Instrumental variables regression with triple interaction term

[Christos Mavrovitis]

Hello Everyone

I would like to estimate the following equation (triple interaction with its main effects and controls). The main variable of concern is the triple interaction term:

Y = b0+ b1*X1 +b2*X2 + b3*X3 + b4*X1*X2 + b5X2*X3 + b6*X1*X3 + b7*X1*X2*X3 + Controls + ε, where X1 is the endogenous binary variable (with Z1 being its instrument), while X2, X3, and Controls are exogenous and control variables, respectively.

My understanding is that since X1 is endogenous then all interacted variables with X1 (e.g., X1*X2, X1*X3, and X1*X2*X3) are considered to be endogenous as well (see relevant post here https://www.stata.com/statalist/arch.../msg01239.html).

Obviously the choice of command to run the IV has its pros and cons. See for example the post here https://www.stata.com/statalist/arch.../msg00710.html. More specifically:

• If one wishes to estimate only the outcome equation, the post suggests using -ivreg- or -ivreg2-.
• If one wishes to estimate it as a system, the post suggests using -treatreg- (renamed to -etregress-).
• Alternatively, one could use the Wooldridge procedure as described here https://www.stata.com/statalist/arch.../msg00339.html.

With these in mind, I tried to use the following commands to run the IV for the equation above. So:

• My syntax using -ivreg2-: ivreg2 Y X2 X3 X2*X3 Controls (X1 X1*X2 X1*X3 X1*X2*X3 = Z1 Z1*X2 Z1*X3 Z1*X2*X3 Controls).
• Using -etregress- Stata does not allow me to use more than one endogenous variable in the treat part of the command. It says invalid syntax. Just to remind that the official basic syntax of the command is etregress y x, treat(treat = x w).
• Not sure how to approach the Wooldridge procedure either (i.e, because of the interaction variables).

To sum up, I am still trying to get my head around the following and I would appreciate your input:

1. Is the process as described above (the syntax of the commands) appropriate for performing an IV analysis for an equation with an endogenous triple interaction term?
2. If, so, I am required to use more than one instrument? The post here (https://www.stata.com/statalist/arch.../msg01239.html) implies that I need at least as many instruments as the endogenous terms (4 in my case). Or does my solution above (with Z1 interacted) solves the issue (at least econometrically speaking)? Please note that this post also suggests that “…interactions of instruments might be appropriate…”.
3. When I run the -ivreg2- command (as outlined above), the Stock-Yogo weak ID test critical values are not available (Stata output says “not available”). Not sure why this is the case, but I assume it has to do with how I syntax the command (Stata output, however, reports the coefficients of the regression).
4. How would I have to adjust the Wooldridge procedure to make it work with interacted endogenous variables (the triple interaction variable, in my case, including its double interaction effects)? Or does this procedure not apply when the equation has endogenous interacted variables?

Thank you in advance for your response(s)

Christos

[Jeff Wooldridge]

Your proposal using ivreg2 is sound. I don't discuss the triple interaction case in my MIT Press book but I do discuss, in Chapter 8, the case of interactions (and then from there the interaction of two exogenous variables is just another exogenous variable). As I emphasize, given an IV for X1, interactions with exogenous variables come with the own IVs.

I'm curious as to what you mean by the "Wooldridge procedure."

BTW, the weak IV problem is a bit tricky. I would study it in the context of the simple equation without interactions, so just X1. If Z1 is strong for X1, the interactions involving Z1 should also be strong.

JW

[Christos Mavrovitis]

Thank you for your advice, Jeff Wooldridge. I wasn’t precise enough earlier. I was referring to the Procedure 21.1 in Wooldridge (2010), where one can use the fitted probabilities of the first stage as an instrument in the second stage. In my understanding, Procedure 21.1 for a single endogenous variable would look as follows:

probit X1 Z1 X2 X3 Controls, robust
predict phat, p
ivregress 2sls Y (X1 = phat) X2 X3 Controls, robust

Assuming I have correctly coded Procedure 21.1, I would then attempt coding it for an interaction term as follows:

probit X1 Z1 X2 X3 Controls, robust
predict phat, p
ivregress 2sls Y (X1 X1*X2= phat phat*X2) X2 X3 Controls, robust

Am I constructing this appropriately?

Something else I noted regarding -ivregress- is the fact that the command does not allow you to include variables in the IV (2SLS) regression that are not included in the first-stage regressions.

Is there a 2SLS assumption dictating this? Can/should one bypass this?

Thank you

Christos

Reference
Wooldridge, J.M., 2010. Econometric analysis of cross section and panel data. MIT press.