An application of the chi-squared goodness-of-fit test to discrete common stock returns

This article presents the “centered” method for establishing cell boundaries in the y2 goodness- of-fit test, which when applied to common stock returns significantly reduces the high bias of the test statistic associated with the traditional Mann-Wald equiprobable approach. A modified null hypothesis is proposed to incorporate explicitly the usually implicit assumption that the observed discrete returns are “approximated” by the hypothesized continuous density. Simulation results indicate extremely biased y2 values resulting from the traditional approach, particularly for low-priced and low volatile stocks. Daily stock returns for 114 firms are tested to determine whether they are approximated by a normal or one of several normal mixture densities. Results indicate a significantly higher degree of fit than that reported elsewhere to date.