Performing Truncated Regression with Endogenous Variable

Hello,

I am trying to estimate the following simultaneous equation model:

X is endogenous, but I have a credible instrument, Z. So if all my variables were continuous, I could just use 2SLS (ivreg2) to obtain consistent coefficient estimates. However, I am not dealing with a random sample. Data for my dependent variable is truncated from below. In other words, observations that had a Y value below a certain level, L, were not included in the data set (in still other words: Y=Y if Y>L and Y=. if Y<L).

If I didn't have an endogenous variable, I would just estimating a truncated regression using truncreg (details on truncated regression provided in Chapter 17.3 of Wooldridge (2002)). But since I *do* have an endogenous variable, I am stuck.

My first idea for a solution is to basically do the a two-step procedure similar to 2SLS, but to use tuncreg in the second step.

1. Step 1: Obtain predicted values for X (i.e. estimate X = G0 + G1*Z + e using OLS and obtain predicted values).
2. Step 2: Substitute predicted values for X (call them X_hat) and substitute into main equation and estimate using truncated regression.

But I am not sure if this is kosher. I think I can make a reasonable argument the coefficient estimates will be unbiased, but what about the standard errors? The literature dealing with endogenous variables in truncated regressions seems pretty light. The only paper I can find that directly addresses this issue is Hausman and Wise (1977), but their solution would require a bit more work than what I am proposing. It is okay if that is what I have to do, but I am hoping there is a simpler solution. Any help would be much appreciated!

References:
Hausman, Jerry A., and David A. Wise. "Social experimentation, truncated distributions, and efficient estimation." Econometrica: Journal of the Econometric Society (1977): 919-938.

Wooldridge, Jeffrey M. Econometric analysis of cross section and panel data. MIT press, 2002.