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Journal ArticleDOI

The Nobel Memorial Prize for Robert F. Engle

AbstractEngle's footsteps range widely. His major contributions include early work on band-spectral regression, development and unification of the theory of model specification tests (particularly Lagrange multiplier tests), clarification of the meaning of econometric exogeneity and its relationship to causality, and his later stunningly influential work on common trend modeling (cointegration) and volatility modeling (ARCH, short for AutoRegressive Conditional Heteroskedasticity). More generally, Engle's cumulative work is a fine example of best-practice applied time-series econometrics: he identifies important dynamic economic phenomena, formulates precise and interesting questions about those phenomena, constructs sophisticated yet simple econometric models for measurement and testing, and consistently obtains results of widespread substantive interest in the scientific, policy, and financial communities.

Summary (2 min read)

1. Introduction

  • And the authors use neither the no-arbitrage approach nor the equilibrium approach.
  • Instead, the authors use the Nelson-Siegel (1987) exponential components framework to distill the entire yield curve, period-by-period, into a threedimensional parameter that evolves dynamically.

2. Modeling and Forecasting the Term Structure I: Methods

  • Here the authors introduce the framework that they use for fitting and forecasting the yield curve.
  • The authors argue that the well-known Nelson-Siegel (1987) curve is well-suited to their ultimate forecasting purposes, and they introduce a novel twist of interpretation, showing that the three coefficients in the Nelson-Siegel curve may be interpreted as latent level, slope and curvature factors.
  • The authors also argue that the nature of the factors and factor loadings implicit in the Nelson-Siegel model facilitate consistency with various empirical properties of the yield curve that have been cataloged over the years.
  • Finally, motivated by their interpretation of the Nelson-Siegel model as a three-factor model of level, slope and curvature, the authors contrast it to various multi-factor models that have appeared in the literature.

Constructing "Raw" Yields

  • Let us first fix ideas and establish notation by introducing three key theoretical constructs and the relationships among them: the discount curve, the forward curve, and the yield curve.
  • A second discount-curve approach to yield construction is due to Vasicek and Fong (1982) , who fit exponential splines to the discount curve, using a negative transformation of maturity instead of maturity itself, which ensures that the forward rates and zero-coupon yields converge to a fixed limit as maturity increases.
  • In their approach, the unit loading on the first factor is imposed from the beginning, which potentially enables us to estimate the other factors more efficiently.
  • The long-term factor , for example, governs the yield curve level.
  • In closing this sub-section, it is worth noting that what the authors have called the "Nelson-Siegel curve" is actually a different factorization than the one originally advocated by Nelson and Siegel (1987) , who used .

The Data

  • The authors use end-of-month price quotes (bid-ask average) for U.S. Treasuries, from January 1985 through December 2000, taken from the CRSP government bonds files.
  • Fama-Bliss forward rates into unsmoothed Fama-Bliss zero yields.
  • The variation in slope and curvature is less strong, but nevertheless apparent.

Fitting Yield Curves

  • As discussed above, the authors fit the yield curve using the three-factor model, 9 Other weightings and loss functions have been explored by Bliss (1997b) , Soderlind and Svensson (1997), and Bates (1999) .
  • In Figure 5 the authors dig deeper by plotting the raw yield curve and the threefactor fitted yield curve for some selected dates.
  • As noted in Bliss (1997b) , regardless of the term structure estimation method used, there is a persistent discrepancy between actual bond prices and prices estimated from term structure models.
  • The authors include the VAR forecasts for completeness, although one might expect them to be inferior to the AR forecasts for at least two reasons.
  • In Figure 8 (right column) the authors provide some evidence on the goodness of fit of the AR(1) models fit to the estimated level, slope and curvature factors, showing residual autocorrelation functions.

Out-of-Sample Forecasting Performance of the Three-Factor Model

  • A good approximation to yield-curve dynamics should not only fit well in-sample, but also forecast well out-of-sample.
  • The authors model's 6-monthahead forecasting results, reported in Table 5 , are noticeably improved, and their model's 12-month-ahead forecasting results, reported in Table 6 , are much improved.
  • The strong yield curve forecastability at the 12-month-ahead horizon is, for example, very attractive from the vantage point of active bond trading and the vantage point of credit portfolio risk management.
  • A 17 We note, however, that their enthusiasm must be tempered by the fact that their in-sample and out-of-sample periods are not identical to Duffee's, so definitive comparisons can not be made.the authors.the authors.
  • 17 Finally, the authors note that although their approach is closely related to direct principal components regression, neither their approach nor their results are identical.

Notes to Table 4:

  • The authors present the results of out-of-sample 1-month-ahead forecasting using eight models, as described in detail in the text.
  • The authors estimate all models recursively from 1985:1 to the time that the forecast is made, beginning in 1994:1 and extending through 2000:12.

Notes to Table 5:

  • The authors present the results of out-of-sample 6-month-ahead forecasting using eight models, as described in detail in the text.
  • The authors estimate all models recursively from 1985:1 to the time that the forecast is made, beginning in 1994:1 and extending through 2000:12.

Notes to Table 6:

  • The authors present the results of out-of-sample 12-month-ahead forecasting using twelve models, as described in detail in the text.
  • The authors estimate all models recursively from 1985:1 to the time that the forecast is made, beginning in 1994:1 and extending through 2000:12.

Notes to Table 7:

  • The authors present Diebold-Mariano forecast accuracy comparison tests of their three-factor model forecasts (using univariate AR(1) factor dynamics) against those of the Random Walk model (RW) and the Fama-Bliss forward rate regression model (FB).
  • The null hypothesis is that the two forecasts have the same mean squared error.
  • - Notes to Figure 7 : We define the level as the 10-year yield, the slope as the difference between the 10-year and 3-month yields, and the curvature as the twice the 2-year yield minus the sum of the 3-month and 10year yields.the authors.the authors.

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Content maybe subject to copyright    Report

No. 2004/09
Forecasting the Term Structure
of Government Bond Yields
Francis X. Diebold and Canlin Li

Center for Financial Studies
The Center for Financial Studies is a nonprofit research organization, supported by an
association of more than 120 banks, insurance companies, industrial corporations and
public institutions. Established in 1968 and closely affiliated with the University of
Frankfurt, it provides a strong link between the financial community and academia.
The CFS Working Paper Series presents the result of scientific research on selected top-
ics in the field of money, banking and finance. The authors were either participants in
the Center´s Research Fellow Program or members of one of the Center´s Research Pro-
jects.
If you would like to know more about the Center for Financial Studies, please let us
know of your interest.
Prof. Dr. Jan Pieter Krahnen Prof. Volker Wieland, Ph.D.

Acknowledgments: The National Science Foundation, the Wharton Financial Institutions Center, and the
Guggenheim Foundation provided research support. For helpful comments we are grateful to the Editor (Arnold
Zellner), the Associate Editor, and three referees, as well as Dave Backus, Rob Bliss, Michael Brandt, Todd
Clark, Qiang Dai, Ron Gallant, Mike Gibbons, David Marshall, Monika Piazzesi, Eric Renault, Glenn
Rudebusch, Til Schuermann, and Stan Zin, and seminar participants at Geneva, Georgetown, Wharton, the
European Central Bank, and the National Bureau of Economic Research. We, however, bear full responsibility
for all remaining flaws.
Copyright © 2000-2003 F.X. Diebold and C. Li. This paper is available on the World Wide Web at
http://www.ssc.upenn.edu/~diebold and may be freely reproduced for educational and research purposes, so long
as it is not altered, this copyright notice is reproduced with it, and it is not sold for profit.
University of Pennsylvania, and NBER, University of Pennsylvania and NBER, fdiebold@wharton.upenn.edu
University of California, Riverside
CFS Working Paper No. 2004/09
Forecasting the Term Structure
of Government Bond Yields
*
Francis X. Diebold
and Canlin Li
This Draft: October 14, 2003
Abstract:
Despite powerful advances in yield curve modeling in the last twenty years, comparatively
little attention has been paid to the key practical problem of forecasting the yield curve. In this
paper we do so. We use neither the no-arbitrage approach, which focuses on accurately fitting
the cross section of interest rates at any given time but neglects time-series dynamics, nor the
equilibrium approach, which focuses on time-series dynamics (primarily those of the
instantaneous rate) but pays comparatively little attention to fitting the entire cross section at
any given time and has been shown to forecast poorly. Instead, we use variations on the
Nelson-Siegel exponential components framework to model the entire yield curve, period-by-
period, as a three-dimensional parameter evolving dynamically. We show that the three time-
varying parameters may be interpreted as factors corresponding to level, slope and curvature,
and that they may be estimated with high efficiency. We propose and estimate autoregressive
models for the factors, and we show that our models are consistent with a variety of stylized
facts regarding the yield curve. We use our models to produce term-structure forecasts at both
short and long horizons, with encouraging results. In particular, our forecasts appear much
more accurate at long horizons than various standard benchmark forecasts.
JEL Code: G1, E4, C5
Key Words: Term structure, yield curve, factor model, Nelson-Siegel curve

1
The empirical literature that models yields as a cointegrated system, typically with one
underlying stochastic trend (the short rate) and stationary spreads relative to the short rate, is similar in
spirit. See Diebold and Sharpe (1990), Hall, Anderson, and Granger (1992), Shea (1992), Swanson and
White (1995), and Pagan, Hall and Martin (1996).
2
For comparative discussion of point and density forecasting, see Diebold, Gunther and Tay
(1998) and Diebold, Hahn and Tay (1999).
1
1. Introduction
The last twenty-five years have produced major advances in theoretical models of the term
structure as well as their econometric estimation. Two popular approaches to term structure modeling
are no-arbitrage models and equilibrium models. The no-arbitrage tradition focuses on perfectly fitting
the term structure at a point in time to ensure that no arbitrage possibilities exist, which is important for
pricing derivatives. The equilibrium tradition focuses on modeling the dynamics of the instantaneous
rate, typically using affine models, after which yields at other maturities can be derived under various
assumptions about the risk premium.
1
Prominent contributions in the no-arbitrage vein include Hull and
White (1990) and Heath, Jarrow and Morton (1992), and prominent contributions in the affine
equilibrium tradition include Vasicek (1977), Cox, Ingersoll and Ross (1985), and Duffie and Kan
(1996).
Interest rate point forecasting is crucial for bond portfolio management, and interest rate density
forecasting is important for both derivatives pricing and risk management.
2
Hence one wonders what the
modern models have to say about interest rate forecasting. It turns out that, despite the impressive
theoretical advances in the financial economics of the yield curve, surprisingly little attention has been
paid to the key practical problem of yield curve forecasting. The arbitrage-free term structure literature
has little to say about dynamics or forecasting, as it is concerned primarily with fitting the term structure
at a point in time. The affine equilibrium term structure literature is concerned with dynamics driven by
the short rate, and so is potentially linked to forecasting, but most papers in that tradition, such as de Jong
(2000) and Dai and Singleton (2000), focus only on in-sample fit as opposed to out-of-sample
forecasting. Moreover, those that do focus on out-of-sample forecasting, notably Duffee (2002),
conclude that the models forecast poorly.
In this paper we take an explicitly out-of-sample forecasting perspective, and we use neither the
no-arbitrage approach nor the equilibrium approach. Instead, we use the Nelson-Siegel (1987)
exponential components framework to distill the entire yield curve, period-by-period, into a three-
dimensional parameter that evolves dynamically. We show that the three time-varying parameters may

3
Classic unrestricted factor analyses include Litterman and Scheinkman (1991) and Knez,
Litterman and Scheinkman (1994).
2
be interpreted as factors. Unlike factor analysis, however, in which one estimates both the unobserved
factors and the factor loadings, the Nelson-Siegel framework imposes structure on the factor loadings.
3
Doing so not only facilitates highly precise estimation of the factors, but, as we show, it also lets us
interpret the factors as level, slope and curvature. We propose and estimate autoregressive models for
the factors, and then we forecast the yield curve by forecasting the factors. Our results are encouraging;
in particular, our models produce one-year-ahead forecasts that are noticeably more accurate than
standard benchmarks.
Related work includes the factor models of Litzenberger, Squassi and Weir (1995), Bliss (1997a,
1997b), Dai and Singleton (2000), de Jong and Santa-Clara (1999), de Jong (2000), Brandt and Yaron
(2001) and Duffee (2002). Particularly relevant are the three-factor models of Balduzzi, Das, Foresi and
Sundaram (1996), Chen (1996), and especially the Andersen-Lund (1997) model with stochastic mean
and volatility, whose three factors are interpreted in terms of level, slope and curvature. We will
subsequently discuss related work in greater detail; for now, suffice it to say that little of it considers
forecasting directly, and that our approach, although related, is indeed very different.
We proceed as follows. In section 2 we provide a detailed description of our modeling
framework, which interprets and extends earlier work in ways linked to recent developments in multi-
factor term structure modeling, and we also show how it can replicate a variety of stylized facts about the
yield curve. In section 3 we proceed to an empirical analysis, describing the data, estimating the models,
and examining out-of-sample forecasting performance. In section 4 we offer interpretive concluding
remarks.
2. Modeling and Forecasting the Term Structure I: Methods
Here we introduce the framework that we use for fitting and forecasting the yield curve. We
argue that the well-known Nelson-Siegel (1987) curve is well-suited to our ultimate forecasting purposes,
and we introduce a novel twist of interpretation, showing that the three coefficients in the Nelson-Siegel
curve may be interpreted as latent level, slope and curvature factors. We also argue that the nature of the
factors and factor loadings implicit in the Nelson-Siegel model facilitate consistency with various
empirical properties of the yield curve that have been cataloged over the years. Finally, motivated by our
interpretation of the Nelson-Siegel model as a three-factor model of level, slope and curvature, we
contrast it to various multi-factor models that have appeared in the literature.

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Cites background or methods from "The Nobel Memorial Prize for Robert..."

  • ...F-ARCH (Factor ARCH) The multivariate factor ARCH model developed by Diebold and Nerlove (1989) (see also Latent GARCH) and the factor GARCH model of Engle, Ng and Rothschild (1990) assumes that the temporal variation in the N×N conditional covariance matrix for a set of N returns can be described by univariate GARCH models for smaller set of K<N portfolios,...

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  • ...A leading example is the N-dimensional factor ARCH model of Diebold and Nerlove (1989), , where and denote N×1 vectors of factor loadings and i.i.d. innovations respectively, and the conditional variance of is determined by an ARCH model in lagged squared values of the latent factor (see also F-ARCH). Models in which the innovations are subject to censoring is another example (see Tobit-GARCH). In contrast to standard ARCH and GARCH models, for which the likelihood functions are readily available through a prediction error decomposition type argument (see ARCH), the likelihood functions for latent GARCH models are generally not available in closed form. General estimation and inference procedures for latent GARCH models based on Markov Chain Monte Carlo methods have been developed by Fiorentini, Sentana and Shephard (2004) (see also SV)....

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References
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Abstract: Traditional econometric models assume a constant one-period forecast variance. To generalize this implausible assumption, a new class of stochastic processes called autoregressive conditional heteroscedastic (ARCH) processes are introduced in this paper. These are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances. For such processes, the recent past gives information about the one-period forecast variance. A regression model is then introduced with disturbances following an ARCH process. Maximum likelihood estimators are described and a simple scoring iteration formulated. Ordinary least squares maintains its optimality properties in this set-up, but maximum likelihood is more efficient. The relative efficiency is calculated and can be infinite. To test whether the disturbances follow an ARCH process, the Lagrange multiplier procedure is employed. The test is based simply on the autocorrelation of the squared OLS residuals. This model is used to estimate the means and variances of inflation in the U.K. The ARCH effect is found to be significant and the estimated variances increase substantially during the chaotic seventies.

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  • ...For the next twenty years volatility clustering generated little interest; instead, the literature focused 8 See, for example, Fama (1965), Blattberg and Gonnedes (1974), and the references therein....

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  • ...9 For interesting commentary on the furious pace of development, both retrospective and prospective, see Engle (2002b)....

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Frequently Asked Questions (1)
Q1. What are the contributions in this paper?

In this paper the authors do so. The authors use neither the no-arbitrage approach, which focuses on accurately fitting the cross section of interest rates at any given time but neglects time-series dynamics, nor the equilibrium approach, which focuses on time-series dynamics ( primarily those of the instantaneous rate ) but pays comparatively little attention to fitting the entire cross section at any given time and has been shown to forecast poorly. The authors show that the three timevarying parameters may be interpreted as factors corresponding to level, slope and curvature, and that they may be estimated with high efficiency. The authors propose and estimate autoregressive models for the factors, and they show that their models are consistent with a variety of stylized facts regarding the yield curve.