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?

Greatly appreciate any answers

Quartile sorts on beta

Quartile sorts on Skew then beta

  • $\begingroup$ 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?" $\endgroup$ – Matthew Gunn May 26 '16 at 23:02
  • $\begingroup$ 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. $\endgroup$ – Nicolai May 26 '16 at 23:31
  • $\begingroup$ 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? $\endgroup$ – Matthew Gunn May 27 '16 at 3:44
  • $\begingroup$ That is correct $\endgroup$ – Nicolai May 27 '16 at 19:22

Caveat, I haven't thought about skewness issues much, so there's a chance I'm missing something. Several comments:

  1. A t-stat of 1.72 does not allow you to reject the null at the standard 5 percent confidence level. It's suggestive, you're close to the edge of the 10% significance level, but there's a huge concern in finance of fishing expeditions that find spurious results. T-stat of 1.72 is not convincing, and many believe that you need t-stats closer to 3 to have a strong argument you've found something real.
  2. The pattern of alphas in the second graph isn't a clean, monotonic pattern, further suggesting it looks a bit random.
  3. On the other hand, the t-stat from the first table for lo - hi is economically large and statistically significant (t-stat > 3). Furthermore, the pattern of alphas is monotonically decreasing in skewness.

    • This suggests that sorting portfolios on skewness may be capturing some risk factor compensated for in expected returns (or behavioral bias against holding skewness leading to higher average returns) that's not explained by the Carhart 4-factor model.
  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.

  • $\begingroup$ Thank you very much for the answer Matthew, appreciate it! I have an additional question that you might have an answer to. I would like to further test if variable b (one of several skewness variables) can explain variable a (beta) (for consistency). Would making a time series index/factor (applying FF factor methodology) of each explanatory variable and then include it to a Carhart model have any advantages over the double sorting approach I have conducted? If it matters, my sample contains a reasonably small number of stocks (200-400 avg. monthly over 25 years) $\endgroup$ – Nicolai May 27 '16 at 19:40

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