Stephen A. BuserG. Andrew KarolyiAnthony 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 spotrates is weak over long sample periods and typically varies dramatically acrossdifferent subperiods. Fama (1976, 1984) conjectures that the low forecast power is dueto a failure to control for the term premium embedded in forward rates. We show thatFama’s conjecture is consistent with the data using any of four different models of theterm premium. We measure the term premium using a variety of ex ante instruments,including the junk bond premium, bid-ask spreads in Treasury bills, the Standard &Poor’s 500 stock index’s dividend yield and the conditional volatility of interest ratechanges using an Autoregressive Conditionally Heteroscedastic (ARCH) process.Forward rates adjusted for the term premium are reliable predictors of future spot ratesover the entire 1963-1993 period.

Version: April, 1996 (fourth). Comments welcome.

*Authors are, respectively, Professor, Associate Professor and Professor of Finance at the FisherCollege of Business at the Ohio State University, 1775 College Road, Columbus, OH 43210-1399.Seminar participants at Indiana University, University of Illinois and Southern Methodist Universityprovided helpful comments. The authors are particularly grateful for clarifying discussions withEugene Fama, Wayne Ferson, Robert Hodrick and Avi Kamara. Karolyi and Sanders acknowledge theDice Center of Financial Economics for support. All remaining errors are our own.

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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 one-month forward rate has the power to predict

the spot rate one month ahead, but finds little evidence that two- to five-month 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 1984. 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 1982.

Fama (1976, 1984) conjectures that the weakness of the forecast power stems

from model mis-specification 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

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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

(1959-82) and re-appears in other subperiods (1982-93). 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 mis-specified 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 bid-ask spread in the

yields of one- to six-month Treasury bills, the junk bond premium, measured as the

returns spread between Moody’s Baa corporate bonds and long-term 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 hypothesisof 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 smallsamples. Kamara (1988, 1996) hypothesizes that biases in Treasury spread forecasts are due to adefault premium from short sellers, which is not evident in futures-implied Treasury bill spreadsbecause of the existence of the clearing association that eliminates the cost of default on futurescontracts. Finally, Hein, Hafer and MacDonald (1995) also demonstrate that the bias in Treasuryspread forecasts may be due to time-varying term premia which can be extracted from survey data andTreasury bill futures prices.

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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 six-months and for all subperiods from 1959-93. For

example, for the four-month Treasury bills, the slope coefficients on the forward

spread are adjusted upwards from 0.38 to 1.07 and associated R2 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 Vτ, 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 one-month rate can be written as,

V1, t = exp(-rt+1).

(1)

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Similarly, the price of longer maturity bills can be expressed as

Vτ, t = exp(-rt+1 - F2, t - ... -Fτ, t),

(2)

where Fτ, t , the forward rate for month t + τ observed at t, is

Fτ, t = ln (V(τ-1), t / Vτ, t).

(3)

We can also define the one-month holding period return from t to t+1 on a bill with τ

months to maturity as,

Hτ,t+1 = ln(Vτ,t+1 / Vτ,t), (4)

and the term premium on that τ-month bill as its holding period return in excess of that

for the one-month spot rate as,

Pτ,t+1 = Hτ,t+1 - rt+1.

(5)

Fama (1976) shows that equation (3) for the forward rate can be decomposed into

components that relate to market expectations about τ-period ahead spot rates, Et(rt+τ),

the expected premium in the one-month return on the τ-period bill and expected

changes in the series of future expected premiums,

Fτ,t = Et (Pτ,t+1) + [ Et(P(τ-1),t+2)-Et(P(τ-1),t+1) ] + ... + [ Et(P2,t+τ-1)-Et(P2,t+τ-2) ] + Et(rt+τ).

(6)

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Since all expectations are found on the right-hand-side of the equation, this expression

is an identity; however, it is the variation in the expected premiums that obscure the

predictive power of Fτ,t for future spot rates, Et (rt+τ). Assuming these adjacent-period

changes in expected premiums are negligible, and rearranging the expression in terms

of future spot rates, we can simplify to:

Et(rt+τ) = Fτ,t + Et (Pτ,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, rt+τ - rt+1 , on the current forward-spot

differential or (slope of the term structure), Fτ, t - rt+1 ,

rt+τ - rt+1 = α + β ( Fτ, t - rt+1 ) + εt+τ-1.

(8)

For example, consider the forward rate implied by a two-month Treasury bill observed

at month t, F2, t. The spot rate observed at month t for the upcoming month is rt+1 .

The future spot rate that is relevant for the test is rt+2 . Therefore, the change in the

spot rate, rt+2 - rt+1 , is regressed against the slope of the term structure, F2, t - rt+1 .

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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 market-specific 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 one-year 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

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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 1990.

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 first-differenced form, the autocorrelation problem is

less severe.

Table 2 shows the estimated βs, associated robust t-statistics and R2s 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 Newey-West (1987) residual covariance matrixestimator.

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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 six-month

forward rate spreads. Furthermore, the R2 measures for the regression tests are much

higher than in any previous subperiod. For example, the R2 are less than 1% for the

four-month 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 post-1982 period has greater forecast

power than the pre-1982 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 time-varying 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

two-month bills to only 0.62% during 1988-93. 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

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basis points. Similarly, bid-ask spreads on end-of-month quotes on Treasury bills

ranging from two- to six-months 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 1988. 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 τ-period bills, Et

(Pτ,t+1), has been omitted from the regression model. Combining (7) and (8)

produces generalized forms of Fama's test equation:

rt+τ - rt+1 = α + β ( Fτ, t - rt+1 ) + γ Et (Pτ,t+1) + εt+τ-1. (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.

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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, Zt. For example, we specify:

Pτ,t+1 = δ0 + Σk δk zk,t + ηt+τ-1, (10)

where zk,t are the components of the information set available at the time the forward

rate forecasts are made, δk are parameters, and ηt+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 zk,t, we estimate:

rt+τ - rt+1 = α + β ( Fτ, t - rt+1 ) + γ zk,t (Fτ, t - rt+1 ) + εt+τ-1.

(11)

to evaluate the extent to which the bias in β is adjusted by the introduction of the

product of zk,t and (Fτ,t - rt+1). In this way, we are able to determine how the bias

changes with different market-specific 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 issuesbetter.

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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 long-term 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 PREMt,, monthly from Ibbotson Associates (1995) for

the period 1963-1993.

A second proxy is the bid-ask spread for bills of the corresponding maturities.

According to the existing market microstructure literature, the quoted bid-ask 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 bid-ask spread proxy primarily in

terms of its third component in that it measures mostly interest rate uncertainty. The

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spreads, denoted SPRτ,t, are obtained separately for each maturity to six-months 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, Dt /Pt.

The fourth risk premium proxy is measured as the level of the spot rate, rt+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 one-month 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 time-varying conditional second order moments and risk premia in

the term structure of interest rates. Our model extends the earlier work of Engle, Lilien

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and Robins (1987), Engle, Ng and Rothschild (1990) and Longstaff and Schwartz

(1992). Specifically, we posit:

rt+τ - rt+1 = α + β ( Fτ, t - rt+1 ) + γ ht+τ-1 (Fτ, t - rt+1 ) + εt+τ-1, (12a)

εt+τ-1 ~ N(0,ht+τ-1), ht+τ-1 = a + Σj bj ht+τ-j + Σk ck εt+τ-k2 + d rt+1

3/2,

(12b)

where the residuals are assumed to be conditionally zero-mean and independently

Gaussian distributed with variance, ht+τ-1 . 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 quasi-

maximum 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 1993. The bid-ask spreads vary by maturity,

increasing from on average 2.43 basis points for two-month bills to 3.29 basis points

for six-month bills. The default spread, PREMt, 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

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with first- to third-order autocorrelation coefficients ranging from 0.66 to 0.97. 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 6.

5. 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 bid-ask

spread (SPRτ, t), the junk bond premium (PREMt) and the S&P dividend yield (Dt /Pt).

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 1993. 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 six-months 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 1982-93. The β

coefficients increase from 0.1 to 0.2 across maturities up to 0.6 and 0.8 post 1982; the

R2 estimates jump from less than 1% in the period before 1982 up to 35-40%, after

1982.

5a. Bid-Ask Spreads

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The introduction of an interactive variable between the forward spread and the

bid-ask 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 four-month Treasury bill regression in the first subperiod

increases from 0.17 up to 0.93 and the R2 increases over ten-fold 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 bid-ask 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 bid-ask spread variable. Finally, it is important to note that γ coefficients are

smaller, negative values in the post-1982 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 R2

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 long-term government bonds from Ibbotson

and Associates, denoted PREMt. We expect the junk bond premium to be positively

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related to the term premium. The results are very similar to those obtained with the

bid-ask 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 pre-1982 period than in the later subperiod. Finally, the

R2 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

three-month bills and as high as 0.92 for the four-month bills. The adjustments are

generally more dramatic for the longer maturity, five- and six-month bills, and even in

the post-1982 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 time-varying

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

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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 four-month bills,

for example, the β adjusts from 0.36 to 1.51, which is insignificantly different from

one. The adjusted R2 increases from 4.8% to 17.0%. The weakest results occur for the

two-month 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 0.15. 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 pre-1982 and

post-1982 subperiods are higher than without the interactive variable. The degree of

adjustment is smallest in magnitude, however, compared to the other term premium

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proxies in Table 4. The γ coefficients are all largely negative across maturities, but

again largely insignificantly different from zero. Finally, the R2 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 four-month

forecast model in the pre-1982 subperiod, the β coefficient in the basic model is 0.17

with an R2 of 0.5%. The extended model adjusts the β coefficient to 2.59 - although

not reliably different from one - and the R2 value jumps to 21.3%. The γ coefficient for

the bid-ask spread variable, SPRτ, 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

pre-1982 period - as are the R2 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 four-month 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

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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 1976-77 and 1982-83. 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 1979-82 period is

substantial compared to the periods of relatively low interest rate uncertainty during

the 1965-70 and 1985-93 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 serially-correlated 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 R2 measures were also less dramatically

affected.

6b. “Peso” Problems

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Bekaert, Hodrick and Marshall (1995, 1996) document extreme bias and

dispersion in the small sample distributions in regression-based 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 regime-switching model dependent on the level of the spot rate; the forward

spread forecasts of future spot rate changes are allowed to be different in low-interest-

rate and high-interest-rate 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

six-month 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 mis-specification or measurement error that is introduced by

a failure to control for the term premium embedded in forward rates. We show that

21

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 1963-1993 period. We measure the term premium

using a variety of ex ante instruments, including the junk bond premium, bid-ask

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.

22

References

Bekaert, Geert, Robert Hodrick and David Marshall, 1995, “Peso ProblemExplanations for Term Structure Anomalies,” Northwestern University workingpaper.

Bekaert, Geert, Robert Hodrick and David Marshall, 1996, “On Biases in Tests of theExpectations Hypothesis of the Term Structure of Interest Rates,” NorthwesternUniversity working paper.

Bollerslev, Tim and Jeffrey Wooldridge, 1992, “Quasi-maximum LikelihoodEstimation and Inference in Dynamic Models with Time-varying Covariances,”Econometric Reviews 11, 143-172.

Campbell, John, and Robert Shiller, 1991, “Yield Spreads and Interest RateMovements: A Bird’s Eye View,” Review of Financial Studies 58, 495-514.

Chan, K.C., G. Andrew Karolyi, Francis A. Longstaff and Anthony B. Sanders, 1992,“An Empirical Comparison of Alternative Models of the Short-term InterestRate,” Journal of Finance 47, 1209-1227.

Chen, Nai-fu, Richard Roll and Stephen A. Ross, 1986, “Economic Forces and theStock Market,” Journal of Business 59, 383-403.

Copeland, Thomas and Dan Galai, 1983, “Information Effects on the Bid-ask Spread,”Journal of Finance 38, 1457-1469.

Cox, John C., Jonathan E. Ingersoll, Jr. and Stephen A. Ross, 1981, “A Re-examination of Traditional Hypotheses about the Term Structure of InterestRates,” Journal of Finance 36, 769-799.

Cox, John C., Jonathan E. Ingersoll, Jr. and Stephen A. Ross, 1985, “A Theory of theTerm Structure of Interest Rates,” Econometrica 53, 385-407.

Culbertson, J. W., 1957, “The Term Structure of Interest Rates,” Quarterly Journal ofEconomics, 485-517.

Engle, Robert, 1982, “Autoregressive Conditional Heteroskedasticity with Estimatesof the Variance of UK Inflation,” Econometrica 50, 987-1008.

Engle, Robert, David Lilien and Russell Robins, 1987, “Estimating Time-varying RiskPremia in the Term Structure: The ARCH-M Model,” Econometrica 55, 391-407.

23

Engle, Robert, Victor Ng and Michael Rothschild, 1990, “Asset Pricing with a FactorARCH Covariance Structure: Empirical Estimates for Treasury Bills.” Journalof Econometrics 45, 213-238.

Fama, Eugene F., 1976, “Forward Rates as Predictors of Future Spot Rates,” Journalof Financial Economics 3, 361-377.

Fama, Eugene F., 1984, “The Information in the Term Structure,” Journal of FinancialEconomics 13, 509-576.

Fama, Eugene F., 1990, “Term-Structure Forecasts of Interest Rates, Inflation, andReal Returns,” Journal of Monetary Economics, 25, 59-76.

Fama, Eugene F. and Robert R. Bliss, 1987, “The Information in Long-MaturityForward Rates,” American Economic Review 77, 680-692.

Fama, Eugene F. and Kenneth R. French, 1988, “Dividend Yields and Expected StockReturns,” Journal of Financial Economics 22, 3-25.

Fama, Eugene F. and Kenneth R. French, 1989, “Business Conditions and ExpectedReturns on Stocks and Bonds,” Journal of Financial Economics 25, 23-49.

Glosten, Lawrence and Lawrence Harris, 1988, “Estimating the Components of theBid-ask Spread,” Journal of Financial Economics, 21, 123-142.

Glosten, Lawrence, Ravi Jagannathan, and David Runkle, “Seasonal Patterns in theVolatility of Stock Index Excess Returns,” Journal of Finance 48, 1779-1801.

Hafer, R., Scott Hein and Scott MacDonald, 1995, “Predicting the Ex Post TermPremium in Treasury Bill Rates,” Texas Tech University working paper.

Hardouvelis, Gikas A., 1988, “The Predictive Power of the Term Structure duringRecent Monetary Regimes,” Journal of Finance 43, 339-356.

Hasbrouck, Joel, 1988, “Trades, Quotes, Inventories and Information,” Journal ofFinancial Economics 22, 229-252.

Hickman, W. Braddock, 1942, “The Term Structure of Interest Rates: An ExploratoryAnalysis,” National Bureau of Economic Research, mimeo.

Ibbotson & Associates, 1995, Stocks, Bonds, Bills and Inflation: 1995 Yearbook andMarket Results for 1926-1994, Chicago, IL.

Kamara, Avi, 1988, “Market Trading Structure and Asset Pricing: Evidence from theTreasury Bill Markets,” Review of Financial Studies 1, 357-375.

24

Kamara, Avi, 1996, “The Relation between Default-free Interest Rates and ExpectedEconomic Growth is Stronger than You Think,” University of Washingtonworking paper.

Longstaff, Francis and Eduardo Schwartz, 1992, “Interest Rate Volatility and the TermStructure: A Two-factor General Equilibrium Model,” Journal of Finance 47,1259-1281.

Macauley, Frederick R., 1938, The Movements of Interest Rates, Bond Yields, BondYields and Stock Prices in the United States Since 1859, National Bureau ofEconomic Research (New York: Columbia University Press).

Mankiw, N. Gregory and Jeffrey A. Miron, 1986, “The Changing Behavior of theTerm Structure of Interest Rates,” Quarterly Journal of Economics 101, 211-228.

Mishkin, Frederic S., 1988, “The Information in the Term Structure: Some FurtherResults,” Journal Applied Econometrics 3, 307-314.

Newey, Whitney and Kenneth West, 1987, “A Simple Positive, Semi-definite,Heteroskedasticity Consistent Covariance Matrix,” Econometrica 55, 703-708.

Shiller, Robert, 1979, “The Volatility of Long-term Interest Rates and ExpectationsModels of the Term Structure,” Journal of Political Economy 87, 1190-1219.

Startz, Richard, 1982, “Do Forecast Errors or Term Premia Really Make theDifference Between Long and Short Rates?,” Journal of Financial Economics10, 323-329.

Stoll, Hans, 1989, “Inferring the Components of the Bid-ask Spread,” Journal ofFinance 44, 115-134.

Table 1. Summary statistics for the change in future spot rate and forward spread for Treasury bills in selected subperiods fromFebruary 1959 through December 1993. Fτ, t is the 1-month forward rate observed at t for τ months ahead and r t+1 is the one-month spotrate at t. All statistics are computed for the full sample from February 1959 to December 1993 and various subperiods. Data is from theGovernment Bond Files of the Center for Research in Security Prices (CRSP). ρk denotes the k-th order autocorrelation coefficient.

Statistic rt+2-rt+1 rt+3-rt+1 rt+4-rt+1 rt+5-rt+1 rt+6-rt+1 F2,t-rt+1 F3,t-rt+1 F4,t-rt+1 F5,t-rt+1 F6,t - rt+1

1959: 2 - 1993:12Mean 0.0013 0.0023 0.0026 0.0020 0.0025 0.6634 0.9117 0.7410 1.1075 1.2755Standard Deviation 0.7983 1.0532 1.2506 1.3883 1.4668 0.7660 0.8225 0.7568 0.9526 1.0053t-statistic (Mean=0) 0.03 0.04 0.04 0.03 0.04 17.73 22.69 20.04 23.80 25.97Skewness -0.87 -1.05 -0.81 -0.70 -0.35 2.37 1.71 1.36 1.42 1.07Kurtosis 9.82 8.58 10.29 7.38 6.42 7.76 5.05 4.76 4.59 3.28ρ1 -0.132 0.415 0.561 0.630 0.694 0.108 0.348 0.222 0.343 0.461ρ2 -0.015 -0.135 0.178 0.316 0.444 0.172 0.212 0.255 0.269 0.347ρ3 -0.073 -0.156 -0.255 0.019 0.142 0.184 0.131 0.108 0.283 0.2371959: 2 - 1982: 7Mean 0.0144 0.0307 0.0486 0.0660 0.0857 0.5706 0.8929 0.7288 1.0779 1.3121Standard Deviation 0.8262 1.1230 1.3510 1.4952 1.5912 0.6532 0.8132 0.7789 0.8651 1.0170t-statistic (Mean=0) 0.29 0.46 0.60 0.74 0.90 14.67 18.44 15.71 20.92 21.66Skewness -1.15 -1.25 -1.06 -0.95 -0.55 2.19 1.58 1.33 0.62 0.80Kurtosis 11.88 9.59 11.01 7.99 6.64 6.44 5.21 5.45 2.30 2.52ρ1 -0.070 0.451 0.587 0.673 0.722 0.115 0.271 0.122 0.238 0.395ρ2 0.003 -0.094 0.202 0.351 0.463 0.235 0.201 0.228 0.238 0.329ρ3 -0.103 -0.154 -0.219 0.023 0.131 0.171 0.151 0.105 0.228 0.2391982: 8 - 1993: 12Mean -0.0259 -0.0569 -0.0943 -0.1337 -0.1751 0.8543 0.9504 0.7661 1.1684 1.2002Standard Deviation 0.7393 0.8909 1.0049 1.1219 1.1423 0.9312 0.8428 0.7114 1.1124 0.9800t-statistic (Mean=0) -0.41 -0.74 -1.09 -1.37 -1.76 10.73 13.19 12.60 12.29 14.33Skewness -0.08 -0.27 0.34 0.34 0.43 2.21 1.96 1.46 2.09 1.71Kurtosis 3.14 1.64 1.51 1.12 1.32 6.16 4.90 2.75 5.34 5.61ρ1 -0.276 0.287 0.462 0.474 0.566 0.037 0.427 0.419 0.403 0.476ρ2 -0.020 -0.206 0.126 0.215 0.382 0.050 0.129 0.203 0.206 0.295ρ3 0.026 -0.098 -0.325 0.052 0.197 0.150 0.059 0.022 0.296 0.201

1

Table 2. Regression model estimates of the change in the future spot rate on the forward spread . Fτ, t is the one-month forward rateobserved at t for τ months ahead and r t+1 is the one-month spot rate at t. Estimates are computed for the full sample from February 1959 toDecember 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 Newey-West correction for heteroscedasticity and serial correlation and associated t-statistics arereported in parentheses.

rt+τ - rt+1 = α + β ( Fτ, t - rt+1 ) + εt+τ-1 ( 5)

rt+2 - rt+1 rt+3 - rt+1 rt+4 - rt+1 rt+5 - rt+1 rt+6 - rt+1

Models β t (β=0) Adj.R2 β t (β=0) Adj.R2 β t (β=0) Adj.R2 β t (β=0) Adj.R2 β t (β=0) Adj.R2

Overall Period:1959: 2 - 1993:12 0.481 8.39 0.211 0.372 3.41 0.082 0.377 2.54 0.050 0.312 4.07 0.044 0.289 3.78 0.037Fama Subperiod:1959: 2 - 1982: 7 0.459 4.46 0.129 0.233 1.44 0.025 0.189 0.96 0.008 0.107 0.92 0.000 0.132 1.34 0.004Subperiods:1959: 2 - 1964: 1 0.444 3.94 0.223 0.299 3.26 0.127 0.490 3.34 0.134 0.064 0.81 -0.010 0.053 0.46 -0.0101964: 2 - 1969: 1 0.500 3.96 0.368 0.387 3.41 0.224 0.356 3.17 0.116 0.324 4.05 0.183 0.265 3.41 0.1731969: 2 - 1974: 1 0.158 1.97 0.018 0.050 0.31 -0.016 0.212 1.24 0.010 -0.029 -0.22 -0.017 0.032 0.21 -0.0171974: 2 - 1979: 1 0.591 3.88 0.116 0.081 0.57 -0.012 0.379 1.48 0.038 0.126 0.63 -0.011 0.073 0.25 -0.0161979: 2 - 1982: 7 0.694 3.65 0.177 0.418 1.26 0.029 0.183 0.58 -0.018 0.213 0.71 -0.018 0.308 1.24 -0.0071982: 8 - 1988: 1 0.609 12.64 0.504 0.637 5.83 0.356 0.751 5.20 0.294 0.515 5.68 0.308 0.543 3.90 0.2351988: 2 - 1993:12 0.471 3.84 0.418 0.786 8.40 0.452 1.192 10.48 0.548 1.096 11.86 0.561 1.140 7.41 0.535

2

Table 3. Summary statistics on Informational Variables for Term Premium Proxy from January 1963 through December 1993. BAk

denotes the bid-ask spread (in percent per month) on the k-month Treasury bill from the Government Bond Files of the Center for Researchin Security Prices (CRSP). PREM is the yield spread between Moody’s Baa and Aaa corporate bonds, D t/Pt is the Standard and Poor’s 500stock index dividend yield, r t+1 is the level of the one-month spot rate at t. h t+τ-1 denotes the conditional volatility from ARCH modelestimates of the τ-period difference in the spot rates (see Table 5). ρk denotes the k-th order autocorrelation coefficient.

Statistic BA2,t BA3,t BA4,t BA5,t BA6,t

PREMt

Dt/Pt rt+1 ht+1 ht+2 ht+3 ht+4 ht+5

Mean 0.0243 0.0187 0.0359 0.0413 0.0329 0.0108 0.0371 0.0604 0.0578 0.0586 0.0764 0.0795 0.0841Standard Deviation 0.0222 0.0175 0.0276 0.0305 0.0295 0.0047 0.0081 0.0261 0.0329 0.0317 0.0537 0.0474 0.0628t-statistic (Mean=0) 21.11 20.57 25.13 26.09 21.49 44.24 22.69 20.04 23.80Skewness -0.87 -1.05 -0.81 -0.70 -0.35 2.37 1.71 1.36 1.42Kurtosis 9.82 8.58 10.29 7.38 6.42 7.76 5.05 4.76 4.59Autocorrelations:ρ1

ρ2

ρ3

Differences:ρ1 -0.132 0.415 0.561 0.630 0.694 0.108 0.348 0.222 0.343ρ2 -0.015 -0.135 0.178 0.316 0.444 0.172 0.212 0.255 0.269ρ3 -0.073 -0.156 -0.255 0.019 0.142 0.184 0.131 0.108 0.283

3

Table 4. Regression estimates of the change in the future spot rate on the forward spread and a risk premium in ττ-period bonds.The risk premium is measured by an interactive term with an instrumental variable with the forward spread, F τ, t - rt+1. The instrumentalvariables include: the bid/ask spread of τ-period bonds, SPR τ, the junk bond premium measured by the returns spread between Moody’s Baacorporate bonds and government bonds, PREM t, the Standard and Poor 500 dividend yield, D t/Pt, the level of the short rate, r t+1, and theconditional variance of interest rate changes estimated from ARCH models of Table 5. Estimates are computed for January 1963 toDecember 1993 and two subperiods from January 1963 to July 1982 and August 1982 to December 1993. Standard errors are computedusing Newey-West correction for heteroscedasticity and serial correlation with six lagged autocovariances. The t-statistic is computed underthe null hypothesis that β=1. The basic model is based on equation (5) in Table 2: rt+τ - rt+1 = α + β ( Fτ, t - rt+1

) + εt+τ-1 ( 5)

The adjusted model augments (5) by including interactive term with the instrumental variable, X t: rt+τ - rt+1 = α + β ( Fτ, t - rt+1 ) + γ [ Xt ( Fτ, t - rt+1 ) ] + εt+τ-1

(10)rt+2 - rt+1 rt+3 - rt+1 rt+4 - rt+1 rt+5 - rt+1 rt+6 - rt+1

Models β t (β=1) Adj.R2 β t (β=1) Adj.R2 β t (β=1) Adj.R2 β t (β=1) Adj.R2 β t (β=1) Adj.R2

Overall PeriodUnadjusted 0.481 -10.40 0.210 0.374 -5.22 0.080 0.376 -4.50 0.048 0.327 -7.29 0.046 0.317 -6.69 0.040SPRt 0.539 -7.25 0.212 0.520 -3.68 0.093 0.820 -0.90 0.093 0.552 -3.46 0.061 0.479 -4.25 0.052PREMt: 0.576 -2.70 0.209 0.700 -1.08 0.093 0.918 -0.22 0.070 0.621 -1.27 0.053 0.550 -1.45 0.045Dt/Pt 0.520 -1.81 0.208 0.915 -0.24 0.089 0.889 -0.22 0.053 0.653 -1.06 0.048 0.666 -0.98 0.044ht+1 0.453 -6.92 0.208 0.621 -1.68 0.086 0.689 -0.83 0.061 0.437 -1.94 0.045 0.338 -2.50 0.038rt+1 0.589 -4.13 0.211 0.879 -0.66 0.121 1.511 1.33 0.170 1.066 0.26 0.122 1.010 0.05 0.1171963: 1 - 1982:7Unadjusted 0.456 -5.48 0.121 0.221 -4.25 0.020 0.169 -4.75 0.005 0.110 -6.37 0.001 0.145 -6.77 0.003SPRt 1.144 0.50 0.172 0.423 -2.09 0.029 0.932 -0.15 0.054 0.635 -1.48 0.019 0.364 -3.81 0.016PREMt: 0.663 -1.36 0.122 0.559 -1.18 0.030 0.645 -0.76 0.017 0.356 -1.58 0.001 0.315 -1.70 0.003Dt/Pt 0.607 -0.93 0.118 0.825 -0.34 0.027 0.661 -0.55 0.006 0.449 -1.09 0.001 0.444 -1.14 0.003ht+1 0.419 -5.48 0.118 0.421 -1.88 0.021 0.427 -1.25 0.010 0.218 -2.51 0.003 0.111 -3.05 0.001rt+1 0.680 -1.69 0.124 0.784 -0.69 0.049 1.486 0.82 0.117 0.864 -0.35 0.059 0.814 -0.57 0.0571982: 8 - 1993:12Unadjusted 0.543 -11.02 0.466 0.646 -4.80 0.372 0.852 -1.37 0.366 0.581 -4.47 0.335 0.625 -2.81 0.292SPRt 0.430 -8.83 0.474 0.744 -2.77 0.373 0.915 -0.53 0.364 0.710 -1.97 0.343 0.746 -1.51 0.296PREMt: 0.362 -3.97 0.467 0.803 -1.05 0.372 1.122 0.39 0.369 0.979 -0.08 0.357 1.097 0.32 0.318Dt/Pt 0.387 -2.48 0.463 0.777 -0.58 0.369 0.734 -0.53 0.362 1.147 0.57 0.354 1.317 1.19 0.318

4

ht+1 0.424 -4.11 0.465 0.737 -1.85 0.368 0.942 -0.29 0.363 0.774 -1.16 0.335 0.888 -0.38 0.298rt+1 0.565 -7.26 0.462 0.842 -1.04 0.377 1.325 1.31 0.392 1.211 1.05 0.409 1.306 1.52 0.388Table 5. ARCH model estimates of the change in the future spot rate on the forward spread and a risk premium in ττ-period bonds.The risk premium is measured by interactive term of conditional volatility of interest rate changes, h t+τ-1, with the forward spread, F τ, t - rt+1.Estimates are computed from January 1963 to December 1993 and two subperiods from January 1963 to July 1982 and August 1982 toDecember 1993. The model is estimated using quasi-maximum likelihood methods based on Bollerslev and Wooldridge (1992). Robust t-statistics are reported in parentheses. R 2 are computed from second-pass regressions of the spot rate changes on the forward spread and theinteractive variable. The The ARCH model augments equation (5) of Table 2 by including interactive term with conditional volatility, h t+τ-1,using GARCH(1,1) specification including level of the short-rate, r t+1: rt+τ - rt+1 = α + β ( Fτ, t - rt+1 ) + γ [ ht+τ-1 ( Fτ, t - rt+1 ) ] + εt+τ-1 + θ εt+τ-2

(11) εt+τ-1 ~ N(0, ht+τ-1) ht+τ-1 = a + b ht+τ-2 + c εt+τ-22 + d rt+1

3/2

(12)

Overall Period 1963:1 - 1993:12 Subperiod I 1963:1 - 1982:7 Subperiod II1982:8 - 1993:12Coefficients rt+2-rt+1 rt+3-rt+1 rt+4-rt+1 rt+5-rt+1 rt+6-rt+1 rt+2-rt+1 rt+3-rt+1 rt+4-rt+1 rt+5-rt+1 rt+6-rt+1 rt+2-rt+1 rt+3-rt+1 rt+4-rt+1 rt+5-rt+1 rt+6-rt+1

β 0.453 0.621 0.689 0.437 0.338 0.419 0.421 0.427 0.218 0.111 0.424 0.737 0.942 0.774 0.888t (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 0.003 -0.024 -0.021 -0.009 -0.001 0.003 -0.018 -0.016 -0.009 0.002 0.018 -0.010 -0.008 -0.014 -0.018

(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 -.282 0.973 0.406 0.761 0.526 -0.246 0.981 0.405 0.945 0.517 -0.341 0.756 0.369 0.563 0.466

(-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) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 -0.001 0.000 0.000 0.000

(-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 0.793 0.773 0.674 0.752 0.82 0.761 0.736 0.669 0.805 0.855 0.828 0.292 0.731 0.764 0.843

(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 0.123 0.127 0.251 0.174 0.133 0.194 0.192 0.292 0.136 0.122 0.056 0.323 0.186 0.152 0.100

(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) 1.094 1.342 2.039 1.951 1.560 0.790 1.101 1.503 1.599 1.315 1.751 7.027 2.153 2.255 1.761

(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 -0.024 -0.028 -0.025 -0.031 -0.021 -0.064 -0.059 -0.067 -0.067 -0.069 -0.033 -0.025 0.011 -0.005 -0.002Std. Dev. 1.003 1.001 0.999 0.999 1.001 1.000 1.007 1.006 0.993 0.978 1.001 1.002 1.015 1.005 1.001Skewness -0.534 -0.505 -0.281 -0.045 -0.249 -0.476 -0.422 -0.243 -0.101 -0.233 -0.315 -0.334 -0.049 -0.016 -0.188

5

Kurtosis 2.711 2.443 2.118 0.718 1.143 2.509 2.188 2.009 0.903 1.242 1.669 1.504 1.415 0.901 1.302

Log Likelihood 2981.52 2963.11 2870.15 2830.68 2791.58 2037.07 2030.95 1948.46 1970.44 1921.09 924.82 933.14 888.56 894.80 880.60

Table 6. Regression estimates of the change in the future spot rate on the forward spread and a risk premium in ττ-period bonds.The risk premium is measured by a joint interactive term of all of the instrumental variable with the forward spread, F τ, t - rt+1. Theinstrumental variables include: the bid/ask spread of τ-period bonds, SPR τ, the junk bond premium measured by the returns spread betweenMoody’s Baa corporate bonds and government bonds, PREM t, and the Standard and Poor’s dividend yield, D t/Pt, the conditional volatilityof interest rate changes estimated from the GARCH models in Table 5, h t+τ-1, and the level of the short rate of interest, r t+1 . Estimates arecomputed for January 1963 to December 1993 and two subperiods from January 1963 to July 1982 and August 1982 to December 1993.Standard errors are computed using Newey-West correction for heteroscedasticity and serial correlation and associated t-statistics arereported in parentheses. The basic model is based on equation (5) in Table 2: rt+τ - rt+1 = α + β ( Fτ, t - rt+1

) + εt+τ-1 (5) The expanded model augments (5) by including multiple interactiveterms with all four instrumental variables, X j, t: rt+τ - rt+1 = α + β ( Fτ, t - rt+1

) + Σj γj [ Xj, t ( Fτ, t - rt+1 ) ] + εt+τ-1 (10)

Overall Period 1963:1 - 1993:12 Subperiod I 1963:1 - 1982:7 Subperiod II1982:8 - 1993:12Coefficients rt+2-rt+1 rt+3-rt+1 rt+4-rt+1 rt+5-rt+1 rt+6-rt+1 rt+2-rt+1 rt+3-rt+1 rt+4-rt+1 rt+5-rt+1 rt+6-rt+1 rt+2-rt+1 rt+3-rt+1 rt+4-rt+1 rt+5-rt+1 rt+6-rt+1

β 0.481 0.374 0.376 0.327 0.317 0.456 0.221 0.169 0.110 0.145 0.543 0.646 0.852 0.581 0.625t (β=1) (-10.40) (-5.22) (-4.50) (-7.29) (-6.69) (-5.48) (-4.25) (-4.75) (-6.37) (-6.77) (-11.02) (-4.80) (-1.37) (-4.47) (-2.81)Adj R2 0.210 0.080 0.048 0.046 0.040 0.121 0.020 0.005 -0.001 0.003 0.466 0.372 0.366

0.335 0.291β 0.599 1.161 1.069 0.989 1.484 1.679 1.789 2.586 2.147 1.847 0.620 0.763 0.422 1.196 1.310t (β=1) (-1.24) (0.36) (0.11) (-0.03) (1.00) (1.27) (1.06) (1.53) (1.37) (1.34) (-1.39) (-0.63) (-1.24) (0.50) (1.12)γ1 (SPRt) -0.027 -0.059 -0.057 -0.038 -0.044 -0.165 -0.106 -0.147 -0.122 -0.059 0.165 -0.087 -0.071 -0.017 -0.026

(-1.35) (-2.01) (-2.05) (-1.65) (-2.18) (-3.04) (-2.15) (-2.59) (-2.50) (-2.32) (1.32) (-1.21) (-0.69) (-0.27) (-0.68)γ2 (PREMt) -0.265 -0.123 -0.324 -0.227 -0.383 -0.744 -0.104 -0.366 -0.425 -0.443 -0.002 -0.122 -0.416 -0.178 -0.096

(-1.27) (-0.44) (-0.92) (-0.97) (-1.58) (-2.10) (-0.28) (-0.87) (-1.33) (-1.38) (-0.01) (-0.55) (-1.19) (-0.50) (-0.27)γ3 (Dt/Pt) 0.045 -0.050 0.245 0.078 -0.056 0.092 -0.177 0.013 -0.037 -0.090 -0.074 0.104 0.640 0.250 0.089

(0.51) (-0.34) (1.16) (0.56) (-0.38) (0.88) (-0.90) (0.05) (-0.20) (-0.51) (-0.71) (0.65) (4.13) (1.33) (0.52)γ4 (ht+ τ -1) 0.035 0.020 0.029 0.038 0.072 0.066 0.034 0.069 0.107 0.107 0.018 -0.006 -0.035 -0.042 -0.011

(1.39) (0.42) (0.72) (1.05) (2.50) (1.70) (0.55) (1.45) (2.16) (3.24) (0.72) (-0.18) (-1.36) (-1.26) (-0.30)

6

γ5 (rt+ 1) -0.338 -0.675 -2.105 -1.646 -1.921 -0.600 -0.803 -2.640 -2.631 -2.767 -0.045 -0.550 -2.363 -1.728 -1.457(-1.51) (-1.97) (-2.78) (-3.09) (-3.31) (-1.95) (-1.49) (-2.80) (-2.81) (-2.83) (-0.20) (-2.01) (-4.35) (-3.09) (-2.79)

Adj R2 0.219 0.128 0.210 0.149 0.200 0.214 0.072 0.213 0.153 0.207 0.460 0.367 0.451 0.412 0.378