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I am using panel data, running a fixed effects regression using xtreg Y X1 X2 X3, fe. I suspect, however, that past values of Y are correlated with future values of one of the regressors e.g. X1. This means that the strict exogeneity assumption of fixed effects regression will be violated.

To correct for this, can I use an IV? I have another variable Z1 which is correlated with X1 but uncorrelated with Y once all other regressors are controlled for.

Could I, for example, run ivreg Y X2 X3 ... Xn (X1=Z1)? How would I interpret my results?

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I think you might be able to use a first difference estimator here since it only needs weak exogeneity:

$$E[\varepsilon_{it} \vert \alpha_{i},x_{i1},...,x_{it}]=0.$$

This permits future values of the regressors to be correlated with the error. If there is feedback from today's idiosyncratic shock to a covariate tomorrow, you should still be OK with FD.

In Stata, you can estimate FD with:

D.(y x1 x2 x3), vce(cluster id) noconstant

Omit the noconstant if a linear trend in levels makes sense in your undifferenced model.

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