# Interpreting Time series regression/bivariate sorts

I am somewhat unsure how to interpret some result from an analysis that I have done on two independent variables and a dependent variable. My goal is to test whether the abnormal return difference on low and high market beta stocks can be explained by skewness.

To do this I have used double sorts, first sort on beta variable, then on skewness variable – calculated the value weighted returns to the portfolios then ran a time series regression of the portfolio returns on factors from an asset pricing model (Fama French + Carhart). The results are in the first picture: Q1:Q4 portfolios of low to high beta (average skewness), R-RF is excess returns, a is the time series intercept, B the realized beta of the portfolio.

I have also added in the second picture the same analysis only with beta as a sort variable.

How should I interpret the effect on beta from controlling for skewness? Is the difference portfolio of alpha (Low - High) with t-statistic of t=1.72 and below t=2 enough power to reject a null hypothesis of no difference in abnormal returns between Low and High beta portfolios?

• Is the first column average return in excess of risk free rate? Are these alphas with respect to the Carhart 4-factor model? Is the $\beta$ referring to $\beta_{rmrf}$, $\beta_{smb}$, $\beta_{hml}$, or $\beta_{mom}$? What do you intend by saying "controlling for skewness?" Commented May 26, 2016 at 23:02
• The first column (R-RF) is the average monthly returns in excess of risk free rate, the alpha reported is relative to the Carhart model, the β is referring to βrmrf. With "controlling for skewness" I mean that the average stock skewness within portfolios, as you move from Q1 to Q4 is relatively constant (controlling), while the portfolio betas are changing. Commented May 26, 2016 at 23:31
• And the beta that you sort on to produce portfolio comes from a standard market model regression of excess returns on excess returns of the market? Commented May 27, 2016 at 3:44
• That is correct Commented May 27, 2016 at 19:22

4. I assume you used prior skewness to form and sort portfolios (eg. use skewness from year $t-1$ to form portfolios in June of year $t$)? When forming portfolios, it's a huge no-no to use any variable that peaks into the future. You undoubtedly already know this and didn't do it, but for completeness, I thought I'd mention it.