Introductory Econometrics
Endogeneity
by Hieu Nguyen
Fall 2024
1.
Suppose that you wish to estimate the effect of class attendance on student performance. A model
to explain standardized outcome on a final exam (stndfnl) in terms of percentage of classes attended
(attnd), prior college Grade Point Average (priGPA), and American College Testing score (ACT) is:
stndfnl = β0 + β1attnd + β2priGPA + β3ACT + ϵ.
(a) Why might attnd be suspected to be endogenous in the model?
(b) Let dist be the distance from the students’ living quarters to the lecture hall. Do you think dist is
uncorrelated with ϵ?
(c) Assuming that dist and ϵ are uncorrelated, what other assumption must dist satisfy in order to be
a good instrument for attnd?
(d) Suppose we add the interaction term priGPA · attnd to the model:
stndfnl = β0 + β1attnd + β2priGPA + β3ACT + β4priGPA · attnd + u.
If attnd is correlated with ϵ, then, in general, so is priGPA · attnd. What might be a good
instrument candidate for priGPA · attnd?
2.
The data in fertil2.gdt includes, for a sample of women in Botswana during 1988, information on the
number of children, years of education, age, and religious and economic status variables.
(a) Estimate this model by OLS and briefly comment on results:
children = β0 + β1educ + β2age + β3age2
+ ϵ.
If 100 women receive another year of education, how many fewer children are they expected to
have?
(b) In lecture #10, we discussed why we might suspect educ to be endogenous in this model. We also
suggested frsthalf (a dummy variable equal to one if the woman was born in the first six months
of a year) to be a good candidate for an instrument for educ. Show its relevance via a first stage
regression. Assume that frsthalf is uncorrelated with the error term ϵ. Now estimate the model
from part (a) by using frsthalf as an instrument for educ (= IV estimator, 2SLS). Compare the
estimated effect of education with the OLS estimate. Which of the estimators is consistent?
(c) Add the binary explanatory variables electric, tv, and bicycle to the model and assume these
are exogenous as well. Estimate the equation by 2SLS directly in Gretl and compare the estimated
coefficient of educ with part (b) and with the OLS estimate. Interpret the output of the Hausman
test.
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3.
A researcher estimated by OLS two specifications of a regression model:
y = α + βx1 + ϵ,
y = ˜α + ˜βx1 + ˜γx2 + ˜ϵ
Explain theoretically under what circumstances the following will be true. If some case cannot be
true, explain why.
(a) ˆβ =
ˆ˜β.
(b) β is statistically significant (at the 5
(c) ˜β is statistically significant (at the 5% level) but β is not.
(d) If ˆϵi and ˆ˜iϵ are the estimated residuals from the two equations,
n
i=1
ˆϵi
2
≥
n
i=1
ˆ˜iϵ2
.
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