ivregress and etregress with endogenous dichotomous variable

[Roberto Torrado]

Good morning, everyone. I've been stuck with an estimation for a few days now. I've read a lot on this forum, but I'm currently unable to proceed.
My goal is to estimate a model with a continuous dependent variable (Y). My main variables of interest are an endogenous dichotomy (X1) and a factor variable (X2). Additionally, I'm introducing other covariates into the model (X3, X4, X5, X6... Xn) and two instruments for the endogenous variable: Z and its square.
The first step was to try a 2SLS ivregress model:

ivregress 2sls Y X3 X4 (X1 i.X2 = Z Z^2 X5 X6)

However, I believe this regression is what Wooldridge refers to as a "forbidden regression."
So, the next step was to estimate a model using the "etregress" command, with one of the independent variables as endogenous and the other as a covariate:

etregress Y X2 X3 X4, treat (X1 = Z Z^2 X5 X6)

The literature doesn't mention instrumental variables with the "etregress" command. I'm not sure if my last estimation are correct. Could anyone shed some light on this?

[George Ford]

etregress estimates an average treatment effect and the other parameters of a linear regression model augmented with an endogenous binary-treatment variable.

[Jeff Wooldridge]

I’ll try to provide a full response tomorrow. There are several issues here, starting with which variable(s) are thought to be endogenous. And you can’t do a forbidden regression with ivregress.

[Roberto Torrado]

Dear George Ford and Jeff Wooldregie,

Many thanks for your responses. I have continued to work on this topic. After reading Ruonan Xu's work, which was recommended here precisely by Prof. Wooldridge, and taking into account the comments from the former and other comments from here, I have come to the conclusion that one potential solution to the issue I set out could involve the preliminary estimation of a logit model for each endogenous variable. Subsequently, I utilized the estimations from these logit models in the ivregress estimation:

Logit X1 X3 X4 X5 X6 Z Z^2

Predict X1^, p

Logit X2 X3 X4 X5 X6 Z Z^2

Predict X2^, p

ivregress 2sls Y X3 X4 X5 X6 (X1 i.X2 = X1^ X2^)

Is this corrrect?

[Jeff Wooldridge]

This could work, but you're asking your single IV, Z, to do a lot of work. Ideally, Z^2 would be a strong predictor in one equation but not necessarily the other. The nonlinearity in the logit can give a false sense of security. If you were doing linear first stages, and Z^2 didn't help to predict X1 or X2, you'd lose identification. Or, if the pattern of coefficients on Z, Z^2 are the same in the two first stages, that could also spell trouble. So you need to look at the two first stages and convince yourself that Z, Z^2 are causing enough separate variation in X1, X2.

[Israel Escudero]

Hello everyone,

I have applied the same model as the ones explained here, but in my case, I use X2 in the logit where I regress X1, and I have used X1 in the logit where I regress X2. Adapting to the specification that appears above, my model would be:

Logit X1 X2 X3 X4 X5 X6 Z Z^2

Logit X2 X1 X3 X4 X5 X6 Z Z^2

Then I estimated a ivreg2 model:

ivreg2 Y X3 X4 X5 X6 (X1 X2=X1^ X2^ Z), robust endog(X1 X2)

Now that I have searched for this information, I have doubts about whether my model is econometrically well-designed. I would appreciate any information on this. Thank you very much.

IE