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King, Robert G.; Andre Kurmann,. "Expectations and the term structure of interest rates: evidence and implications." Economic Quarterly. Federal Reserve Bank of Richmond. 2002. HighBeam Research. 19 Oct. 2018 <https://www.highbeam.com>.
King, Robert G.; Andre Kurmann,. "Expectations and the term structure of interest rates: evidence and implications." Economic Quarterly. 2002. HighBeam Research. (October 19, 2018). https://www.highbeam.com/doc/1G1-95490452.html
King, Robert G.; Andre Kurmann,. "Expectations and the term structure of interest rates: evidence and implications." Economic Quarterly. Federal Reserve Bank of Richmond. 2002. Retrieved October 19, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-95490452.html
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Interest rates on long-term bonds are widely viewed as important for many economic decisions, notably business plant and equipment investment expenditures and household purchases of homes and automobiles. Consequently, macroeconomists have extensively studied the term structure of interest rates. For monetary policy analysis this is a crucial topic, as it concerns the link between short-term interest rates, which are heavily affected by central bank decisions, and long-term rates.
The dominant explanation of the relationship between short- and long-term interest rates is the expectations theory, which suggests that long rates are entirely governed by the expected future path of short-term interest rates. While this theory has strong implications that have been rejected in many studies, it nonetheless seems to contain important elements of truth. Therefore, many central bankers and other practitioners of monetary policy continue to apply it as an admittedly imperfect yet useful benchmark. In this article, we work to quantify both the dimensions along which the expectations theory succeeds in describing the link between expectations and the term structure and those along which it does not, thus providing a better sense of the utility of this benchmark.
Following Sargent (1979) and Campbell and Shiller (1987), we focus on linear versions of the expectations theory and linear forecasting models of future interest rate expectations. In this context, we reach five notable conclusions for the period since the Federal Reserve-Treasury Accord of March 1951. (1)
First, cointegration tests confirm that the levels of both long and short interest rates are driven by a common stochastic trend. In other words, there is a permanent component that affects long and short rates equally, which accords with one of the basic predictions of the expectations theory.
Second, while changes in this stochastic trend dominate the month-to-month changes in long-term interest rates, the same changes affect the short-term rate to a much less important degree. We summarize our detailed econometric analysis with a useful rule of thumb for applied researchers: it is optimal to infer that the stochastic trend in interest rates has varied by 97 percent of any change in the long-term interest rate. (2) In this sense, the long-term interest rate is a good indicator of the stochastic trend in interest rates in general. (3)
Third, according to cointegration tests, the spread between long and short rates is not affected by the stochastic trend, which is consistent with the expectations theory. Rather, the spread is a reasonably good indicator of changes in the temporary component of short-term interest rates. Developing a similar rule of thumb, we compute that on average, a 1 percent increase in the spread indicates a 0.71 percent decrease in the temporary component of the short rate, i.e., in the difference between the current short rate and the stochastic trend.
Fourth, the expectations theory imposes important rational expectations restrictions on linear time series models in the spread and short-rate changes. Like Campbell and Shiller (1987), who pioneered testing of the expectations theory in a cointegration framework, we find that these restrictions are decisively rejected by the data. But our work strengthens this conclusion by using a longer sample period and a better testing methodology. (4) We interpret the rejection as arising from predictable time-variations in term premia. Under the strongest form of the expectations theory, term premia should be constant and fluctuations in the spread should be entirely determined by expectations about future short-rate changes. However, our calculations indicate that--as another rule of thumb--a 1 percent deviation of the spread from its mean signals a 0.69 percent fluctuation of the expectations component with the remainder viewed as arising from shifts in the term premia.
Fifth, based on the work by Sargent (1979), we show how to adapt the restrictions implied by the expectations theory to a situation where term premia are time-varying but unpredictable over some forecasting horizons. Our tests indicate that these modified restrictions continue to be rejected with forecasting horizons of up to a year. Thus, departures from the expectations theory in the form of time-varying term premia are not simply of a high frequency form, although the cointegration results indicate that the term premia are stationary.
Our empirical findings should provide some guidance for macroeconomic modeling, including work on small-scale econometric models and on monetary policy rules. In particular, our results suggest that the presence of a common stochastic trend in short and long nominal rates is a feature of post-Accord history that deserves greater attention. Furthermore, the detailed empirical results and the summary rules-of-thumb can be considered as a useful guide for monetary policy discussions. As an example, we ask whether the general patterns in the 50-year sample hold up over the period 1986-2001. Interestingly, we find a reduced variability in the interest rate stochastic trend: it is only about half as volatile as during the entire sample period. Nevertheless, the appropriate rule of thumb is still to view 85 percent of any change in the long rate as reflecting a shift in the stochastic trend. Our analysis also indicates that the expectations component of the spread (the discounted sum of expected short-rate changes) is of larger importance in the more recent sample, justifying an increase of the relevant rule-of-thumb coefficient from 69 percent to 77 percent. One interpretation of these different results is that they indicate increased credibility of the Federal Reserve System over the last decade and a half, which Goodfriend (1993) describes as the Golden Age of monetary policy because of enhanced credibility.
1. HISTORICAL BEHAVIOR OF INTEREST RATES
The historical behavior of short-term and long-term interest rates during the period April 1951 to November 2001 is shown in Figure 1. The two specific series that we employ have been compiled by Ibbotson (2002) and pertain to the 30-day T-bill yield for the short rate and the long-term yield on a bond of roughly twenty years to maturity for the long rate. One motivation for our use of this sample period is that the research of Mankiw and Miron (1986) suggests that the expectations theory encounters particular difficulties after the founding of the Federal Reserve System, particularly during the post-Accord period, because of the nonstationarity of short-term interest rates.
In this section, we start by discussing some key stylized facts that have previously attracted the attention of many researchers. We then conduct some basic statistical tests on these series that provide important background to our subsequent analysis.
Basic Stylized Facts
We begin by discussing three important facts about the levels and comovement of short-term and long-term interest rates and then discuss two additional important facts about the predictability of these series.
Wandering levels: The levels of short-term and long-term interest rates vary substantially through time, as shown in Figure 1. Table 1 reports the very different average values over subsamples: in the 1950s, the short rate averaged 1.85 percent and the long rate averaged 3.02 percent; in the 1970s, the short rate averaged 6.13 percent and the long rate averaged 7.57 percent; and in the 1990s, the short rate averaged 4.80 percent and the long rate averaged 7.10 percent. These varying averages suggest that there are highly persistent factors that affect interest rates.
Comovement: While the levels of interest rates wander through time, subperiods of high average short rates are also periods of high average long rates. Symmetrically, short-term and long-term interest rates have a tendency to simultaneously display low average values within subperiods. This suggests that there may be common factors affecting long and short rates.
Relative stability of the spread: The spread between long- and short-term interest rates is much more stable over time, with average values of 1.17 percent, 1.45 percent, and 2.30 percent over the three decades discussed above. This again suggests that there is a common source of persistent variation in the two rates.
Predictability of the spread: While apparently returning to a more or less constant value, the spread between long and short rates appears relatively forecastable, even from its own past, because it displays substantial autocorrelation. This predictability has made the spread the focus of many empirical investigations of interest rates.
Changes in short-term and long-term interest rates: Figure 2 shows that changes in short and long rates are much less auto correlated. The two plots also highlight the changing volatility of short-term and long-term interest rates, which has been the subject of a number of recent investigations, including that of Watson (1999).
Basic Statistical Tests
The behavior of short-term and long-term interest rates displayed in Figures 1 and 2 has led many researchers to model the two series as stationary in first differences rather than in levels.
Unit root tests for interest rates: Accordingly, we begin by investigating whether there is evidence against the assumption that each series is stationary in differences rather than in levels. For this purpose, the first two columns of Table 2 report regressions of the augmented Dickey-Fuller (ADF) form. Specifically, the regression for the short rate [R.sub.t] takes the form
[DELTA][R.sub.t] = [a.sub.0] + [a.sub.1][DELTA][R.sub.t-1] + [a.sub.2] [DELTA][R.sub.t-2] + .... [a.sub.p][DELTA][R.sub.t-p] + f [R.sub.t-1] + [e.sub.Rt].
Our null hypothesis is that the short-term interest rate is difference stationary and that there is no deterministic trend in the level of the rate. In particular, stationarity in first differences implies that f = 0; if a deterministic trend is also absent, then [a.sub.0] = 0 as well. The alternative hypothesis is that the interest rate is stationary in levels (f < 0); in this case, a constant term is not generally zero because there is a non-zero mean to the level of the interest rate. The relevant test is reported in Table 2 for a lag length of p = 4. (5) It involves a comparison of fit of the constrained regression in the first column and the unconstrained regression in the second column, with the former appropriate under the null hypothesis of a unit root and the latter appropriate under the alternative of stationarity. There is no strong evidence against the null, since the Dickey-Fuller F-statistic of 2.94 is less than the 10 percent critical value of 3.78. (6) Looking at comparable results for the lon g rate [R.sup.L.sub.t] we find even less evidence against the null hypothesis. (7) The value of the Dickey-Fuller F-statistic is even smaller. (8) We therefore model both interest rates as first difference stationary throughout our analysis.
In these regressions, we also find the first evidence of different predictability of short-term and long-term interest rates, a topic that will be a focus of much discussion below. Foreshadowing this discussion, we will find in every case that long-rate changes are less predictable than short-rate changes. In Table 2, the unconstrained regression for changes in the long rate accounts for about 3.5 percent of its variance, and the unconstrained regression for changes in the short rate accounts for about 8 percent of its variance. (9)
A simple cointegration test: Since we take the long-term and short-term rate as containing unit roots, the spread [S.sub.t] = [R.sup.L.sub.t] - [R.sub.t] may either be nonstationary or stationary. If the spread is stationary, then the long-term and short-term interest rates are cointegrated in the terminology of Engle and Granger (1987), since a linear combination of the variables is stationary. One simple test for cointegration when the cointegrating vector is known, discussed for example in Hamilton (1994, 582-86), is based on a Dickey-Fuller regression. In our context, we run the regression
[DELTA][S.sub.t] = [a.sub.0] + [a.sub.1][DELTA][S.sub.t-1] + [a.sub.2] [DELTA][S.sub.t-2] + .... [a.sub.p][DELTA][S.sub.t-p] + f [S.sub.t-1] + [e.sub.St].
As above, we take the null hypothesis to be that the spread is nonstationary, but that there is no deterministic trend in the level of the spread. The alternative of stationarity (cointegration) is a negative value of f; the value of [a.sub.0] then captures the non-zero mean of the spread. The results in Table 2 show that we can reject the null at a high critical level: the value of the Dickey-Fuller F-statistic is 9.67, which exceeds the 5 percent critical level of 4.59.
Thus, we tentatively take the short-term and long-term interest rate to be cointegrated, but we will later conduct a more powerful test of cointegration. The regression results in Table 2 also highlight the fact that the spread is more predictable from its own past than are either of its components. In the unconstrained regression, 16 percent of month-to-month changes in the spread can be forecast from past values.
Cointegration of short-term and long-term interest rates is a formal version of the second stylized fact above: there is comovement of short and long rates despite their shifting levels. It is based on the third stylized fact: the spread appears relatively stationary although it is variable through time.
2. THE EXPECTATIONS THEORY
The dominant economic theory of the term structure of interest rates is called the expectations theory, as it stresses the role of expectations of future short-term interest rates in the determination of the prices and yields on longer-term bonds. There are a variety of statements of this theory in the literature that differ in terms of the nature of the bond which is priced and the factors that enter into pricing. We make use of a basic version of the theory developed in Shiller (1972) and used in many subsequent studies. (10) This version is suitable for empirical analyses of yields on long-term coupon bonds such as those that we study, since it delivers a simple linear formula for long-term yields. The derivation of this formula, which is reviewed in Appendix A, is based on the assumption that investors equate the expected holding period yield on long-term bonds to the short-term interest rate [R.sub.t], plus a time-varying excess holding period return [k.sub.t], which is not described or restricted by the model but could represent variation in risk premia, liquidity premia and so forth. It is based on a linear approximation to this expected holding period condition that neglects higher order terms. More specifically, the theory indicates that
[R.sup.L.sub.t] = [beta][E.sub.t][R.sup.L.sub.t+1] + (1 - [beta])([R.sub.t] + [k.sub.t]) (1)
where [beta] = 1/(1 + [R.sup.L]) is a parameter based on the mean of the long-term interest rate around which the approximation is taken. (11)
This expectational difference equation can be solved forward to relate the current long-term interest rate to a discounted value of current and future R and k:
[R.sup.L.sub.t] = (1 - [beta]) [summation over ([infinity]/j=0)][[beta].sup.j][[E.sub.t][R.sub.t+j] + [E.sub.t][k.sub.t+j]]. (2)
Various popular term-structure theories can be accommodated within this framework. The pure expectations theory occurs when there are no k terms, so that [R.sup.L.sub.t] = (1 - [beta]) [summation over ([infinity]/j=0)][[beta].sup.j][E.sub.t][R.sub.t+j]. This is a useful form for discussing various propositions about long-term and short-term interest rates that also arise in richer theories.
Implication for permanent changes in interest rates: Notably, the pure expectations theory predicts that if interest rates increase at date tin a manner which agents expect to be permanent, then there is a one-for-one effect of such a permanent increase on the level of the long rate because the weights sum to one, i.e., (1 - [beta]) [summation over ([infinity]/j=0)][[beta].sup.j] = (1 - [beta])/(1 - [beta]) = 1. This is a basic and important implication of the expectations theory long stressed by analysts of the term structure and that appears capable of potentially explaining the comovement of short-term and long-term interest rates that we discussed above.
Implications for temporary changes in interest rates: Temporary changes in interest rates have a smaller effect under the pure expectations theory, with the extent of this effect depending on how sustained the temporary changes are assumed to be. Supposing that the short-term interest rate is governed by the simple autoregressive process [R.sub.t] = [rho][R.sub.t-1] + [e.sub.Rt] with the error term being unforecastable, it is easy to see that E [R.sub.t+j] = [[rho].sup.j][R.sub.t]. It follows that a rational expectations solution for the long-term rate is
[R.sup.L.sub.t] = (1 - [beta]) [summation over ([infinity]/j=0)][[beta].sup.j][E.sub.t][R.sub.t+j]
= (1 - [beta]) [summation over ([infinity]/j=0)][[beta].sup.j] [[rho].sup.j] [R.sub. …
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