Instrumental Variable First Stage Regression and Testing with interaction terms in STATA

[Thomas Kurian]

Hi everyone. I'm working on my thesis and I am using an instrumental variable approach. I have a main explanatory variable (childhood malnutrition) and I am investigating the effect of this across multiple survey rounds in a panel survey with pooled OLS. My main regression is like this:

Yit= a + b1CHILDHOODMALNUTRITIONindicator + b2FEMALE+ b3FEMALE*CHILDHOODMALNUTRITIONindicator

B2 captures the effect of childhood malnutrition on Y if male and b3 captures the additional effect of childhood malnutiriton if female.

I have an instrumental variable, z1, to proxy for changes in childhood malnutrition. As my childhood malnutrition indicator is interacted with an exogenous term, female, in the above regression, I have included two instrumental variables which are: alpha1z1 + alpha2z1*female.

My question is in relation to representing the first-stage regression equation in STATA. As in STATA, when using IV, you enter all of your instrumental variables together in a single 2SLS model. This means that the instrument z1*female will enter the equation of Childhood malnutrition, when it is actually intended as an instrument for childhoodmalnutrition*Female. Based on this insight, I have 2 questions regarding STATA:

1. If I am testing instrumental relevance i.e a F Test of joint significance of the instruments on my main explanatory variable, should I just test for alpha1=0 in one equation (where the main dependent variable is childhood nutrition) and alpha2=0 in a separate equation (where the main endogenous variable is gender*childhoodnutrition? Or should I test both instruments equal to 0 even though the instruments are for different instrumental variables?
2. How should I represent the first-stage equation in STATA when I have interaction terms?

[Joro Kolev]

2SLS does not work the way how you imply you think it works.

In the first stage all instruments are used in each equation. You do not have specific instruments for specific equations.

If you want to test for instrument relevance, you need to do it in each first stage equation.

You have two first stage equations.

[Thomas Kurian]

Hi Joro, many thanks for the reply. I understand now how STATA operates with 2SLS. My only concern is say, if I am testing instrument relevance in a first-stage equation, is the correct (theoretical) point to set the joint coefficient of all instruments in each first-stage equation equal to 0? I understand that STATA automatically includes all instruments to each first-stage equation, but when testing for relevance should I manually set the coefficient as 0 for all instruments?

Please let me know if this question does not make sense.

[Joro Kolev]

The way how I understand your question it does make sense.

Let us say that you are running

ivregress 2sls y w (x1 x2 = z1 z2 z3)

If you want to test relevance, you need to test whether z1 z2 z3 help predict both x1 and x2 in their respective equations, and after controlling for the included exogenous variable w.

Hi Joro, many thanks for the reply. I understand now how STATA operates with 2SLS. My only concern is say, if I am testing instrument relevance in a first-stage equation, is the correct (theoretical) point to set the joint coefficient of all instruments in each first-stage equation equal to 0? I understand that STATA automatically includes all instruments to each first-stage equation, but when testing for relevance should I manually set the coefficient as 0 for all instruments?

Please let me know if this question does not make sense.