Interpret/evaluate effect of including variables in regression analysis

Question

I am running an asset pricing test (Fama MacBeth); regressing one month ahead excess stock returns on market beta and some firm-level variables (e.g. MAX and EISKEW shown below). My object is to evaluate the (if any) effect of controlling for two variables MAX and/or EISKEW on the third variable market beta (theory predicts a positve coefficient on market beta and returns).

I have included a picture of my results, where model (1) exclude the control variables and model (8) include them. My interpretation is that they both seem to have a meaningful effect as the coefficient is shrinking towards zero, but not in a substantial way.

Are there any more formal way of analyzing/interpreting the effect of control variables on another independent variable?

Answer 1

Using results from regression (7) or (8), you want to run an $\chi^2$ or F-test on the hypothesis that the coefficients on MAX and EISKEW are jointly zero. If you can reject the hypothesis that they are both zero, you can reject the testable implication of the CAPM asset pricing model that variation in market beta explains all cross-sectional variation in expected returns.

My own personal reference is liked below, but this is entirely standard stuff and you should be able to find page after page after page on the Internet explaining how to do this explicitly: Testing linear restriction in R

(I'm not sure the difference between regression 7 or 8?)