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Derivation of the Formula for the TSLS Estimator in Equation (12.4) 513

which contains 200 observations on (Yi, Xi, Zi) for the instrumental regres-
sion Yi = b0 + b1Xi + ui.
a. Construct bn1TSLS, its standard error, and the usual 95% confidence

interval for b1.
b. Compute the F-statistic for the regression of Xi on Zi. Is there evidence

of a “weak instrument” problem?
c. Compute a 95% confidence interval for b1, using the Anderson–Rubin

procedure. (To implement the procedure, assume that -5 … b1 … 5.)
d. Comment on the differences in the confidence intervals in (a) and (c).

Which is more reliable?

A p p e n di x

12.1 The Cigarette Consumption Panel Data Set

The data set consists of annual data for the 48 contiguous U.S. states from 1985 to 1995.
Quantity consumed is measured by annual per capita cigarette sales in packs per fiscal year,
as derived from state tax collection data. The price is the real (that is, inflation-adjusted)
average retail cigarette price per pack during the fiscal year, including taxes. Income is real
per capita income. The general sales tax is the average tax, in cents per pack, due to the
broad-based state sales tax applied to all consumption goods. The cigarette-specific tax is
the tax applied to cigarettes only. All prices, income, and taxes used in the regressions in
this chapter are deflated by the Consumer Price Index and thus are in constant (real) dollars.
We are grateful to Professor Jonathan Gruber of MIT for providing us with these data.

A p p e n di x

12.2 Derivation of the Formula

for the TSLS Estimator in Equation (12.4)

The first stage of TSLS is to regress Xi on the instrument Zi by OLS and then compute
the OLS predicted value Xn i; the second stage is to regress Yi on Xn i by OLS. Accordingly,
the formula for the TSLS estimator, expressed in terms of the predicted value Xn i, is the
formula for the OLS estimator in Key Concept 4.2, with Xn i replacing Xi. That is,