Introduction to Econometrics 3rd Edition by James Stock, James Stock

تمرين 11

Consider TSLS estimation with a single included endogenous variable and a single instrument. Then the predicted value from the first-stage regression is Use the definition of the sample variance and covariance to show that Use this result to fill in the steps of the derivation in Appendix of Equation (12.4).
Appendix
Derivation of the Formula for the TSLS Estimator in Equation (12.4)
The first stage of TSLS is to regress X i on the instrument Z i by OLS and then compute the OLS predicted value , and the second stage is to regress Y i on by OLS. Accordingly, the formula for the TSLS estimator, expressed in terms of the predicted value is the formula for the OLS estimator in Key Concept 4.2, with replacing X i. That is, , where , is the sample variance of and is the sample covariance between Y i and
Because X i is the predicted value of X i from the first-stage regression, , the definitions of sample variances and covariances imply that (Exercise 12.4). Thus, the TSLS estimator can be written as . Finally, is the OLS slope coefficient from the first stage of TSLS, so Substitution of this formula for into the formula yields the formula for the TSLS estimator in Equation (12.4).