# 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?" – Matthew Gunn May 26 '16 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. – Nicolai May 26 '16 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? – Matthew Gunn May 27 '16 at 3:44
• That is correct – Nicolai May 27 '16 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.