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BEBR 
 
 FACULTY WORKING 
 PAPER NO. 731 
 
 Further Evidence of the Beta Stability and 
 Tendency: An Application of a Variable 
 Mean Response Regression Model 
 
 Cheng F. Lee 
 Carl R. Chen 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois, Urbana-Champaign 
 
AW'' 
 
 FACULTY WORKING PAPER 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana-Champaign 
 December 15, 1980 
 
 FURTHER EVIDENCE ON 1 
 TENDENCY: AN APPLIC^ 
 RESPONSE REGRESSION J 
 
 Cheng F. Lee, Profess 
 Carl R. Chen, Univers 
 
 #: 
 
 Summary 
 
 A Variable Mean Response Regres 
 the stability and tendency of beta ( 
 model is a better model than the tr; 
 model in investigating the beta stal 
 error criteria is also used to show 
 from the VMRR model are generally mc 
 future betas than those obtained frc 
 model. 
 
 The person charging this material is re- 
 sponsible for its return to the library from 
 which it was withdrawn on or before the 
 Latest Date stamped below. 
 
 Theft, mutilation, ond underlining of books are reasons 
 for disciplinary action and may result in dismissal from 
 the University. 
 To renew call Telephone Center, 333-8400 
 
 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 
 
 JUN 8 ^ 
 
 me 
 
 R 
 
 on 
 
 e 
 
 L161— O-1096 
 
 ill?D^7!IV II 
 
I, Introduction 
 
 The stability of beta coefficients is of interest to both security 
 and portfolio analysis. Levy [1971] has investigated short-term sta- 
 tionarity of beta coefficients; Blume [1971, 1975] and Klemkosky and 
 Martin [1975] have studied the regression tendency of beta coefficients; 
 and Fabozzi and Francis [1978] have conducted a random coefficient test 
 on over 700 stocks. However, none of these studies has explicitly de- 
 rived a model to allow both the regression tendency and the degree of 
 uncertainty for a beta coefficient to be tested empirically. The main 
 purpose of this paper is, therefore, to derive a variable mean response 
 regression capital asset pricing model for testing simultaneously the 
 stability and the regression tendency of beta coefficients. 
 
 Based upon either a linear or a quadratic time trend specification, 
 it will be shown that the model presented in this paper allows for the 
 co-existence of a random beta and a shifting beta. 
 
 In the second section, the problems of testing the mean-variance 
 type of capital asset pricing model (CAPM) is discussed in accordance 
 with the papers by Roll [1977, 1978]; the arbitrage pricing theory (APT) 
 is reviewed in terms of the papers by Ross [1976, 1977, 1978]. In the 
 third section, the model to be used in this study is developed; its re- 
 lationship to previous studies in investigating the regression tendency 
 and the stability of beta coefficients is explored; and the theoretical 
 and empirical reasons for using the model are discussed. In the fourth 
 section, the method for estimating the variable cean response regression 
 market model is presented and asymptotic estimators are derived. In the 
 
-2- 
 
 fifth section, monthly data for 363 firms from January, 1965 to 
 September, 1979 is used to test the stability of betas and the existence 
 of regression tendency for beta coefficients in accordance with the 
 model developed in this paper. Some of the empirical results are re- 
 ported, and the implications of random and trend components on previous 
 empirical risk decomposition analysis and the heteroscedasticity test 
 of the CAPM are also analyzed. Finally, the results of the study are 
 summarized in the last section. 
 
 II. Capital Asset Pricing Model and Arbitrage Pricing Theory 
 
 Professor Roll [1977, 1978] has statistically and theoretically 
 criticized the mean-variance type of CAPM developed by Sharpe [1964], 
 Lintner [1965] and Mossin [1966]. Statistically, a researcher is gen- 
 erally unable to accurately measure the true market portfolio. Theoret- 
 ically, there exist some tautological relationships in determining the 
 efficient set and testing efficient market hypothesis. Statistical 
 problems can be resolved by modern econometric methods. However, the 
 tautological theoretical relationship is a much more serious problem 
 than the statistical problem. As an alternative, Professor Roll sug- 
 gests the APT theory developed by Ross [1976, 1977 and 1978]. The APT 
 theory is basically a multi-index model developed under much less strin- 
 gent assumptions than those of the CAPM. In the following section, the 
 variable mean response regression model is defined. It will empirically 
 be shown that the variable mean regression response methodology will be 
 
 This section is based upon one of the anon37Tiious reviewer's con- 
 structive suggestions. 
 
-3- 
 
 Qore applicable to the APT than either the fixed coefficient or the 
 standard random coefficient regression model. 
 
 III. A Variable Mean Response Regression Capital Asset Pricing Model 
 
 Following Singh, Nagan, Choudhry, and Raj [SNCR, 1976], a variable 
 mean response regression model for estimating the parameters of the cap- 
 ital asset pricing model developed by Sharpe [1964], Lintner [1965-a], 
 and Mossin [1966] is defined as follows: 
 
 ^^^ ^t) = ^t)^t) -^ ^ity 
 
 where Y^^^ = R^^ - R^^, X^^^ = R^^, 
 
 R. = 1 + rates of return on. jth asset, 
 
 R . = 1 + market rate of return, 
 mt 
 
 R^ = 1 + market rate of return, 
 e(t) = disturbance term, and 
 
 (2) 6(t)=^+ V^V^^^t)- 
 
 In equation (2), 8 represents the constant component of the beta coeffi- 
 cient; t represents the time trend; r\. . is the random stock associated 
 
 with 6, .; and a. and a„ are parameters associated with the time trend. 
 
 2 
 The estimated variance of H/. x, o , can be used to measure the degree 
 
 of uncertainty for 6^ x; and the signs and magnitudes of the estimated 
 
 values for a. and a^ will test for the existence of beta tendency. It 
 
 should also be noted that the specification of equation (2) can be used 
 
 to make better estimates of the beta coefficient and the random risk. 
 
-4- 
 
 The specification of equation (2) can be considered as a general 
 
 case of previous research. There are three reasons why this is so. 
 
 2 
 
 First, if the estimated values for a., a„, and a are not significantly 
 
 different from zero, then the fixed coefficient ordinary least squares 
 (OLS) estimator will be an acceptable method for estimating the betas. 
 
 This is because equation (2) can be reduced to 6- . = B. Secondly, if 
 
 2 
 the estimated value of a is significantly different from zero while 
 
 the trend components are not, the random coefficient model developed by 
 
 Theil and Mennes [1959], and Hildreth and Houcks [1968] could be used 
 
 to analyze the degree of uncertainty and to estimate beta. Fabozzi and 
 
 Francis [1978] have utilized this approach to investigate the stability 
 
 2 
 of betas. And thirdly, if the estimated values for a,, a., and a are 
 
 all statistically different from zero, this may indicate that beta co- 
 efficients are moving randomly around a trend line. As a result, the 
 SNCR's variable mean response regression model could be used to estimate 
 explicitly the constant component, the trend component, and the random 
 component of beta coefficients. These three components can be used to 
 describe the arbitrage process of capital asset pricing determination. 
 This has not been considered in previous studies. 
 
 Further, the necessity of using a variable mean response regression 
 model instead of fixed coefficient model to estimate the beta coeffi- 
 cients can be justified on both empirical and theoretical grounds. Em- 
 pirically, Cohen and Pogue [1967] and Lee and Lloyd [1977] have found 
 that the multi-index model can generally be applied to improve the ex- 
 planatory power of the market model. If a single index model is adopted 
 instead of a multi-index model, the necessity of using the variable mean 
 
-5- 
 
 response regression model can be justified by the impact of omitted 
 
 2 
 variables. This is consistent with Hildreth and Houck's argument that 
 
 existence of population variances associated with the regression coeffi- 
 cients can be explained by the impact of omitted variables. Cooley and 
 Prescott [1973] have also argued that sequential parameter variation may 
 arise because of structural change, mis-specifications, and the problem 
 of aggregation. Theoretically, Merton [1973] has shown that the invest- 
 ment opportunity set generally shifts over time unless the interest rate 
 is constant over time. Black [1976] has argued that shocks in the cap- 
 ital market should be regarded as random fluctuations in the beta coef- 
 ficient of a dynamic capital asset pricing model. These arguments, 
 obviously, strongly support the hypothesis advanced in this paper. 
 
 IV. Estimation Methodology 
 
 To test equation (2), SNCR's study proposed alternative estimation 
 methods. In this study, the Hildreth-Houck method is used. Substitut- 
 ing equation (2) into equation (1), the following is obtained: 
 
 where 
 
 (4) X*^^ = tX^^^ and X*^^ = t^X^^^, and 
 
 (5) w^^^ = n(^)X^^) + e^^). 
 
 2 
 Turnbull [1977] has theoretically shown how a firm's systematic 
 
 risk can be affected by both the firm-related variables and by general 
 
 economic variables. 
 
-6- 
 
 The reason for using quadratic functional form of time trend is that the 
 quadratic form is used by most researchers and it also allows us to test 
 the regression tendency. To estimate equation (3), the Hildreth-Houck 
 two-step estimator as extended by SNCR is applied. For simplicity, 
 equation (3) can be expressed in matrix notation as: 
 
 (6) Y = Z6 + w. 
 
 where Y is a Txl vector of observations on the dependent variable Y ; 
 and 
 
 (7) Z = [X X*], 
 
 * *2 
 X and X* are Txl and Tx2 matrices of regressors, X, . and [X, vX-. vj 
 
 respectively; and 
 
 (8) e' = [I'a']. 
 
 where 6 and a are column vectors such that a' = [ct a^] is a row vector. 
 The distribution of w is assumed to be: 
 
 (9.1) E(w) = 0, and 
 
 (9.2) E(w'w) = E(XTiri'X') + E(ee') 
 
 = XAX' + a"! 
 
 e 
 
 = Q. 
 To estimate g, the following is first obtained: 
 
 (10) w = ^fw, 
 
-7- 
 
 where M is a syranetric, idempotent roatrix such that: 
 
 (11) M = I - Z(Z'Z)~^Z. 
 
 Next, OLS is applied to: 
 
 A 
 
 (12) w=MX +u=GA+u, 
 
 where w, M, and X are the vector and matrices of the squared elements 
 of w, M and X, respectively; and u is a vector of random error. The 
 A estimator is thus: 
 
 (13) A = (G'G)~-'-G'w. 
 
 With A estimated, Q, can be constructed following equation (9.2). 
 Finally, the generalized least squares estimator for g can be written 
 as: 
 
 (lA) 6 = iz'n'hrh'n'h, 
 
 with the variance-covariance matrix: 
 
 (15) Var(6) = (Z'^'^Z)"-"". 
 
 V. Empirical Results and Their Implications 
 
 The monthly rates of return for 363 New York Stock Exchange (NYSE) 
 companies from January, 1965 to September, 1979 are selected for this 
 study in accordance with the model developed in the previous section. 
 Both stock dividends and splits are adjusted for proper rates of return. 
 The Standard and Poor (S&P) composite index is employed to calculate the 
 
-8- 
 
 monthly market rate of return. The monthly treasury bill rate is used 
 to proxy the risk-free rates of return. To compare the results obtained 
 from the variable mean response regression CAPM with those obtained from 
 the traditional and fixed coefficient CAPM, the OLS method is first ap- 
 plied to estimate the traditional fixed coefficient betas (2.) and the 
 
 unsystematic risk. Secondly, the model of equation (3) without the term 
 
 2 — 2 2 3 
 
 at X> . is used to estimate 6 , a., a , and a . The generalized least 
 
 squares (GLS) estimated constant component of systematic risk (B) and 
 
 the GLS estimated a. are calculated as defined in equation (lA); and the 
 
 asymptotic variance estimators associated with 3 and a. are calculated 
 
 2 
 as defined in equation (15). The least squares estimators for both 
 
 2 4 
 
 and a are found in equation (9.2). 
 
 To measure the degree of uncertainty, an index of uncertainty (lOU) 
 
 is defined following the coefficient of variation concept as: 
 
 (16) lOUj = %j/|6j!. 
 
 where a is the estimated population standard deviation associated with 
 the beta coefficient for the jth firm. 6, is the estimated constant 
 component of the beta coefficient for the jth firm. This index 
 
 3 2 
 t X> . was added to test the convergence of beta tendency. How- 
 ever, due to the high degree of multicollinearity, this variable was 
 dropped in the final model. 
 
 There exists about twenty percent of stocks with negative esti- 
 
 2 
 mated a . As Fabozzi and Francis [1978] pointed out, the restricted GLS 
 
 emcloying quadratic programming can be used to deal with this kind of 
 problem. 
 
 This index has been used by Kau and Lee [1977] to measure the de- 
 gree of stability of the density gradient for an urban structure study. 
 
-9- 
 
 provides a criteria for security analysts to decide whether the histori- 
 cal beta coefficient of a firm is an acceptable predictor for the future 
 beta of the firm. The mean and standard deviation of the estimated lOU 
 for 363 firms are 1.209 and 1.671 respectively. It is obvious that the 
 estimated population standard deviation associated with the beta coeffi- 
 cient (a ) can also be used to formulate the interval estimate for a 
 firm's beta coefficient. An index of uncertainty can also be an indi- 
 cator of the marginal benefit of utilizing the multi-index CAPM. As the 
 index approaches zero, the implication is that 3. is an efficient pop- 
 ulation estimate and requires no additional data to estimate the multi- 
 index CAPM. According to the estimates derived in this study, sixty- 
 five percent of the sample firms have o values which are significantly 
 different from zero. This suggests that the residuals associated with 
 these fiinns are not homoscedastic. One hundred five firms (29%) have 
 an estimated a value that is significantly different from zero. This 
 indicates that a significant portion of the sample firms have shifting 
 
 betas during the period from 1965 to 1975. This result, in addition to 
 
 2 
 the significant a values, could well indicate that a variable mean re- 
 
 n 
 
 sponse regression market model better describes the beta coefficient 
 for some firms than does the constant beta and/or pure random beta mod- 
 
 A 
 
 els. From the signs of 6. and a^^, it is found that there exists some 
 regression tendency as established earlier by Blume [1975]. This argu- 
 ment is based upon the fact that the estimated value of a is a factor 
 for adjusting the beta coefficient toward unity. In other words, the 
 sign of a. is generally positive when the magnitude of of the estimated 
 beta is less than one, and negative if the magnitude of beta is greater 
 
-10- 
 
 than one. Hence the sign, magnitude, and degree of significance of a. 
 provides information for security analysts to improve their forcasting 
 of future beta coefficients. 
 
 To investigate the possible success of beta predictions of variable 
 mean response regression model relative to the traditional fixed coeffi- 
 cient regression model, the mean squared error (MSE) technique developed 
 by Mincer and Zarnowitz [1969] as indicated in equation (17) will be 
 used to do empirical analysis. 
 
 (17) MSE = (\^^-\)' + a~\)^S^l 4. (1-r2,^i.3,)s2^, 
 
 bias inefficiency random error 
 
 where 6 . and g are the means of all beta in period t+1 and t, respec- 
 tively, b- is the slope coefficient of 6 . regressed on 3 , S„,^ and 
 S„^ are the sample variance of 6^., and 6^, respectively. R.,, ^^ is 
 
 Pt t+X t PT'J.jPt 
 
 A A 
 
 the coefficient of determination for regression of 6 . ^ or 6 . Note 
 that Klemkosky and Martin [1975] have used this kind of technique to 
 analyze the performance of alternative beta forcasting technique. 
 To estimate the MSE as indicated in equation (17), data from 
 January, 1965 to March, 1975 are used to estimate the related informa- 
 tion of period t and the data from April, 1975 - September, 1979 are 
 used to estimate the related information in period t+1. The results 
 associated with equation (17) are listed in Table 1. It is found that 
 the MSE of forcasting obtained from the time-varying coefficient beta 
 estimates is smaller than that obtained from constant coefficient beta 
 estimates. The reduction of MSE is essentially due to the reduction of 
 
-11- 
 
 Table 1 
 
 MSE of Constant Coefficient Betas 
 and Time-Varying Betas 
 
 ^t+1 
 
 Bt 
 
 8t+l 
 
 et+1, et 
 
 bias 
 
 In efficiency 
 random error 
 MSE 
 
 Constant Coefficient 
 1.01325 
 .94682 
 .30780 
 .1355 
 .2396 
 .03030 
 .00441 
 .00880 
 .05570 
 .06892 
 
 Time-Varying Coefficient 
 .99928 
 .96648 
 .42850 
 .18913 
 .2448 
 .10957 
 .00108 
 .01168 
 .05338 
 .06613 
 
-12- 
 
 bias. Note that the inefficiency from time-varying coefficient estimate 
 is larger than that for constant coefficient estimates. In sum, it 
 seems reasonable to conclude that the main contribution of time-varying 
 beta estimates in forcasting betas is to reduce the bias. Besides this 
 kind of advantages, the time-varying coefficient can also be used to im^ 
 prove the risk-return trade-off test as shown in the latter portion of 
 this section. 
 
 Finally, the findings of this paper have other Implications for 
 previous research. Following equation (3), the total risk of each in- 
 dividual security can be decomposed as : 
 
 (18) VarCY^) = [CF^ + ah^ + 2"Ba, t)Var (X^) ] + [aV + o^]. 
 
 t 1 1 t n t e 
 
 2 
 If the estimates for both a. and o approach zero, equation (17) can 
 
 be reduced to: 
 
 (19) Var(Y^) = J^ Var(X^) + a^. 
 
 t t e 
 
 Equation (18) indicates that total risk can be decomposed into system- 
 atic and unsystematic risk by the OLS regression method as discussed by 
 Francis [1979] and others. However, this result does not hold unless 
 g is a deterministic variable. As the beta coefficient is stochastic, 
 
 the OLS estimate of nonsystematic risk actually has two components, i.e., 
 
 2 2 
 
 cr X + c . Therefore, the partition of systematic risk and unsystematic 
 
 risk is not possible since the random risk is confounded with "noise" 
 
 Fabozzi and Francis (1978) have similar arguments. However, 
 their decomposition is only a special case of our decomposition as In- 
 dicated in equation (18). 
 
-13- 
 
 from shifting betas. Both Lintner [1956] and Douglas [1969] found that the 
 rates of return on individual stocks are strongly correlated with random 
 risk and this is contrary to the capital asset pricing theory. Subse- 
 quently, Miller and Scholes [197] carefully reexamined this issue and 
 still failed to provide a satisfactory explanation. 
 
 This issue is reexamined in the present study by running two tests. 
 First, the OLS residual variance is used as a measure of random risk, 
 and a simple linear regression is run with the monthly rates of return 
 for each firm as the dependent variable. The results are consistent 
 with Lintner and Douglas's finding that rates of return are strongly 
 
 correlated with random risk. However, if the "pure residual" variance 
 
 2 
 o is utilized in place of the OLS residual variance, it is found that 
 
 the relationship between rates of return and random risk is not statis- 
 tically significant. This finding could imply that both Lintner and 
 Douglas's result may be due to the problem of using the fixed beta coef- 
 ficient to decompose the total risk while, in fact, beta is not a con- 
 stant. 
 
 V. Summary and Concluding Remarks 
 
 This paper has pointed out some of the problems of previous re- 
 search in the investigation of beta stability and beta tendency. None 
 of the previous studies explicitly derived a model that allows for co- 
 existence of both beta instability and beta tendency. With the applica- 
 tion of the variable mean response regression model, beta can be decom- 
 posed into constant, trend, and random components. The model developed 
 in this study can therefore be considered as a general case of the 
 
-14- 
 
 previous studies. Monthly data for 363 firms was used in this paper to 
 test beta stability and beta tendency. The empirical results revealed 
 that sixty-five percent of the sample firms had significant a values, 
 which indicates that more than half of the firms had an unstable beta 
 over the ten year period. In addition, 105 firms (29%) had significant 
 a. values which implies that a significant portion of the sample firms 
 had shifting betas. The sign and magnitude of ct. is consistent with 
 Blume's [1975] finding that there exists some regression tendency of 
 beta coefficients over time. 
 
 The MSE of forecasting obtained from the model developed in this 
 paper is smaller than that obtained from the traditional fixed coeffi- 
 cient beta estimates.' This study also reexamined the Lintner and Douglas 
 "paradox." After the elimination of both trend and random components of 
 beta risk, the residual variance was found not to be correlated with the 
 security return. This suggests that the risk partition method that has 
 been used by previous studies may have a fundamental flaw, i.e., random 
 risk can be confounded with "noise" from random betas. To improve esti- 
 mates of beta risk and/or random risk, the model present in this study 
 is a viable alternative. The causes of beta tendency, however, are still 
 unknox^n and future research in this area is needed. 
 
-15- 
 
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-16- 
 
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-17- 
 
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