I am using panel data, and would like to evaluate whether coefficients from different fixed effects specifications are statistically significantly different from one another. Specifically, I estimate models a), b) and c) with -reghdfe-, such that

a) ln(y) = βx + γz + id (reghdfe lny i.x c.z, absorb(id) cluster(id))

b) ln(y) = βx+ γz + id + firm (reghdfe lny i.x c.z, absorb(id firm) cluster(id))

c) ln(y) = βx+ γz + id*firm (reghdfe lny i.x c.z, absorb(id#firm) cluster(id))

ln(y) is a natural logarithm of the dependent variable, x is a time-varying categorical variable, z is a time-varying continuous variable, id are individual fixed effects, firm are firm fixed effects and id*firm are the interaction of individual and firm fixed effects. y and x are the same variables in each specification, and the estimations are run on the same sample (singletons are obviously dropped in each case). I want to test, whether β (for each category of x) estimated from a) is statistically significantly different from β estimated from b), or from β estimated from c), .

Do formulas suggested for nested multiple regressions in Clogg et al. (1995) apply in the comparison of β's estimated from a) and b) (why/why not)? What about in the comparison of β's estimated from a) and c), where the models are non-nested (why/why not)? Or would the solution be to demean the variables by hand, run specifications with -regress- on demeaned data, and then apply -suest- and -test- (why/why not)? In addition to an appropriate solution, references to other posts and econometrics papers discussing this particular issue would be much appreciated - thus far I have not been able to find ones myself.

Clifford C. Clogg, Eva Petkova and Adamantios Haritou (1995). Statistical Methods for Comparing Regression Coefficients Between Models. American Journal of Sociology, Vol. 100, No. 5, pp. 1261-1293.