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12.1 The IV Estimator with a Single Regressor and a Single Instrument 473

The idea behind TSLS is to use the problem-free component of Xi, p0 + p1Zi,
and to disregard vi. The only complication is that the values of p0 and p1 are
unknown, so p0 + p1Zi cannot be calculated. Accordingly, the first stage of TSLS
applies OLS to Equation (12.2) and uses the predicted value from the OLS regres-
sion, Xn i = pn0 + pn1Zi, where pn0 and pn1 are the OLS estimates.

The second stage of TSLS is easy: Regress Yi on Xn i using OLS. The resulting
estimators from the second-stage regression are the TSLS estimators, bnT0 SLS and
bn1TSLS.

Why Does IV Regression Work?

Two examples provide some intuition for why IV regression solves the problem
of correlation between Xi and ui.

Example #1: Philip Wright’s problem. The method of instrumental variables esti-
mation was first published in 1928 in an appendix to a book written by Philip G.
Wright (Wright, 1928), although the key ideas of IV regression appear to have
been developed collaboratively with his son, Sewall Wright (see the box). Philip
Wright was concerned with an important economic problem of his day: how to
set an import tariff (a tax on imported goods) on animal and vegetable oils and
fats, such as butter and soy oil. In the 1920s, import tariffs were a major source
of tax revenue for the United States. The key to understanding the economic
effect of a tariff was having quantitative estimates of the demand and supply
curves of the goods. Recall that the supply elasticity is the percentage change in
the quantity supplied arising from a 1% increase in the price and that the demand
elasticity is the percentage change in the quantity demanded arising from a 1%
increase in the price. Philip Wright needed estimates of these elasticities of supply
and demand.

To be concrete, consider the problem of estimating the elasticity of demand
for butter. Recall from Key Concept 8.2 that the coefficient in a linear equation
relating ln(Yi) to ln(Xi) has the interpretation of the elasticity of Y with respect to
X. In Wright’s problem, this suggests the demand equation

ln(Qbi utter) = b0 + b1ln(Pibutter) + ui, (12.3)

where Qibutter is the ith observation on the quantity of butter consumed, Pibutter is its
price, and ui represents other factors that affect demand, such as income and
consumer tastes. In Equation (12.3), a 1% increase in the price of butter yields a
b1 percent change in demand, so b1 is the demand elasticity.