Hi everyone

Short form of question
Can you use standard corrections for generated regressors when the regressor does not enter the 2nd equation directly, but only as an interaction term?

Context
Model 1 predicts the unanticipated part of a shock (time series). That is,
reg y1 <indepvars1>
predict res1, r

Note that res only varies over time.

Model 2 is a panel model with time and group fixed effects that tries to estimate the effect of X, and of X interacted with the unanticipated part of the shock (res1).
reghdfe y2 x2 x2#res1, absorb(time group) cluster(time)

Long form of question
Model two has two pecularities: a) it uses clustered standard errors and b) the generated regressor only enters in an interaction term c) the generated regressor is the residual, not the predicted value.

a) I believe is okay as long as you use the procedure described in Hardin 2002 (The robust variance estimated for two-stage models).
b) I'm not sure what sort of adjustment needs to be done to accommodate this. Hardin's formula uses the coefficient of the generated regressor in the second stage, should this then be the coefficient of the interaction term times x2 (the other variable in the interaction term)?
c) The examples almost always use y_hat as generated regressor, whereas I would like to use y-y_hat. I think that doesn't matter, but I haven't seen that explicitly written out anywhere?

Many thanks for any help you can provide
(reghdfe is a user-written command, available on ssc and github)