پیش بینی ساختار دوره بازدهی اوراق قرضه دولتی

Despite powerful advances in yield curve modeling in the last 20 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 nor the equilibrium approach. 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.

The last 25 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 et al. (1992), and prominent contributions in the affine equilibrium tradition include Vasicek (1977), Cox et al. (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 and 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 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 et al. (1995), Bliss, 1997a and Bliss, 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 et al. (1996), Chen (1996), and especially the Andersen and 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 multifactor 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.

We have re-interpreted the Nelson–Siegel yield curve as a dynamic model that achieves dimensionality reduction via factor structure (level, slope and curvature), and we have explored the model's performance in out-of-sample yield curve forecasting. Although the 1-month-ahead forecasting results are no better than those of random walk and other leading competitors, the 1-year-ahead results are much superior. A number of authors have proposed extensions to Nelson–Siegel to enhance flexibility, including Bliss (1997b), Soderlind and Svensson (1997), Björk and Christensen (1999), Filipovic, 1999 and Filipovic, 2000, Björk (2000), Björk and Landén (2000) and Björk and Svensson (2001). From the perspective of interest rate forecasting accuracy, however, the desirability of the above generalizations of Nelson–Siegel is not obvious, which is why we did not pursue them here. For example, although the Bliss and Soderlind–Svensson extensions can have in-sample fit no worse than that of Nelson–Siegel, because they include Nelson–Siegel as a special case, there is no guarantee of better out-of-sample forecasting performance. Indeed, accumulated experience suggest that parsimonious models are often more successful for out-of-sample forecasting.17 Some of the extensions alluded to above are designed to make Nelson–Siegel consistent with no-arbitrage pricing. It is not obvious to us, however, that use of arbitrage-free models is necessary or desirable for producing good forecasts.18 Indeed we have shown that our model, which is not arbitrage-free, can produces good forecasts. In closing, we would like to elaborate on the likely reason for the forecasting success of our approach, which relies heavily on a broad interpretation of the shrinkage principle. The essence of our approach is intentionally to impose substantial a priori structure, motivated by simplicity, parsimony, and theory, in an explicit attempt to avoid data mining and hence enhance out-of-sample forecasting ability. This includes our use of a tightly parametric model that places strict structure on factor loadings in accordance with simple theoretical desiderata for the discount function, our decision to fix λλ, our emphasis on simple univariate modeling of the factors based upon our theoretically derived interpretation of the model as one of approximately orthogonal level, slope and curvature factors, and our emphasis on the simplest possible AR(1) factor dynamics. All of this is in keeping with a broad interpretation of the “shrinkage principle,” which has a firm foundation in Bayes–Stein theory, in empirical intuition, and in an accumulated track record of good performance (e.g., Garcia-Ferrer et al., 1987, Zellner and Hong, 1989 and Zellner and Min, 1993). Here we interpret the shrinkage principle as the insight that imposition of restrictions, which will of course degrade in-sample fit, may nevertheless be helpful for out-of-sample forecasting, even if the restrictions are false. The fact that the shrinkage principle works in the yield-curve context, as it does in so many other contexts, is precisely what theory and empirical experience would lead one to expect. This is not to say, of course, that our specification is in any sense uniquely best, and we make no claims to that effect. Rather, the broad lesson of the paper is to show in the yield-curve context that the shrinkage perspective, which tends to produce seemingly naive but truly sophisticatedly simple models (of which ours is one example), may be very appealing when the goal is forecasting. Put differently, the paper emphasizes in the yield curve context Zellner's (1992) “KISS principle” of forecasting —“Keep it sophisticatedly simple.”