I am trying to estimate a causal effect using a difference in difference estimator. I suspect there will be a different effect for small and large firms. I am interested in this different effect. I want to know if it is statistically significantly different.
$$ y_it =d_t* \beta1+Treated_i* \beta2+Treated_i *d_t *\beta3+big_i*d_t*\beta4+big_i *Treated_i*\beta5+Treated_i*d_t*big_i*\beta6+\epsilon $$ Where big is a dummy indicating whether the firm is small or big. 'd' is an indicator for the moment of treatment and treated is an dummy for being in the treatment group. The function will include some control variables, to ensure the trend in the residuals is similar.
This was the formula I was thinking of using. Or do I only need to interact the actual treatment effect with the interaction term (rather than the treatment moment and treatmentgroup dummy as well)? Furthermore am I correct in thinking that this will satisfy the assumptions underlying the DiD estimator ? The trends in the pre-treatment period will be the same, conditional on being a large or small firm, which is captured by this setup. The treatment is an exogeneous shock.
If this is not a valid approach, what is a suitable alternative ? I thought of running the DiD separately for the small and big sample, but this will not tell me anything about which group faces a bigger effect. At least not in a way that I can statistically test which one is stronger.