Visualizing mean-reversion in characteristics

In the context of generating coefficients that explain mean-reversion in correlation, mean return, and volatility in stocks, it is useful to visualise how the relationships are formed over time. Below are cross sectional ordinary least squares regressions performed several hundred times on monthly return data. These form the first step in the Fama-MacBeth regressions I use to create estimates for future correlation, mean return, and volatility. The historical period is looking back 60 months, and 12 months into the future at time t. When I construct portfolios, I use the coefficients estimated until time t - 12months to avoid look-ahead bias. I do the same exercise for weekly and daily data.

Fama-MacBeth regression coefficients over time. Monthly iterations, weekly iterations, then daily iterations.

I describe the series of return made by the portfolio by the components: Mean return, standard deviation, skewness, and kurtosis. These are coincidentally the ingredients of a Pearson-distribution. If the components really describe the portfolio-returns i.e. the mean return, standard deviation, skewness, and kurtosis are not random, it might be interesting to compare the Pearson distribution to an otherwise normal distribution. Below are one million random draws of a Pearson-distribution with the four components of each portfolio, compared to one million random draws of a normal distribution. The normal distribution only takes into account the mean return and standard deviation.