TIME-SERIES ANALYSIS OF RETURN AND BETA IN U.S.
Kwang Woo (Ken) Park, Minnesota State University, Mankato, Minnesota, USA
ABSTRACT
Most CAPM tests following Pettengill et al. procedure (1995) have focused on the cross-sectional aspects
of data. However, it is more appropriate to examine the conditional relationship between beta and return
by using time-series analysis, as it is well known that the beta is not stable over time. This paper
examines the conditional relationship between returns and betas by using Kalman filter technique that is
the special case of the general state-space model. This paper provides another evidence from U.S. stock
market that there is a significant and systematic relationship between return and Kalman filtered beta
when the conditional nature of the CAPM is analyzed.
1. INTRODUCTION
The capital asset pricing model (CAPM), first introduced by Sharpe (1964) and Lintner (1966), has made
a profound impact on the way investors understand the relationship between price and risk of capital
assets. The CAPM simply states that the systematic difference in security returns can be explained by a
single measure of risk, beta. According to the CAPM, the expected return on any risky security or
portfolio of risky securities can be measured by the risk-free rate and the market risk premium multiplied
by the beta coefficient (β).
In spite of this straightforward relationship between expected return on an asset and market risk premium,
previous empirical tests of the CAPM are often questionable due to many obstacles, including nonstationarity of beta coefficient and risk premium, inadequate proxy of the market portfolio, and joint
hypothesis test problems associated with unobservable expected returns. As a result, many of the tests
fail to give a strong basis for evaluating beta as a reliable measure of systematic risk. (Fama and French,
1992; Davis, 1994; He and Ng, 1994; Burnie and Gunay, 1993; Pettengill et al, 1995), even though some
of the earlier empirical tests conclude in favor of the CAPM (Black et al., 1973; Fama and Macbeth,
1973). In particular, some recent results from cross-sectional tests of the CAPM indicate that the crosssectional variation in expected returns cannot be explained by the market beta alone (Fama and French,
1992; Chan et al., 1991).
On the other hand, other studies show that the market beta has a substantial explanatory power on the
market return (Pettengill et al, 1995). Chan and Chen (1988), using both time-varying and stationary
assumption of portfolio betas, conclude that the unconditional single-factor market model is a better
alternative to the pricing model with a size variable. These mixed empirical results found for the CAPM
are interpreted in the literature as either evidence against the CAPM itself (Fama and French, 1992) or
indication that the testing methodology is inappropriate (Calvet and Lefoll, 1989; Roll and Ross, 1994).
In particular, most of CAPM tests have focused on the cross-sectional aspects of data holding beta
coefficients constant in the sense that CAPM was originally developed to explain differences in risks
across capital assets. (Jaganathan and McGrattan, 1995) However, it is well known that firms frequently
change their risk structures in conjunction with the macroeconomic environment, that is, the risk structure
of any given firm will vary over time. Hence, it is more appropriate to examine the relationship between
return and beta using time series tests in the sense that time-varying properties of beta coefficients
seems more realistic than the non-stochastic beta assumption. Jagannathan and Wang (1994) point out
that the constant beta assumption is not reasonable. (For the other researches on time varying beta, refer
to Ferson and Harvey,1991; Fama and French, 1992; Chan and Chen,1988; Groenwold and Fraser,
1999; Black and Fraser, 2000; Fraser et al., 2000) As Longstaff (1989) points out, the model allowing
time-varying expected returns and betas can improves the description of return behavior.
Following time series properties of non-stochastic time-varying beta, this paper examines the conditional
relationship between returns and betas by using a time series analysis, Kalman filter technique. Since the
change of the systematic risk, or beta, is the fundamental premise of the time-series analysis, the timeseries test is less affected by data selection bias than the case for the related cross-sectional studies.
Black (1993) argues that previous findings of a flat relationship between beta and return (Fama and
French, 1992) may be attributed to data mining bias.
In particular, this paper is closely related to the notion of conditional testing approach proposed by
Pettengill et al (1995) and the subsequent works by Fletzer (1997), Hodoshima (2000), Elsas et al (1999)
in the light of having the conditional relation between beta and return. However, the present paper
departs from these in considering beta as time-varying coefficient using the estimated Kalman Filter
technique. In addition, one advantage of using time series for CAPM tests is that returns and betas do not
need to be averaged out over time to compare among different portfolios, as is the case for the most of
cross-sectional studies. Another advantage is that one can avoid using too few observations. With crosssectional studies, this often becomes problematic especially when researchers try to draw a conclusion
from examining the cross-sectional relationships among less than fifty portfolios. Fama and French (1992)
used the twenty-two portfolio average returns to conclude that there is a flat relationship between return
and beta.
The rest of the paper is organized into the following sections: The Empirical Framework, The Data,
Empirical Results and Conclusion.
2. THE EMPIRICAL FRAMEWORK
The empirical test is composed of two parts: estimating time-varying beta coefficients using the Kalman
filter technique and investigating the conditional relationship between return and beta over time.
2.1 Time-Varying Parameter Model
The Kalman filter is a special case of the general state-space model that is composed of the
measurement equation and transition equation. Consider the following measurement equation of statespace model.
ERpt=αpt + βptERmt + et ,
et ~ i.i.d N(0, σe2)
(1)
where ERpt is the excess return observed at time t for portfolio p and time varying parameter vectors, αpt
and βpt, are unobserved state variables for portfolio p; ERmt is the excess market return that links the
observed ERpt and the unobserved βpt.
The transition equations that describe the evolution of the time-varying state vector assume the following
simple form of a first-order difference equation in the state vector.
αt=αt-1 + vt
βt=βt-1 + wt
E(et vt’) = E(et wt’) =0
vt ~ i.i.d. N(0, σv2)
wt ~ i.i.d. N(0, σw2)
(2)
(3)
(4)
The Kalman filter estimates the unobserved state variables (αit, βit) through recursive procedure using
MLE method based on the prediction and updating. The maximized log likelihood function is represented
by
1 n
1 n '
ln L = − ∑ ln(2π f t |t −1 ) − ∑ηt |t −1 f t |−t 1−1ηt |t −1
2 t =1
2 t =1
(5)
where ηt|t-1 is the prediction error and ft|t-1 is the conditional variance of the prediction error. The prediction
error is the difference between actual value, ERpt and the fitted value of ERpt given information up to t-1,
yt|t-1. Thus, we have
ηt |t −1 = ER pt − ER pt |t −1
(6)
and the conditional variance of the prediction error is calculated as
f t|t −1 = E[ηt2|t −1 ]
(7)
Since the Kalman filter estimates the entire series in a Bayesian fashion when new information is
available in a world of uncertainty, it brings the uncertainty about the future states as well as the
uncertainty about the current states into the model. In addition, the Kalman filter can capture the
uncertainty about the unobserved current state through the changing conditional variance of excess
return of each portfolio. The variance of the conditional forecast error in the Kalman Filter is given by
'
2
f t |t −1 = ERmt |t −1Pt |t −1ERmt
−1 + σ e
(8)
where Pt|t-1 is the covariance matrix of βpt conditional on information up to time t.
2.2 Time Series Methodology between return and beta
The empirical tests begin with the conventional method of examining the simple unconditional relation
between returns and time-varying betas over time.
ERpt=γp0 +γp1 βpt-1 + εt
(9)
where ERpt is the excess return observed at time t for portfolio p and βpt-1 is the systematic risk estimated
at time t-1. ERpt is regressed on βpt-1, since the realized, rather than expected, returns are used.
According to the CAPM theory, the regression coefficient, γp1 must be significant and positive. Some
previous studies (Fama and French, 1992) found a flat relationship between return and risk, based on the
unconditional test in equation (9). However, the above equation does not consider the conditional nature
of the relation between return and beta. As Pettengill et al. (1995) argue, the above traditional studies
focusing on the relationship between return and beta does not consider the ex-post negative market risk
premium. While the portfolios with the higher systematic risk should yield higher return in the case of the
positive market premium, they should have lower returns when the market risk premium is negative.
Otherwise, no investor would hold the less risky asset (Elsas et. al., 1999). The implication is that there
should be a positive relationship between return and beta when the excess market return is positive, and
a negative relationship when the excess market return is negative. Pettengill et al (1995) argued that ‘the
existence of a large number of negative market excess return periods suggests that previous studies that
test for unconditional positive correlation between beta and realized returns are biased against finding a
positive relationship.’ This cross-sectional relationship between return and beta should also hold in a
specific portfolio over time. To test the conditional relationship, the following relation is estimated over
time:
ERpt=γp0 +γp1 Dt βpt-1 +γp2 (1-Dt) βpt-1+ εt
(10)
where Dt =1 if realized excess market return is positive and Dt =0 if realized excess market return is
negative. The coefficients γp1 and γp2 capture the relation between return and beta conditional on market
risk premium. Thus, the coefficient γp1 is expected to be positive conditional on the up-market and the
coefficient γp2 is expected to be negative conditional on the down-market. As a result, we test two sets of
hypotheses (Ho: γp1=0, Ha:γp1>0) and ( Ho: γp2=0, Ha:γp2<0) which can be tested by simple t-statistics.
Econometrically, we need to check the spurious regression problem proposed by Granger and Newbold
(1974) since the integrated orders between the return and time-varying beta are different. As Banerjee et
al. (1993) point out; the spurious problem would occur not only when the orders of integrated processes
are the same, but also when the orders of integrated processes are different. However, this problem is
relevant mainly in the regression of higher order integrated processes. In equation (10), the return is an
I(0) process and beta is an I(1) process. In this case, we can rule out the spurious regression problem.
Marmol (1996) shows that the problem of spurious regression would not occur in regressions that include
one variable distributed as an I(0) stochastic process, since the correlation coefficient when one of the
series is I(0) is similar to the distribution of this statistic when both series are I(0). Thus, the residual of
equation (10), εt, follows a stationary process.
3. THE DATA
We construct two sets of portfolios based on size and industry. The empirical results are based on
monthly returns calculated from ten size portfolios and twelve industry portfolios obtained from the Center
for Research in Security Prices (CRSP) for the period January 1960 to December 1997. The size portfolio
is constructed following Longstaff (1989), Fama and MacBeth (1973) and the industry portfolio is based
on the two-digit standard industrial classification (SIC). The firms listed on New York Stock Exchange are
used for the monthly return and size (market capitalization) data. The formation of the portfolios and the
industry classification follow Breeden et al. (1989) and Ferson and Harvey (1991). For the risk free rates
of return, three-month Treasure bill rates from CRSP are used.
Table 1 demonstrates the summary statistics for the excess returns on the size portfolios by selected and
all estimation periods. The whole data is divided into two sub-periods considering the stock market crash
in October 1987. The results from the whole sample period (1960/01-1997/12) and the first sub period
(1960/01-1987/09) confirm the previous research. Chan and Chen (1988), Ferson and Harvey (1991),
and Longstaff (1989) report that smaller portfolios tend to give bigger average portfolio returns. In
addition, as the size increases, the variation of the excess returns decrease, meaning less risk for the
excess portfolio returns. However, the trends on return are not consistent across the second sub period
as it shows the opposite effects of size on the average excess returns.
TABLE 1: SUMMARY STATISTICS FOR THE EXCESS RETURNS ON THE SIZE PORTFOLIOS
Size portfolio
Size 1
S2
S3
S4
S5
S6
S7
S8
S9
S10
1960/01 – 1997/12
Mean [Std. Dev.]
0.603 [6.036]
0.661 [5.379]
0.617 [5.202]
0.557 [5.159]
0.600 [4.731]
0.535 [4.702]
0.587 [4.630]
0.542 [4.545]
0.490 [4.302]
0.461 [4.042]
1960/01 – 1987/09
Mean [Std. Dev.]
0.803 [6.460]
0.737 [5.747]
0.685 [5.567]
0.597 [5.438]
0.622 [4.963]
0.504 [4.905]
0.573 [4.799]
0.478 [4.658]
0.461 [4.390]
0.326 [4.077]
1987/10-1997/12
Mean [Std. Dev.]
0.060 [4.683]
0.455 [4.237]
0.434 [4.067]
0.448 [4.333]
0.541 [4.052]
0.618 [4.119]
0.623 [4.159]
0.713 [4.240]
0.569 [4.069]
0.826 [3.939]
Monthly Excess Returns of 10 Value-Weight NYSE Portfolios Based on Size from 1960:01 to 1997:12 (in percentile).
Size 1 represents the smallest portfolio of NYSE list. Size 10 represents the largest. All returns are in excess of 3-month
Treasury bill rate.
Table 2 shows the summary statistics for the excess returns of the 12 industry portfolios by selected and
all estimation periods. Results from the first and second sub period show a clear asymmetry of returns
and risk. Between the two sub periods, the average of excess returns on 11 out of 12 industry portfolios
increased while only 2 out of 12 industries show the increase in variation. This result again implies that
there may be a structural break in the stock market around the crash of October 1987
TABLE 2: SUMMARY STATISTICS FOR THE EXCESS RETURNS OF THE INDUSTRY PORTFOLIOS
Industry Portfolio
I 1. Petroleum
I 2. Finance/real estate
I 3. Consumer durables
I 4. Basic industries
I 5. Food/tobacco
I 6. Construction
I 7. Capital goods
I 8. Transportation
I 9. Utilities
I 10. Textiles/trade
I 11. Services
I 12. Leisure
1960/01 – 1997/12
Mean [Std. Dev.]
0.655 [4.897]
0.550 [4.868]
0.502 [5.169]
0.483 [4.645]
0.760 [4.443]
0.347 [5.821]
0.433 [4.980]
0.507 [6.098]
0.403 [3.638]
0.545 [5.491]
0.630 [5.684]
0.635 [5.906]
1960/01 – 1987/09
Mean [Std. Dev.]
0.638 [5.187]
0.408 [4.928]
0.404 [5.211]
0.327 [4.649]
0.609 [4.378]
0.279 [5.837]
0.411 [5.033]
0.452 [6.224]
0.330 [3.709]
0.497 [5.441]
0.713 [5.943]
0.659 [6.082]
1987/10-1997/12
Mean [Std. Dev.]
0.702 [4.025]
0.938 [4.700]
0.770 [5.065]
0.904 [4.628]
1.168 [4.608]
0.532 [5.797]
0.492 [4.853]
0.655 [5.763]
0.603 [3.447]
0.677 [5.644]
0.404 [4.931]
0.638 [5.424]
Monthly Excess Returns of 12 Value-Weight NYSE Portfolios Based on Industry Classification from 1960:01 to 1997:12
(in percentile). Industry formation follows Sharpe (1982)’s two-digit standard industrial classification (SIC). All returns
are in excess of 1 month Treasury bill rate.
4. EMPIRICAL RESULTS
4.1 Time-varying betas
Initially the standard market model was estimated for the 10 size portfolios and 12 industry portfolios
between 1965 and 1997. The standard market model assumes a constant beta coefficient.
TABLE 3: CONSTANT AND TIME-VARYING BETA ESTIMATES OF NYSE, 1965-1997.
Size or
Industry
S1
S2
S3
S4
S5
S6
S7
S8
S9
S 10
I1
I2
I3
I4
I5
I6
I7
I8
I9
I 10
I 11
I 12
Constant Beta
(Standard Error)
Kalman Beta
(high/low)
1.146(0.047)
1.092(0.037)
1.077(0.032)
1.127(0.029)
1.059(0.023)
1.071(0.021)
1.063(0.019)
1.069(0.016)
1.019(0.012)
0.951(0.011)
0.868(0.042)
1.079(0.025)
1.149(0.027)
1.051(0.019)
0.932(0.027)
1.225(0.037)
1.045(0.029)
1.242(0.043)
0.649(0.030)
1.100(0.039)
1.133(0.041)
1.239(0.037)
1.157(1.745/0.624)
1.102(2.141/0.281)
1.093(1.744/0.427)
1.122(1.628/0.630)
1.065(1.314/0.786)
1.060(1.642/0.632)
1.080(1.642/0.617)
1.058(1.135/0.035)
1.016(1.098/0.920)
0.951(1.091/0.820)
0.853(1.499/0.582)
1.090(1.397/0.775)
1.121(1.191/0.981)
1.063(1.176/0.919)
0.911(1.200/0.515)
1.215(1.483/0.796)
1.082(1.280/1.002)
1.241(1.615/0.993)
0.661(0.818/0.433)
1.087(1.357/0.658)
1.096(1.457/0.824)
1.261(1.888/0.628)
Note: Kalman betas are the mean values over the entire sample period.
The first column of table 3 presents the results of the unconditional constant beta coefficients. The result
shows that each of the constant beta coefficients is significantly different from zero. In addition, the result
shows that industry 7 (Capital goods) and size 9 portfolio beta coefficients are not significantly different
from the unity; thus, the systematic risk of both portfolios can be approximated by the market risk. The
highest constant beta among industry portfolios is 1.242 of I8 (Transportation) and the lowest beta is
0.649 of I9 (Utilities). For the size portfolios, the small sized portfolios have higher beta than large sized
portfolio and the standard errors are clearly declining as the size of capitalization increases. This is
consistent with the previous findings that there is an inverse relationship between size and beta (Chan
and Chen, 1988; Ferson and Harvey, 1991).
In order to see the stability of risk structure in the unconditional classical regression, the cumulative of
sum of squares (CUSUMSQ) test of Brown et al (1975) was used. Indeed, the test results confirm the
general result that the beta coefficients are not stable over time. Figure I the result of CUSUMSQ test for
the size 1 portfolio. In the figure, it can be seen that the plot of the recursive residuals first breaches the
5% boundary in approximately 1975 and stay outside until 1987, indicating a shift in regime. All the beta
coefficients of other size portfolios showed the similar instability over time. The exceptions that did not
exhibit such parameter instability were found in two of the industry portfolios: industry seven (Capital
goods) and nine (Utilities).
FIGURE 1: THE CUSUMSQ TEST OF BETA STABILITY FOR SIZE 1 PORTFOLIO
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
70
75
80
CUSUM of Squares
85
90
95
5% Significance
Given the evidence of beta instability, it is appropriate to allow the time-varying assumption on systematic
risk. The estimation of time-varying betas is based on Kalman filter approach. The second column of
Table 3 presents the mean value of beta estimates generated by Kalman filter technique, as well as the
range of beta observations (in parentheses). The betas for all sectors are clearly non-stationary. Both
ADF and Phillips-Perron test showed that the Kalman betas of all the portfolios are I(0) integrated. Even
though their means are similar to constant betas, most of Kalman beta estimations show a substantial
variation around their means. This implies that use of constant beta estimates may cause serious pricing
error when the time-variation in beta is substantial
The results show that the means of the Kalman beta processes are roughly same as the means of the
systematic risk under the assumption of constant risk. For example, the mean of constant betas is 1.063
and the mean of rolling betas over portfolios is 1.063. The correlation between Kalman beta and constant
beta is more than 0.99. The largest range of the observations generated by Kalman regression was
found in industry 12 (Leisure).
For the industry portfolios, I1 (Petroleum), I6 (Construction), and I12 (Leisure) demonstrate the relatively
highest volatility in beta movement in both estimations of time-varying beta. On the other hand, I4 (Basic
Industry) and I7 (Capital Goods) show relatively stable beta movement.
For the size portfolios, the difference between the highest and the lowest beta estimates declines as the
capital size increases, which shows that the larger size portfolio has smaller volatility of beta movement,
hence a greater degree of stability. Small sized portfolios show greater time-variation in betas than large
sized ones. Berglund and Knif (1999) explain that small size is often associated with greater time
variation in betas due to relatively low degree of diversification and resulting large fluctuation in the
market value of equity. Indeed, the lowest standard deviation was found in the size 10 portfolio and the
highest in the size 1 portfolio for both time-varying beta estimations. This issue can be examined more
clearly in Figure 2.
FIGURE 2: SELECTED KALMAN BETA AND ALPHA
1.8
1.6
1.4
1.2
1.0
0.8
0.6
65
70
75
80
KB_S1
85
90
KB_S10
95
KB_S5
0.015
0.010
0.005
0.000
-0.005
-0.010
65
70
75
KA_S1
80
85
KA_S5
90
95
KA_S10
Figure 2 (top) represents some selected time-varying Kalman beta. The betas for all sectors are clearly
non-stationary. The bottom part of Figure 2 demonstrates that the time-varying alphas are moving around
zero. In particular, the time varying alpha of size portfolio 10 shows that the time-varying alpha is virtually
indistinguishable from zero. Since alphas represent the difference between the expected excess return
and the actual return on the portfolio, the plot indicates that the abnormal return becomes negligible as
the size of portfolio increases.
Figure 3 depicts the changing conditional variance or uncertainty underlying excess return, as analytically
given in (1). In particular, the figure is consistent with the substantially higher uncertainty of conditional
variance around the Crash of 1987 (Bertero and Mayer, 1990). Thus, higher conditional variances are
consistent with high volatility periods, being associated with downturns in the business cycle.
FIGURE 3: CONDITIONAL VARIANCE OF EXCESS RETURN OF SIZE PORTFOLIO 2
0.008
0.006
0.004
0.002
0.000
65
70
75
80
85
90
95
CONV_S2
4.2 Conditional Relation between Return and Beta
Table 4 reports the test results of unconditional relationship between realized return and systematic risk.
According to the single factor CAPM, the slope coefficient of equation (9) should be significantly positive.
For the whole sample period, the hypothesis test of flat relationship between return and risk can be
rejected for over one-half of total (14 out of 22) portfolios at the 5% level. However, two sub periods show
inconsistent pattern of the relationship between returns and risk. In the first sub period, only one portfolio
(I 11) rejects the hypothesis of flat relationship between risk and return at the 5% level. In the second sub
period, four portfolios (i.e., S 10, I 2, I 4, I 5) can reject the null hypothesis of flat relationship. This
generally flat and inconsistent, unconditional relationship confirms much existing literature, including
Fama and French (1992) and Pettengill et al (1995).
2
In addition, although not reported here, none of the regression R values exceed 0.00 in absolute value.
This seems too low even if we consider the time-series nature of the regression. Hence, this may lead to
a stronger interpretation of the results in table 5, if we accept the argument of Hodoshima et al (2000) that
the strength of the relationship between return and risk is more appropriately measured by the goodness
of fit measure. That is, none of the slope estimates in table 4 cannot reject the null hypothesis of flat
relationship between returns and risk.
The intertemporal inconsistency in table 4 is largely due to the joint hypothesis problem and reflects the
argument of Pettengill et al (1995) and others (Fletcher, 1997; Elsas et al., 1999; Hodoshima et al., 2000)
that the relationship between risk and returns is conditional on the excess market return. Since realized
excess market return, not expected return, is used for testing CAPM, high beta portfolios will have higher
return when market is up (i.e., excess market return is positive) and will have lower return than lower beta
portfolios when market is down. By using Kalman filtered beta, the present paper tests the hypothesis of
conditional correlation of risk and returns in equation (10).
TABLE 4: TESTS OF THE UNCONDITIONAL RETURN AND BETA RELATIONSHIP, 1965-1997
Total Period
Jan. 1965 – Dec.1997
S1
S2
S3
S4
S5
S6
S7
S8
S9
S 10
I1
I2
I3
I4
I5
I6
I7
I8
I9
I 10
I 11
I 12
0.005(0.003)
0.006(0.002)*
0.005(0.002)*
0.005(0.002)*
0.005(0.002)*
0.004(0.002)*
0.005(0.002)*
0.005(0.002)*
0.005(0.002)*
0.005(0.002)*
0.006(0.003)
0.005(0.002)*
0.004(0.002)
0.005(0.002)*
0.008(0.002)*
0.004(0.002)
0.004(0.002)
0.003(0.003)
0.004(0.003)
0.004(0.003)
0.006(0.003)*
0.004(0.002)*
Period 1
Jan. 1965 – Sep.1987
0.005(0.003)
0.005(0.003)
0.005(0.003)
0.005(0.003)
0.006(0.003)
0.004(0.003)
0.004(0.003)
0.004(0.003)
0.004(0.003)
0.003(0.003)
0.004(0.004)
0.003(0.003)
0.003(0.003)
0.003(0.003)
0.006(0.003)
0.003(0.003)
0.004(0.003)
0.002(0.003)
0.002(0.003)
0.003(0.003)
0.007(0.003)*
0.004(0.003)
Period 2
Oct. 1987 – Dec.1997
0.001(0.005)
0.007(0.004)
0.004(0.004)
0.006(0.004)
0.005(0.004)
0.006(0.004)
0.006(0.004)
0.007(0.004)
0.005(0.004)
0.008(0.004)*
0.009(0.005)
0.009(0.004)*
0.007(0.004)
0.008(0.004)*
0.011(0.004)*
0.004(0.004)
0.005(0.004)
0.005(0.004)
0.009(0.005)
0.006(0.004)
0.004(0.004)
0.005(0.004)
Note: In order to test the unconditional relationship between return and beta, the following model is estimated over time.
ERpt=γp0 +γp1 βpt-1 + εt
where ERpt is the excess return observed at time t for portfolio p and βpt-1 is the systematic risk estimated at time t-1.
Standard errors are between parentheses. * represents a point estimate that is significant at the 5% level of confidence.
Table 5 supports the expectation that there exists a positive significant relationship between the beta and
return during the up markets, while there is a negative significant relationship during the down markets.
γp1, or the slope coefficient of Dt βpt-1, represents the market risk premium during the period when the
excess market return is positive (up markets). For the entire sample period shown in the first column of
table 5, all the estimates of γp1 are positive and significant. When the excess market return is negative
(down market), conversely, all the estimates of γp2 are negative and significant. This is in essence
consistent with previous findings that there is a conditional correlation between beta and returns. All the
slope coefficients, γp1 and γp2, during the whole sampling period were significant at the 5% level.
In addition, the second and third columns of Table 5 report the estimates of γp1 and γp2 for the two sub
periods. Similar to the previous finding of Pettingill et al (1995), there is no intertemporal inconsistency
across the time period. Both sub periods show that there is a significant relationship between realized
return and estimated betas when excess market return is positive. Similarly, for both sub periods, the null
hypothesis of flat relationship between realized return and risk can be rejected at the 5% level when
excess market return is negative. As in the total sampling period, all the slope coefficients, γp1 and γp2,
during the sub periods were significant at the 5% level. Hence, this result reconfirms the previous crosssectional empirical results (Pettengill et al., 1995; Hodoshima et al., 2000) that when the conditional
relationship between return and beta is considered, there is consistent and significant relationship
between returns and beta regardless of sub periods. Additionally, note in table 5 that the estimates of γp1
generally offset the corresponding estimates of γp2. This can partly explain why our earlier result of
unconditional test is both relatively weak and inconsistent across the sub periods.
TABLE 5: THE CONDITIONAL RELATIONSHIP BETWEEN RETURN AND BETA, 1965-1997
Total Period
Jan. 1965 – Dec.1997
γp2
γp1
S1
S2
S3
S4
S5
S6
S7
S8
S9
S 10
I1
I2
I3
I4
I5
I6
I7
I8
I9
I 10
I 11
I 12
0.035 (0.003) -0.029 (0.003)
0.033 (0.002) -0.028 (0.003)
0.034 (0.002) -0.029 (0.003)
0.034 (0.002) -0.029 (0.003)
0.034 (0.002) -0.029 (0.002)
0.034 (0.002) -0.031 (0.002)
0.034 (0.002) -0.030 (0.002)
0.034 (0.002) -0.032 (0.002)
0.033 (0.002) -0.031 (0.002)
0.033 (0.002) -0.031 (0.002)
0.034 (0.003) -0.029 (0.004)
0.034 (0.002) -0.031 (0.002)
0.034 (0.002) -0.033 (0.003)
0.032 (0.002) -0.031 (0.002)
0.036 (0.002) -0.028 (0.003)
0.032 (0.002) -0.032 (0.003)
0.032 (0.002) -0.032 (0.003)
0.033 (0.003) -0.032 (0.003)
0.030 (0.003) -0.029 (0.004)
0.033 (0.003) -0.029 (0.004)
0.036 (0.003) -0.030 (0.003)
0.034 (0.002) -0.031 (0.003)
Period 1
Jan. 1965 – Sep.1987
γp1
γp2
0.038 (0.003)
0.034 (0.003)
0.036 (0.003)
0.037 (0.003)
0.039 (0.003)
0.036 (0.003)
0.037 (0.002)
0.038 (0.003)
0.037 (0.002)
0.035 (0.002)
0.037 (0.004)
0.037 (0.003)
0.037 (0.003)
0.035 (0.003)
0.039 (0.003)
0.036 (0.003)
0.034 (0.003)
0.036 (0.003)
0.031 (0.004)
0.036 (0.003)
0.042 (0.003)
0.038 (0.003)
-0.028 (0.004)
-0.028 (0.003)
-0.028 (0.003)
-0.029 (0.003)
-0.029 (0.003)
-0.030 ( 0.003)
-0.030 (0.003)
-0.032 (0.003)
-0.031 (0.003)
-0.032 (0.003)
-0.028 (0.004)
-0.032 (0.003)
-0.033 (0.003)
-0.032 (0.003)
-0.031 (0.003)
-0.032 (0.003)
-0.030 (0.003)
-0.033 (0.003)
-0.029 (0.004)
-0.033 (0.004)
-0.030 (0.004)
-0.032 (0.003)
Period 2
Oct. 1987 – Dec.1997
γp1
γp2
0.022 (0.005)
0.026 (0.005)
0.026 (0.004)
0.026 (0.004)
0.025 (0.003)
0.028 (0.004)
0.028 (0.003)
0.027 (0.003)
0.026 (0.003)
0.029 (0.003)
0.029 (0.005)
0.028 (0.004)
0.028 (0.004)
0.028 (0.004 )
0.030 (0.004 )
0.025 (0.004)
0.029 (0.004)
0.026 (0.004)
0.028 (0.005)
0.026 (0.004)
0.022 (0.005)
0.026 (0.004)
-0.038 (0.007)
-0.030 (0.006)
-0.035 (0.006)
-0.031 (0.005)
-0.031 (0.005)
-0.032 (0.005)
-0.032 (0.004)
-0.031 (0.004)
-0.032 (0.004)
-0.030 (0.004)
-0.029 (0.007)
-0.028 (0.005)
-0.034 (0.005)
-0.028 (0.005)
-0.024 (0.006)
-0.033 (0.006)
-0.039 (0.005)
-0.032 (0.006)
-0.028 (0.007)
-0.032 (0.006)
-0.031 (0.006)
-0.033 (0.005)
Note: In order to test the conditional relationship between return and beta, the following model is estimated over time.
ERpt=γp0 +γp1 Dt βpt-1 +γp2 (1-Dt) βpt-1+ εt
where Dt =1 if realized excess market return is positive and Dt =0 if realized excess market return is negative. Standard
errors are between parentheses. All coefficients are significant at the 5% level of confidence.
FIGURE 4: SCATTER PLOT OF MEANS OF BETA AND RETURN, 1965-1997
0.05
0.04
0.03
0.02
Return
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
Beta
Figure 4 shows the average return and estimated betas of 22 portfolios during up markets and down
markets. The conditional relationship reconfirms the result from table 6. When excess market return is
positive, a portfolio with a higher average beta gains a higher average return. On the other hand, high
beta portfolios bring lower average return than lower beta portfolios when the market is down.
V. CONCLUSION
This paper investigates the conditional relationship between beta and return under the assumption of
time-varying betas. Unlike previous studies, the paper employs time-series analysis to focus on the
behavior of beta and the corresponding return for each selected portfolio over time. Time-series testing
of the CAPM for the relationship between beta and return is attractive for the following reasons. First,
unlike the cross-sectional studies, the returns and/or beta do not need to be averaged out over time to
compare among different portfolios. Second, one do not need to be concerned with the change of risk
characteristics, hence beta, of a given portfolio over time. Third, relatively more observations can be used
under time-series methodology. Hence, the time-series test is less likely to be influenced by data
selection bias and is more likely to give robust results than the standard cross-sectional test.
Initial tests of beta stability show that the beta is non-stationary. This paper estimates the time-varying
betas for twenty-two selected portfolios based on size and industry. Comparison among the traditional
constant betas and Kalman filtered time-varying betas indicates that use of constant beta estimates may
cause serious pricing error.
From the initial examination of unconditional relationship between return and beta, we find weak and
inter-temporarily inconsistent results. However, when positive and negative realized market returns are
considered, the results shows that there is a significant and systematic relationship between beta and
return. That is, empirical results are consistent with the implication that beta is a useful measure of
systematic risk over time. Hence, the conditional cross-sectional relationships found in previous studies
also hold for the time-series methodological framework.
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Dr. Kwang Woo (Ken) Park earned his Ph. D. at Claremont Graduate University in 2002. Currently he is
an assistant professor of economics at Minnesota State University, Mankato.