Control Function for IV Poisson Regression

Question

I need to estimate a regression in which the DV is Poisson distributed and one the independent variables of interest is endogenous due to measurement errors.

One could employ estimation strategies relying on GMM to estimate Poisson regressions in presence of endogeneity (e.g via ivpoisson in Stata). The shortcoming is that in presence of a relative high number of dummies the model fails to converge (especially if one requires the robust cluter s.e. to be computed).

Wondering about alternatives I came up with the idea of employing a control function approach to estimate my coefficient of interest in presence of endogeneity. Rivers and Vuong, 1988 propesed this strategy to take into account for endogeneity in probit models.

On the first stage you regress your endogenous variable on a set of exogenous variables including the selected instrument. The residuals are then used as an additional regressor in the main model of interest. In such a way the residuals from the first stage control for the endogenous part of the endogenous regressor, leaving the latter unbiased.

Then, is it right to apply the same methodology with the only difference that the second stage is not a probit, but a poisson regression?

Answer 1

The answer is yes. The procedure you are describing is summarized in Section 10.4.2 in "Regression Analysis of Count Data" (2nd Edition) by Cameron and Trivedi (2013, pp. 401 and 402). The procedure was first proposed by Wooldridge (1997) in his paper "Quasi-Likelihood Methods for Count Data." To test endogeneity, you simply do a t-test on the coefficient multiplying your residuals in the second stage (using the robust standard errors the computer gives you), but if you reject the null hypothesis, then you'll need to use the correct standard errors.