Is this exclusion restriction?

[Mahtab Karimi]

Hi all,

I have a question and would appreciate your help.

I am estimating a probit model of firms being affected by a shock, it is:

Probit(shock=1 or 0)_it=ratio_i(t-1)+X_i(t-1)+ year FE+Industry FE+e_it

my hypothesis is that the ratio is important in getting hit by the shock. So all the control variables are estimated one period before the shock. The result of the base model aligns with my hypothesis. Notice that I am not controlling for firm FE because if I do the firms that have never experienced the shock in the time horizon of the sample will be dropped from the sample. There are variables in X that are mostly constant for a firm in time, though.

I am thinking of using the lag of ratio as an IV for the ratio in the base model. So I estimate the following model

ratio_it=ratio_i(t-1)+X+year FE+Industry FE+firm FE+er_it, the point is I should control for firm FE in this one, because otherwise, the coefficient on lag will be close to one and basically has no intuition.

My question is: can I do this? can I control for firm FE in the first stage, while there is no firm FE in the base model? Does this violate the exclusion restriction?
My concern is: what if there is something time-invariant or fundamental about some firms that makes them more exposed to the shock after controlling for the covariates?

Thank you,
Mahtab

[George Ford]

You should first look into the estimation of fixed effects probit models. It is not straightforward. Might look at clogit.