The macroeconomy and the yield curve: a dynamic latent factor approach
Introduction
Macroeconomists, financial economists, and market participants all have attempted to build good models of the yield curve, yet the resulting models are very different in form and fit. In part, these differences reflect the particular modeling demands of various researchers and their different motives for modeling the yield curve (e.g., interest rate forecasting or simulation, bond or option pricing, or market surveillance). Still, an unusually large gap is apparent between the yield curve models developed by macroeconomists, which focus on the role of expectations of inflation and future real economic activity in the determination of yields, and the models employed by financial economists, which eschew any explicit role for such determinants. This paper takes a step toward bridging this gap by formulating and estimating a yield curve model that integrates macroeconomic and financial factors.
Many other recent papers have also modeled the yield curve, and they can be usefully categorized by the extent and nature of the linkages permitted between financial and macroeconomic variables. Many yield curve models simply ignore macroeconomic linkages. Foremost among these are the popular factor models that dominate the finance literature—especially those that impose a no-arbitrage restriction. For example, Knez et al. (1994), Duffie and Kan (1996), and Dai and Singleton (2000) all consider models in which a handful of unobserved factors explain the entire set of yields. These factors are often given labels such as “level,” “slope,” and “curvature,” but they are not linked explicitly to macroeconomic variables.
Our analysis also uses a latent factor model of the yield curve, but we also explicitly incorporate macroeconomic factors. In this regard, our work is more closely related to Ang and Piazzesi (2003), Hördahl et al. (2002), and Wu (2002), who explicitly incorporate macro determinants into multi-factor yield curve models. However, those papers only consider a unidirectional macro linkage, because output and inflation are assumed to be determined independently of the shape of the yield curve, but not vice versa. This same assumption is made in the vector autoregression (VAR) analysis of Evans and Marshall, 1998, Evans and Marshall, 2001 where neither contemporaneous nor lagged bond yields enter the equations driving the economy. In contrast to this assumption of a one-way macro-to-yields link, the opposite view is taken in another large literature typified by Estrella and Hardouvelis (1991) and Estrella and Mishkin (1998), which assumes a yields-to-macro link and focuses only on the unidirectional predictive power of the yield curve for the economy. The two assumptions of these literatures—one-way yields-to-macro or macro-to-yields links—are testable hypotheses that are special cases of our model and are examined below. Indeed, we are particularly interested in analyzing the potential bidirectional feedback from the yield curve to the economy and back again. Some of the work closest to our own allows a feedback from an implicit inflation target derived from the yield curve to help determine the dynamics of the macroeconomy, such as Kozicki and Tinsley (2001), Dewachter and Lyrio (2002), and Rudebusch and Wu (2003). In our analysis, we allow for a more complete set of interactions in a general dynamic, latent factor framework.
Our basic framework for the yield curve is a latent factor model, although not the usual no-arbitrage factor representation typically used in the finance literature. Such no-arbitrage factor models often appear to fit the cross-section of yields at a particular point in time, but they do less well in describing the dynamics of the yield curve over time (e.g., Duffee, 2002, Brousseau, 2002). Such a dynamic fit is crucial to our goal of relating the evolution of the yield curve over time to movements in macroeconomic variables. To capture yield curve dynamics, we use a three-factor term structure model based on the classic contribution of Nelson and Siegel (1987), interpreted as a model of level, slope, and curvature, as in Diebold and Li (2002). This model has the substantial flexibility required to match the changing shape of the yield curve, yet it is parsimonious and easy to estimate. We do not explicitly enforce the no-arbitrage restriction. However, to the extent that it is approximately satisfied in the data—as is likely for the U.S. Treasury bill and bond obligations that we study—it will also likely be approximately satisfied in our estimates, as our model is quite flexible and gave a very good fit to the data. Of course, there may be a loss of efficiency in not imposing the restriction of no arbitrage if it is valid, but this must be weighed against the possibility of misspecification if transitory arbitrage opportunities are not eliminated immediately.
In Section 2, we describe and estimate a basic “yields-only” version of our model—that is, a model of just the yield curve without macroeconomic variables. To estimate this model, we introduce a unified state-space modeling approach that lets us simultaneously fit the yield curve at each point in time and estimate the underlying dynamics of the factors. This one-step approach improves upon the two-step estimation procedure of Diebold and Li (2002) and provides a unified framework in which to examine the yield curve and the macroeconomy.
In Section 3, we incorporate macroeconomic variables and estimate a “yields-macro” model. To complement the nonstructural nature of our yield curve representation, we also use a simple nonstructural VAR representation of the macroeconomy. The focus of our examination is the nature of the linkages between the factors driving the yield curve and macroeconomic fundamentals.
In Section 4, we relate our framework to the expectations hypothesis, which has been studied intensively in macroeconomics. The expectation hypotheses of the term structure is a special case of our model that receives only limited support.
We offer concluding remarks in Section 5.
Section snippets
A yield curve model without macro factors
In this section, we introduce a factor model of the yield curve without macroeconomic variables, which is useful for two key reasons. First, methodologically, such a model proves to be a convenient vehicle for introducing the state-space framework that we use throughout the paper. Second, and substantively, the estimated yields-only model serves as a useful benchmark to which we subsequently compare our full model that incorporates macroeconomic variables.
A yield curve model with macro factors
Given the ability of the level, slope, and curvature factors to provide a good representation of the yield curve, it is of interest to relate them to macroeconomic variables. This can be done readily in an expanded version of the above state-space framework. In the next five subsections, we analyze the dynamic interactions between the economy and the yield curve, and we assess their importance.
Examining the expectations hypothesis
It is useful to contrast our representation of the yield curve with others that have appeared in the literature. Here we relate our yield curve modeling approach to the traditional macroeconomic approach based on the expectations hypothesis.
The expectations hypothesis of the term structure states that movements in long rates are due to movements in expected future short rates. Any term or risk premia are assumed to be constant through time. In terms of our notation above, which pertains to the
Summary and conclusions
We have specified and estimated a yield curve model that incorporates both yield factors (level, slope, and curvature) and macroeconomic variables (real activity, inflation, and the stance of monetary policy). The model's convenient state-space representation facilitates estimation, the extraction of latent yield-curve factors, and testing of hypotheses regarding dynamic interactions between the macroeconomy and the yield curve. Interestingly, we find strong evidence of macroeconomic effects on
Acknowledgements
We thank the Editor (Giampiero Gallo) and two referees for insightful guidance. We thank the Guggenheim Foundation, the National Science Foundation, and the Wharton Financial Institutions Center for research support. Finally, we thank Andrew Ang, Pierluigi Balduzzi, Todd Clark, Ron Gallant, Ken Nyholm, Monika Piazzesi, Mike Wickens, Scott Weiner, and discussants and participants at many seminars and conferences, for helpful comments. The views expressed in this paper do not necessarily reflect
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