Testing the Multivariate Normality of Australian Stock Returns.

Abstract:

The multivariate normality of stock returns is a crucial assumption in many tests of assets pricing models. While past Australian research has examined the univariate normality of returns, univariate test statistics are unreliable for testing multivariate normality since they ignore the contemporaneous correlation between asset returns. This paper utilises a multivariate test procedure, based on the generalised method of moments, to test whether residuals from market model regressions are multivariate normal. The results suggest violations of the multivariate normality assumption which cast doubt over the validity over inferential procedures commonly used in the extant empirical literature.

Keywords:

MULTIVARIATE NORMALITY; GMM; SKEWNESS; KURTOSIS.

1. Introduction

Whether or not stock returns conform to a multivariate normal distribution is an issue of general importance to researchers in finance. From a theoretical perspective, the mean-variance framework which underlies the Sharpe-Lintner capital asset pricing model (CAPM) is predicated on certain assumptions required for rational agents to hold mean-variance efficient portfolios. A sufficient assumption is that asset returns are distributed multivariate normal. Since a normal distribution is described completely by its first two moments (mean and variance), the assumption of normality fits neatly with the mean-variance framework of CAPM.

From an empirical perspective, the multivariate normality assumption is a feature of several well-known tests of asset pricing models. For example, based on the assumption of multivariate normality, Gibbons, Ross and Shanken (1989) and MacKinlay (1987) derive a finite sample test of the mean-variance efficiency of a portfolio, while Gibbons (1982), Kandel (1984), Shanken (1985, 1986) and Zhou (1991) provide tests of Black's zero-beta CAPM. In the Australian context, Stokie (1982a), Faff (1991) and Wood (1991) have tested asset pricing models relying on the multivariate normality assumption.

Given the importance of the assumption, it is surprising how little research has been conducted on the multivariate normality of stock returns. While the distribution of stock returns has been studied for decades,(1) the focus of this research has largely been univariate normality. On face value, a univariate test of normality might appear sufficient to test multivariate normality: if asset returns are multivariate normal, then each asset's marginal distribution is normal. Conversely, if an asset's return is not univariate normally distributed, then the joint distribution of all assets' returns cannot be multivariate normal. Hence, rejection of univariate normality implies returns are not multivariate normal. However, Richardson and Smith (1993) note that univariate tests ignore contemporaneous correlation between assets and consequently tests based on univariate statistics may be misleading. By allowing for cross-sectional dependence in returns, Richardson and Smith provide the first multivariate test of the normality of stock returns.

In the Australian context, only a handful of papers have examined the distribution of stock returns, and these have been exclusively univariate tests. This paper provides a test of the multivariate normality of Australian stock returns. Following Richardson and Smith (1993), we employ the distribution theory for parameter estimates under Hansen's (1982) generalised method of moments (GMM) to derive test statistics which explicitly accommodate cross-correlations in returns.

The remainder of the paper is structured as follows. Section 2 discusses univariate testing methodology commonly employed and reviews several prior Australian studies of the distribution of stock returns. Section 3 outlines the multivariate test procedure advocated by Richardson and Smith (1993). In particular, the asymptotic distribution of parameter estimates under GMM is employed to derive joint tests of skewness and kurtosis which explicitly account for contemporaneous correlation between asset returns. This approach is also extended to utilise the multivariate structure of asset returns in test statistics by calculating several cross-moments of returns. To assess the validity of the multivariate normality assumption in tests of asset pricing models, section 4 applies the multivariate tests to the residuals from market model regressions for ten size portfolios. Section 5 concludes the paper.

2. Background

2.1 Univariate Test Statistics

Before reviewing previous research on the distribution of Australian stock returns, it is useful to outline the univariate test statistics which have been commonly employed in testing for normality.

The (univariate) normal distribution is completely described by its first two moments--mean ([Mu]) and variance ([[Sigma].sup.2]). All odd moments higher than one are zero and all even moments are a function of variance.(2) Testing whether a variable is normally distributed requires comparison of sample moments with their theoretical values under the normal null hypothesis. Denote by [R.sub.it] the return on asset i in period t. The sample mean and variance of [R.sub.it] are calculated:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the sample third moment (skewness) and sample fourth moment (kurtosis) of asset i respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Univariate tests of skewness and kurtosis are based on standardised measures of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which we denote [S.sub.i] and [K.sub.i] respectively. Namely, under the normal null:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Hence, a univariate test for the normality of returns to asset/portfolio i will involve testing whether sample values of [S.sub.i] and [K.sub.i] differ significantly from zero. In the following section, we note that these tests have been the primary method of testing the normality of Australian stock returns in previously published research.

2.2 Extant Australian Evidence

Several studies have examined the normality of Australian stock returns, either directly or indirectly. Ball, Brown and Officer (1976) primarily test several asset pricing models, but in doing so, consider the normality of returns on beta-sorted portfolios during the 1958-1970 period. They calculate [S.sub.i] for each portfolio and finding only two out of 20 test statistics differ significantly from zero, conclude there is little evidence of skewness.(3)

Praetz and Wilson (1978) calculate [S.sub.i] and [K.sub.i] for 578 stocks listed between 1958 and 1973, as well as portfolios of various sizes. They find that 72% of individual stocks are not skewed, and that of the remaining 28%, the majority are positively skewed. Nearly 70% of individual stocks exhibit excess kurtosis. Over 76% of stocks have an acceptable goodness-of-fit to a t-distribution, a result consistent with US findings that a fat-tailed distribution describes stock returns well (see Blattberg & Gonedes 1974).

Stokie (1982b) argues that much of the non-normality reported in Praetz and Wilson (1978) can be attributed to the high incidence of zero returns, especially in the less frequently traded smaller companies. Using the Praetz and Wilson dataset, Stokie examines the top 40 stocks (by market capitalisation) which have few zero returns and finds that the percentage of top 40 stocks which fail skewness (13%) and kurtosis (56%) tests is only slightly less than that of all listed firms (28% and 63% respectively). Hence, Stokie's (1982b, p. 168) conclusion that `[there are] no conclusive grounds for rejecting the normal distribution as a representation of the monthly log-returns' is not entirely consistent with his findings.

Beedles (1986) notes that the said findings evidence more asymmetry in return distributions than Stokie recognises. Over the 1974-1980 period, Beedles examines the cross-sectional and time-series properties of return asymmetry. He finds that 81% of stocks exhibit significant skewness, with positive skewness predominating. Splitting the sample into industrial and resource stocks, Beedles finds little difference in the total rate of rejection of normality (for either positive or negative skewness), but notes that positive skewness is more frequent in resource stocks.(4)

In seeking an explanation for the size effect in share returns, Beedles, Dodd and Officer (1988) examine the possibility that distributional asymmetry in returns is priced. Acknowledging the potential problem with univariate skewness statistics caused by cross-correlation between asset returns, Beedles, Dodd and Officer calculate [S.sub.i] for residuals from market model regressions which they argue neutralizes the cross-correlation's.(5) Their findings indicate statistically significant positive skewness in residuals on the smallest three size portfolios.

Finally, in a recent study examining higher moments of equity returns, Alles and Spowart (1995) calculate [S.sub.i] and [K.sub.i] for 144 stocks listed over the 1984-1990 period. They find around 80% of stocks exhibit significant skewness, with approximately equal numbers positively and negatively skewed. Alles and Spowart document positive kurtosis on average, but do not report the rate of statistically significant departures from zero.

2.3 Limitations of Univariate Tests

The introduction noted that rejection of univariate normality is generally a sufficient condition for rejecting multivariate normality. However, it can be misleading to draw inferences regarding the multivariate distribution of asset returns from univariate tests of normality such as those outlined in section 2.1. Since asset returns are contemporaneously correlated, the univariate test statistics will also be dependent.

To clarify this point, re-consider the finding of Stokie (1982b) that 13% of the top 40 stocks exhibit significant skewness. Even if these stock returns were distributed multivariate normal, we would expect some of our sample statistics to exhibit (spurious) skewness.(6) Given that returns on the top 40 stocks are likely to be correlated, any spurious skewness that occurs is likely to be evident in several stocks even if the joint distribution is in fact multivariate normal. Hence, based on univariate test statistics alone, it is difficult if not impossible to know whether the sample incidence of skewness is spurious or represents a statistically significant departure from multivariate normality.

This important point is emphasized by Richardson and Smith (1993) and motivates their development of a multivariate test procedure which recognizes the cross-correlation's between individual asset returns by allowing for the correlation's between univariate test statistics. The following section outlines this multivariate methodology.

3. Multivariate Test Procedure

3.1 Overview of GMM

Richardson and Smith (1993) derive tests of multivariate normality using the generalized method of the moments. The asymptotic distribution theory for parameter estimates under GMM, developed by Hansen (1982), can be employed to derive a test methodology which allows for cross-correlation's between univariate test statistics, as well as exploiting the multivariate structure of assets returns. A brief overview of GMM follows.

Denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the n-dimensional time series of asset returns. A null hypothesis (e.g. that stock returns are multivariate normal) places restrictions on the moments of the data, which are written as:

(3) E [h ([R.sub.t], [Theta])] = 0,

where [Theta] is an m-vector of parameters governing the distribution of R and h(.) is an r-vector of functional forms.(7) Asymptotically, under the null hypothesis, the sample counterparts of equation 3 converge to zero:

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

GMM finds the values of the unknown parameters ([Theta]) which set the vector of sample moments in equation 4 to zero. Using only mild regularity assumptions,(8) Hansen (1982) derives the asymptotic distribution of the parameter estimates as:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where:

[D.sub.0] = E [[Delta]h ([R.sub.t], [Theta])]/[[Delta][Theta]'] and

[S.sub.0] = E [h ([R.sub.t], [Theta]) h ([R.sub.t], [Theta])'].

In many applications of GMM, there are more moment conditions than unknown parameters (r [is greater than] m) and the system is `over-identified'. In such a case, m linear combinations of gT([Theta]) can be set to zero, and the null hypothesis is tested by determining whether the remaining r-m `over-identifying restrictions' are (close to) zero.(9) In empirical applications, while So is usually unknown, any consistent estimator suffices and the heteroscedasticity- and autocorrelation-consistent estimators of Newey and West (1987) or Andrews (1991) are often used.

In this paper, however, the vector of functional forms h(.) is exactly identified (r = m). That is, there are m unknown parameters and m moment restrictions. An analytic expression for the variance-covariance matrix of the parameters [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]] can be derived and this is central to the development of a test for multivariate normality.

In the next section, the moment restrictions h(.) under the multivariate normal null are specified and the test statistics applied in this paper are derived.

3.2 Estimating the Joint Distribution of Univariate Test Statistics

Testing the multivariate normality of stock returns using GMM requires specification of the r-vector of functional forms h(.) from equation 3. Following Richardson and Smith (1993), and without loss of generality, consider the case where the returns on two assets ([R.sub.it] and [R.sub.jt]) are distributed bivariate normal. From the bivariate normal moment generating function,

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we can specify as many moments as required.(10) To illustrate the procedure, the following set of moment conditions represents the first four (univariate) moments of assets i and j:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Theta] [equivalence] ([[Mu].sub.i], [[Mu].sub.j], [[Sigma].sub.i], [[Sigma].sub.j], [S.sub.i], [S.sub.j], [K.sub.i], [K.sub.j]). The first two restrictions pertain to means, and the next two variances. The next two restrictions identify skewness and follow from equation 1, whilst the last two restrictions identify kurtosis and follow from equation 2.

Using the distributional properties of GMM estimators (equation 5), the joint asymptotic distribution of univariate skewness and kurtosis statistics can be analytically obtained. Richardson and Smith (1993) show that the joint distribution of [S.sub.i] and [K.sub.i] for assets i and j is:

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[Rho].sub.ij] is the correlation between assets i and j. Note that result 7 contains the distributional results 1 and 2 underlying popular univariate test procedures. However, it can also be seen from 7 that the asymptotic covariance of these univariate statistics is non-zero [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respectively), suggestive of the danger of drawing inference about multivariate normality from univariate tests.

Given the joint distribution of (say) the univariate skewness statistics, the interdependence of univariate statistics can be explicitly accommodated in testing whether [S.sub.i] = 0, [inverted A] i = 1, ..., n with an appropriate multivariate test. For example, a Wald test of this hypothesis is:

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where S is the n-vector of skewness measures and Vs is the variance-covariance matrix of these measures shown in the upper-left block of the covariance matrix in equation 7. Similarly, a Wald test that the kurtosis measures [K.sub.i] = 0, [inverted A] i = 1, ..., n can be conducted.

Since the above procedure utilizes the correlation between univariate test statistics, it overcomes the fundamental problem with univariate tests of normality outlined in section 2.3. Hence, the Wald test is sufficient to test the multivariate normality hypothesis. Note, however, that this test has utilized only the marginal distributions of the assets. It is also possible to exploit the multivariate structure of returns by estimating the cross-moments implied under the multivariate normal null. This avenue is explored next.

3.3 Utilizing Cross-Moments in Test Procedures

In addition to restrictions on the univariate distribution of each asset, the multivariate normal null places restrictions on the cross-moments of asset returns. Define the cross-skewness and cross-kurtosis between assets i and j respectively as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[Rho].sub.ij] is the sample correlation coefficient between assets i and j. Again, using the moment generating function, the first step is to specify an exactly identified vector of functional forms:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Theta] [equivalence] ([[Mu].sub.i], [[Mu].sub.j], [[Mu].sub.k], [[Mu].sub.l], [[Sigma].sub.i], [[Sigma].sub.j], [[Sigma].sub.k], [[Sigma].sub.l], [S.sub.ij], [S.sub.kl], [K.sub.ij], [K.sub.kl] [[Sigma].sub.ij], [[Sigma].sub.kl]). Each asset has a mean and variance restriction. Two covariance, two cross-skewness and two cross-kurtosis restrictions are specified giving an exactly identified system of 14 equations and 14 parameters.

The joint distributions of the cross-skewness and cross-kurtosis measures follow from equation 5. Richardson and Smith (1993) report the following extracts from the variance-covariance matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]:

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Wald tests of the joint restrictions on [S.sub.ij] or [K.sub.ij], [inverted A]i, j can be conducted once the relevant correlation coefficients have been calculated.

4. Empirical Results

Many previous papers have examined the distributional properties of individual stock returns.(11) Tests of asset pricing models, however, often utilise size and/or industry portfolios to test the mean-variance efficiency of the market proxy. For example, the popular Gibbons, Ross and Shanken (1989) test of the Sharpe-Lintner CAPM involves running market model regressions for size portfolios and calculating a test statistic, the distribution of which is based on the assumption that market model residuals are multivariate normal. Therefore, while the multivariate normality of raw stock returns is of some interest, the multivariate normality of market model residuals is of direct relevance to several popular tests of asset pricing models.

This paper examines the latter issue with a view to assessing the reasonableness of the multivariate normality assumption in tests of financial models. Hence, in the previous discussion of restrictions under the multivariate normal null, asset returns [R.sub.it] are replaced with market model residuals [[Epsilon].sub.it].

Data consist of all stocks listed on the Australian Stock Exchange from January 1974 to December 1997 obtained from the Centre for Research in Finance (CRIF) at the Australian Graduate School of Management. In each calendar year, stocks were ranked according to market capitalisation in the previous December and ten size portfolios were formed each containing an equal number of stocks. Monthly portfolio returns were calculated by weighting stock returns equally.

For size portfolio p, market model residuals [[Epsilon].sub.pt] were estimated by OLS regression:

[R.sub.pt] - [R.sub.ft] = [[Alpha].sub.p] + [[Beta].sub.p] ([R.sub.mt] - [R.sub.ft]) + [[Epsilon].sub.pt], [inverted A]p = 1, ..., 10.

Table 1 reports the correlation matrix for market model residuals on the ten size portfolios. The highest and lowest correlations are 0.76 and -0.57 between portfolios 9 and 10 and portfolios 5 and 7 respectively. Table 1 also reveals distinctive patterns in the correlations. The residuals of small market capitalisation portfolios are positively correlated, as are the residuals of large portfolios. However, the correlations between small and large market capitalisation portfolios are negative.(12) Given the joint distribution of univariate skewness and kurtosis measures in equation 7 and the high correlation between certain portfolios, the univariate skewness and/or kurtosis measures for these portfolios are clearly not independent.

Table 1
Correlation between Market Model Residuals(1)

Portfolio     1       2       3       4       5

    1        1.00
    2        0.29    1.00
    3        0.10    0.40    1.00
    4        0.07    0.20    0.31    1.00
    5       -0.11   -0.06    0.12    0.09    1.00
    6       -0.35   -0.27   -0.11   -0.06   -0.04
    7       -0.35   -0.43   -0.27   -0.17   -0.01
    8       -0.44   -0.57   -0.48   -0.38   -0.13
    9       -0.44   -0.56   -0.55   -0.40   -0.24
   10       -0.40   -0.43   -0.50   -0.46   -0.37

Portfolio      6      7       8      9     10

    1
    2
    3
    4
    5
    6         1.00
    7         0.29   1.00
    8         0.25   0.38   1.00
    9         0.08   0.23   0.67   1.00
   10         0.00   0.18   0.47   0.76   1.00

Note:

(1.) Correlation coefficients are for market model residuals on ten size portfolios constructed from monthly returns on stocks from the AGSM database from January 1974 to December 1997 (288 observations). The market model residuals for portfolio p are calculated using the regression: [R.sub.pt] - [R.sub.ft] = [[Alpha].sub.p] + [[Beta].sub.p] ([R.sub.mt] - [R.sub.ft]) + [[Epsilon].sub.pt], [inverted A] p = 1, ..., 10.

4.1 Tests Utilising Univariate Normal Moments

Table 2 reports univariate tests for skewness and kurtosis on individual size portfolios as per equations 1 and 2 respectively. The probability values of test statistics reported in this paper are based on two measures. First, p-values are based on the theoretical distribution of the statistics under the null. Since these follow from the distributional results in Hansen (1982), they are valid only asymptotically. Second, Monte Carlo p-values are calculated by simulating residuals for ten size portfolios from a multivariate normal distribution with a variance-covariance matrix equal to the empirical variance-covariance matrix, and calculating the relevant test statistic for that simulation.(13) Ten thousand repetitions of the simulations are performed and the Monte Carlo p-value is the proportion of the simulated test statistics which exceed the reported statistic.

Table 2
Univariate Skewness and Kurtosis Tests(1)

                    Univariate              Monte     Univariate
Portfolio            Skewness    p-value   Carlo p     Kurtosis

Smallest                1.61     0.0000     0.0000        4.07
    2                   0.86     0.0000     0.0000        1.79
    3                   1.58     0.0000     0.0000        6.28
    4                   0.85     0.0000     0.0000        2.72
    5                   0.26     0.0732     0.0676        0.82
    6                  -0.14     0.3383     0.3362        1.51
    7                  -0.03     0.8445     0.8392        1.10
    8                  -0.24     0.1011     0.1014        0.78
    9                  -0.15     0.3027     0.2908        0.92
 Largest                0.04     0.8058     0.8005        2.37
 [MATHEMATICAL        318.34     0.0000     0.0000      869.46
 EXPRESSION NOT
 REPRODUCIBLE
 IN ASCII]

                             Monte
Portfolio         p-value   Carlo p

Smallest          0.0000     0.0000
    2             0.0000     0.0002
    3             0.0000     0.0000
    4             0.0000     0.0000
    5             0.0047     0.0092
    6             0.0000     0.0004
    7             0.0001     0.0028
    8             0.0072     0.0121
    9             0.0014     0.0047
 Largest          0.0000     0.0001
 [MATHEMATICAL    0.0000     0.0000
 EXPRESSION NOT
 REPRODUCIBLE
 IN ASCII]

Note:

(1.) These tests utilise market model residuals for ten size portfolios constructed from monthly returns on stocks from the AGSM database from January 1974 to December 1997 (288 observations). Columns 2 and 3 report the univariate skewness statistic and asymptotic p-value respectively as per equation 1. Columns 5 and 6 report the univariate kurtosis statistic and asymptotic p-value respectively as per equation 2. Monte Carlo p-values for each portfolio are based on a simulation of 288 returns from a multivariate normal distribution with a variance-covariance matrix equal to the empirical variance-covariance matrix of the ten size portfolios. Ten thousand repetitions of the simulation are performed. The Monte Carlo p-value is the proportion of simulated test statistics which exceed the reported statistic.

In table 2, only the four smallest portfolios exhibit significant positive skewness. However, by taking correlations between skewness measures into account, the Wald test rejects the null hypothesis that [S.sub.i] = 0, [inverted A] i = 1, ..., 10. Univariate kurtosis tests indicate that all portfolios are significantly leptokurtic and the Wald test reinforces this conclusion. In summarising table 2, the tests which utilise the moments of a univariate normal distribution provide strong evidence that residuals from market model regressions are not multivariate normal.

4.2 Tests Utilising the Multivariate Structure of Returns

Tests exploiting the multivariate structure of asset returns under the null hypothesis are outlined in equations 9 and 10. Following Richardson and Smith (1993), we select portfolio 1 (the smallest decile of firms ranked by market capitalisation) as the benchmark portfolio and calculate all cross-skewness and cross-kurtosis measures relative to that portfolio.

Table 3 reports tests of cross-skewness [S.sub.1,p] = 0, [inverted A] p = 2, ..., 10 and cross-kurtosis [K.sub.1,p] = 0, [inverted A] p = 2, ..., 10. In terms of cross-skewness, the restrictions of the bivariate normal null hypothesis are violated for all combinations except [S.sub.1,3] and [S.sub.1,4]. Cross-kurtosis restrictions are violated for all pairs except [K.sub.1,7]. These findings once again suggest that market model residuals are not multivariate normal and the Wald tests of [S.sub.1,p] = 0 and [K.sub.1,p] = 0, [inverted A] p = 2, ..., 10 both support this conclusion.

Table 3
Cross-Skewness and Cross-Kurtosis Tests

Portfolio 1                        Cross-                   Monte
with                             Skewness        p-value    Carlo
Portfolio                        [S.sub.1,p]                p-value

              2                       0.44       0.0000     0.0000
              3                       0.07       0.4360     0.4310
              4                      -0.07       0.4207     0.4128
              5                      -0.24       0.0045     0.0045
              6                      -0.45       0.0000     0.0000
              7                      -0.44       0.0000     0.0000
              8                      -0.55       0.0000     0.0000
              9                      -0.53       0.0000     0.0000
           Largest                   -0.48       0.0000     0.0000
[MATHEMATICAL EXPRESSION             78.32       0.0000     0.0000
NOT REPRODUCIBLE IN ASCII]

Portfolio 1                       Cross-                    Monte
with                             Kurtosis       p-value     Carlo
Portfolio                        [K.sub.1,p]                p-value

              2                      0.99        0.0000     0.0000
              3                      1.04        0.0000     0.0000
              4                      1.56        0.0000     0.0000
              5                      1.06        0.0000     0.0000
              6                      0.33        0.0208     0.0176
              7                      0.18        0.2103     0.1897
              8                      0.64        0.0000     0.0013
              9                      0.79        0.0000     0.0004
           Largest                   0.79        0.0000     0.0002
[MATHEMATICAL EXPRESSION           336.40        0.0000     0.0000
NOT REPRODUCIBLE IN ASCII]

Note: 1. These tests utilise market model residuals for ten size portfolios constructed from monthly returns on stocks from the AGSM database from January 1974 to December 1997 (288 observations). Columns 2 and 3 report the cross-skewness statistic and asymptotic p-value respectively as per equation 9. Columns 5 and 6 report the cross-kurtosis statistic and asymptotic p-value respectively as per equation 10. Monte Carlo p-values for each portfolio are based on a simulation of 288 returns from a multivariate normal distribution with a variance-covariance matrix equal to the empirical variance-covariance matrix of the ten size portfolios. Ten thousand repetitions of the simulation are performed. The Monte Carlo p-value is the proportion of simulated test statistics which exceed the reported statistic.

4.3 Sub-Period Analysis

In empirical studies, it is common to estimate market model regressions over periods of approximately 60 observations and assume that regression residuals are multivariate normal over this period. To assess the reasonableness of such assumptions, the sample is broken into four non-overlapping 6-year sub-periods each containing 72 observations. For each sub-period, Table 4 reports how many of the univariate skewness and kurtosis statistics were rejected (max = 10), as well as the Wald tests which take contemporaneous correlations into account. Table 5 reports, for each sub-period, how many of the cross-skewness and cross-kurtosis tests were rejected (max = 9), as well as the Wald tests.

Table 4
Sub-Period Analysis--Skewness and Kurtosis Tests(1)

                              Skewness

            Univariate   Wald Test              Monte
Period      Rejections   Skewness    p-value    Carlo
            (max = 10)                          p-value

1974-1979       2          48.44     0.0000     0.0000
1980-1985       1          32.35     0.0003     0.0006
1986-1991       5          84.97     0.0000     0.0000
1992-1997       3          91.90     0.0000     0.0000

                              Kurtosis

            Univariate   Wald Test              Monte
Period      Rejections   Kurtosis    p-value    Carlo
            (max = 10)                          p-value

1974-1979       5          94.04     0.0000     0.0003
1980-1985       1          44.58     0.0000     0.0033
1986-1991       3         128.66     0.0000     0.0002
1992-1997       3         176.46     0.0000     0.0000

Note: (1) These tests utilise market model residuals for ten size portfolios constructed from monthly returns on stocks from the AGSM database over four 6-year sub-periods (72 observations each). Column 2 reports how many of the univariate skewness statistics were significant (max = 10) at the 1% level. Column 3 contains the multivariate Wald test for skewness, and columns 4 and 5 report p-values on this test. Column 6 reports how many of the univariate kurtosis statistics were significant (max = 10) at the 1% level. Column 7 contains the multivariate Wald test for kurtosis, and columns 8 and 9 report p-values. Monte Carlo p-values are based on a simulation of 72 returns from a multivariate normal distribution with a variance-covariance matrix equal to the empirical variance-covariance matrix of the ten size portfolios. Ten thousand repetitions of the simulation are performed. The Monte Carlo p-value is the proportion of simulated test statistics which exceed the reported statistic.

The results in table 4 illustrate the shortcomings of inference based on univariate test statistics which are contemporaneously correlated. In the 1974-1979, 1980-1985 and 1992-1997 sub-periods, very few of the univariate skewness tests reject the null hypothesis and the researcher may be inclined to conclude that multivariate normality is a reasonable assumption for conducting market model regressions. However, by accounting for contemporaneous correlations, the Wald test strongly rejects the multivariate normal null in all sub-periods. Similarly, in the 1980-1985, 1986-1991 and 1992-1997 sub-periods, very few univariate kurtosis tests reject the null hypothesis. The Wald test for kurtosis rejects the null in all sub-periods.

In table 5, the tests which examine the cross-moments of portfolio returns relative to the benchmark portfolio 1 have mixed results. The cross-skewness tests reject the majority of portfolios in the 1974-1979 and 1992-1997 sub-periods. The cross-kurtosis tests, however, seem to have little power. The Wald tests for cross-skewness and cross-kurtosis reject the null in every sub-period.

Table 5
Sub-Period Analysis--Cross-Skewness and Cross-Kurtosis Tests

                                Skewness

            Rejections   Wald Test    p-value     Monte
Period      (max = 9)    for Cross-               Carlo
                         Skewness                 p-value

1974-1979       6          30.59       0.0003     0.0009
1980-1985       1          19.11       0.0242     0.0232
1986-1991       4          21.33       0.0113     0.0136
1992-1997       5          54.81       0.0000     0.0000

                                Kurtosis

            Rejections   Wald Test    p-value     Monte
Period      (max = 9)    for Cross-               Carlo
                         Kurtosis                 p-value

1974-1979       2          39.81       0.0000      0.0035
1980-1985       1          20.52       0.0150      0.0276
1986-1991       7          61.74       0.0000      0.0011
1992-1997       2         124.01       0.0000      0.0001

Note: These tests utilise market model residuals for ten size portfolios constructed from monthly returns on stocks from the AGSM database over four 6-year sub-periods (72 observations each). Market model residuals for portfolio I were used as the benchmark for cross-skewness and cross-kurtosis tests. Column 2 reports the how many of the cross-skewness statistics were significant (max = 9) at 1% level. Column 3 contains the multivariate Wald test for cross-skewness, and columns 4 and 5 report p-values. Column 6 reports how many of the cross-kurtosis statistics were significant (max = 9) at 1% level. Column 7 contains the multivariate Wald test for cross-kurtosis, and columns 8 and 9 report p-values. Monte Carlo p-values are based on a simulation of 72 returns from a multivariate normal distribution with a variance-covariance matrix equal to the empirical variance-covariance matrix of the ten size portfolios. Ten thousand repetitions of the simulation are performed. The Monte Carlo p-value is the proportion of simulated test statistics which exceed the reported statistic.

5. Conclusion

From both a theoretical and empirical perspective, the multivariate normality of stock returns is an important issue to researchers in finance. While univariate tests of normality are commonly employed, they are unreliable since they fail to accommodate cross-correlations between test statistics. Using the distribution theory relating to GMM estimates, the joint distribution of univariate test statistics is derived and, by allowing for the covariance between test statistics, a multivariate test of normality is conducted. This approach is extended to exploit the multivariate structure of asset returns by calculating several cross-moments implied under the multivariate normal null hypothesis.

The multivariate tests are applied to the residuals from market model regressions for ten size portfolios. High levels of correlation between portfolios are documented reinforcing the need to accommodate cross-correlations in univariate test statistics. While univariate skewness and kurtosis test statistics do not provide an unambiguous result (particularly in sub-periods of the length typical in empirical studies), the multivariate skewness and kurtosis test statistics provide strong evidence that market model residuals are not multivariate normal. This is true for the entire 1974-1997 sample, as well as in all sub-periods.

These documented violations of the multivariate normality assumption for market model residuals have important implications for researchers using regression techniques for capital markets and asset pricing studies. For example, in testing an asset pricing model such as the Sharpe-Lintner CAPM, the researcher may have more confidence in tests which are based on distribution-free procedures rather than the strong and seemingly inappropriate assumption of multivariate normal market model residuals.

(1.) For example, see Fama (1965) and Officer (1972).

(2.) In general, for a variable X ~ N([Mu], [[Sigma].sup.2] the moment generating function shows that, for all integers n [is greater than or equal to] 1, any odd or even moment is easily calculated as:

E [(X - [Mu])[.sup.2n-1]] = 0, and

E [(X - [Mu])[.sup.2n]] = [[Sigma].sup.2n](2n)!/[2.sup.n]n!

(3.) Ball, Brown and Officer also used normal probability plots to show that the sample data was an adequate approximation of a normal distribution. However, this procedure was only used for one of the 20 portfolios, as well as the market index.

(4.) Beedles argues that Stokie's focus on industrial stocks, which are less frequently positively skewed, is the reason he concluded skewness was not significant.

(5.) In section 4 of this paper, however, we show that market model residuals are themselves highly correlated. Hence, univariate skewness tests of market model residuals are also susceptible to misleading inference.

(6.) Note that Stokie used a 1% level of significance in his tests for skewness.

(7.) The r rows of equation 3 are a set of orthogonality conditions. In sections 3.2 and 3.3, the moment conditions comprising h(.) are specified.

(8.) Namely, the time series of returns [R.sub.t] is stationary and ergodic with finite fourth moments.

(9.) The over-identifying test statistic is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(see Hansen 1982).

(10.) The joint moment [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is calculated by differentiating M ([t.sub.i], [t.sub.j]) p times with respect to [t.sub.i] and q times with respect to [t.sub.j] and setting [t.sub.i] and [t.sub.j] equal to zero. See, for example, Mood, Graybill and Boes (1974, p. 164).

(11.) For example, see Praetz and Wilson (1978), Stokie (1982b) and Beedles (1986).

(12.) These patterns are consistent with previous findings (see Brown, Kleidon and Marsh 1983, p. 47).

(13.) For each size portfolio, 288 residuals are simulated for statistics covering the whole sample, while 72 residuals are simulated for sub-period analyses.

(Date of receipt of final typescript: June 1998 Accepted by Garry Twite, Area Editor.)

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Philip Gray, Australian Graduate School of Management, The University of New South Wales, Sydney NSW 2052; E-mail: [e-mail]

Egon Kalotay, School of Economic and Financial Studies, Macquarie University, Ryde NSW 2109.

Julie McIvor, School of Economics and Finance, Queensland University of Technology, Gardens Point Campus, GPO Box 2434, Brisbane QLD 4001.

The authors are grateful for the comments of Garry Twite (Deputy General Editor), Tom Smith (Finance Area Editor) and an anonymous referee.3