[Nitin Jain]

Not a problem at all. In fact, it helps with the identification argument. Provided Z2 strongly predicts X2, you’re all set. Don’t impose exclusion restrictions on the first stages. Just do what you’re doing.

Prof. Wooldridge, Could you please tell if we can replicate xtivreg as below for our model Y = a + bX1 + cX2 + dZ1 + eZ2 + fX1X3 + gX2X3 + C.

Suppose Y is the dv, X1, X2 are the indep vars, Z1 is an instrument for X1 and Z2 for X2. C is the set of controls and X3 is an exogenous moderator.

Step 1: X1 = f(Z1, C) to get X1_hat

Step 2: X2 = f(Z2, C) to get X2_hat

Step 3: X1X3 = f(Z1X3, C) to get X1X3_hat

Step 4: X2X3 = f(Z2X3, C) to get X2X3_hat

Step 5: Y = f(X1_hat, X2_hat, X1X3_hat, X2X3_hat, C)

[Nitin Jain]

Not a problem at all. In fact, it helps with the identification argument. Provided Z2 strongly predicts X2, you’re all set. Don’t impose exclusion restrictions on the first stages. Just do what you’re doing.

Ok, thanks, Prof. Wooldridge.

[Jeff Wooldridge]

Not a problem at all. In fact, it helps with the identification argument. Provided Z2 strongly predicts X2, you’re all set. Don’t impose exclusion restrictions on the first stages. Just do what you’re doing.

[Nitin Jain]

Fabio: I'm not sure what the issue is. I think the command you want is

ivregress 2sls Y X2 X3 (X1 c.X1#c.X2 c.X1#c.X3 = Z1 Z2 c.Z1#c.X2 c.Z2#c.X2 c.Z1#c.X3 c.Z2#c.X3)

I'm actually not sure that ivregress allows factor notation. If not, you need to construct the interactions yourself, preferably centered so that the level effects have meaning.

The variables X1, X1#X2, X1#X3 are all endogenous. So you need IVs for all of them. And the interaction are the very natural IVs to use.

JW

Dear Prof. Wooldridge,

Suppose we have two endogenous vars X1, X2 and their respective instruments Z1 and Z2. In the ivreg specification, we write as (X1 X2 = Z1 Z2), so the first stage models keep both Z1, Z2 for X1 and X2 even though Z1 is instrument for X1 only and Z2 for X2. If Z2 is not coming significant in the first stage model for X1, is it fine?

Thanks.

[Devon Smith]

Yes, that's correct. In fact, using the result on optimal instruments, if it happens that E(X1|X2,X3,Z1,Z2) is linear and the structural error is homoskedastic, you can show that using the interactions is the optimal choice of IVs. It is also legitimate to interact first stage fitted values for X1 with X2 and X3 and use those as IVs (no regressors -- that would be the forbidden regression!).

Hi Jeff:

Are you aware of any paper where they interacted first-stage fitted values with some other variable like X2 above to use as instruments?

[Fabio Semmel]

Excellent. Thanks to your very helpful input, I am now comfortable enough with my methodology to proceed with my work.

Thank you very much for your time, Jeff!

[Jeff Wooldridge]

Yes, that's correct. In fact, using the result on optimal instruments, if it happens that E(X1|X2,X3,Z1,Z2) is linear and the structural error is homoskedastic, you can show that using the interactions is the optimal choice of IVs. It is also legitimate to interact first stage fitted values for X1 with X2 and X3 and use those as IVs (no regressors -- that would be the forbidden regression!).

[Fabio Semmel]

Many thanks to both of you, Phil and Jeff!

I think the command you want is

ivregress 2sls Y X2 X3 (X1 c.X1#c.X2 c.X1#c.X3 = Z1 Z2 c.Z1#c.X2 c.Z2#c.X2 c.Z1#c.X3 c.Z2#c.X3)

As per my first post, I have tried virtually the exact same approach you proposed, yet with the -ivreg2- command instead of -ivregress-. It worked; in that it provided an output that appears to produce somewhat reasonable results after double-checking.

I was just wondering, if the code in that form would take care of every aspect of the problem (i.e. correct standard errors, centered interaction terms). So the notion I get from your answer is that this is a reasonable method of coding my model (i.e. put it in one -ivreg2- command vs. constructing the interaction terms manually first).
Plus, given the absence of any theoretical objections in your two responses, I assume this is generally a valid way to go about this from a methodological point of view.

The variables X1, X1#X2, X1#X3 are all endogenous. So you need IVs for all of them. And the interaction are the very natural IVs to use.

I understand that every interaction term that contains X1 is endogenous. I believe what you mean is that given Z1 and Z2 are relevant and valid instruments for X1, I should also use them as instruments for the interaction terms with X1 (which is also reflected in your code example). Am I understanding this part correctly?


Best

[Jeff Wooldridge]

Fabio: I'm not sure what the issue is. I think the command you want is

 
 ivregress 2sls Y X2 X3 (X1 c.X1#c.X2 c.X1#c.X3 = Z1 Z2 c.Z1#c.X2 c.Z2#c.X2 c.Z1#c.X3 c.Z2#c.X3)

I'm actually not sure that ivregress allows factor notation. If not, you need to construct the interactions yourself, preferably centered so that the level effects have meaning.

The variables X1, X1#X2, X1#X3 are all endogenous. So you need IVs for all of them. And the interaction are the very natural IVs to use.

JW

[Phil Bromiley]

You'll increase your chances of a useful answer by following the FAQ on asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex.

The normal practice is to calculate the interactions first and then for any interactions that include the endogenous variables, include them in the list of endogenous variables for ivreg or ivreg2.

There is some question about this: Maurice J. G. Bun & Teresa D. Harrison (2019) OLS and IV estimation of
regression models including endogenous interaction terms, Econometric Reviews, 38:7, 814-827,
DOI: 10.1080/07474938.2018.1427486