Testing for instrument relevance with 2 endogenous variables and 2 instruments

[Claire James]

Hi all,

I would like to test for instrument relevance with 2 endogenous variables (UEduration and UEduration^2), and 2 instruments (averagestateUEduration and averagestateUEduration^2). I understand that the normal procedure in the case of 1 endogenous variable would be to regress the endogenous variable on the instruments and all of the other (non-endogenous) explanatory variables. However, I'm not sure as to whether I have done the right procedure for my case of 2 endogenous variables with exact identification. I have done:

*test for instrument relevance*

*testing for relevance of avgstateUEduration*
xtreg UEduration avgstateUEduration EBreceived stateUErate i.mdate, fe
test avgstateUEduration
*F>10, instrument is relevant*

*testing for relevance of avgstateUEduration2*
xtreg UEduration2 avgstateUEduration2 Ebreceived stateUErate i.mdate, fe
test avgstateUEduration2
*F>10, instrument is relevant*

My confusion stems from doubt as to whether I should be omitting the other instrument that is not being tested for relevance (i.e. when testing for avgstateUEduration's relevance, whether to omit avgstateUEduration2 from the regression), and whether I should be omitting the other endogenous variable from the regression (as you cannot regress two endogenous variables on the instruments and other explanatory variables). I do realise that this is more of an econometric question than a Stata question, but any help would be very much appreciated!