[Devon Smith]

..

[Devon Smith]

Jeff:

Just had a follow-up question:

If the homoskedasticity assumption in the structural equation does not hold, does using fitted values as instrument lead to more efficient estimates than using the actual instrument, Z, itself? I have a situation where my instrument also happens to be binary.

[Jeff Wooldridge]

Devon: There are no guarantees with heteroskedasticity. It could still be more efficient. Use robust standard errors in both cases. With binary Z you’re counting on variation in X to strengthen the IV.

[Devon Smith]

Got it! Thanks, Jeff! Appreciate your help.

Best, Devon.

[Mohammed Omran]

You may want to check the following from Angrist & Krueger (2001):

We conclude our review of pitfalls with a discussion of functional form issues for both the first and second stages in two-stage least squares estimation. Researchers are sometimes tempted to use probit or logit to generate first-stage predicted values in applications with a dummy endogenous regressor. But this is not necessary and may even do some harm. In two-stage least squares, consistency of the second-stage estimates does not turn on getting the first-stage functional form right (Kelejian, 1971). So using a linear regression for the first-stage estimates generates consistent second-stage estimates even with a dummy endogenous variable. Moreover, using a nonlinear first stage to generate fitted values that are plugged directly into the second-stage equation does not generate consistent estimates unless the nonlinear model happens to be exactly right, a result which makes the dangers of misspecification high.¹⁰

Angrist, J. D., & Krueger, A. B. (2001). Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments. Journal of Economic Perspectives, 15(4), 69–85. https://doi.org/10.1257/jep.15.4.69

[Mohammed Omran]

You may want to check the following from Angrist & Krueger (2001):

We conclude our review of pitfalls with a discussion of functional form issues for both the first and second stages in two-stage least squares estimation. Researchers are sometimes tempted to use probit or logit to generate first-stage predicted values in applications with a dummy endogenous regressor. But this is not necessary and may even do some harm. In two-stage least squares, consistency of the second-stage estimates does not turn on getting the first-stage functional form right (Kelejian, 1971). So using a linear regression for the first-stage estimates generates consistent second-stage estimates even with a dummy endogenous variable. Moreover, using a nonlinear first stage to generate fitted values that are plugged directly into the second-stage equation does not generate consistent estimates unless the nonlinear model happens to be exactly right, a result which makes the dangers of misspecification high.¹⁰

Angrist, J. D., & Krueger, A. B. (2001). Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments. Journal of Economic Perspectives, 15(4), 69–85. https://doi.org/10.1257/jep.15.4.69

[Thaer Alhalabi]

Hello Jef, I have a question that is related to this thread, is it okay if my ivreg2 shows that my instruments are strong but then when I do the first step -probit- only one instrument is significant (out of three). Can I still use the estimated probability as an instrument or does it mean that the instruments are questionable?
Thanks