Adjusted Forward Rates as Predictors of Future Spot Rates. Abstract


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1 Stephen A. Buser G. Andrew Karolyi Anthony B. Sanders * Adjusted Forward Rates as Predictors of Future Spot Rates Abstract Prior studies indicate that the predictive power of implied forward rates for future spot rates is weak over long sample periods and typically varies dramatically across different subperiods. Fama (1976, 1984) conjectures that the low forecast power is due to a failure to control for the term premium embedded in forward rates. We show that Fama s conjecture is consistent with the data using any of four different models of the term premium. We measure the term premium using a variety of ex ante instruments, including the junk bond premium, bidask spreads in Treasury bills, the Standard & Poor s 500 stock index s dividend yield and the conditional volatility of interest rate changes using an Autoregressive Conditionally Heteroscedastic (ARCH) process. Forward rates adjusted for the term premium are reliable predictors of future spot rates over the entire period. Version: April, 1996 (fourth). Comments welcome. * Authors are, respectively, Professor, Associate Professor and Professor of Finance at the Fisher College of Business at the Ohio State University, 1775 College Road, Columbus, OH Seminar participants at Indiana University, University of Illinois and Southern Methodist University provided helpful comments. The authors are particularly grateful for clarifying discussions with Eugene Fama, Wayne Ferson, Robert Hodrick and Avi Kamara. Karolyi and Sanders acknowledge the Dice Center of Financial Economics for support. All remaining errors are our own.
2 Adjusted Forward Rates as Predictors of Future Spot Rates 1. Introduction According to the pure expectations hypothesis, forward rates provide unbiased predictions about future spot rates. Early tests reject this pure form of the expectations hypothesis (see, for example, Macauley (1938), Hickman (1942), and Culbertson (1975)). However, even if the pure expectations hypothesis is rejected, there are varying degrees of support for weaker forms of the expectations hypothesis. For example, Fama (1984) finds that the onemonth forward rate has the power to predict the spot rate one month ahead, but finds little evidence that two to fivemonth forward rates can predict future spot rates (see also Shiller (1979) and Campbell and Shiller (1991)). The evidence is further mixed by the dramatic variations in forecast power across different subperiods. For example, Mankiw and Miron (1986) find strong forecast power from 1890 through 1914, weaker forecast power from 1914 to 1933 and no power at all from 1933 through Hardouvelis (1988) finds that forward rates have predictive power up to six weeks ahead prior to October 1979, but that it diminishes substantially during the October 1979 through August 1982 period. Finally, Mishkin (1988) finds that the forecast power of forward rates is generally higher after August Fama (1976, 1984) conjectures that the weakness of the forecast power stems from model misspecification or measurement error. That is, since the forward spread (implied forward rate net of spot interest rate) incorporates both a forecast of future spot rate changes and a premium for risk, failure to control for this risk premium in 1
3 the predictive regression models of future spot rate changes on the forward spread could lead to specification bias. Specifically, in the regression forecasts of future spot rate changes on the forward spread, omitting the term premium biases the slope coefficient estimates toward zero, reduces their overall precision, and decreases the power of the forecasts. 1 Our paper investigates whether Fama s conjecture about the weakness of the forecast power can reconcile why the forecast power is invisible in some subperiods ( ) and reappears in other subperiods ( ). Specifically, we examine empirically a series of ex ante economic variables that proxy for a term premium in bond yields and allow them to interact with the forward spread in the regression forecasts. If the regression model is misspecified because the term premium is omitted, then by extracting the component of the forward spread due to the term premium, we can show how to adjust forward rates to be better predictors for future spot rate changes. The proxy variables we study include the bidask spread in the yields of one to sixmonth Treasury bills, the junk bond premium, measured as the returns spread between Moody s Baa corporate bonds and longterm government bonds, the Standard & Poor s 500 stock index dividend yield, the level of the spot rate itself, and a measure of the conditional volatility of spot rate changes, measured using 1 There are, of course, other potential explanations for the biases in tests of the expectations hypothesis of the term structure of interest rates. For example, Bekaert, Hodrick and Marshall (1995, 1996) demonstrate that large biases and dispersion in the regression test statistics are likely to arise in small samples. Kamara (1988, 1996) hypothesizes that biases in Treasury spread forecasts are due to a default premium from short sellers, which is not evident in futuresimplied Treasury bill spreads because of the existence of the clearing association that eliminates the cost of default on futures contracts. Finally, Hein, Hafer and MacDonald (1995) also demonstrate that the bias in Treasury spread forecasts may be due to timevarying term premia which can be extracted from survey data and Treasury bill futures prices. 2
4 an Autoregressive Conditionally Heteroskedastic (ARCH) model. Our results show that each of the term premium proxies interact significantly with the forward spread. These variables have the predicted effect on the coefficient estimates and the power of the tests for maturities up to sixmonths and for all subperiods from For example, for the fourmonth Treasury bills, the slope coefficients on the forward spread are adjusted upwards from 0.38 to 1.07 and associated R 2 measures increase from less than 5% on average to almost 21%. Finally, we show how to extract the component of the forward spread that is due to the term premium and how to adjust the forward spread to forecast future spot rate changes. Section 2 provides variable definitions and outlines hypotheses to be tested. Data and preliminary results are described in Section 3. We discuss the implications of our preliminary findings for supplementary tests that measure the term premium in Section 4. Section 5 provides the main results and robustness checks are discussed in Section 6. Conclusions follow. 2. Definitions and Hypotheses 2a. Definitions Following Fama (1984), we use V τ, t to denote the price at the end of month t of a unit discount bond (bill) that matures and pays $1 for certain at the end of month t+τ. The continuously compounded onemonth rate can be written as, V 1, t = exp(r t+1 ). (1) 3
5 Similarly, the price of longer maturity bills can be expressed as V τ, t = exp(r t+1  F 2, t F τ, t ), (2) where F τ, t, the forward rate for month t + τ observed at t, is F τ, t = ln (V (τ1), t / V τ, t ). (3) We can also define the onemonth holding period return from t to t+1 on a bill with τ months to maturity as, H τ,t+1 = ln(v τ,t+1 / V τ,t ), (4) and the term premium on that τmonth bill as its holding period return in excess of that for the onemonth spot rate as, (5) P τ,t+1 = H τ,t+1  r t+1. Fama (1976) shows that equation (3) for the forward rate can be decomposed into components that relate to market expectations about τperiod ahead spot rates, E t (r t+τ ), the expected premium in the onemonth return on the τperiod bill and expected changes in the series of future expected premiums, (6) F τ,t = E t (P τ,t+1 ) + [ E t (P (τ1),t+2 )E t (P (τ1),t+1 ) ] [ E t (P 2,t+τ1 )E t (P 2,t+τ2 ) ] + E t (r t+τ ). 4
6 Since all expectations are found on the righthandside of the equation, this expression is an identity; however, it is the variation in the expected premiums that obscure the predictive power of F τ,t for future spot rates, E t (r t+τ ). Assuming these adjacentperiod changes in expected premiums are negligible, and rearranging the expression in terms of future spot rates, we can simplify to: E t (r t+τ ) = F τ,t + E t (P τ,t+1 ), (7) which is the forecast relation we test in this study. 2b. Hypotheses We study the forecast power of forward rates for future spot rates using regression analysis. However, early tests of the corresponding version of (7) reveal substantial autocorrelation in the yields. To correct for this autocorrelation, Fama (1984) regresses the change in the spot rate, r t+τ  r t+1, on the current forwardspot differential or (slope of the term structure), F τ, t  r t+1, r t+τ  r t+1 = α + β ( F τ, t  r t+1 ) + ε t+τ1. (8) For example, consider the forward rate implied by a twomonth Treasury bill observed at month t, F 2, t. The spot rate observed at month t for the upcoming month is r t+1. The future spot rate that is relevant for the test is r t+2. Therefore, the change in the spot rate, r t+2  r t+1, is regressed against the slope of the term structure, F 2, t  r t+1. 5
7 In the pure form of the expectations hypothesis the forward rate should be exactly equal to the expected future spot rate which suggests the null hypothesis α equals zero and β equals one. Fama's response to concerns about term premia is to generalize the investigation of equation (8) to determine whether the slope of the term structure has power to predict future spot rates. If the coefficient β is equal to zero, there is no predictive power in the slope. If β is equal to one (and α is zero), there is evidence for the pure expectations hypothesis. If β lies between zero and one, then there is indirect evidence in favor of the expectations hypothesis, but forecasts embedded in forward rates are systematically biased upward because of the existence of a term premium. The empirical problem, however, surrounds the term premiums in longer maturity yields which can cause forward rates to exceed subsequent spot rates and exhibit less variation. Our goal is to offer marketspecific variables and macroeconomic variables that can proxy for the term premium. Moreover, we show how to extract these components to adjust the forward rate forecasts and reduce the resulting systematic bias. 3. Data and Preliminary Results 3a. Data. The U.S. Treasury bill and oneyear bond data were obtained from the Center for Research on Security Prices (CRSP) at the University of Chicago. On the last trading day of each month, a Treasury bill is chosen that has the maturity closest to six 6
8 months. After one month the same bill is chosen as the five month bill. Since bills with exact maturities are rarely available, the exact number of days to maturity is used to compute the spot or forward rate. The daily value is then multiplied by 30.4 to generate a uniform monthly series. 3b. Preliminary Results for Treasury Bills. Table 1 shows the means and standard deviations for the actual spot rate changes and forward spreads, measured as the forward rate minus the future spot rate for various maturities and subperiods from February 1959 through December The results indicate that forward rates are consistently higher than observed future spot rates. This indicates that there is likely a liquidity premium embedded in forward rates. However, it is interesting to note that while the means and standard deviations of the differences between the forward rates and the spot rates increase with maturity for most of the subperiods, the means and standard deviations for the February 1988 through December 1993 subperiod are constant across maturities. We also show that the autocorrelations in forward and spot rates is large and indicative of close to an integrated time series process. In firstdifferenced form, the autocorrelation problem is less severe. Table 2 shows the estimated βs, associated robust tstatistics and R 2 s for regression equation (5). The standard errors are adjusted for possible heteroskedasticity and serial correlation generated by overlapping data using Newey and West s (1987) procedures. 2 The results are virtually the same as the second half of 2 We chose to use 6 lags in the construction of the NeweyWest (1987) residual covariance matrix estimator. 7
9 Fama's (1984) Table 3 and Mishkin's (1988) Table 1. The primary differences between our results and Fama's (1984) is that the CRSP data has been updated and corrected for data errors. The startling finding is for the February 1988 to December 1993 subperiod. The β coefficient is not statistically significant from one for the four to sixmonth forward rate spreads. Furthermore, the R 2 measures for the regression tests are much higher than in any previous subperiod. For example, the R 2 are less than 1% for the fourmonth forward rate spreads during Fama s 1959 to 1982 subperiod, but increases to 54.8% during the 1988 to 1993 period. Although the β coefficient for 2 and 3 month forward rate spreads are not significantly different from the previous subperiod (October 1982 to June 1986), it is clear that the post1982 period has greater forecast power than the pre1982 period. The results in Table 2 are consistent with a constant (and low) premium in the forwards rates during the latter part of the 1980s and 1990s while the results for the 1982 to June 1988 period are consistent with a timevarying premium. The source of this premium could be lower inflation expectations, which declined in the latter part of the 1980s. We also have other indicators of the low premium. For example, in Table 1, the standard deviation of Treasury yield changes fell from about 0.80% per month for twomonth bills to only 0.62% during Similar declines in volatility were observed in other maturities, as well. Consider also the default spread between yields on bonds with Moody s ratings of Aaa and Baa, which averaged only 94 basis points from 1988 to 1993 whereas over the early 1980s this spread reached as high as 130 8
10 basis points. Similarly, bidask spreads on endofmonth quotes on Treasury bills ranging from two to sixmonths in maturity were as low as 5 to 7.5 basis points during the 1988 to 1993 period, although they averaged between 7 to 10.5 points from 1982 to We show how the term premium could be modeled as a function of such indicators in the next section. 4. Implications of Preliminary Findings 4a. Modeling the Term Premium The time pattern in our preliminary findings suggests potential problems with Fama's specification of test equation (8). The term premium for τperiod bills, E t (P τ,t+1 ), has been omitted from the regression model. Combining (7) and (8) produces generalized forms of Fama's test equation: r t+τ  r t+1 = α + β ( F τ, t  r t+1 ) + γ E t (P τ,t+1 ) + ε t+τ1. (9) Fama (1976) and Startz (1982) report evidence of significant temporal variation in term premia. Failure to control for this variation can produce inefficient and potentially biased forecast results. In effect, by omitting the term premium variable in equation (9), Fama's procedure forces the average effect of the term premia into the intercept (α) and adds the variable effect to the residual, likely due to a possible systematic link via the forward rate which would show up as a bias in the coefficient of interest, β. 9
11 We attempt to shore up the bias in the slope coefficient and the forecast power of the forward spread by introducing a linear forecast model of the term premium conditional on public information variables, Z t. For example, we specify: P τ,t+1 = δ 0 + Σ k δ k z k,t + η t+τ1, (10) where z k,t are the components of the information set available at the time the forward rate forecasts are made, δ k are parameters, and η t+τ+1 is the forecast error. A direct approach would integrate the term premium model of (10) into the forecast model of (9). However, since both are linked by the identity in equations (6) and (7), we are not restricted to any one particular specification for the term premium. Because our focus is on the bias in forward rate forecasts for future spot rates, we employ a specification in which the term premium model in (9) is defined as a series of interactive variables between the forward spread and the information variables that proxy for the term premium. For example, for an information variable z k,t, we estimate: r t+τ  r t+1 = α + β ( F τ, t  r t+1 ) + γ z k,t (F τ, t  r t+1 ) + ε t+τ1. (11) to evaluate the extent to which the bias in β is adjusted by the introduction of the product of z k,t and (F τ,t  r t+1 ). In this way, we are able to determine how the bias changes with different marketspecific or macroeconomic proxy variables. We perform this extended regression for each of a series of information variables and a combination of all of these variables. The next section describes the information variables and associated proxies for the term premium model. 3 3 The authors are grateful to Wayne Ferson for providing the framework to understand these issues better. 10
12 4b. Proxies for Term Premium As a check for potential specification bias, we consider several crude proxies for possible term premia effects. Each variable is measured as at the beginning of the month in the forecast regressions so that it is a genuine ex ante measure. The first proxy is the bond quality spread which we measure as the difference between the returns on Moody s Baa corporate bonds and longterm government bonds. The idea behind the use of such a proxy is that the term premia of interest may vary systematically with measures of risk premia and/or liquidity premia as reflected in the quality spread. This risk premium proxy has been used in previous empirical asset pricing studies, including Chen, Roll and Ross (1986) and Fama and French (1989). We obtain this series, denoted PREM t,, monthly from Ibbotson Associates (1995) for the period A second proxy is the bidask spread for bills of the corresponding maturities. According to the existing market microstructure literature, the quoted bidask spread has three components (Copeland and Galai, 1983; Glosten and Harris, 1988; Hasbrouck, 1988; and Stoll, 1989): the component due to order processing costs for market makers, a second reflecting their inventory holding costs, and finally the adverse selection component, which represents compensation for market makers risk in dealing with informed traders. We interpret this bidask spread proxy primarily in terms of its third component in that it measures mostly interest rate uncertainty. The 11
13 spreads, denoted SPR τ,t, are obtained separately for each maturity to sixmonths from the Treasury bill files of CRSP. The third proxy we employ is the dividend yield of the Standard & Poor s stock index. It is computed as the ratio of sum of the monthly dividends of the index during the month and dividing by the value of the portfolio at the end of the month. Fama and French (1988, 1989) have demonstrated the predictive power of the dividend yield for stock and bond returns. The intuition is that stock prices are low in relation to dividends when discount rates and expected returns are high, so the dividend yield varies positively with the market risk premium. This dividend yield series is obtained from the monthly stock master of CRSP and measured as the difference between the S&P returns with and without dividends, D t /P t. The fourth risk premium proxy is measured as the level of the spot rate, r t+1. Numerous models of the term structure of interest rates model the conditional volatility of spot interest rate changes as a function of the level of the spot rate (Cox, Ingersoll and Ross, 1985). Empirical studies have shown that the conditional volatility of bonds and stocks are predictable from the level of the spot rate, including Longstaff and Schwartz (1992), and Glosten, Jagannathan and Runkle (1993). The data is obtained directly from the CRSP bond files for the onemonth yield. Our final risk premium proxy is estimated using the family of Autoregressive Conditionally Heteroscedastic (ARCH) models (Engle, 1982). ARCH models can be used to capture the timevarying conditional second order moments and risk premia in the term structure of interest rates. Our model extends the earlier work of Engle, Lilien 12
14 and Robins (1987), Engle, Ng and Rothschild (1990) and Longstaff and Schwartz (1992). Specifically, we posit: r t+τ  r t+1 = α + β ( F τ, t  r t+1 ) + γ h t+τ1 (F τ, t  r t+1 ) + ε t+τ1, (12a) ε t+τ1 ~ N(0,h t+τ1 ), h t+τ1 = a + Σj bj h t+τj + Σ k c k ε 2 t+τk + d r 3/2 t+1, (12b) where the residuals are assumed to be conditionally zeromean and independently Gaussian distributed with variance, h t+τ1. The ARCH process projects the conditional variance of interest rate changes linearly on past squared residuals and past estimates of the conditional variance. We also add a term that allows the conditional variance to be dependent on the level of the interest rate, consistent with the findings of Chan, Karolyi, Longstaff and Sanders (1992). The model is estimated using the quasimaximum likelihood techniques of Bollerslev and Wooldridge (1992) and impose the lag structure to be GARCH (1,1) although a number of extended lag specifications were attempted. Table 3 provides summary statistics on each of these information variables from January 1963 to December The bidask spreads vary by maturity, increasing from on average 2.43 basis points for twomonth bills to 3.29 basis points for sixmonth bills. The default spread, PREM t, averages about 108 basis points with a relatively low standard deviation of 47 points. Finally, the conditional volatility estimates from the GARCH model of equation (12) reveal increasing average volatility with longer maturity bills and also higher variation in the conditional volatilities. The common feature of these information variables is that they are highly autocorrelated 13
15 with first to thirdorder autocorrelation coefficients ranging from 0.66 to An innovations series of the variables constructed as their first differences dampens down the autocorrelations considerably. We examine the sensitivity of our tests to using innovations series of the information variables in Section Extended Results Table 4 presents the results of the regressions of the extended model in equation (11) for the various term premium proxy variables, including the bidask spread (SPR τ, t ), the junk bond premium (PREM t ) and the S&P dividend yield (D t /P t ). For each variable, the regressions are run for each of three periods: the overall period from January 1963 to December 1993, the first subperiod that corresponds most closely to that of Fama (1984), or January 1963 to July 1982, and the second subperiod, August 1982 to December The first panel of the table highlights the findings in Table 2 of the simple forward spread regression on the future spot rate changes for each Treasury bill from two to sixmonths in maturity. The results again indicate that the forecast power is weaker and bias in the slope coefficients more evident in the first subperiod, in contrast with the period from The β coefficients increase from 0.1 to 0.2 across maturities up to 0.6 and 0.8 post 1982; the R 2 estimates jump from less than 1% in the period before 1982 up to 3540%, after a. BidAsk Spreads 14
16 The introduction of an interactive variable between the forward spread and the bidask spread in Treasury bill prices measurably increases the slope coefficient on the forward spread across all maturities and in all three subperiods studied. For example, the β estimate for the fourmonth Treasury bill regression in the first subperiod increases from 0.17 up to 0.93 and the R 2 increases over tenfold to about 6%. The reason for this is the usually statistically significant, negative γ coefficient on the interactive term (not reported). If we interpret the bidask spread, and especially its adverse selection component, to proxy interest rate uncertainty, then this negative coefficient implies that the forward spread forecast needs to be adjusted downward in those months in when there is greater uncertainty. This adjustment is necessary because the term premium comprises a larger component of the forward rate measure in those months. The γ coefficient is mostly significant across the different maturities, although the estimates have no patterns. This could arise because each maturity uses a different bidask spread variable. Finally, it is important to note that γ coefficients are smaller, negative values in the post1982 period, as would be expected in a relatively more stable interest rate environment. As a result, smaller adjustments would appear to be necessary; the β coefficient estimates are adjusted upward very little and the R 2 measures are largely the same with or without the interactive terms. 5b. Junk Bond Premium Table 4 also presents the extended estimates for equation (11) using the returns spread on the Baa corporate bonds and the longterm government bonds from Ibbotson and Associates, denoted PREM t. We expect the junk bond premium to be positively 15
17 related to the term premium. The results are very similar to those obtained with the bidask spreads across all maturities. For the first subperiod, the γ coefficients (not reported) are all negative although not reliably different from zero. The β coefficient estimates are higher and statistically not different than one, but our inference draws from much larger standard errors. As expected, the magnitude of the increase of the β estimates is much larger in the pre1982 period than in the later subperiod. Finally, the R 2 measures are only slightly higher than the basic forward spread regressions. 5c. Dividend Yield Table 4 indicates that the interactive variable with the forward spread and the S&P 500 dividend yield plays the same role in the spot rate forecast regressions. The γ coefficient is negative for all maturities, although not significantly different from zero (again, not reported). This interactive term allows the slope coefficient on the forward spread variable (around 0.15) to increase to values close to as low as 0.70 for the threemonth bills and as high as 0.92 for the fourmonth bills. The adjustments are generally more dramatic for the longer maturity, five and sixmonth bills, and even in the post1982 period. 5d. Level of the Spot Rate A subset of the extended regression results of Table 4 use the level of the spot rate of interest as the information variable. In a number of empirical studies, such as Glosten, Jagannathan and Runkle (1993), estimates of the conditional timevarying risk premium and volatility in S&P 500 stock returns have been shown to be dependent on the level of the spot rate of interest. The results in Table 4 show that 16
18 similar patterns obtain as for the other information variables. The γ coefficients are reliably negative (not reported) and, as a result, the β coefficients in the forecast regressions are systematically adjusted upward toward one. For the fourmonth bills, for example, the β adjusts from 0.36 to 1.51, which is insignificantly different from one. The adjusted R 2 increases from 4.8% to 17.0%. The weakest results occur for the twomonth bills, although the correction is in the right direction. 5e. Conditional Volatility Forecasts using ARCH Models Table 4 summarizes the results for the conditional volatility proxy, but for this model, we also report in Table 5 the detailed ARCH model estimates by Treasury bill maturity that were employed for the conditional volatility forecasts. We present the basic forward spread regression forecasts in the first panel (identical to those of Table 4), the ARCH model estimates for equation (12) in the second panel, and, finally, the standardized residuals for each model in the third panel. The ARCH model estimates demonstrate the type of persistence in the conditional volatilities processes in a number of financial time series: the b coefficient estimate on the lagged conditional volatility measure has a value close to 0.80 and the c coefficient on the lagged squared residual is close to The sum of the coefficients is close to 0.95 which indicates how close to integrated the series is. The dependence of the conditional volatility on the level of the interest rate, measured by the d coefficient, is also important, as demonstrated by Chan et al. (1992). The adjusted β slope coefficients in pre1982 and post1982 subperiods are higher than without the interactive variable. The degree of adjustment is smallest in magnitude, however, compared to the other term premium 17
19 proxies in Table 4. The γ coefficients are all largely negative across maturities, but again largely insignificantly different from zero. Finally, the R 2 measures (presented in Table 4) indicate greater forecast power with the adjustment terms but not as great as with the other risk premium proxies. 5f. Term Premium Model with All Information Variables Table 6 provides estimates of the extended model with all four of the term premium proxies included in the regression forecast. Several interesting patterns arise in how the various interactive variables influence each other and the forward spread variable in both subperiods and across all maturities. For example, for the fourmonth forecast model in the pre1982 subperiod, the β coefficient in the basic model is 0.17 with an R 2 of 0.5%. The extended model adjusts the β coefficient to although not reliably different from one  and the R 2 value jumps to 21.3%. The γ coefficient for the bidask spread variable, SPR τ, t, is negative and typically significant; those γ coefficients for the other risk premium proxies are negative but with larger standard errors. The main difference in the extended model combining all information variables is that the β slope coefficients are now reliably adjusted upward  particularly in the pre1982 period  as are the R 2 measures across all maturities. Figures 1 and 2 illustrate the extent of the forward bias correction that our models yield. Figure 1 shows the adjusted forward spread for fourmonth Treasury bills (solid line), which is the fitted series from the regression model in Table 6. We contrast this adjusted forward spread forecast with that of the unadjusted forecast using 18
20 the raw forward spread (dotted line). The forward spread tends to be positive over the entire sample whereas the adjusted forward spread can become negative at times, such as for example during the deflationary periods of and Figure 2 presents the adjustment factor as the difference between the two series in Figure 1. We can see that the level of adjustment required during the period is substantial compared to the periods of relatively low interest rate uncertainty during the and periods. 6. Robustness Checks 6a. Spurious Association Table 3 revealed that a common feature of the information variables used in our analysis is their high level of persistence. Serial correlation coefficients up to three lags ranged from 0.6 to 0.9. One possibility is that the bias correction in the forward spread regressions is an artifact of these seriallycorrelated time series. To gauge the sensitivity of our conclusions to this issue, we replicated our experiments in Table 6 using a crude measure of the innovations in these information variables. Table 3 showed that the differenced series yielded much lower autocorrelations. The results (available from the authors) were similar though somewhat weaker. The β coefficients were adjusted upward, as before, but not as dramatically, and statistically we would reject that they were equal to one. The R 2 measures were also less dramatically affected. 6b. Peso Problems 19
21 Bekaert, Hodrick and Marshall (1995, 1996) document extreme bias and dispersion in the small sample distributions in regressionbased tests of the expectations hypothesis, as in this study. They argue that these biases derive from the extreme persistence in short term interest rates. To illustrate this phenomenon, they estimate a regimeswitching model dependent on the level of the spot rate; the forward spread forecasts of future spot rate changes are allowed to be different in lowinterestrate and highinterestrate states. This model is similar in spirit to our extended regression tests in Table 4 in which the term premium proxy variable is the level of the spot rate. Their tests show that the expectations hypothesis is more strongly rejected when these small sample biases are corrected; that is, the β coefficient should approach perhaps as high as 1.25 or 1.50 in our regressions. Our results in Table 4 are somewhat consistent with this premise in that we show that the β estimates with the level of the spot rate variable can even adjust above one for each of the four, five and sixmonth Treasury bill regressions. However, future research should explore the implications of small sample biases for our regressions. 7. Summary and Implications Prior studies indicate that the predictive power of implied forward rates for future spot rates is weak over long sample periods and typically varies dramatically across different subperiods. Fama (1976, 1984) conjectures that the low forecast power in general is due to model misspecification or measurement error that is introduced by a failure to control for the term premium embedded in forward rates. We show that 20
22 Fama s conjecture is consistent with the data using any of four different models of the term premium. Forward rates adjusted for the term premium are reliable predictors of future spot rates over the entire period. We measure the term premium using a variety of ex ante instruments, including the junk bond premium, bidask spreads in Treasury bills, the Standard & Poor s 500 stock index s dividend yield and conditional volatility of interest rate changes using an Autoregressive Conditionally Heteroscedastic (ARCH) process. Using these proxies, we show how to quantify the magnitude of the bias in the forward rate forecasts introduced by the term premium and how to adjust for it. 21
23 References Bekaert, Geert, Robert Hodrick and David Marshall, 1995, Peso Problem Explanations for Term Structure Anomalies, Northwestern University working paper. Bekaert, Geert, Robert Hodrick and David Marshall, 1996, On Biases in Tests of the Expectations Hypothesis of the Term Structure of Interest Rates, Northwestern University working paper. Bollerslev, Tim and Jeffrey Wooldridge, 1992, Quasimaximum Likelihood Estimation and Inference in Dynamic Models with Timevarying Covariances, Econometric Reviews 11, Campbell, John, and Robert Shiller, 1991, Yield Spreads and Interest Rate Movements: A Bird s Eye View, Review of Financial Studies 58, Chan, K.C., G. Andrew Karolyi, Francis A. Longstaff and Anthony B. Sanders, 1992, An Empirical Comparison of Alternative Models of the Shortterm Interest Rate, Journal of Finance 47, Chen, Naifu, Richard Roll and Stephen A. Ross, 1986, Economic Forces and the Stock Market, Journal of Business 59, Copeland, Thomas and Dan Galai, 1983, Information Effects on the Bidask Spread, Journal of Finance 38, Cox, John C., Jonathan E. Ingersoll, Jr. and Stephen A. Ross, 1981, A Reexamination of Traditional Hypotheses about the Term Structure of Interest Rates, Journal of Finance 36, Cox, John C., Jonathan E. Ingersoll, Jr. and Stephen A. Ross, 1985, A Theory of the Term Structure of Interest Rates, Econometrica 53, Culbertson, J. W., 1957, The Term Structure of Interest Rates, Quarterly Journal of Economics, Engle, Robert, 1982, Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of UK Inflation, Econometrica 50, Engle, Robert, David Lilien and Russell Robins, 1987, Estimating Timevarying Risk Premia in the Term Structure: The ARCHM Model, Econometrica 55,
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25 Kamara, Avi, 1996, The Relation between Defaultfree Interest Rates and Expected Economic Growth is Stronger than You Think, University of Washington working paper. Longstaff, Francis and Eduardo Schwartz, 1992, Interest Rate Volatility and the Term Structure: A Twofactor General Equilibrium Model, Journal of Finance 47, Macauley, Frederick R., 1938, The Movements of Interest Rates, Bond Yields, Bond Yields and Stock Prices in the United States Since 1859, National Bureau of Economic Research (New York: Columbia University Press). Mankiw, N. Gregory and Jeffrey A. Miron, 1986, The Changing Behavior of the Term Structure of Interest Rates, Quarterly Journal of Economics 101, Mishkin, Frederic S., 1988, The Information in the Term Structure: Some Further Results, Journal Applied Econometrics 3, Newey, Whitney and Kenneth West, 1987, A Simple Positive, Semidefinite, Heteroskedasticity Consistent Covariance Matrix, Econometrica 55, Shiller, Robert, 1979, The Volatility of Longterm Interest Rates and Expectations Models of the Term Structure, Journal of Political Economy 87, Startz, Richard, 1982, Do Forecast Errors or Term Premia Really Make the Difference Between Long and Short Rates?, Journal of Financial Economics 10, Stoll, Hans, 1989, Inferring the Components of the Bidask Spread, Journal of Finance 44,
26 Table 1. Summary statistics for the change in future spot rate and forward spread for Treasury bills in selected subperiods from February 1959 through December F τ, t is the 1month forward rate observed at t for τ months ahead and r t+1 is the onemonth spot rate at t. All statistics are computed for the full sample from February 1959 to December 1993 and various subperiods. Data is from the Government Bond Files of the Center for Research in Security Prices (CRSP). ρ k denotes the kth order autocorrelation coefficient. Statistic r t+2 r t+1 r t+3 r t+1 r t+4 r t+1 r t+5 r t+1 r t+6 r t+1 F 2,t r t+1 F 3,t r t+1 F 4,t r t+1 F 5,t r t+1 F 6,t  r t : :12 Mean Standard Deviation tstatistic (Mean=0) Skewness Kurtosis ρ ρ ρ : : 7 Mean Standard Deviation tstatistic (Mean=0) Skewness Kurtosis ρ ρ ρ : : 12 Mean Standard Deviation tstatistic (Mean=0) Skewness Kurtosis ρ ρ ρ
27 Table 2. Regression model estimates of the change in the future spot rate on the forward spread. F τ, t is the onemonth forward rate observed at t for τ months ahead and r t+1 is the onemonth spot rate at t. Estimates are computed for the full sample from February 1959 to December 1993 and various subperiods. Data is from the Government Bond Files of the Center for Research in Security Prices (CRSP). Standard errors are computed using NeweyWest correction for heteroscedasticity and serial correlation and associated tstatistics are reported in parentheses. r t+τ  r t+1 = α + β ( F τ, t  r t+1 ) + ε t+τ1 ( 5) r t+2  r t+1 r t+3  r t+1 r t+4  r t+1 r t+5  r t+1 r t+6  r t+1 Models β t (β=0) Adj.R 2 β t (β=0) Adj.R 2 β t (β=0) Adj.R 2 β t (β=0) Adj.R 2 β t (β=0) Adj.R 2 Overall Period: 1959: : Fama Subperiod: 1959: : Subperiods: 1959: : : : : : : : : : : : : :
28 Table 3. Summary statistics on Informational Variables for Term Premium Proxy from January 1963 through December BA k denotes the bidask spread (in percent per month) on the kmonth Treasury bill from the Government Bond Files of the Center for Research in Security Prices (CRSP). PREM is the yield spread between Moody s Baa and Aaa corporate bonds, D t /P t is the Standard and Poor s 500 stock index dividend yield, r t+1 is the level of the onemonth spot rate at t. h t+τ1 denotes the conditional volatility from ARCH model estimates of the τperiod difference in the spot rates (see Table 5). ρ k denotes the kth order autocorrelation coefficient. Statistic BA 2,t BA 3,t BA 4,t BA 5,t BA 6,t PREM t D t /P t r t+1 h t+1 h t+2 h t+3 h t+4 h t+5 Mean Standard Deviation tstatistic (Mean=0) Skewness Kurtosis Autocorrelations: ρ 1 ρ 2 ρ 3 Differences: ρ ρ ρ
29 Table 4. Regression estimates of the change in the future spot rate on the forward spread and a risk premium in τperiod bonds. The risk premium is measured by an interactive term with an instrumental variable with the forward spread, F τ, t  r t+1. The instrumental variables include: the bid/ask spread of τperiod bonds, SPR τ, the junk bond premium measured by the returns spread between Moody s Baa corporate bonds and government bonds, PREM t, the Standard and Poor 500 dividend yield, D t /P t, the level of the short rate, r t+1, and the conditional variance of interest rate changes estimated from ARCH models of Table 5. Estimates are computed for January 1963 to December 1993 and two subperiods from January 1963 to July 1982 and August 1982 to December Standard errors are computed using NeweyWest correction for heteroscedasticity and serial correlation with six lagged autocovariances. The tstatistic is computed under the null hypothesis that β=1. The basic model is based on equation (5) in Table 2: r t+τ  r t+1 = α + β ( F τ, t  r t+1 ) + ε t+τ1 ( 5) The adjusted model augments (5) by including interactive term with the instrumental variable, X t : r t+τ  r t+1 = α + β ( F τ, t  r t+1 ) + γ [ X t ( F τ, t  r t+1 ) ] + ε t+τ1 (10) r t+2  r t+1 r t+3  r t+1 r t+4  r t+1 r t+5  r t+1 r t+6  r t+1 Models β t (β=1) Adj.R 2 β t (β=1) Adj.R 2 β t (β=1) Adj.R 2 β t (β=1) Adj.R 2 β t (β=1) Adj.R 2 Overall Period Unadjusted SPR t PREM t : D t /P t h t r t : :7 Unadjusted SPR t PREM t : D t /P t h t r t : :12 Unadjusted SPR t PREM t : D t /P t
30 h t r t Table 5. ARCH model estimates of the change in the future spot rate on the forward spread and a risk premium in τperiod bonds. The risk premium is measured by interactive term of conditional volatility of interest rate changes, h t+τ1, with the forward spread, F τ, t  r t+1. Estimates are computed from January 1963 to December 1993 and two subperiods from January 1963 to July 1982 and August 1982 to December The model is estimated using quasimaximum likelihood methods based on Bollerslev and Wooldridge (1992). Robust t statistics are reported in parentheses. R 2 are computed from secondpass regressions of the spot rate changes on the forward spread and the interactive variable. The The ARCH model augments equation (5) of Table 2 by including interactive term with conditional volatility, h t+τ1, using GARCH(1,1) specification including level of the shortrate, r t+1 : r t+τ  r t+1 = α + β ( F τ, t  r t+1 ) + γ [ h t+τ1 ( F τ, t  r t+1 ) ] + ε t+τ1 + θ ε t+τ2 (11) ε t+τ1 ~ N(0, h t+τ1 ) h t+τ1 = a + b h t+τ2 + c ε 2 3/2 t+τ2 + d r t+1 (12) Overall Period 1963:11993:12 Subperiod I 1963:11982:7 Subperiod II 1982:81993:12 Coefficients r t+2 r t+1 r t+3 r t+1 r t+4 r t+1 r t+5 r t+1 r t+6 r t+1 r t+2 r t+1 r t+3 r t+1 r t+4 r t+1 r t+5 r t+1 r t+6 r t+1 r t+2 r t+1 r t+3 r t+1 r t+4 r t+1 r t+5 r t+1 r t+6 r t+1 β t (β=1) (4.76) (3.00) (2.41) (2.26) (2.06) (3.06) (1.48) (1.26) (1.05) (0.61) (3.00) (4.33) (4.75) (3.82) (3.81) γ (0.27) (1.13) (0.90) (0.53) (0.12) (0.27) (0.73) (0.62) (0.43) (0.17) (1.14) (0.65) (0.56) (1.26) (1.72) θ (1.03) (13.51) (1.14) (3.84) (1.78) (0.86) (14.64) (1.00) (7.90) (1.30) (0.79) (1.97) (0.63) (2.51) (1.25) a (0000s) (0.04) (0.04) (0.05) (0.05) (0.06) (0.02) (0.03) (0.03) (0.04) (0.05) (0.06) (0.12) (0.06) (0.04) (0.06) b (14.42) (16.39) (6.91) (12.22) (25.50) (7.96) (7.32) (4.54) (11.95) (20.36) (15.94) (1.37) (14.24) (8.68) (24.52) c (1.90) (2.41) (1.79) (2.09) (2.81) (1.90) (2.28) (1.50) (1.88) (2.30) (0.76) (1.01) (2.26) (1.30) (2.23) d (0000s) (2.67) (3.24) (2.01) (2.18) (2.83) (1.35) (1.52) (1.09) (1.92) (2.00) (3.49) (3.43) (12.73) (1.52) (2.87) Std. Residuals: Mean Std. Dev Skewness
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