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