What about the control function approach to estimation? It's a powerful approach for both cross section and panel applications. I'm a fan for sure.
However, the CF approach can impose more assumptions than approaches that use generated IVs.
#metricstotheface
However, the CF approach can impose more assumptions than approaches that use generated IVs.
#metricstotheface
In such cases, we have a clear tradeoff between consistency and efficiency.
In models additive in endogenous explanatory variables with constant coefficients, CF reduces to 2SLS or FE2SLS -- which is neat. Of course, the proof uses Frisch-Waugh.
In models additive in endogenous explanatory variables with constant coefficients, CF reduces to 2SLS or FE2SLS -- which is neat. Of course, the proof uses Frisch-Waugh.
The equivalence between CF and 2SLS implies a simple, robust specification test of the null that the EEVs are actually exogenous. One can use "robust" or Newey-West or "cluster robust" very easily. The usual Hausman test is not robust, and suffers from degeneracies.
In addition, that CF = 2SLS nicely highlights the pretesting problem in deciding between OLS and 2SLS: leave the CF in the regression and its 2SLS. Drop it and we do OLS. The pretesting problem is compounded by the CF being a "generated regressor" from the first stage.
Here is the basic approach. Start with w (scalare) potentially endogenous in
y = a + b*w + x*g + u
The linear projection first stage is
w = f + x*d + z*h + v
The error v is the control function. We estimate it by OLS on the first stage and obtain the residuals, vh.
y = a + b*w + x*g + u
The linear projection first stage is
w = f + x*d + z*h + v
The error v is the control function. We estimate it by OLS on the first stage and obtain the residuals, vh.
Second stage: Run the regression
y on 1, w x, vh
Coefficients on 1, w, x are all 2SLS. Adding vh controls for the endogeneity of w. A robust (as you want) t statistic on vh tests Ho: w is exogenous.
This method is all based on LPs. It works for discrete w without change.
y on 1, w x, vh
Coefficients on 1, w, x are all 2SLS. Adding vh controls for the endogeneity of w. A robust (as you want) t statistic on vh tests Ho: w is exogenous.
This method is all based on LPs. It works for discrete w without change.
If the coefficient on v in projecting u onto v is not zero (w is endogenous), the standard errors for the CF approach are not valid. Because it's 2SLS, go back to 2SLS software. One could bootstrap both steps, and that's useful for more complicated settings.
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