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, bid-ask 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 1963-1993 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 43210-1399.
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.
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
1
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 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 futures-implied 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 time-varying term premia which can be extracted from survey data and
Treasury bill futures prices.
2
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)
3
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)
4
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 .
5
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
6
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 matrix
estimator.
7
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 3month 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
8
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.
9
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 issues
better.
10
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
11
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
12
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,
εt+τ-1 ~ N(0,ht+τ-1),
(12a)
ht+τ-1 = a + Σj bj ht+τ-j + Σk ck εt+τ-k2 + d rt+13/2,
(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 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 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
13
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
14
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
15
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
16
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
17
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
18
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
19
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-interestrate 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
20
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.
21
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24
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 1993. Fτ, t is the 1-month forward rate observed at t for τ months ahead and r t+1 is the one-month 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). ρk denotes the k-th order autocorrelation coefficient.
Statistic
1959: 2 - 1993:12
Mean
Standard Deviation
t-statistic (Mean=0)
Skewness
Kurtosis
ρ1
ρ2
ρ3
1959: 2 - 1982: 7
Mean
Standard Deviation
t-statistic (Mean=0)
Skewness
Kurtosis
ρ1
ρ2
ρ3
1982: 8 - 1993: 12
Mean
Standard Deviation
t-statistic (Mean=0)
Skewness
Kurtosis
ρ1
ρ2
ρ3
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
0.0013
0.7983
0.03
-0.87
9.82
-0.132
-0.015
-0.073
0.0023
1.0532
0.04
-1.05
8.58
0.415
-0.135
-0.156
0.0026
1.2506
0.04
-0.81
10.29
0.561
0.178
-0.255
0.0020
1.3883
0.03
-0.70
7.38
0.630
0.316
0.019
0.0025
1.4668
0.04
-0.35
6.42
0.694
0.444
0.142
0.6634
0.7660
17.73
2.37
7.76
0.108
0.172
0.184
0.9117
0.8225
22.69
1.71
5.05
0.348
0.212
0.131
0.7410
0.7568
20.04
1.36
4.76
0.222
0.255
0.108
1.1075
0.9526
23.80
1.42
4.59
0.343
0.269
0.283
1.2755
1.0053
25.97
1.07
3.28
0.461
0.347
0.237
0.0144
0.8262
0.29
-1.15
11.88
-0.070
0.003
-0.103
0.0307
1.1230
0.46
-1.25
9.59
0.451
-0.094
-0.154
0.0486
1.3510
0.60
-1.06
11.01
0.587
0.202
-0.219
0.0660
1.4952
0.74
-0.95
7.99
0.673
0.351
0.023
0.0857
1.5912
0.90
-0.55
6.64
0.722
0.463
0.131
0.5706
0.6532
14.67
2.19
6.44
0.115
0.235
0.171
0.8929
0.8132
18.44
1.58
5.21
0.271
0.201
0.151
0.7288
0.7789
15.71
1.33
5.45
0.122
0.228
0.105
1.0779
0.8651
20.92
0.62
2.30
0.238
0.238
0.228
1.3121
1.0170
21.66
0.80
2.52
0.395
0.329
0.239
-0.0259
0.7393
-0.41
-0.08
3.14
-0.276
-0.020
0.026
-0.0569
0.8909
-0.74
-0.27
1.64
0.287
-0.206
-0.098
-0.0943
1.0049
-1.09
0.34
1.51
0.462
0.126
-0.325
-0.1337
1.1219
-1.37
0.34
1.12
0.474
0.215
0.052
-0.1751
1.1423
-1.76
0.43
1.32
0.566
0.382
0.197
0.8543
0.9312
10.73
2.21
6.16
0.037
0.050
0.150
0.9504
0.8428
13.19
1.96
4.90
0.427
0.129
0.059
0.7661
0.7114
12.60
1.46
2.75
0.419
0.203
0.022
1.1684
1.1124
12.29
2.09
5.34
0.403
0.206
0.296
1.2002
0.9800
14.33
1.71
5.61
0.476
0.295
0.201
Table 2. Regression model estimates of the change in the future spot rate on the forward spread . Fτ, t is the one-month forward rate
observed 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 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 Newey-West correction for heteroscedasticity and serial correlation and associated t-statistics are
reported in parentheses.
rt+τ - rt+1 = α + β ( Fτ, t - rt+1 ) + εt+τ-1
Models
Overall Period:
1959: 2 - 1993:12
Fama Subperiod:
1959: 2 - 1982: 7
Subperiods:
1959: 2 - 1964: 1
1964: 2 - 1969: 1
1969: 2 - 1974: 1
1974: 2 - 1979: 1
1979: 2 - 1982: 7
1982: 8 - 1988: 1
1988: 2 - 1993:12
β
rt+2 - rt+1
2
t (β=0) Adj.R
β
rt+3 - rt+1
2
t (β=0) Adj.R
β
( 5)
rt+4 - rt+1
2
t (β=0) Adj.R
β
rt+5 - rt+1
2
t (β=0) Adj.R
β
rt+6 - rt+1
2
t (β=0) Adj.R
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
0.037
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
0.004
0.444
0.500
0.158
0.591
0.694
0.609
0.471
3.94
3.96
1.97
3.88
3.65
12.64
3.84
0.223
0.368
0.018
0.116
0.177
0.504
0.418
0.299
0.387
0.050
0.081
0.418
0.637
0.786
3.26
3.41
0.31
0.57
1.26
5.83
8.40
0.127
0.224
-0.016
-0.012
0.029
0.356
0.452
0.490
0.356
0.212
0.379
0.183
0.751
1.192
3.34
3.17
1.24
1.48
0.58
5.20
10.48
0.134
0.116
0.010
0.038
-0.018
0.294
0.548
0.064
0.324
-0.029
0.126
0.213
0.515
1.096
0.81
4.05
-0.22
0.63
0.71
5.68
11.86
-0.010
0.183
-0.017
-0.011
-0.018
0.308
0.561
0.053
0.265
0.032
0.073
0.308
0.543
1.140
0.46
3.41
0.21
0.25
1.24
3.90
7.41
-0.010
0.173
-0.017
-0.016
-0.007
0.235
0.535
1
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 Research
in 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 500
stock 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 model
estimates 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
Dt/Pt
rt+1
ht+1
ht+2
ht+3
ht+4
ht+5
PREMt
Mean
Standard Deviation
t-statistic (Mean=0)
Skewness
Kurtosis
Autocorrelations:
ρ1
ρ2
ρ3
Differences:
ρ1
ρ2
ρ3
0.0243
0.0222
21.11
-0.87
9.82
0.0187
0.0175
20.57
-1.05
8.58
0.0359
0.0276
25.13
-0.81
10.29
0.0413
0.0305
26.09
-0.70
7.38
0.0329
0.0295
21.49
-0.35
6.42
0.0108
0.0047
44.24
2.37
7.76
0.0371
0.0081
22.69
1.71
5.05
0.0604
0.0261
20.04
1.36
4.76
-0.132
-0.015
-0.073
0.415
-0.135
-0.156
0.561
0.178
-0.255
0.630
0.316
0.019
0.694
0.444
0.142
0.108
0.172
0.184
0.348
0.212
0.131
0.222
0.255
0.108
0.0578
0.0329
0.0586
0.0317
0.0764
0.0537
0.0795
0.0474
0.0841
0.0628
23.80
1.42
4.59
0.343
0.269
0.283
2
Table 4. Regression estimates of the change in the future spot rate on the forward spread and a risk premium in τ-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 instrumental
variables include: the bid/ask spread of τ-period 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/Pt, 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 1993. Standard errors are computed
using Newey-West correction for heteroscedasticity and serial correlation with six lagged autocovariances. The t-statistic is computed under
the 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)
Models
Overall Period
Unadjusted
SPRt
PREMt:
Dt/Pt
ht+1
rt+1
1963: 1 - 1982:7
Unadjusted
SPRt
PREMt:
Dt/Pt
ht+1
rt+1
1982: 8 - 1993:12
Unadjusted
SPRt
PREMt:
Dt/Pt
β
rt+2 - rt+1
t (β=1) Adj.R2
β
rt+3 - rt+1
t (β=1) Adj.R2
β
rt+4 - rt+1
t (β=1) Adj.R2
β
rt+5 - rt+1
t (β=1) Adj.R2
β
rt+6 - rt+1
t (β=1) Adj.R2
0.481
0.539
0.576
0.520
0.453
0.589
-10.40
-7.25
-2.70
-1.81
-6.92
-4.13
0.210
0.212
0.209
0.208
0.208
0.211
0.374
0.520
0.700
0.915
0.621
0.879
-5.22
-3.68
-1.08
-0.24
-1.68
-0.66
0.080
0.093
0.093
0.089
0.086
0.121
0.376
0.820
0.918
0.889
0.689
1.511
-4.50
-0.90
-0.22
-0.22
-0.83
1.33
0.048
0.093
0.070
0.053
0.061
0.170
0.327
0.552
0.621
0.653
0.437
1.066
-7.29
-3.46
-1.27
-1.06
-1.94
0.26
0.046
0.061
0.053
0.048
0.045
0.122
0.317
0.479
0.550
0.666
0.338
1.010
-6.69
-4.25
-1.45
-0.98
-2.50
0.05
0.040
0.052
0.045
0.044
0.038
0.117
0.456
1.144
0.663
0.607
0.419
0.680
-5.48
0.50
-1.36
-0.93
-5.48
-1.69
0.121
0.172
0.122
0.118
0.118
0.124
0.221
0.423
0.559
0.825
0.421
0.784
-4.25
-2.09
-1.18
-0.34
-1.88
-0.69
0.020
0.029
0.030
0.027
0.021
0.049
0.169
0.932
0.645
0.661
0.427
1.486
-4.75
-0.15
-0.76
-0.55
-1.25
0.82
0.005
0.054
0.017
0.006
0.010
0.117
0.110
0.635
0.356
0.449
0.218
0.864
-6.37
-1.48
-1.58
-1.09
-2.51
-0.35
0.001
0.019
0.001
0.001
0.003
0.059
0.145
0.364
0.315
0.444
0.111
0.814
-6.77
-3.81
-1.70
-1.14
-3.05
-0.57
0.003
0.016
0.003
0.003
0.001
0.057
0.543
0.430
0.362
0.387
-11.02
-8.83
-3.97
-2.48
0.466
0.474
0.467
0.463
0.646
0.744
0.803
0.777
-4.80
-2.77
-1.05
-0.58
0.372
0.373
0.372
0.369
0.852
0.915
1.122
0.734
-1.37
-0.53
0.39
-0.53
0.366
0.364
0.369
0.362
0.581
0.710
0.979
1.147
-4.47
-1.97
-0.08
0.57
0.335
0.343
0.357
0.354
0.625
0.746
1.097
1.317
-2.81
-1.51
0.32
1.19
0.292
0.296
0.318
0.318
3
ht+1
rt+1
0.424
0.565
-4.11
-7.26
0.465
0.462
0.737
0.842
-1.85
-1.04
0.368
0.377
0.942
1.325
-0.29
1.31
0.363
0.392
0.774
1.211
-1.16
1.05
0.335
0.409
0.888
1.306
-0.38
1.52
0.298
0.388
Table 5. ARCH model estimates of the change in the future spot rate on the forward spread and a risk premium in τ-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 to
December 1993. The model is estimated using quasi-maximum likelihood methods based on Bollerslev and Wooldridge (1992). Robust tstatistics are reported in parentheses. R 2 are computed from second-pass 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+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, h t+τ-1)
ht+τ-1 = a + b h t+τ-2 + c εt+τ-22 + d rt+13/2
(12)
Overall Period 1963:1 - 1993:12
Subperiod I 1963:1 - 1982:7
Subperiod II
1982:8 - 1993:12
Coefficients
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
t (β=1)
γ
0.453
(4.76)
0.003
(0.27)
-.282
(-1.03)
0.000
(-0.04)
0.793
(14.42)
0.123
(1.90)
1.094
(2.67)
θ
a (0000s)
b
c
d (0000s)
Std. Residuals:
Mean
Std. Dev.
Skewness
-0.024
1.003
-0.534
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 0.888
(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)
-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
(-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)
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
(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)
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) (-0.05) (-0.05) (-0.06) (-0.02) (-0.03) (-0.03) (-0.04) (-0.05) (-0.06) (-0.12) (-0.06) (-0.04) (-0.06)
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
(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)
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
(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)
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
(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)
-0.028
1.001
-0.505
-0.025
0.999
-0.281
-0.031
0.999
-0.045
-0.021
1.001
-0.249
-0.064
1.000
-0.476
-0.059
1.007
-0.422
-0.067
1.006
-0.243
-0.067
0.993
-0.101
-0.069
0.978
-0.233
-0.033
1.001
-0.315
-0.025
1.002
-0.334
0.011
1.015
-0.049
-0.005
1.005
-0.016
-0.002
1.001
-0.188
4
Kurtosis
Log Likelihood
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
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 τ-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. The
instrumental variables include: the bid/ask spread of τ-period bonds, SPR τ, the junk bond premium measured by the returns spread between
Moody’s Baa corporate bonds and government bonds, PREM t, and the Standard and Poor’s dividend yield, D t/Pt, the conditional volatility
of 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 are
computed 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 are
reported 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 interactive
terms 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 II
1982:8 - 1993:12
Coefficients
β
t (β=1)
Adj R2
β
t (β=1)
γ1 (SPRt)
γ2 (PREMt)
γ3 (Dt/Pt)
γ4 (ht+
τ -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 rt+2-rt+1 rt+3-rt+1 rt+4-rt+1 rt+5-rt+1 rt+6-rt+1
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 0.625
(-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)
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 0.291
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 1.310
(-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)
-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)
-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)
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)
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)
5
γ5 (rt+ 1)
Adj R2
-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)
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
6