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Expectations and the Term
Structure of Interest Rates:
Evidence and Implications
Robert G. King and Andr´e Kurmann
I
nterest 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. Con- sequently, macroeconomists have extensively studied the term structure of interest rates. For monetary policy analysis this is a crucial topic, as it con- cerns 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
The authors would like to thank Michael Dotsey, Huberto Ennis, Pierre-Daniel Sarte, and Mark Watson for helpful comments. The views expressed in this article are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of Richmond or the Federal Reserve System. Robert G. King: Professor of Economics, Boston University, and consultant to the Research Department of the Federal Reserve Bank of Richmond. Andr´e Kurmann: Department of Economics, University of Qu´ebec at Montr´eal.
Federal Reserve Bank of Richmond Economic Quarterly Volume 88/4 Fall 2002 49
50 Federal Reserve Bank of Richmond Economic Quarterly
future interest rate expectations. In this context, we reach five notable con- clusions 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 expec- tations 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 deci- sively 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
(^1) See Hetzel and Leach (2001) for an interesting recent account of the events surrounding the Accord. (^2) The sense in which this measure is optimal is discussed in more detail below, but it is based on minimizing the variance of prediction errors over our sample period of 1951 to 2001. (^3) By contrast, a similar calculation indicates that changes in short-term interest rates are a much less strong indicator of changes in the stochastic trend: the comparable adjustment coefficient is 0.17 rather than 0.97. This finding is consistent with other evidence of important temporary variations in short-term interest rates, presented in this article and other studies. (^4) We impose the cross-equation restrictions on the VAR and calculate a likelihood ratio test that compares the fit of the constrained and unconstrained VAR, while Campbell and Shiller (1987) use a Wald-type test of the restrictions on an estimated unrestricted VAR. It is now understood that Wald tests of nonlinear restrictions are sensitive to the details of how such tests are set up and suffer from much more severe small-sample bias than the method we employ here (see Bekaert and Hodrick [2001]).
52 Federal Reserve Bank of Richmond Economic Quarterly
Figure 1 The Post-Accord History of Interest Rates
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.
R. G. King and A. Kurmann: Expectations and the Term Structure 53
Table 1 Decade Averages
Short Rate Long Rate Spread
1950s 1.85 3.02 1. 1960s 3.81 4.63 0. 1970s 6.13 7.57 1. 1980s 8.54 10.69 2. 1990s 4.80 7.10 2. Full Sample 5.13 6.67 1.
Notes: All values are in percent per annum.
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. 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 autocor- relation. 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
R. G. King and A. Kurmann: Expectations and the Term Structure 55
Figure 2 The History of Interest Rate Changes
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
St = a 0 + a 1 St− 1 + a 2 St− 2 +.... ap St−p + f St− 1 + e (^) 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 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
56 Federal Reserve Bank of Richmond Economic Quarterly
Table 2 Unit Root Tests
Full Sample Estimates (1951.4–2001.11)
Rt R (^) tL St− 1 con. uncon. con. uncon. con. uncon. constant 0 0.0123 0 0.0043 0 0. (0.0057) (0.0025) (0.0043) lagged 0 −0.0283 0 −0.0068 0 −0. level (0.0116) (0.0042) (0.0261) lag 1 −0.2151 −0.0198 0.0896 0.0918 −0.3256 −0. (0.0406) (0.0411) (0.0409) (0.0409) (0.0404) (0.0437) lag 2 −0.1649 −0.1499 −0.0441 −0.0418 −0.2610 −0. (0.0415) (0.0419) (0.0407) (0.0407) (0.0425) (0.0444) lag 3 −0.0082 0.0037 −0.1390 −0.1369 -0.0759 −0. (0.0416) (0.0417) (0.0407) (0.0407) (0.0425) (0.0433) lag 4 −0.1193 −0.1094 0.0384 0.0398 −0.1521 −0. (0.0406) (0.0407) (0.0409) (0.0409) (0.0404) (0.0407) R-square 0.0721 0.0811 0.0301 0.0348 0.1322 0. F-value 2.9352 1.4415 9.
Notes: Numbers in parentheses represent standard errors. The critical 5 percent (10 per- cent) value for the Adjusted Dickey-Fuller F-test is 4.59 (3.78).
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
58 Federal Reserve Bank of Richmond Economic Quarterly
unforecastable, it is easy to see that ERt+j = ρ j^ Rt. It follows that a rational expectations solution for the long-term rate is
R (^) tL = ( 1 − β)
∑^ ∞
j = 0
β j^ Et Rt+j
= ( 1 − β)
∑^ ∞
j = 0
β j^ ρ j^ Rt =
1 − β 1 − βρ
R (^) t = θRt.
This solution can be used to derive implications for temporary changes in short rates. If these are completely transitory, so that ρ = 0, there is a minimal effect on the long rate, since θ = 1 − β ≈ 0 .005. On the other hand, as the changes become more permanent (ρ approaches one) the θ coefficient approaches the one-for-one response previously discussed as the implication for fully permanent changes in the level of rates. Accordingly, the response of the long rate under the expectations theory depends on the degree of persistence that agents perceive in short-term interest rates, a property that Mankiw and Miron (1986) and Watson (1999) have exploited to derive interesting implications of the term structure theory that accord with various changes in the patterns of short-term and long-term interest rates in different periods of U.S. history. The spread as an indicator of future changes: There has been much interest in the idea that the expectations theory implies that the long-short spread is an indicator of future changes in short-term interest rates. With a little bit of algebra, as in Campbell and Shiller (1987), we can rewrite (2) as
R (^) tL − R (^) t = ( 1 − β)
∑^ ∞
j = 0
β j^ [(Et Rt+j − Rt )] =
j = 1
β j^ Et Rt+j ,
when there are no term premia.^12 Hence, the spread is high when short-term interest rates are expected to increase in the future, and it is low when they are expected to decrease. Further, permanent changes in the level of short-term interest rates, such as those considered above, have no effect on the spread because they do not imply any expected future changes in interest rates. While these three implications can easily be derived under the pure ex- pectations theory, they carry over to other more general theories so long as the changes in interest rates do not affect ( 1 − β)
j = 0 β^
j (^) Et kt+j in (2). Further,
while the pure expectations theory is a useful expository device, it is simply rejected: one of the stylized facts is that long rates are generally higher than short rates (there is a positive average value to the term spread). For this reason, all empirical studies of the effects of expectations on the long rate
(^12) To undertake this derivation, note that R (^) t+j −Rt = Rt+j −Rt+j − 1 +... (Rt+ 1 −Rt ). Hence,
each expected change enters many times in the sum, with a total effect of ∑∞ h=j β h^ E (^) t (R (^) t+j −
Rt+j − 1 ) = 1 β−^ jβ Et (R (^) t+j − R (^) t+j − 1 ).
R. G. King and A. Kurmann: Expectations and the Term Structure 59
minimally use a modified form
R Lt = ( 1 − β)
∑^ ∞
j = 0
β j^ Et Rt+j + K,
where K is a parameter capturing the average value of the term spread that comes from assuming that k (^) t is constant.^13
The Efficient Markets Test
As exemplified by the work of Roll (1969), one strategy is to derive testable im- plications of the expectations theory that (i) do not require making assumptions about the nature of the information set that market participants use to forecast future interest rates and that (ii) impose restrictions on a single linear equation. In the current setting, such an efficient markets test is based on manipulating (1) so as to isolate a pure expectations error, R Lt = (^1) β R Lt− 1 −( 1 −β β)(Rt− 1 +K)+ξ (^) t , where ξ (^) t = R (^) tL − Et− 1 R Lt. As in Campbell and Shiller (1987, 1991), this con- dition may be usefully reorganized to indicate that the long-short spread (and only the spread) should forecast long-rate changes,
R Lt − R Lt− 1 = (
β
− 1 )(R Lt− 1 − R (^) t− 1 − K) + ξ (^) t ,
which is a form that is robust to nonstationarity in the interest rate. The essence of efficient markets tests is to determine whether any vari- ables that are plausibly in the information set of agents at time t − 1 can be used to predict ξ (^) t = R Lt − R (^) tL− 1 − ( (^1) β − 1 )(R Lt− 1 − Rt− 1 − K). The forecasting relevance of any stationary variable can be tested with a standard t-statistic and the relevance of any group of p stationary variables can be tested by a likelihood ratio test, which has an asymptotic χ^2 p distribution. Table 3 re- ports a battery of such efficient markets tests. The first regression simply is a benchmark, relating R Lt − R Lt− 1 to a constant and to ( (^1) β − 1 )St− 1 in the manner suggested by the efficient markets theory. The second regression frees up the coefficient on St− 1 and finds its estimated value to be negative rather than positive. The t-statistic for testing the hypothesis that the coefficient equals ( (^) β^1 − 1 ) = 0 .005 takes on a value of 2.345, which exceeds the standard 95 percent critical level. This finding has been much discussed in the context of long-term bonds and some other financial assets, in that financial markets spreads have a “wrong-way” influence on future changes relative to the pre- dictions of basic theory.^14 At the same time, the low R^2 of 0.0051 indicates that the prediction performance of the regression is very modest.
(^13) Below, we use the notation K (^) t = ( 1 − β) ∑∞ j = 0 β j (^) E (^) t kt+j. But if k (^) t = k, then K = k. (^14) See, for example, Campbell and Shiller (1991) for the term structure of interest rates or Bekaert and Hodrick (2001) for foreign exchange rates.
R. G. King and A. Kurmann: Expectations and the Term Structure 61
table indicate that adding lagged variables does not significantly increase the explanatory power compared to the original efficient markets regression.^15 The efficient markets regression again highlights that there is a substantial amount of unpredictable variation in changes in long bond yields, which makes it difficult to draw strong conclusions about the nature of predictable variations in these returns.^16 One measure of the degree of this unpredictable variation is presented in panel B of Figure 2, where there is a very smooth and apparently quite flat line that is labelled as the “predicted changes in long rates.” Those predicted changes are ( (^) β^1 − 1 )(R (^) tL− 1 − Rt− 1 ) with a value of β suggested by the average level of long rates over our sample period. Panel B of Figure 2 highlights the fact that the expectations theory would explain only a tiny portion of interest rate variation if it were exactly true. Sargent (1979) refers to this as the “near-martingale property of long-term rates” under the expectations hypothesis. But it would not look very different if the fitted values of the other specifications in Table 3 were employed. Changes in the long rate are quite hard to predict and their predictable components are inconsistent with the efficient markets hypothesis.
Where Do We Go from Here?
Given that the efficient markets restriction is rejected, some academics sim- ply conclude we know nothing about the term structure.^17 However, central bankers and other practitioners actually do seem to employ the expectations theory as a useful yet admittedly imperfect device to interpret current and his- torical events (examples in this review are Dotsey [1998], Goodfriend [1993], and Owens and Webb [2001]). In the remainder of this analysis, we recognize that the expectations theory is not true but instead of simply rejecting it, we use modern time series methods to understand the dimensions along which it appears to succeed and those along which it does not. Section 3 develops and tests the common stochastic trend/cointegration restrictions that the expecta- tions theory imposes. Consistent with earlier studies, we find that U.S. data
(^15) One potential explanation for the failure of the efficient markets tests—highlighted in Fama (1977)—is that there may be time-variation k (^) t in the equilibrium returns, which investors require to hold an asset. Then the theory predicts that
R Lt − R Lt− 1 = ( (^) β^1 − 1 )(R Lt− 1 − R (^) t− 1 − k (^) t− 1 ) + ξ (^) t.
But the researcher conducting the test does not observe time variation in k, which may give rise to a biased estimate on the spread. Fama stresses that efficient markets tests involve a joint hypothesis about the efficient use of information and a model of equilibrium returns, so that a rejection of the theory may arise from either element. (^16) See the discussion of Nelson and Schwert (1977) on testing for a constant real rate. (^17) For example, at a recent macroeconomics conference, one prominent monetary economist argued that the expectations theory of the term structure has been rejected so many times that it should never be built into any model.
62 Federal Reserve Bank of Richmond Economic Quarterly
do not allow us to reject these restrictions and, thus, that the theory appears to contain an important element of truth as far as the common stochastic trend implication is concerned. Section 4 then follows Sargent (1979) in developing and testing a variety of cross-equation restrictions that the expectations theory implies. These restrictions are rejected in the data. Finally, in Section 5, we build on the approach by Campbell and Shiller (1987) to extract estimates of changes in market expectations, which also allows us to extract estimates of time-variation in term premia.
3. COINTEGRATION AND COMMON TRENDS
A basic implication of the expectations theory is that an unexpected and per- manent change in the level of short rates should have a one-for-one effect on the long rate. In other words, the theory implies that there is a common trend for the short and the long rate. This idea can be developed further using the concept of cointegration and related methods can be used to estimate the common trend. The starting point is Campbell and Shiller’s (1987) observation that present value models have cointegration implications, if the underlying series are nonstationary in levels, and that these implications survive the introduction of stationary deviations from the pure expectations theory such as time-varying term premia. In the context of the term structure, we can rewrite the long-rate equation (2) as
R Lt − Rt = ( 1 − β)
∑^ ∞
j = 0
β j^ [(Et Rt+j − R (^) t ) + E (^) t kt+j ] (3)
j = 1
β j^ Et Rt+j + ( 1 − β)
∑^ ∞
j = 0
β j^ Et kt+j (4)
so that the expectations theory stipulates that the spread is stationary so long as (i) first differences of short rates are stationary and (ii) the expected deviations from the pure expectations theory are stationary. Thus, cointegration tests are one way of assessing this implication of the theory. In Section 1, we found evidence against the hypothesis that the spread contains a unit root and suggested that a stationary spread was a better de- scription of the U.S. data. That is, we found some initial evidence consistent with modeling the short rate and the long rate as cointegrated. Here, in Section 3, we confirm that the spread also passes a more rigorous cointegration test. Given this result, we then define and estimate the common stochastic trend for the short rate and the long rate. We also present an easy-to-use rule of thumb that decomposes fluctuations of the short and the long rate into fluctuations in the common trend and fluctuations in the temporary components.
64 Federal Reserve Bank of Richmond Economic Quarterly
To test for cointegration, we estimate both the VAR and the VECM and compare their respective fit. A substantial increase in the log likelihood of the VECM over the VAR signals that the cointegration terms aid in the prediction of interest rate changes. More specifically, a large likelihood ratio results in a rejection of the null hypothesis in favor of the alternative of cointegration. In particular, we follow the testing procedure by Horvath and Watson (1995) and assume a priori that the cointegrating relationship is given by the spread St = R (^) tL − Rt rather than estimating the cointegrating vector.^19 Table 4 reports estimates of the VAR and VECM models for the lag length of p = 4, which we choose as the reference lag length throughout. Before discussing the cointegration test results in detail, it is worthwhile looking at a few elements that the VAR and VECM regressions have in common. First, changes in short rates are somewhat predictable from past changes in short rates, as was previously found with the Dickey-Fuller regression in Table 2. In addition, past changes in long rates are important for predicting changes in short rates in both the VAR and the VECM.^20 Finally, changes in short rates are predicted by the lagged spread: if the long rate is above the short rate, then short rates are predicted to rise. Second, changes in long rates are still fairly hard to predict with either the VAR or the VECM. Moving to the cointegration test, the likelihood ratio between the VECM and the VAR equals 2 ∗ (LV ECM − LV AR ) = 27 .67, which exceeds the 5 percent critical level of 6.28 calculated by the methods of Horvath and Watson (1995).^21 In other words, we can comfortably reject the hypothesis of no cointegration between R (^) t and R Lt , which is consistent with earlier studies
(^19) This type of test is somewhat more powerful than the unit root test on the spread reported in Table 2, which may be revealed by taking the difference between the two VECM equations and reorganizing the results slightly to obtain
S (^) t =
∑^ p j = 1
(c (^) i − a (^) i )R (^) tL−i +
∑^ p j = 1
(d (^) i − b (^) i )R (^) t−i + (g − f )S (^) t− 1 + (e (^) Lt − e (^) Rt ),
which can be further rewritten as
St =
∑^ p j = 1
(c (^) i − a (^) i )S (^) t−i +
∑^ p j = 1
(c (^) i + d (^) i − a (^) i − b (^) i )R (^) t−i + (g − f )S (^) t− 1 + (e (^) Lt − e (^) Rt ).
That is, the Horvath-Watson test essentially introduces some additional stationary regressors to the forecasting equation for changes in the spread that was used in the DF test. Adding these regressors can improve the explanatory power of the regression, resulting in a more powerful test. (^20) That is, long rates Granger-cause short rates. (^21) As Horvath and Watson (1995) stress, the relevant critical values for the likelihood ratio must take into account that the spread is nonstationary under the null. Thus, we cannot refer to a standard chi-square table. We estimate the VAR and VECM without constant terms, since we are assuming no deterministic trends in interest rates. However, we allow for a mean value of the spread, which is not zero as shown in (7) and (8). Unfortunately, this combination of assumptions means that we cannot use the tables in Horvath and Watson (1995), but must conduct the Monte Carlo simulations their method suggests to calculate the critical values reported in the text. Details are contained in replication materials available at http://people.bu.edu/rking.
R. G. King and A. Kurmann: Expectations and the Term Structure 65
Table 4 VAR/VECM Estimates
Full Sample Estimates (1951.4–2001.11)
Rt R Lt VAR VECM VAR VECM
St− 1 0.1101 −0. (0.0237) (0.0094) Rt− 1 −0.3095 −0.2382 0.0001 −0. (0.0408) (0.0430) (0.0160) (0.0171) R (^) t− 2 −0.1997 −0.1393 −0.0038 −0. (0.0427) (0.0440) (0.0168) (0.0175) R (^) t− 3 −0.0051 0.0426 −0.0151 −0. (0.0417) (0.0423) (0.0164) (0.0168) R (^) t− 4 −0.0879 −0.0466 0.0387 0. (0.0377) (0.0382) (0.0148) (0.0152)
R (^) tL− 1 0.8712 0.8209 0.0943 0. (0.1038) (0.1026) (0.0408) (0.0409) R (^) tL− 2 0.6250 0.5954 −0.0397 −0. (0.1095) (0.1078) (0.0430) (0.0429) R (^) tL− 3 0.1791 0.1698 −0.1347 −0. (0.1123) (0.1104) (0.0441) (0.0440)
R (^) tL− 4 0.0430 0.0520 0.0526 0. (0.1133) (0.1114) (0.0445) (0.0443)
R-square 0.2220 0.2492 0.0459 0. F-statistic 21.2172 21.9030 3.5785 3.
Notes: Numbers in parentheses represent standard errors. The likelihood ratio statistic of the VECM against the VAR is 27.6704. Comparing this value to the corresponding critical value in Horvath and Watson’s tables leads to strong rejection of null of two unit roots (p-value higher than 0.01).
and reinforces the statistical support for the common trend implication of the expectations theory. Therefore, the data is consistent with the basic implication of cointegration of the expectations theory and we thus view the VECM as the preferred specification and assume cointegration for the remainder of our analysis.^22
(^22) An alternative approach in this section would be to estimate the cointegrating vector and use the well-known testing method of Johansen (1988). Horvath and Watson (1995) establish that their procedure is more powerful if the cointegrating vector is known.
R. G. King and A. Kurmann: Expectations and the Term Structure 67
= R (^) tL− 1 +
k= 0
hL H M k^ xt = R Lt− 1 + h (^) L H [I − M]−^1 xt ,
where hL = [0 1 0] such that R (^) tL = h (^) L zt. Finally, the difference between R^ ¯ Lt and R¯t is the limit forecast of the spread. By definition of cointegration, the spread is stationary and therefore its limit forecast must be a constant:^24 K = lim k→∞
Et St+k = lim k→∞
Et R Lt+k − lim k→∞
Et Rt+k = ¯R Lt − ¯R (^) t.
Thus, the trends for the long rate and the short rate differ only by the constant K: in other words, the long rate and the short rate have a common stochastic trend component. Since this is sometimes termed the permanent component, deviations from it are described as temporary components. Using this lan- guage, the temporary component of the short rate is Rt − ¯Rt and that of the long rate is R Lt − ¯R Lt.
A Stochastic Trend Estimate: 1951–
Figure 3 shows the common stochastic trend in long and short rates based on the VECM from Table 3, constructed using the method that we just discussed. In line with the expectations theory, we interpret this stochastic trend as de- scribing permanent changes in the level of the short rate, which are reflected one-for-one in the long rate. Short rates and the stochastic trend: In panel A, we see that the short rate fluctuates around its stochastic trend. There are some lengthy periods, such as the mid-1960s, where the short rate is above the stochastic trend for a lengthy period and others, such as the mid-1990s, where the short rate is below the stochastic trend. The vertical distance is a measure of the temporary component to short rates, which we will discuss in greater detail further below. Long rates and the stochastic trend: In panel B, we see that the long rate and the stochastic trend correspond considerably more closely. This result accords with a very basic implication of the expectations theory: long rates should be highly responsive to permanent variations in the short-term interest rate. 25
(^24) Under the expectations theory with a constant term premium, the average value of the spread must be the term premium K. So, to avoid proliferation of symbols, we use that notation here. (^25) To understand the sensitivity of the trend to the form of the estimated equation for the long rate, we compared three alternative measures of the trend. The first was the test measure based on the estimated VECM (i.e., the one reported in this section); the second was based on replacing the long-rate equation with the result of a simple regression of long-rate changes on the spread (i.e., the specification that we used for testing the efficient markets restriction above) so that there was a small negative weight on the spread in the long-rate equation; and the third
68 Federal Reserve Bank of Richmond Economic Quarterly
Figure 3 Interest Rates and the Common Stochastic Trend
Variance Decompositions
It is useful to consider a decomposition of the variance of short-rate and long-rate changes into contributions in terms of changes in the temporary and permanent components. For the short-rate changes, since var(Rt ) = var(R (^) t + (Rt − R (^) t )), this decomposition takes the form
var(Rt ) = var( R¯t ) + var((Rt − ¯R (^) t ))
- 2 ∗ cov( R¯t , (Rt − ¯R (^) t ))
- 656 = 0. 105 + 0. 544 + 2 ∗ ( 0. 004 )
was based on the efficient markets restriction (i.e., placed a small positive weight on the lagged spread). While there were some differences in these trend estimates on a period-by-period basis, they tell the same basic story in terms of the general pattern of rise and fall in the stochastic trend.
