2SLS with subsamples

[Claudio Mora]

Hi.

I am trying to estimate a 2sls regression for different subsamples but using the same first stage. So, for example, if the structural equation is

y = a + bX + u

and the first stage is

X = c + dZ + e

I want to know the size of b (hat) among two subsample, for example among big and small schools, but using just ONE first stage. I understand that this is possible, since other papers present estimates this way when testing for heterogeneous effects. But I do not know how to implement it in Stata.

Any ideas?

Let me know if you need more info!

Thanks!

[Sean Lambert]

Although the approach you are discussing did receive some attention, I believe there has been discussion that a 2 stage approach is better. One advantage when you separate the stages is that it is possible to evaluate the goodness of fit using the adjusted r-squared of the first stage. If you combine them, the r-squared is meaningless.

Technically, I think the example you have there is just a simple instrumental variables regression. 2SLS is usually when you have instruments AND exogenous variables in the first stage.

ex. Y1=a + b1Y2 + b2X1 + u

The Y1 and Y2 are both endogenous. X1 is exogenous.

In a regular IV approach, you would find an instrument to correct for the endogeneity problem.

ex. Y2 = c+ dZ + e

Where Z is uncorrelated with u, and Y1 but is correlated with Y2.

The problem is that if Z is correlated with Y2, by transitivity it is also correlated with Y1, which is correlated with X1. To correct for this, 2SLS includes X1 as a regressor in the first stage.

ex. Y2 = c + d1Z + d2X1 + e

In stata this would be:

ivreg 2sls Y1 (Y2 = Z X1) , robust first

It's a very interesting topic certainly, one which is thoroughly discussed in the literature

EDIT:
Recommended Reading:
Angrist and Krueger (2001) -Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments