IV estimation with multiple interaction terms and binary outcome variable

[Gokay Uzun]

Hello Everyone,

First of all, I think I should note that I have read all relevant posts and Prof. Wooldridge comments on this issue, but I am still not sure if I fully understand how to proceed.

Here is my problem:

My dependent variable is a binary outcome: Y

My variable of interests are: X*A and X*B, where X is a binary variable and A and B are continuous variables. Let's also say W is the set of control variables.

My OLS estimation as follows: Y = i.X*A + i.X*B + W + e (excuse the non-econometric writing please). Here, A and B are endogenous variables so I constructed two instruments Z₁ (for A) and Z₂ (for B).

However, I am not sure about the right IV specification. I tried to estimate:

ivreg2 Y (i.X*A i.X*B = i.X*Z₁ i.X*Z₂) + W + e

but STATA gives me the following error: "equation not identified; must have at least as many instruments not in the regression as there are instrumented variables". Clearly, this specification suffers from collinearity.

Any suggestions? For example, should I create interaction terms manually and estimate again? If so, what would be the correct specification and should I correct SEs manually?

Many thanks for your help in advance, much appreciated.

[Philip Gigliotti]

I believe your problem is that you need an instrument for X as well. If you have an interaction you need an instrument for each term in the interaction. If you have a 3 way interaction you need an instrument for all 3 terms.

My guess is that a third instrument will not be forthcoming, as I find it implausible that you conveniently found 2 instruments in the same data set as is. However, if we assume your instruments are valid, you can still identify the differing effects of the two variables in each binary treatment group, by simply stratifying your model by group, and doing a direct two stage model.using your instruments.

I'm also a little confused as to why you are trying to identify two unrelated variables at once. Using two instruments is necessary for interactions of two endogenous variables, but you are not directly interacting A and B. Sometimes people will instrument a second variable if the first instrument is only exogenous conditional on the second. But I don't see why you would try to make two identifications in one model. Either one would be a paper in itself. Better to break this model down into minimal publishable units.

[Phil Bromiley]

Gokay - you'll want to set up your interactions before the ivreg2 statement. While I suspect ivreg will handle interactions in the main equation, I don't think it allows them on the lhs of the instrument equation. But I don't think ivreg2 even allows them in the main equation. Look at the command xi to create the interactions.