Interaction of 2 endogenous variables in 2SLS

[Rene Ceipek]

Dear members of the STATA forum,

I am currently working on a paneldata 2SLS model with the xtivreg2 command. I have the following set of equations:

x1 = x + x2 + x3 + c.x2##c.x3
x2 = x + x1 + x3 + c.x1##c.x3
x3 = x + x1 + x2 + c.x1##c.x2

According to the theory, all three x1, x2, x3 are endogenous variables. Therefore I created instrumental variables for all: x1 = l1a & I1b, x2= I2a & I2b, x3= I3a & I3b.

My question is directed to the interaction effects. Do I create correct interaction effects if I multiply the instrumental variables? For instance: c.x2##c.x3 = I2a * I2b * I3a * I3b. Do I need to use this term as an instrument in the 2SLS for the original interaction effect?

Thank you for your support!

René

[Phil Bromiley]

This has been discussed repeatedly on this listserve. As I understand it, you need to include the interaction term as an endogenous variable and instrument it directly. You don't necessarily need to have a multiplicative function on the rhs of the instrumental variable equation, indeed that would be tricky in many of the iv routines.

[Rene Ceipek]

Thank you Phil! This would mean that I could instrument the endogenous interaction term just with a single instrumental variable?

[Jeff Wooldridge]

I discuss this a bit in Chapter 8 of my MIT Press Book, "Econometric Analysis of Cross Section and Panel Data," 2010. Include all levels of exogenous variables as IVs (that's pretty clear) and then squares and cross products of some or all. I would actually generate some overidentification so that you get a test of functional form. Whatever you do, don't omit the levels of the exogenous variables. You should include all IVs in the general model that you would in a pure linear model.

[Rene Ceipek]

Dear Mr. Woolridge,

thank you for your comment and help!

René