Time-series analysis of return and beta in U.S
Kwang Woo Park
Kwang Woo (Ken) Park, Minnesota State University, Mankato, Minnesota, USA
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 genera/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.
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 (beta).
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 non-stationarity 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 cross-sectional 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 time-series 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 cross-sectional 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 state-space model.
(1) E[R.sub.pt]=[[alpha].sub.pt] + [[beta].sub.pt]E[R.sub.mt] + [e.sub.t], [e.sub.t] ~ i.i.d. N(0, [[sigma].sub.e.sup.2]
where E[R.sub.pt] is the excess return observed at time t for portfolio p and time varying parameter vectors, [[alpha].sub.pt] and [[beta].sub.pt], are unobserved state variables for portfolio p; E[R.sub.mt] is the excess market return that links the observed E[R.sub.pt] and the unobserved [[beta].sub.pt].
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.
(2) [[alpha].sub.t]=[[alpha].sub.t-1] + [v.sub.t] [v.sub.t] ~ i.i.d. N(0, [[sigma].sub.v.sup.2])
(3) [[beta].sub.t] = [[beta].sub.t-1] + [w.sub.t] [w.sub.t] ~ i.i.d. N(0, [[sigma].sub.w.sup.2])
(4) E([e.sub.t] [v.sub.t]) = E([e.sub.t] [w.sub.t’]) = 0
The Kalman filter estimates the unobserved state variables ([alpha]it, [beta]it) through recursive procedure using MLE method based on the prediction and updating. The maximized log likelihood function is represented by
(5) 1n L = -1/2 [n.summation over t=1]1n(2[pi] [f.sub.t|t-1])-1/2 [n.summation over t=1][[eta]’.sub.t|t-1][f.sup.-1.sub.t|t-1][[eta].sub.t|t-1]
where [[eta].sub.t|t-1] is the prediction error and [f.sub.t|t-1] is the conditional variance of the prediction error. The prediction error is the difference between actual value, E[R.sub.pt] and the fitted value of E[R.sub.pt] given information up to t-1, [y.sub.t|t-1]. Thus, we have
(6) [[eta].sub.t|t-1] = E[R.sub.pt] – E[R.sub.pt|t-1]
and the conditional variance of the prediction error is calculated as
(7) [f.sub.t|t-1] = E[[[eta].sup.2.sub.t|t-1]]
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
(8) [f.sub.t|t-1] = E[R.sub.mt|t-1][P.sub.t|t-1]E[R’.sub.mt-1] + [[sigma].sup.2.sub.e]
where [P.sub.t|t-1] is the covariance matrix of [[beta]pt] 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.
(9) E[R.sub.pt]=[[gamma].sub.p0] + [[gamma].sub.p1] [[beta].sub.pt-1] + [[epsilon].sub.t]
where E[R.sub.pt] is the excess return observed at time t for portfolio p and [[beta].sub.pt-1] is the systematic risk estimated at time t-1. E[R.sub.pt] is regressed on [[beta].sub.pt-1], since the realized, rather than expected, returns are used. According to the CAPM theory, the regression coefficient, [[gamma].sub.p1] 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:
(10) E[R.sub.pt]=[[gamma].sub.p0] + [[gamma].sub.p1] [D.sub.t] [[beta].sub.pt-1] + [[gamma].sub.p2] (1-[D.sub.t]) [[beta].sub.pt-1] + [[epsilon].sub.t]
where [D.sub.t] =1 if realized excess market return is positive and [D.sub.t] =0 if realized excess market return is negative. The coefficients [[gamma].sub.p1] and [[gamma].sub.p2] capture the relation between return and beta conditional on market risk premium. Thus, the coefficient [[gamma].sub.p1] is expected to be positive conditional on the up-market and the coefficient [[gamma].sub.p2] is expected to be negative conditional on the down-market. As a result, we test two sets of hypotheses ([H.sub.o]: [[gamma].sub.p1]=0, [H.sub.a]:[gamma]p1>0) and ([H.sub.o]: [[gamma].sub.p2]=0, Ha:[[gamma].sub.p2]<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), [[epsilon].sub.t], 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 (SLC). 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 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
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.
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 18 (Transportation) and the lowest beta is 0.649 of 19 (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 1 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 OMITTED]
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 1(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), 16 (Construction), and I12 (Leisure) demonstrate the relatively highest volatility in beta movement in both estimations of time-varying beta. On the other hand, 14 (Basic Industry) and 17 (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 OMITTED]
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 OMITTED]
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).
In addition, although not reported here, none of the regression R2 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 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. [[gamma].sub.p1], or the slope coefficient of [D.sub.t] [[beta.sub.pt-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 [[gamma].sub.p1] are positive and significant. When the excess market return is negative (down market), conversely, all the estimates of [[gamma].sub.p2] 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, [[gamma].sub.p1] and [[gamma].sub.p2], 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 [[gamma].sub.p1] and [[gamma].sub.p2] 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, [[gamma].sub.p1] and [[gamma].sub.p2] during the sub periods were significant at the 5% level. Hence, this result reconfirms the previous cross-sectional 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 [[gamma].sub.p1] generally offset the corresponding estimates of [[gamma].sub.p2]. This can partly explain why our earlier result of unconditional test is both relatively weak and inconsistent across the sub periods.
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.
TABLE 1: SUMMARY STATISTICS FOR THE EXCESS RETURNS ON THE SIZE
Size 1960/01-1997/12 1960/01-1987/09 1987/10-1997/12
portfolio Mean [Std. Dev.] Mean [Std. Dev.] Mean [Std. Dev.]
Size 1 0.603 [6.036] 0.803 [6.460] 0.060 [4.683]
S2 0.661 [5.379] 0.737 [5.747] 0.455 [4.237]
S3 0.617 [5.202] 0.685 [5.567] 0.434 [4.067]
S4 0.557 [5.159] 0.597 [5.438] 0.448 [4.333]
S5 0.600 [4.731] 0.622 [4.963] 0.541 [4.052]
S6 0.535 [4.702] 0.504 [4.905] 0.618 [4.119]
S7 0.587 [4.630] 0.573 [4.799] 0.623 [4.159]
S8 0.542 [4.545] 0.478 [4.658] 0.713 [4.240]
S9 0.490 [4.302] 0.461 [4.390] 0.569 [4.069]
S10 0.461 [4.042] 0.326 [4.077] 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: SUMMARY STATISTICS FOR THE EXCESS RETURNS OF THE INDUSTRY
Industry Portfolio Mean [Std. Dev.] Mean [Std. Dev.]
I 1. Petroleum 0.655 [4.897] 0.638 [5.187]
I 2. Finance/real estate 0.550 [4.868] 0.408 [4.928]
I 3. Consumer durables 0.502 [5.169] 0.404 [5.211]
I 4. Basic industries 0.483 [4.645] 0.327 [4.649]
I 5. Food/tobacco 0.760 [4.443] 0.609 [4.378]
I 6. Construction 0.347 [5.821) 0.279 [5.837]
I 7. Capital goods 0.433 [4.980] 0.411 [5.033]
I 8. Transportation 0.507 [6.098] 0.452 [6.224]
I 9. Utilities 0.403 [3.638] 0.330 [3.709]
I 10. Textiles/trade 0.545 [5.491] 0.497 [5.441]
I 11. Services 0.630 [5.684] 0.713 [5.943]
I 12. Leisure 0.635 [5.906] 0.659 [6.082]
Industry Portfolio Mean [Std. Dev.]
I 1. Petroleum 0.702 [4.025]
I 2. Finance/real estate 0.938 [4.700]
I 3. Consumer durables 0.770 [5.065]
I 4. Basic industries 0.904 [4.628]
I 5. Food/tobacco 1.168 [4.608]
I 6. Construction 0.532 [5.797]
I 7. Capital goods 0.492 [4.853]
I 8. Transportation 0.655 [5.763]
I 9. Utilities 0.603 [3.447]
I 10. Textiles/trade 0.677 [5.644]
I 11. Services 0.404 [4.931]
I 12. Leisure 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.
TABLE 3: CONSTANT AND TIME-VARYING BETA ESTIMATES OF NYSE, 1965-1997.
Size or Constant Beta Kalman Beta
Industry (Standard Error) (high/low)
S 1 1.146(0.047) 1.157(1.745/0.624)
S 2 1.092(0.037) 1.102(2.141/0.281)
S 3 1.077(0.032) 1.093(1.744/0.427)
S 4 1.127(0.029) 1.122(1.628/0.630)
S 5 1.059(0.023) 1.065(1.314/0.786)
S 6 1.071(0.021) 1.060(1.642/0.632)
S 7 1.063(0.019) 1.080(1.642/0.617)
S 8 1.069(0.016) 1.058(1.135/0.035)
S 9 1.019(0.012) 1.016(1.098/0.920)
S 10 0.951(0.011) 0.951(l.091/0.820)
I 1 0.868(0.042) 0.853(1.499/0.582)
I 2 1.079(0.025) 1.090(1.397/0.775)
I 3 1.149(0.027) 1.121(l.191/0.981)
I 4 1.051(0.019) 1.063(1.176/0.919)
I 5 0.932(0.027) 0.911(1.200/0.515)
I 6 1.225(0.037) 1.215(1.483/0.796)
I 7 1.045(0.029) 1.082(1.280/1.002)
I 8 1.242(0.043) 1.241(1.615/0.993)
I 9 0.649(0.030) 0.661(0.818/0.433)
I 10 1.100(0.039) 1.087(1.357/0.658)
I 11 1.133(0.041) 1.096(1.457/0.824)
I 12 1.239(0.037) 1.261(1.888/0.628)
Note: Kalman betas are the mean values over the entire sample period.
TABLE 4: TESTS OF THE UNCONDITIONAL RETURN AND BETA RELATIONSHIP,
Total Period Period 1 Period 2
Jan. 1965-Dec. 1997 Jan. 1965-Sep.1987 Oct. 1987-Dec. 1997
S 1 0.005(0.003) 0.005(0.003) 0.001(0.005)
S 2 0.006(0.002) * 0.005(0.003) 0.007(0.004)
S 3 0.005(0.002) * 0.005(0.003) 0.004(0.004)
S 4 0.005(0.002) * 0.005(0.003) 0.006(0.004)
S 5 0.005(0.002) * 0.006(0.003) 0.005(0.004)
S 6 0.004(0.002) * 0.004(0.003) 0.006(0.004)
S 7 0.005(0.002) * 0.004(0.003) 0.006(0.004)
S 8 0.005(0.002) * 0.004(0.003) 0.007(0.004)
S 9 0.005(0.002) * 0.004(0.003) 0.005(0.004)
S 10 0.005(0.002) * 0.003(0.003) 0.008(0.004) *
I 1 0.006(0.003) 0.004(0.004) 0.009(0.005)
I 2 0.005(0.002) * 0.003(0.003) 0.009(0.004) *
I 3 0.004(0.002) 0.003(0.003) 0.007(0.004)
I 4 0.005(0.002) * 0.003(0.003) 0.008(0.004) *
I 5 0.008(0.002) * 0.006(0.003) 0.011(0.004) *
I 6 0.004(0.002) 0.003(0.003) 0.004(0.004)
I 7 0.004(0.002) 0.004(0.003) 0.005(0.004)
I 8 0.003(0.003) 0.002(0.003) 0.005(0.004)
I 9 0.004(0.003) 0.002(0.003) 0.009(0.005)
I 10 0.004(0.003) 0.003(0.003) 0.006(0.004)
I 11 0.006(0.003) * 0.007(0.003) * 0.004(0.004)
I 12 0.004(0.002) * 0.004(0.003) 0.005(0.004)
TABLE 5: THE CONDITIONAL RELATIONSHIP BETWEEN RETURN AND BETA,
Total Period Period 1
Jan. 1965-Dec. 1997 Jan. 1965-Sep.1987
[[gamma].sub.p1] [[gamma].sub.p2] [[gamma].sub.p1]
S 1 0.035 (0.003) -0.029(0.003) 0.038 (0.003)
S 2 0.033 (0.002) -0.028(0.003) 0.034 (0.003)
S 3 0.034 (0.002) -0.029(0.003) 0.036 (0.003)
S 4 0.034 (0.002) -0.029(0.003) 0.037 (0.003)
S 5 0.034 (0.002) -0.029(0.002) 0.039 (0.003)
S 6 0.034 (0.002) -0.031(0.002) 0.036 (0.003)
S 7 0.034 (0.002) -0.030(0.002) 0.037 (0.002)
S 8 0.034 (0.002) -0.032(0.002) 0.038 (0.003)
S 9 0.033 (0.002) -0.031(0.002) 0.037 (0.002)
S 10 0.033 (0.002) -0.031(0.002) 0.035 (0.002)
I 1 0.034 (0.003) -0.029(0.004) 0.037 (0.004)
I 2 0.034 (0.002) -0.031(0.002) 0.037 (0.003)
I 3 0.034 (0.002) -0.033(0.003) 0.037 (0.003)
I 4 0.032 (0.002) -0.031(0.002) 0.035 (0.003)
I 5 0.036 (0.002) -0.028(0.003) 0.039 (0.003)
I 6 0.032 (0.002) -0.032(0.003) 0.036 (0.003)
I 7 0.032 (0.002) -0.032(0.003) 0.034 (0.003)
I 8 0.033 (0.003) -0.032(0.003) 0.036 (0.003)
I 9 0.030 (0.003) -0.029(0.004) 0.031 (0.004)
I 10 0.033 (0.003) -0.029(0.004) 0.036 (0.003)
I 11 0.036 (0.003) -0.030(0.003) 0.042 (0.003)
I 12 0.034 (0.002) -0.031(0.003) 0.038 (0.003)
Period 1 Period 2
Jan. 1965-Sep.1987 Oct. 1987-Dec. 1997
[[gamma].sub.p2] [[gamma].sub.p1] [[gamma].sub.p2]
S 1 -0.028(0.004) 0.022 (0.005) -0.038(0.007)
S 2 -0.028(0.003) 0.026 (0.005) -0.030(0.006)
S 3 -0.028(0.003) 0.026 (0.004) -0.035(0.006)
S 4 -0.029(0.003) 0.026 (0.004) -0.031(0.005)
S 5 -0.029(0.003) 0.025 (0.003) -0.031(0.005)
S 6 -0.030(0.003) 0.028 (0.004) -0.032(0.005)
S 7 -0.030(0.003) 0.028 (0.003) -0.032(0.004)
S 8 -0.032(0.003) 0.027 (0.003) -0.031(0.004)
S 9 -0.031(0.003) 0.026 (0.003) -0.032(0.004)
S 10 -0.032(0.003) 0.029 (0.003) -0.030(0.004)
I 1 -0.028(0.004) 0.029 (0.005) -0.029(0.007)
I 2 -0.032(0.003) 0.028 (0.004) -0.028(0.005)
I 3 -0.033(0.003) 0.028 (0.004) -0.034(0.005)
I 4 -0.032(0.003) 0.028 (0.004) -0.028(0.005)
I 5 -0.031(0.003) 0.030 (0.004) -0.024(0.006)
I 6 -0.032(0.003) 0.025 (0.004) -0.033(0.006)
I 7 -0.030(0.003) 0.029 (0.004) -0.039(0.005)
I 8 -0.033(0.003) 0.026 (0.004) -0.032(0.006)
I 9 -0.029(0.004) 0.028 (0.005) -0.028(0.007)
I 10 -0.033(0.004) 0.026 (0.004) -0.032(0.006)
I 11 -0.030(0.004) 0.022 (0.005) -0.031(0.006)
I 12 -0.032(0.003) 0.026 (0.004) -0.033(0.005)
Note: In order to test the conditional relationship between
return and beta, the following model is estimated over time.
E[R.sub.pt]=[[gamma].sub.p0] + [[gamma].sub.p1] [D.sub.t][[beta].sub.pt-1] + [[gamma].sub.p2] (1-[D.sub.t])[[beta].sub.pt-1] + [[epsilon].sub.t]
where [D.sub.t] = 1 if realized excess market return is positive
and [D.sub.t] = 0 if realized excess market return is negative.
Standard errors are between parentheses. All coefficients are
significant at the 5% level of confidence.
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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.
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