A Quantile Regression Analysis of the Cross Section of Stock Market Returns

Traditional methods of modelling returns and testing the Capital Asset Pricing Model (CAPM) do so at the mean of the conditional distribution. Instead, we model returns and test whether the conditional CAPM holds at other points of the distribution by utilizing the technique of quantile regression (Koenker and Bassett 1978). This method allows us to model the performance of firms or portfolios that underperform or overperform in the sense that the conditional mean under- or overpredicts the firm's return. In the context of a conditional CAPM, the market price of beta risk is significant in both tails of the conditional distribution of returns - negative for firms that underperform and positive for firms that overperform - but is insignificant around the median, and the opposite pattern obtains for large firms. Underperforming firms exhibit a positive relationship between size and returns in support of Merton's (1987) prediction, and there is some evidence of a positive relationship between returns and financial paper for overperforming firms. Quantile regression alleviates some of the statistical problems which plague CAPMstudies: errors invariables; omitted variables bias; sensitivity to outliers; and non-normal error distributions