Chi-square tests for multivariate normality with application to common stock prices

The theory of chi-square tests with data-dependent cells is applied to provide tests of fit to the family of p-variate normal distributions. The cells are bounded by hyperellipses (x-[Xbar])'S-1 (x-[Xbar]) = ci centered at the sample mean [Xbar] and having shape deter-mined by the sample covariance matrix S. The Pearson statistic with these cells is affine-invariant, has a null distribution not depending on the true mean and covariance, and has asymptotic critical points between those of x2 (M-1) and x2 (M-2) when M cells are employed. The test is insensitive to lack of symmetry, but peakedness, broad shoulders and heavy tails are easily discerned in the cell counts. Multivariate normality of logarithms of relative prices of common stocks, a common assumption in finan-cial markets theory, is studied using the statistic described here and a large data base.

[1]  David R. Cox,et al.  Testing multivariate normality , 1978 .

[2]  D. S. Moore,et al.  A CHI-SQUARE STATISTIC WITH RANDOM CELL BOUNDARIES' , 1971 .

[3]  D. Pollard.  General chi-square goodness-of-fit tests with data-dependent cells , 1979 .

[4]  M. Osborne.  Brownian Motion in the Stock Market , 1959 .

[5]  E. Fama,et al.  The Theory of Finance , 1974 .

[6]  W. Sharpe.  Portfolio Theory and Capital Markets , 1970 .

[7]  P. Holland,et al.  Discrete Multivariate Analysis. , 1976 .

[8]  Clifford W. Smith,et al.  Option pricing: A review , 1976 .

[9]  G. S. Watson,et al.  On Chi‐Square Goodness‐Of‐Fit Tests for Continuous Distributions , 1958 .

[10]  S. J. Press.  Multivariate stable distributions , 1972 .

[11]  J. L. Warner,et al.  Methods for Assessing Multivariate Normality , 1973 .

[12]  P. S. Dwyer.  Some Applications of Matrix Derivatives in Multivariate Analysis , 1967 .

[13]  David S. Moore,et al.  Unified Large-Sample Theory of General Chi-Squared Statistics for Tests of Fit , 1975 .

[14]  J. Lintner.  THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS , 1965 .

[15]  Geoffrey Stuart Watson,et al.  Some Recent Results in Chi-Square Goodness-of-Fit Tests , 1959 .

[16]  Dennis E. Logue,et al.  Foundations of Finance. , 1977 .

[17]  Ram C. Dahiya,et al.  Pearson chi-squared test of fit with random intervals , 1972 .

[18]  H. Chernoff,et al.  The Use of Maximum Likelihood Estimates in {\chi^2} Tests for Goodness of Fit , 1954 .

[19]  S. J. Press,et al.  Applied Multivariate Analysis. , 1973 .

[20]  J. Tobin.  Liquidity Preference as Behavior towards Risk , 1958 .

[21]  A. Wald,et al.  On the Choice of the Number of Class Intervals in the Application of the Chi Square Test , 1942 .

[22]  David S. Moore,et al.  Generalized Inverses, Wald's Method, and the Construction of Chi-Squared Tests of Fit , 1977 .

[23]  Robert C. Blattberg,et al.  A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices: Reply , 1974 .

[24]  B. Mandelbrot.  The Variation of Certain Speculative Prices , 1963 .

[25]  P. Praetz,et al.  The Distribution of Share Price Changes , 1972 .