while theAR(1) is superior in estimating a diffusion DGP.

Moreover, Yu and Salyards (2009) use the principalcomponent method and find a dichotomy of factor


Table 2Investment-grade corporate yields out-of-sample forecast evaluation: Root Mean Squared Errors (RMSEs).

Bondrating

Model 1-month-ahead

6-month-ahead

12-month-ahead

36-month-ahead

60-month-ahead

Average1, 6, 12 m

Average 36,60 m

AAA

Model 1 0.567 0.879 1.233 4.209 2.105 0.893 3.157Model 2 0.271 0.568 0.761 1.878 1.960 0.533 1.919Model 3 0.230 0.627 1.117 3.688 1.351 0.658 2.520Model 4 0.232 0.576 0.948 1.420 2.392 0.585 1.906Model 5 0.247 0.525 0.960 28.323 1.935 0.577 15.129Model 6 0.294 0.672 0.951 2.744 2.910 0.639 2.827

AA


A+


A


A−


1. See the notes to Table 1. Model 1: NS factors VAR(1); Model 2: NS factors AR(1); Model 3: NS factors state space; Model 4: Random walk;Model 5: Yields level VAR(1); Model 6: NS factors + macroeconomic variables VAR(1).2. Unlike Table 1, to simplify the output, the RMSE in each cell represents the average of the 1, 3, 5, 10, and 30 year maturities of eachinvestment-grade corporate bond.

loadings between investment-grade and speculative-grade yields. This finding supports multiple optimalsolutions for λ across different bond ratings. If themodel was purposely constrained (e.g., for simplicitywe fix λ in this study), then the state space approachwill mitigate the loss of inferior model selections.

In general, we find that the risky corporate bondsare not much harder to predict than Treasury bonds ifwe choose an appropriate model. Model 2 is preferred

to Model 3 for investment-grade bonds. However,Model 3 is robust for non-investment-grade bonds atshort-term forecast horizons.

3.6. Long-term forecasts from the one-step and two-step approaches

It is surprising to see that Model 2 is superiorfor the long-term forecasting of BB+, BB−, B, and


Table 3Non-investment-grade corporate yields out-of-sample forecast evaluation: Root Mean Squared Errors (RMSEs).

RMSE 1-month-ahead

6-month-ahead

12-month-ahead

36-month-ahead

60-month-ahead

Average 1, 6,12 m

Average 36,60 m

BB+

Model 1 0.287 0.572 0.890 2.320 27.319 0.583 14.820Model 2 0.320 0.876 1.465 2.432 2.385 0.887 2.408Model 3 0.286 0.567 0.816 2.397 3.094 0.556 2.746Model 4 0.249 0.589 0.910 2.136 3.298 0.583 2.717Model 5 0.262 0.600 0.922 2.966 13899 0.594 6950.9Model 6 0.311 0.727 0.820 5.795 24.045 0.619 14.920

BB−


B


B−


1. See the notes to Tables 1 and 2.

B− high-yield bonds. Although Model 2 has thelowest RMSE for AA yields among investment-gradeyields, it is the second best model for Treasury,AAA, and A+ yields for long-term forecasts. Incontrast to the reliable forecasting ability of Model3 at the short-term horizon, Model 2 provides morereliable long-term forecasts. A possible reason for thisdichotomy is that the long-term forecasts of yieldsweigh more in the steady state equilibrium levelthan short-term forecasts. The simple AR(1) modelassumes that time series are stationary, and thereforelong-term forecasts revert to long-run levels. On theother hand, the state space approach is based onthe unified structural analysis and recursive one-step-ahead in-sample forecast updates from the Kalmanfilter. It is supposed to forecast well out-of-sample inthe short-run, especially for complex data-generatingstructures, such as high-yield corporate bonds. Thesignal extracted from updating data via the state

space approach is less informative for the long-termhorizons.

3.7. Forecasts from the yields-macro model

We find that introducing macroeconomic variablesinto the yields level VAR improves some yield curveforecasts. That is, Model 6 outperforms Model 1 onthe Treasury yields and many of the corporate yields.For short-term forecasts, Model 6’s RMSEs are lowerthan Model 1’s on Treasury (0.603 vs. 0.658), AAA(0.639 vs. 0.893), A+ (0.621 vs. 0.898), A− (0.567 vs.0.892), BB− (0.652 vs. 1.026), B (0.629 vs. 1.046),and B− (0.794 vs. 1.276) yields. Model 6’s RMSEsare similar to Model 1’s on AA (0.628 vs. 0.612) and A(0.616 vs. 0.615). Finally, Model 6’s RMSE is higherthan Model 1’s on BB+ only (0.619 vs. 0.583). Forlong-term forecasts, the benefits of using additionalmacroeconomic variables are not as clear cut. Model6’s RMSEs are lower than Model 1’s on Treasury


(2.455 vs. 3.693), AAA (2.827 vs. 3.157), A− (2.848vs. 4.127), and BB− (19.539 vs. 105.3) yields, buthigher on A+ (3.199 vs. 2.124), A (3.033 vs. 2.572),B (11.519 vs. 7.721), and B− (9.611 vs. 4.170) yields.This result is consistent with the findings of Dewachterand Lyrio (2006), who provide a macroeconomicinterpretation for the latent factors of yield curves.They argue that the level factor represents the long-run inflation expectation of agents. In other words,in the context of long-term out-of-sample forecasting,macroeconomic variables have been incorporated intothe latent factors well, especially the level factor.Consequently, there will be little information gain andforecast improvement from including macroeconomicvariables.

3.8. Forecast improvements

Some of the long-term forecast errors, especiallyfor 36 months ahead, are extraordinarily large. Forexample, in Table 1, Model 1’s 1- and 3-year RMSEsat the 36-month-ahead horizon are 11.9777 and 6.852,respectively. There are three possible reasons for theseextreme forecasts: (1) the model parameters exhibitstructural breaks during the out-of-sample periods; (2)there are some outliers in the yield data; or (3) thesample size is small.

Stock and Watson (1999) point out that “crazy”forecasts would be noticed and adjusted by humanjudgment. To mimic a forecast adjustment proce-dure, we use an interpolation method to replace ex-treme forecasts (defined as values over some absolutenumber for 36-month-ahead forecasts). Using such amethod, we can reduce the 36-month-ahead RMSEsof Model 1 substantially. For the 1-, 3-, 5-, 10-, and30-year bonds, the RMSEs drop from 11.977, 6.852,4.086, 2.226, and 2.653 to 2.148, 1.691, 1.685, 1.470,and 1.655, respectively.

The interpolation method is ad hoc and requireshuman intervention. As an alternative, we consider apurely statistical approach based on robust regressionmethods to detect and adjust for potential outliersand level shifts.7 Using such a method, the average36-month-ahead RMSE of Model 1 for AAA bonds,which was 4.209 using non-robust methods, declinesto 2.143 using the robust method. Although the robust

7 For details, see Zivot and Wang (2006, Chapter 17).

method reduced the RMSEs of these forecasts, itincreased the RMSEs of the shorter-term forecasts.For example, the 1-month-ahead RMSE rises from0.567 to 1.068. This increase occurs because therobust method modifies some data information as theoutliers and level shifts are adjusted, which is usefulfor short-term forecasts. The ad hoc interpolation androbust methods can be applied to Models 1, 2, 3,5, and 6, and were found to beat the random walkmodel, which produces the best long-term forecastsof Treasury (RMSE: 1.663), AAA (RMSE: 1.906),and A+ (RMSE: 1.962) yields.8 Even with the use ofthese methods, the conclusions we reach regarding themodels’ competitiveness remain the same.

4. Conclusions

We provide a comprehensive short- and long-termforecasting evaluation of the two-step and one-stepapproaches of Diebold and Li (2006) and Dieboldet al. (2006) using Treasury yields and nine differentratings of corporate bonds. We find that the simpletwo-step dynamic Nelson-Siegel factor AR(1) modelsurvives a long list of robustness checks on the out-of-sample forecast accuracy, especially for investment-grade bonds at short-term horizons and for high yieldbonds at long-term horizons. The one-step dynamicNelson-Siegel factor state space model has the bestperformance for high yield bonds at short-termhorizons, in which credit risks have more influenceon the yields dynamics, especially in the level factor,and model parameter instability is more likely to exist.In short, the state space model is more capable ofestimating a jump-diffusion DGP of defaultable bondyields, while the AR(1) is superior for estimating adiffusion DGP of default-free bond yields. We alsoshow that forecasts from the Nelson-Siegel factorVAR(1) model can be improved by incorporatingmacroeconomic variables. Nevertheless, this yields-macro model remains suboptimal compared with theparsimonious NS factors AR(1) model for investment-grade yields in the short run.

Our findings suggest that for real-time forecasts,the multivariate state space model computed by the

8 If the random walk model dominates, this implies that the otherforecasting methods are useless.


Kalman filter is not necessarily better than the simpleunivariate AR(1) model computed by least squares.The preferred model depends on the nature of the dataand the forecast horizons. In the context of the Kalmanfilter, which updates the mean and covariance matrixof the conditional distribution of the state vector asnew observations become available, the state spacemodel can predict more accurately if the data’s impliedstate or hyperparameter is more likely to be time-varying and unstable.

Acknowledgements

We would like to thank Frank Russell InvestmentCompany for providing the data. We would also like tothank the editor, two anonymous referees, Tom Good-win, Steve Murray, Patrick Rowland and MichaelWenz for their helpful comments and discussions. Weare grateful to Christina Flaherty for providing assis-tance.

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Wei-Choun Yu is an Assistant Professor in the Department ofEconomics at Winona State University. He also serves on the boardof directors at the Minnesota Economic Association. He receivedhis Ph.D. in economics at the University of Washington in 2006.Yu’s research interests are in time series econometrics, finance andmacroeconomics, and in particular linear, nonlinear, and volatilitymodeling and forecasting. He has publications in various journalsin the fields of econometrics and finance, including the Journal ofInternational Money and Finance, Journal of Forecasting, AppliedEconomics Letters, Global Economic Review and Economic Affairs.

Eric Zivot is a Professor and Gary Waterman DistinguishedScholar in the Department of Economics, an Adjunct Professor ofStatistics, and an Adjunct Professor of Finance at the University ofWashington. He regularly teaches courses on econometric theory,financial econometrics and time series econometrics, and is arecipient of the Henry T. Buechel Award for Outstanding Teaching.He is an associate editor of Studies in Nonlinear Dynamics andEconometrics. He has published papers in the leading econometricsjournals, including Econometrica, Econometric Theory, Journal ofBusiness and Economic Statistics, Journal of Econometrics, andReview of Economics and Statistics.

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