Long-Memory Inflation Uncertainty: Evidence from the Term Structure of Interest Rates
Backus, David K., Zin, Stanley E., Jefferis, Richard H., Jr., Journal of Money, Credit & Banking
THE TERM STRUCTURE OF INTEREST RATES links the academic fields of macroeconomics and finance. Depending on one's point of view, the level and slope of the yield curve are indicators of the current stance of monetary policy (Bernanke and Blinder 1992), predictors of future movements in real output (Estrella and Hardouvelis 1991), or reflections of the market's assessment of the risk and expected returns of bonds of different maturities (Brennan and Schwartz 1982; Cox, Ingersoll, and Ross 1985; and Vasicek 1977). We continue the tradition of linking finance and macroeconomics by connecting prices of bonds of different maturities to the stochastic process for the short-term rate of interest, as commonly done in finance, then going on to speculate about the macroeconomic origins of interest rate movements.
The primary focus of our analysis, though, is not the relation between interest rates and the macroeconomy, but the dynamics of interest rates. Roughly speaking, there are two dimensions to the dynamics of interest rates: the correlation between short rates at different points in time and the relation between yields on bonds of different maturities at the same date. These "time series" and "cross-section" features of interest rates are not the same, but we show that they are closely related in existing theory. For example, in many of the popular theories of bond pricing, the short rate process has the property that the correlation between short rates n periods apart goes to zero exponentially. We show, in this case, that the yield on an n-period bond coverages to a constant: the variance of the yield on a long bond goes to zero exponentially, as well. In this way the time-series and cross-section properties of interest rates are closely linked.
The implication that long yields are constant seems to us to be at odds with the data. Although the yield curve generally flattens out as the maturity increases, there is considerable variation in long yields, even for yields on bonds with maturities up to ten years. We attempt to reconcile these two properties using the so-called fractional difference process introduced into economics by Granger and Joyeux (1980). With this process the variability of long yields approaches zero, but at a rate slower than exponential. With plausible parameter values, there is substantial variability in yields for maturities up to twenty years.
We develop these points in the remainder of the paper. In the next section we outline a theoretical framework that retains the simplicity of linearity but is general enough to include long memory. We derive, for this framework, formulas for prices and yields of bonds of all maturities. In section 2 we confront the central issue of the paper: the behavior of long yields. We argue that for many common short-rate processes, the theoretical properties of long yields and forward rates differ significantly from what we see in U.S. government bond data for the postwar period. This discrepancy between theory and data motivates the fractional difference model of section 3.
Section 4 is concerned with the ability of the fractional difference model to mimic some of the features of short-term interest rates, inflation, and money growth. With the possible exception of money growth, we find that the fractional difference model performs well relative to stationary ARMA or random walk models. Thus the model is able to reproduce important features of both the long end of the yield curve and the high-order autocorrelations of short rates and inflation. We speculate that the fractional short rate cum inflation process might be the result of heterogeneous agents responding to changes in monetary policy.
1. A THEORETICAL FRAMEWORK
We begin by deriving prices of risk-free bonds in a log-linear theoretical framework. There are two common approaches to theoretical bond pricing. One approach, epitomized by Campbell (1986) and Hansen and Jagannathan (1991), is to start with an equilibrium price measure, or intertemporal marginal rate of substitution: given a stochastic process for one-period state-contingent claims prices, we construct prices of risk-free bonds of different maturities. …