Omitted instrument at the first stage of an IV model

[Emmanouil Avgerinos]

Dear Stata Users,

I am trying to run an IV model with three endogenous variables. Let’s name our endogenous variables A, B and C (which is actually the square term of B). I have a valid instrument for A (let’s name it I1) and a valid instrument for B (let’s name it I2), so I run the first stage for B with these two instruments, take the linear prediction, calculate its square and use it as an instrument for C (a method described by Wooldridge 2002) in order to avoid the forbidden regression. The problem is that when I run the model with these three instruments the endogenous variable B is always omitted. I checked the first stages and saw that in all three first-stage models the instrument I1 is always omitted. Hence there are two instruments so stata omits B. When I run the model without B and C and use only I1 as instrument, it works perfectly. Any ideas what can cause the issue that makes the I1 omitted when I include the other two endogenous variables and the other two instruments? Stata returns a message about collinearities detected but I checked the correlation values of I1 with all other variables and are all below 0.2.

Kind regards,

Emmanouil Avgerinos

Assistant Professor in Decision Sciences
Operations & Technology Area
IE Business School (Instituto de Empresa)
María de Molina, 12 - 5th Floor
28006 Madrid

[Jeff Wooldridge]

As per the FAQ, you have a much better chance at a helpful answer if you show your Stata commands and Stata output. I probably can help if you provide details.

[Emmanouil Avgerinos]

Many thanks for your response! Unfortunately I cannot share any data nor variables but here are my commands and the output.

I have panel data and run a fixed effect model with clustered errors (after I have ran the first stage for B using both I1 ans I2, calculate its linear prediction and then take its square):

xtivreg2 y controls (A B C = I1 I2 linearpredictionsquare), fe r first

And I get the following message:

Warning - collinearities detected
Vars dropped: I2
equation not identified; must have at least as many instruments not in
the regression as there are instrumented variables
r(481);


When I use xtivreg y controls (A B C = I1 I2 linearpredictionsquare), fe vce(r) first

in all three first stages model I1 is omitted (its coefficient is equal to zero and in the sd columns it says (omitted)) and in the second stage A is omitted automatically.

I checked the correlation of I2 with all my variables and they are all below 0.2.

[Eric de Souza]

You should use I1, I2, and I1squared or I2squared or both as instruments for A1, B and Bsquared:
xtivreg2 y controls (A B Bsquared = I1 I2 I1squared I2quared).
Your mistake is in thinking that in the first stage regression you can associate an instrument to an endogeneous variable: the set of instruments as a whole are "associated" with the set of endogenous regressors as a whole.
PS You can even use higher powers of I1 and I2 as instruments.

[Emmanouil Avgerinos]

Many thanks for the response Eric.

I believe that what you are describing (using I1squared or I2squared or both and not the square of the linear prediction of B or that of A or both) is the "forbidden regression" leading to biased results. Also yes-all three instruments are associated with all three endogenous variables-as you pointed out. I do not think that every instrument is associated with one endogenous variable-maybe my description was unclear.

In any case-I found a new instrument that is working fine! Many thanks again!

Best,

Emmanouil