Lets consider the following approach:
(1) We estimate a simple linear regression model y = b0 + b1*x + error. The input data in our model estimation shall be: y = 1 month stock returns (cross sectional) for month n x = market cap for each stock (cross sectional) for month n
(2) We test if our coefficient b1 is statistically significant. And record the result for month n. We also record the R squared.
(3) We do step (1) & (2) for a total of N month.
(4) Finally we have recorded the results of N significance tests for b1, and calculate the percentage of times our coefficient was significant. For R squared we calculate the average of all N R squared.
Are the resulting 'synthetic' measures of significance & average R squared sound from a statistical perspective? Is this approach of aggregating over multiple cross sectional regressions violating any best practices in statistical research?