LONG-TERM BONDS: A Two-Factor Model 4


The term structure of interest rates in the US

We estimate the two-factor term structure model using data on US nominal interest rates, equities and inflation. We use nominal zero-coupon yields at maturities 3 months, 1 year, 3 years, and 10 years from McCulloch and Kwon (1993), updated by Gong and Remolona (1996a,b). We take data on equities from the Indices files on the CRSP tapes. We use the value-weighted return, including dividends, on the NYSE, AMEX and NASDAQ markets. We take data on CPI inflation from the SBBI files on the CRSP tapes. Although the raw data are available monthly, we construct a quarterly data set in order to reduce the influence of high-frequency noise in inflation and short-term movements in interest rates.

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To avoid the implication of the model that bond returns are driven by only two common factors, so that all bond returns can be perfectly explained by any two bond returns, we assume that bond yields are measured with error. The errors in yields are normally distributed, serially uncorrelated, and uncorrelated across bonds. Then the term structure model becomes a classic state-space model in which unobserved state variables xt and zt follow a linear process with normal innovations and we observe linear combinations of them with normal errors.

The model can be estimated by maximum likelihood using a Kalman filter to construct the likelihood function (Berardi 1997, Harvey 1989, Pennacchi 1991, Gong and Remolona 1996a,b, Foresi, Penati, and Pennacchi 1997). This is an attractive alternative to the Generalized Method of Moments used to estimate term structure models by Gibbons and Ra-maswamy (1993) and others.

In Table 1 we report parameter estimates for the period 1952-96 and the period 1983-96. Interest rates were unusually high and volatile in the 1979-82 period, during which the Federal Reserve Board under Paul Volcker was attempting to reestablish the credibility of anti-inflationary monetary policy and was experimenting with monetarist operating procedures. Many authors have argued that real interest rates and inflation have behaved differently in the monetary policy regime established since 1982 by Federal Reserve chairmen Volcker and Alan Greenspan (see for example Clarida, Gali, and Gertler 1998). Accordingly we report separate estimates for the period starting in 1983 in addition to the full sample period.

In earlier versions of this paper we reported completely unrestricted maximum likelihood estimates of the model. In 1952-96 these estimates fit the data well, but in 1983-96 the unrestricted estimates deliver implausibly low means for short-term nominal and real interest rates. (The model does not necessarily fit the sample means because the same parameters are used to fit both time-series and cross-sectional behavior; thus the model can trade off better fit elsewhere for worse fit of mean shortterm interest rates.) Accordingly in this version of the paper we require that the model exactly fit the sample means of nominal interest rates and inflation. This restriction hardly reduces the likelihood at all in 1952-96, and even in 1983-96 it cannot be rejected at conventional significance levels. payday loan lenders only