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

Exercise 6

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).