More than one endogenous variable in a panel data model, with exclusive instruments

[Daipayan Dhar]

Jeff Wooldridge carlo alberto Joro Kolev I am running a fixed effect model y x1 x2 x3. Here, x1 and x2 are endogenous. For x1, I have z1 and z2 as exogenous variable and for x2, I have z3 and z4 as exogenous variable. Now, it will be theoretically wrong to associate z3 with x1 in any equation. To elaborate, suppose x1 and x2 are wheat and rice yields and z1 and z3 are wheat irrigated area and rice irrigated area per 1000 hectares in a district, respectively. In such a situation, how can I run a 2SLS model?

[Kishen Iyengar]

Based on my understanding, you would simply ivreg y x3 (x1 x2 = z1 z2 z3 z4). Your question seems to be based on the assumption that by writing (x1 x2 = z1 z2 z3 z4), you are "associating" x1 with z3 and z4. If z3 and z4 have nothing to do with x1, then the first stage will reflect that fact. If my understanding is correct, first stage is like extracting shared variance. If z3 and z4 have no shared variance with x1, then there is no issue. Theory does not come into it here.

[Daipayan Dhar]

Kishen Iyengar sir, I tried that method. z3 and z4 are significant in explaining x1, but that is a spurious regression. If irrigated area of rice per 1000 hectare is explaining wheat yield, then isn't it problematic? How can I justify such association in my paper, in that case?

[Kishen Iyengar]

Are z1 and z2 also significant, apart from z3 and z4 in explaining x1? If not, then you need better instruments I think. If they are significant, AND z3 and z4 are also significant, can you look at the relative effects of z1 z2 vs. z3 z4. We cannot rule out an unmeasured variable "q" impacting x1 and z3 z4, which shows up as the significance in the first stage.

You could try a stepwise approach: ivreg y x3 (x1 = z1 z2 ) and ivreg y x3 ( x2 = z3 z4) and see if these are supported. If they are, then just go with ivreg y x3 (x1 x2 = z1 z2 z3 z4) in spite of the spurious result of z3 and z4 on x1.

Maybe someone else can comment on whether this issue may have a deeper impact on your theory building and testing.

[Joro Kolev]

Hi Daipayan,

Using different instruments for different endogenous regressors (and not the whole set of instruments for both the endogenous variables jointly) sounds like a good idea on the surface, but turns out to be a bit tricky to justify formally.

I think the conclusion of the literature roughly is that if the system is triangular (as opposed to fully simultaneous) we can do that. If the system is fully simultaneous, in your case meaning that y appears in the system as an explanatory variable for x1 and x2, then we should not be using different instruments for different endogenous variables. I think for triangular systems we can use different instruments. If I am not wrong this is the paper that comes close to what I am saying: Schmidt, Peter. "Three-stage least squares with different instruments for different equations." Journal of econometrics 43.3 (1990): 389-394.
Three-stage least squares with different instruments for different equations - ScienceDirect

I think you should just make your life easy by using the full set of instruments for both endogenous regressors. Something in the lines of what Kishen wrote, -ivreg y x3 (x1 x2 = z1 z2 z3 z4)- or -xtivivreg y x3 (x1 x2 = z1 z2 z3 z4)-.

This approach is easy to justify in your context, again in the lines of what Kishen says: there might be an unobserved variable that is driving both phenomena. E.g., areas/districts which generally irrigate better (or generally have better/more advanced agricultural techniques) deliver better yields both on rice and wheat. Therefore irrigated rice areas appear to explain wheat yields.

[Daipayan Dhar]

Thank you Kishen Iyengar and Joro Kolev .