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ECMT 6002/6702: Econometric Applications
1 Practice problems
1. Suppose that the true model is
yt
wage
= β1 + β2 x2t
education
+ β3x3t + ut, Cov(xjt, ut) = 0, (1.1)
but we consider
yt
wage
= β1 + β2 x2t
education
+ u˜t, (1.2)
(i) If x3t is some variable which is not directly correlated with x2t and ut. Find the OLS
estimator of β̂2 discuss on its consistency.
(ii) If x3t is an academic performance measure of individual t, which is expected to be highly
correlated with x2t. Discuss on potential issues of the OLS estimator obtained from (1.1).
(iii) If x3t is an academic performance measure of individual t, which is expected to be highly
correlated with x2t. Discuss on potential issues of the OLS estimator obtained from (1.1).
In (i) and (iii), you can analyze the above questions based on the lecture slides (Week 5)
where similar examples were considered. The OLS estimator is unbiased and consistent in (i)
and biased and inconsistent in (iii). In case of (ii), there is no endogeneity issue. But high
correlation between x2t and x3t can cause some typical issues related to near-multicollinearity.
2. Suppose that the true model is
yt
wage
= β1 + β2 x2t
ability
+ ut, Cov(x2t, ut) = 0.
But due to data availablity, you consider
yt
wage
= β1 + β2 x2t
test score
+ u˜t,
where the test score is an incomplete measure of ability. In the case where β2 > 0, discuss on
the direction of bias of the OLS estimator.
If β2 > 0, the OLS estimator is downwardly biased, which can be shown as in the lecture
slides (Week 5) where a similar example is considered.
1
2 Empirical application
We will consider the following regression models:
Model 1: yt
log wage
= β1 + β2 x2t
education
+ ut,
Model 2: yt
log wage
= β1 + β2 x2t
education
+ β3 x3t
experience
+ β4 x4t
experience2
+ ut,
Instructions:
1. Compute the IV estimate of β2 from Model 1 using mother’s education as the IV.
- “ivreg” package in R can be used for this excercise (install.pacakges("ivreg"); library(ivreg))
- result = ivreg(wage∼educ|meduc); report=summary(result),
where edu : education, meduc = mother’s education
- The result must be similar to
β̂2,IV = 0.213. (2.1)
2. Compute the standard error of β̂2 from Model 1.
- report$coefficients can be used.
- The result must be similar to
ŜE(β̂2,IV ) = 0.174. (2.2)
3. Compute the IV estimate of β2 from Model 2 using mother’s education as the IV.
- result = ivreg(wage∼educ+exp+exp2|meduc+exp+exp2); report=summary(result),
where edu : education, meduc = mother’s education
- Note : The basic grammar is
ivreg(dependent variable ∼ all the variables in “X” | all the variables in “Z”),
and in the above, the vectors of ones in X and Z should be excluded.
- The result must be similar to
β̂2,IV = 0.232. (2.3)
4. Compute the standard error of β̂2 from Model 2.
2
- report$coefficients can be used.
- The result must be similar to
ŜE(β̂2,IV ) = 0.174. (2.4)
5. I recommend you to directly compute the IV estimator and the standard error by constructing
data matrix y, X and Z as in the lecture.
6. This computing exercise is not mandatory.