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FACULTY WORKING PAPER NO. 1218

Conditional and Unconditional Heteroscedasticity in the Market Model

Anil Bera Edward Bubnys Hun Park

HE

i986

College of Commerce and Business Administration Bureau of Economic and Business Research University of Illinois, Urbana-Champaign

BEBR

FACULTY WORKING PAPER NO 1218 College of Commerce and Business Administration University of Illinois at Ur bana-Champa i gn January 1986

Conditional and Unconditional He t e r os ceda s t i c i t y in the Market Model

Anil Bera, Assistant Professor Department of Economics

Edward Bubnys Memphis State University

Hun Park, Assistant Professor

Department of Finance

This research is partly supported by the Investors in Business Education at the University of Illinois

Conditional and Unconditional Heteroscedasticity in the Market Model

Abstract

Unlike previous studies, this paper tests for both conditional and unconditional heteroscedasticities in the market model, and attempts to provide an alternative estimation of betas based on the autoregres- sive conditional heteroscedastic (ARCH) model introduced by Engle. Using the monthly stock rate of return data secured from the CRSP tape for the period of 1976-1983, it is shown that conditional heterosce- dasticity is more widespread than unconditional heteroscedasticity, suggesting the necessity of the model refinements taking the condi- tional heteroscedasticity into account. In addition, the efficiency of the market model coefficients is markedly improved across all firms in the sample through the ARCH technique.

Conditional and Unconditional Heteroscedasticity in the Market Model

I. Introduction

Over the past two decades, the capital asset pricing model (CAPM) has heen more widely used in finance than any other model. Inherent in the CAPM is that capital assets, under some simple assumptions, are held as functions of the expected means and variances of the rates of return, and that investors pay only for the systematic risks of capi- tal assets which are measured by the relationship between the return on individual assets and the return on the market (e.g., the single- index market model of Sharpe [14]). However, a number of studies have raised questions on the validity of the market model to estimate the systematic risk of financial assets using the ordinary least squares (OLS) technique. In particular, a significant portion of the previous studies have focused on the variance structure of the market model (see for example [1], [2], [4], [5], [6], [8], [91, [12] and [13]).

In the traditional approach, when the mean is assumed to follow a standard Linear regression, the variance is constrained to be constant over time. However, many studies have provided strong evidence of heteroscedasticitv for a number of common stocks not only in the U.S. market but in the Canadian and European markets and attributed in gen- eral to model misspecif ication either by omitted variables or through structural changes. Giaccotto and Ali [10] used two classes of opti- mal nonparametric distribution-free tests for heteroscedasticity and applied them to the market model. They found the assumption of homoscedasticity to be untenable for the majority of stocks analyzed.

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Lehmann and Uarga fll] took issue with Giacotto and Ali's use of recursive residuals in rank tests in order to provide some information concerning the stochastic properties of market model regressions.

Nevertheless, most of the previous studies have heen limited to investigating the existence of heteroscedasticity in the market model using different statistical tests and rarelv considered estimation of betas taking heteroscedasticity explicitly into account. Furthermore, they considered only unconditional variances of the disturbance term, ignoring the possible changes in conditional variances. Some studies attempted to find a form of heteroscedasticity in an ad-hoc fashion. The predominant approach was to introduce an exogeneous variable (the market return in most cases) which may predict the variance. As pointed out by Engle [7], this conventional solution to the problem requires a specification of the causes of the changing variance in an ad-hoc fashion rather than recognizing that both means and variances conditional on the information available may "jointly evolve over time.

The purpose of this study is to test for both conditional and unconditional heteroscedasticities, and provide an alternative estima- tion of betas based on the autoregressive conditional heteroscedastic (ARCH hereafter) model, introduced by Engle [7]. The ARCH process is characterized by mean zero, seriallv uncorrelated processes with non- constant variances conditional on the past but constant unconditional variances. This study is different from the earlier ones in some important aspects. First, using the ARCH model, this study provides a more realistic measure of betas when the underlying conditional

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variance mav change over Lime and is predicted by the past forecast errors rather than making conventional assumptions about the distur- bances. Second, it provides more efficient estimators using the maxi- mum likelihood method. Third, it does not employ an arbitrary exogenous variable such as the market return to explain heterosce- dasticity. Lastly, by the nature of the ARCH process, the effects of omitted variables from the estimated model, and a portion of non- normality of the regression disturbance terms, may be picked up.

Section IT describes the model and section III presents empirical results. Section IV contains a brief summary.

II. Model

Consider the single index market model

R. = a. + 8 .R + E. (1)

it l l mt it

where R. and R , are the random return on security i and the random it mt J

market return, respectively, in period t; a. and 8. are the regression

parameters of security i; E. is the random disturbance term with

it

E(E. ) = 0 for all i and t. The parameter 8. measures the systematic it i J

risk of security i and is defined as Cov(R. , R )/Var(R ). One of the

l m m

many assumptions that model (1) is based on is that the disturbances are homoscedastic.

First, we check for unconditional heteroscedasticity using the White [15] test which does not assume anv specific form of the hetero- scedast icitv. Second, we test for conditional heteroscedasticity and

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simultaneouslv attempt to reestlmate the market model with an ARCH specification.

For ease of exposition, let us put model (1) in a general linear regression framework in matrix notation:

Y = X 8 + E (2)

(Txl)(Txk)(kxl)(Txl)

with k=2. Let 8 be the OLS estimator. If the E's are homoscedastic , then a consistent estimator of the variance-covariance matrix of 8 is given by

VjCS) = a2(X'X)-1 (3)

where a = E'E/T-k and E's are the OLS residuals.

In the presence of heteroscedescity , a consistent estimator of 8 wi 1 1 be

V2(6) = (X'X)"1(X'EX)(X'X)"1 (4)

~ 2 ~ 2 ~2

where E = diag (E , E-, . . . , ET).

Under homosecdasticity , these two estimates V.(8) and V„C8), will converge to the same limit. However, in case of possible heterosce- dasticity, V.(8) is inconsistent and these two estimators will have different limits. White's procedure is based on examining the sta- tistical significance of the limits of the k(k+l)/2 distinct elements of

the matrix V = V (8 ) - V (8). Under the assumption of homoscedast icity ,

2 the test statistic is asymptotically distributed as x with k(k+l)/2

degrees of freedom. One attractive feature of this test is that it

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does not depend on a specific form of heteroscedasticity. Assuming

that the disturbances are homokurtic, the test statistic can be

2 ? calculated as T*R where R is the coefficient of determination

obtained by regressing the square of the residuals on a constant,

- 2 R and R~ . The test statistic will have 2 degrees of freedom (one mt mt

less because of the presence of an intercept term in the regression).

Following Engle [7], we assume that the conditional heterosce- dasticity of the market model can be represented by a first-order ARCH process (suppressing the suffix i) as:

V(VW =°t -*o + *i Et-i • <5)

where Yn > 0, anH 0 < y < 1.

This tvpe of conditional heteroscedasticity has some intuitive appeal since it does not depend on some arbitrarv exogenous variables and can be viewed as some average of the square of the past distur- bances. When Yi = ^» we have conditional homoscedasticity. It can be easily shown that unconditionally E(E ) = 0, V(E ) = Y /(1~Y,) and E(E E ,) = 0 for t * t' (see Engle [7] for details). Since we do not have anv lagged dependent variable in our model, V. (8 ) - V„(B) will have the same probability limit. Therefore, White's procedure will test only for unconditional heteroscedasticity, while the test for Y, = 0 in (5) will provide a test for conditional heteroscedasticity. It is also important to note that under the ARCH model, the E 's are uncorrelated but dependent, which is a property of non-normal dis- tributions .

The log-likelihood function assuming normality is given hv

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T L = Z (a -\ log a1 - \ Z2J°h (6)

t = l

where a is a constant term and E = R - a - 8 R . Using Ouandt's

t t mt

subroutine GRADX of his numerical optimization program 0Q0PT3, we maximize the function L and obtain estimates of a, 3, Yn and y.. These estimates of a and 8 are compared with the OLS estimates. Significance of y, will indicate the presence of conditional heteroscedasticity. This is tested using t-statistics.

We also test for normality of the disturbances of the OLS and ARCH models. The test statistic is calculated as:

m r(skewness) (kurtosis-3) , T [ - + _ ].

Under normality this test statistic asymptotically follows a central

2 X with 2 degrees of freedom (see Bera and Jarque [3] for detaiLs).

III. Empirical Results

Monthly stock rate of return data was secured from the CRSP tapes for the period 1976-83 (96 months) for 35 randomly chosen firms with- out missing values. Market model regressions using the OLS and ARCH methods were run for each firm. The homoscedasticity and normality tests statistics also were calculated. The results based on the OLS regression and the ARCH model are shown in Tables 1 and 2, respec- tively.

First, from the results on the White test we can observe that only for ten firms the test statistics are significant at 1% and 5% levels. This indicates that the unconditional heteroscedasticity may not be very important. However, all of the normality test statistics are

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significant indicating the presence of strong non-normality. The OLS results exhibit the familiar strong significance of the beta coef- ficient, with 33 of the market models having beta t-values significant at the 1% level. The intercept a terms are, for the most part, small and insignificant.

The importance of conditional heteroscedasticity and thus the appropriateness of the ARCH model can be judged from the significance of 26 y, values in Table 2. It should be noted, however, that the significance of y . (a test of conditional heteroscedasticitv) in Table 2 has no relation with the earlier test for unconditional heterosce- dasticitv in Table 1. This is not surprising since they test two completely different hypotheses. For the ARCH model also, the 33 betas are significant and the significance of several of the inter- cepts is enhanced. Of particular interest is the change in beta as a result of the use of ARCH. Fourteen of the 33 firms with significant OLS betas have increased betas when ARCH is used; nineteen others have beta reductions. However, most of the changes are small. Nineteen ARCH model betas change no more than 4 percent up or down from their OLS counterparts. Six firms' betas increase by 5% or more when ARCH is used, while eight betas decrease by at least 5%. Only five firms (14.3% of the sample) have betas change by 10% or more.

The efficiency of the market model coefficients is markedly im- proved across all firms. The t-values for all the ARCH model betas are higher than those for the OLS model due to the lower standard errors of the coefficients. Thus it appears that incorporation of a

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conditional heteroscedastic component results in greater efficiency, though affecting the parameter estimates themselves slightly. In sum, the differences between the OLS and ARCH results reveal the importance of taking account of conditional heteroscedasticity in the market model.

Lastly, the results on normality tests indicate strong non- normality of the disturbance terms in the ARCH model. This is what we should expect because, as pointed out earlier, the ARCH errors are inherently non-normal or mutually dependent. The ARCH procedure takes account of this type of non-normality or the dependent structure, while the OLS procedure (results of Table 1) neglects it. The gain is reflected in the improved standard errors of a and B in Table 2.

IV. Summary

A number of studies have shown that heteroscedasticity in the market model is widespread. However, the previous studies have been limited to investigating the existence of heteroscedasticity using different statistical tests and rarely considered estimation of betas taking heteroscedasticity explicitly into account. Furthermore, they considered only unconditional heteroscedasticity, ignoring conditional heteroscedasticity. This paper tests for both conditional and uncon- ditional heteroscedast icities and provides an alternative estimation of betas based on the autoregressive conditional heteroscedastic (ARCH) model introduced by Engle. Using the monthly stock rate of return data secured from the CRSP tape for the period of 1976-1983, it is shown that conditional heteroscedasticity is more widespread than unconditional heteroscedasticity, suggesting the necessity of the

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model refinements taking the conditional heteroscedasticity into

account. In addition, the efficiency of the market model coefficients

is markedly improved across all firms in the sample through the ARCH technique.

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Ref erences

ll] Barone-Adesi, G. and P. Talwai. "Market Models and Heteroscedasti-

city of Residual Security Returns." Journal of Business and Economic Statistics, 1 (1983), pp. 163-168.

[2] Belkaoui, A. "Canadian Evidence of Heteroscedasticity in the Market Model." Journal of Finance, 32, (September 1977), pp. 1320-1324.

[3] Bera. A. K. and C. M. Jarque. "Model Specification Tests: A

Simultaneous Approach." Journal of Econometrics, 20 (1982), pp. 59-82.

[4] Bey, R. P. and G. E. Pinches. "Additional Evidence of Heterosce- dasticity in the Market Model," Journal of Financial and Quan- titative Analysis, 15, (June, 1980), pp. 299-322.

[5] Brown, S. J. "Heteroscedasticity in the Market Model: A Comment." Journal of Business, 50, (January 1977), pp. 80-83.

[6] Brown, K. C. , Lockwood, L. J. and S. L. Lummer. "An Examination of Event Dependency and Structural Change in Security Pricing Models." Journal of Financial and Quantitative Analysis, 20, (September 1985), pp. 315-334.

[7] Engle, R. F. "Autoregressive Conditional Heteroscedasticity with

Estimates of the Variance of United Kingdom Inflation," Economet rica , Vol. 5, No. 4, (July 1982), pp. 987-1007.

[8] Fabozzi, F. J. and J. C. Francis. "Heteroscedasticity in the Single Index Market Model," Journal of Economics and Business, 32, (Spring 1980), pp. 243-248.

[9] Fisher, L. and J. Kamin. "Forecasting Systematic Risk: Estimates of "Raw" Beta that Take Account of the Tendency of Beta to Change and the Heteroskedast icity of Residual Returns," Journal of Financial and Quantitative Analysis, 20, (June 1985), pp. 127-150.

10] Giaccotto, C. and M. Ali. "Optimal Distribution-Free Tests and

Further Evidence of Heteroscedasticity in the Market Model." Journal of Finance, 37, (December 1982), pp. 1249-1258.

11] Lehmann, B. and A. Warga. "Optimal Distribution-Free Tests and Further Evidence of Heteroscedasticity in the Market Model: A Comment." Journal of Finance, 40, (June 1985), pp. 603-605.

12] Martin, J. D. and R. C. Klemkosky. "Evidence of Heteroscedasticity in the Market Model." Journal of Business, 48, (January 1975), pp. 81-86.

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tis] McDonald, B. and M. Morris. "The Existence of Heteroscedastici ty and its Effect on Estimates of the Market Model Parameters," The Journal of Financial Research, Vol. VI, No. 2, (Summer 1983), pp. 115-126-

[14] Sharpe, W. F. "A Simplified Model for Portfolio Analysis." Manage- ment Science, 9, (January 1963), pp. 277-293.

[15] White, H. "A Heteroskedasticity — Consistent Covariance Matrix

Estimator and a Direct Test for Heteroskedasticity," Econometrica, 48, (1980), pp. 817-838.

D/315

Table 1

Results Based on the Ordinary Least Squares Regression ( t-statistics are in parentheses)

Homoscedasticity*

Firm No.	a	8	(White test)	NormaliLv*
1	-.002 (-.026)	1.22a (7.70)	0.634	36.33a
2	.003 (0.29)	1.06a (4.82)	3.005	22.40a
3	-.003 (-0.66)	1.02a (8.72)	1.152	19. 29a
4	.008b (2.10)	0.69a (8.35)	3.562	21.03a
5	.002 (0.18)	1.75a (6.72)	8.774b	14.72a
6	.021b (1.71)	0.07 (0.24)	0.115	14.473
7	.009 (1.02)	-0.14 (-0.69)	8.304b	22.72a
8	.012 (1.58)	0.75a (4.44)	9.053b	33.81a
9	.021 (1.43)	1.73a (5.31)	0.557	20.793
10	.002 (0.33)	1.02a (6.81)	2.803	23.75a
1 1	-.001	1.05a	16.4353	21.083
	(-0.15)	(7.32)		9.05b
12	.015	1.87a	0.701
	(1.64)	(8.86)
13	.010 (1.34)	0.99a (5.88)	6.682b	42.68a
14	.002 (0.23)	1.07a (6.95)	1.123	16. 31a
15	.010 (1.03)	1.28a (5.69)	4.848	20.39a
16	.016 (1.38)	0.77a (3.04)	2.131	33.26a
17	-.006 (-0.85)	1.26a (8.36)	10.589a	47.57a
*x2	test statistics.

Significant at the VI level; the critical value for t-statistics

2.37 and the critical value for x statistics = 9.21.

Significant at the 5% level; the critical value for t-statistics

1.66 and the critical value for x statistics = 5.99.

Table 1 (con't. )

Results Based on the Ordinary Least Squares Regression ( t-statistics are in parentheses)

			Homoscedasticitv*
Firm No.	a	B	(White test)	Normal i ty
18	-.003 (-0.37)	0.96a (5.92)	3.907	40.c>7a
19	.006	1.08a	0.816	21.74a
	(0.67)	(5.18)		7.73b
20	.010	0.76a	0.701
	(1.20)	(4.25) 0.97a
21	.001	4.080	32.42a
	(0.18)	(6.54)
22	.003 (0.47)	0.88a (5.74)	2.390	44.86a
23	.009 (1.08)	1.95a (10.41)	13.8343	33.27a
24	.009 (1.31)	0.83a (5.13)	11.539a	37.97a
25	-.001 (-0.08)	1.31a (6.21)	2.688	33. 19a
26	.008 (1.25)	0.83a (5.90)	9.907a	13.64a
27	.006 (1.00)	0.89a (6.74)	5.126	25.66a
28	.002 (0.40)	0.58a (4.55)	0.048	19.31a
29	.011 (0.68)	0.94a (2.68)	4.0^0	36.113
30	-.003 (-0.41)	1.05a (6.44)	4.906	34.26a
31	.024b (2.30)	1.31a (5.64)	3.629	17.40a
32	,011b (1.77)	1.07a (7.80)	4.704	29.44a
33	.009 (0.97)	1.1 5a (5.31)	3.725	38.32a
34	.023a (2.79)	1.27a (6.81)	10.0033	18.213
35	.007 (0.76)	1.71a (8.72)	2.803	30.633
*x2	test statistics.

a.

Significant at the L% level; the critical value for t-statistics = 2.37 and the critical value for x2 statistics = 9.21.

Significant at the 5% level; the critical value for t-statistics = 1.66 and the critical value for x2 statistics = 5.99.

Table 2

Results Based on the ARCH Model ( t-stattstics are in parentheses)

	a	a	^	»	Normality
Firm No.	a	8	Y0 .003a	.305a	Test Statistics*
1	-.003	1.33a	33.37a
2	(-.61) .004	(12.30) 0.95*	(5.90) .007a	(2.18) .172	22.56a
3	(0.69) -.003	(6.46) -0.98a	(7.09) .002a	(1.47) .142	19.l4a
4	(-0.095) .008a	(12.51) 0.64a	(7.62) ,008a	(1.59) .275a	20.183
5	(3.61) .002	(12.15) 1.77a	(6.68) .010a	(2.40) .204	14.493
6	(0.23) .018b	(10.48) -0.05	(6.85) .oua	(1.61), .166b	14.23a
7	(2.24) .008	(-0.25) -.216	(7.83) .006a	(1.72) .106	22.70a
8	(1.39), .onb	(-1.59) .784a	(8.14) .003a	(1.50), .310b	33.33a
9	(2.23) .023a	(7.03) 1.74a	(6.11) .017a	(2.29) .077	20.7 53
10	(2.42) .005	(8.14) 1.02a	(8.80) .003a	(1.30), .241b	23.82a
11	(1.10) -.003	(10.44) 1.18a	(6.63) .003a	(1.90), .205b	15.423
12	(-0.82) .016a	(12.69) 1.90a	(7.63) .007a	(2.11) .118	9.09b
13	(2.51) .009	(13.64) 0.98	(7.98) .004a	(1.44) .182b	42.87a
14	(1.76) .001	(8.67) 1.03a	(7.12) .0033	(1.84), .138b	16.113
15	(0.30), .onb	(10.11) 1.55a	(8.22) .005a	(1.83) .452a	19.38a
16	(1.79) .017a	(11.34) 0.72a	(6.01) .008a	(2.99), .254b	33.37a
17	(2.38) -.007	(4.48) 1.23*	(6.77) .003a	(2.15) .149	47.703
	(-1.62)	(12.44)	(6.92)	(1.42)

test statistics.

Significant at the 1% level; the critical values for t-statistics 2.37 and the critical value for x statistics = 9.21.

Significant at the 5% level;„the critical values for t-statistics 1.66 and the critical value for x statistics = 5.99.

Table 2 (cont'd. )

Results Based on the ARCH Model ( t-stat ist ics are in parentheses)

			^	»	Normality
Firm N'o.	a	8	Y0 .004a	Yl .142b	Test Statistics*
18	-.003	0.92a	40.613
	(-0.70)	(8.45) 1.09a	(7.74)	(1.69), .273b
19	.004	.006	21.86a
	(0.75)	(8.11)	(6.91)	(2.18)	7.79b
20	.007	0.753	.003a	.392a
	(1.45)	(6.85)	(6.04)	(2.65), .137h
21	.0008	0.95a	.003a	32.61a
	(0.17)	(9.65) 0.89	(7.88)	(1.72), .243b
22	.003	.003a	44.96a
	(0.70)	(8.98)	(5.68)	(1.69), .221b
23	.006	1.80a	.005a	33.43a
	(1.17)	(14.57)	(7.17)	(2.20) .224h
24	.011b	0.84a	.no3a	37.94a
	(2.29)	(7.86) 1.29a	(6.55)	(1.96) .183
25	-.00003	.006	32.39a
	(-0.005)	(9.39)	(7.25)	(1.81)
26	.004	0.89a	.002	.506a	14.08a
	(1.14)	(10.18)	(6.10)	(3.04) .357b
27	.006	0.82a	.002a	25.69a
	(1.62)	(9.79) 0.68	(6.04)	(2.19), .244b
28	-.0009	.002a	16.43a
	(-0.03)	(8.15)	(7.42)	(2.33)u .138b
29	.006	0.90a	.018a	36.48a
	(0.60)	(3.87)	(7.88)	(1.71)
30	-.002	1.09a	.004a	.117	34.45a
31	(-0.33) .024a	(10.07) 1.20a	(8.12) .008	(1.62), .151b	16.783
	(3.47)	(7.72)	(7.98)	(1.85)
32	.010a	0.95a	.002a	.343a	27.83a
	(2.55)	(11.09)	(6.61)	(2.76) .210
33	.008	1.16a	.006a	38.163
	(1.20)	(8.03) 1.15a	(6.55)	(1.66), .243b
34	.022a	.005a	18.173
	(4.12)	(9.39)	(6.72)	(1.92), .316b
35	.004	1.65a	.005a	29.86a
	(0.77)	(12.64)	(6.13)	(2.34)

test statistics.

Significant at the 1% level; the critical values 2.37 and the critical value for x statistics = 9.21.

for t-statistics =

Significant at the 5% level; the critical values 1.66 and the critical value for x~ statistics = 5.99.

for t-statistics =

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