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.