Page 515 -

514 Chapter 12 Instrumental Variables Regression

bn1TSLS = sXn Y >sX2n, where sX2n is the sample variance of Xn i and sXn Y is the sample covariance
between Yi and Xn i.

Because Xn i is the predicted value of Xi from the first-stage regression, Xn i = pn0 + pn1Zi,
the definitions of sample variances and covariances imply that sXn Y = pn1sZY and sX2n = pn 12sZ2
(Exercise 12.4). Thus, the TSLS estimator can be written as bnT1 SLS = sXn Y >sX2n = sZY> (pn1sZ2 ).
Finally, pn1 is the OLS slope coefficient from the first stage of TSLS, so pn1 = sZX>s2Z. Sub-
stitution of this formula for pn1 into the formula bn1TSLS = sZY>(pn1s2Z) yields the formula for
the TSLS estimator in Equation (12.4).

A p p e n di x

12.3 Large-Sample Distribution

of the TSLS Estimator

This appendix studies the large-sample distribution of the TSLS estimator in the case con-
sidered in Section 12.1—that is, with a single instrument, a single included endogenous
variable, and no included exogenous variables.

To start, we derive a formula for the TSLS estimator in terms of the errors; this formula
forms the basis for the remaining discussion, similar to the expression for the OLS estimator
in Equation (4.30) in Appendix 4.3. From Equation (12.1), Yi - Y = b1(Xi - X ) + (ui - u).
Accordingly, the sample covariance between Z and Y can be expressed as

sZY = n 1 1 n - Z)(Yi - Y )
-
ia=1 (Zi

= n 1 1 n - Z)3b1(Xi - X) + (ui - u)4
-
ia=1 (Zi

= b1sZX + n 1 1 n - Z )(ui - u)
-
ia= 1(Zi

= b1sZX + n 1 1 n - Z )ui, (12.19)
-
ia= 1(Zi

where sZX = [1 > (n - 1)] g n 1(Zi - Z )(Xi - X) and where the final equality follows
i=
n
because g i= 1(Zi - Z) = 0. Substituting the definition of sZX and the final expression in

Equation (12.19) into the definition of bnT1 SLS and multiplying the numerator and denominator

by (n - 1)>n yields

1 n - Z )ui
n
bnT1 SLS = + ia= 1(Zi . (12.20)
b1 n X)
1
n ia= 1(Zi - Z )(Xi -