# Linear first stage, non-linear second stage 2SLS regression

I was wondering if it's possible to perform a 2SLS regression where I first run a regression for my endogenous variable, obtain the predicted values and then use them in my second stage regression (a logit)?

Would this approach yield consistent estimators?

• In general yes, you get a consistent estimator. But there are more efficient ways to obtain the estimates (which are more advanced) Sep 7, 2016 at 18:43
• Hi Repmat, what would these more consistent estimation methods be? Many thanks for the reply! Sep 7, 2016 at 20:35

If I've understood the question correctly, you are manually performing two steps:

1. Use OLS to regress your endogenous variable on your instrumental variable(s) and the exogenous covariates, and save the predicted values
2. Use logit to regress your $Y$ variable on the predicted values and the exogenous covariates.

If this is the case, then there are two potential problems:

1. 2SLS literally refers to 2 Stage (Ordinary) Least Squares regressions, not 1 OLS and then 1 logit. See section 4.8.7 here. The textbook properties of 2SLS need not carry through with 1 OLS, 1 logit. Hence, the answer to your question is that this approach may not yield a consistent estimate.
2. Although 2SLS literally refers to 2 OLS regressions, even if you do that with your statistical software, your standard errors will be incorrect. Here is an illustration using STATA.