ECON7320: Advanced Microeconometrics
Problem Set 1
Fu Ouyang
March 18, 2025
Instruction
Answer all questions and clearly label your answers. For empirical questions, you should show
your R script(s) and outputs (e.g., screenshots for commands, tables, and figures, etc.). You
will lose2 points
whenever you fail to provide R commands and outputs. When you are asked
to explain or discuss something, your response should be brief and compact. You should upload
your assignment (in PDF or Word format) via the “Turnitin” submission link (in the “Problem
Set 1” folder under “Assessment”) by 16:59 on the due date April 1, 2025. Do not hand in a
hard copy. You are allowed to work on this assignment in groups; that is, you can discuss how
to answer these questions with your group members. However, this isnot
a group assignment,
which means that you must answer all the questions in your own words and submit your work
separately. The marking system will check the similarity, and UQ’s student integrity and
misconduct policies on plagiarism apply.
OLS (30 points)
Use thecps09mardataset described in Tutorial 1. Take the sub-sample of non-Hispanic women
to estimate the following wage equation:
log(wage) =β
0
+β
1
education+β
2
experience+β
3
experience
2
/100 +u,(1)
wherewage=earnings/(hours×week) andexperience=age−education−6.
(a) Estimate equation (1) using the described sub-sample and compute theR
2
(5 points).
(b) Include a set of dummy variables for regions and marital status in (1) and estimate the
extended model. For regions, create dummy variables for Northeast, South, and West
so that Midwest is the excluded group. For marital status, create variables for married
(marital≤3), widowed or divorced, and separated, so that single (never married) is
the excluded group (6 points). Calculate standard errors using the HC0, HC1, and HC3
methods (3 points). Are they very different?
(c) In what follows, use the estimation results obtained from (a). Letθbe the ratio of the
return to one year of education to the return to one year of experience forexperience=
10. Writeθas a function of the regression coefficients and compute
ˆ
θfrom the estimated
model (3 points). Suppose the OLS estimator for model (1) is consistent. Is
ˆ
θconsistent
(3 points)? Hint: Apply continuous mapping theorem.
(d) Compute the regression function ateducation= 16 andexperience= 10 (2 points).
Compute a 95% confidence interval for the regression function at this point (3 points).
Hint: You can use theglht()function in themultcomppackage.
1
(e) Consider the same out-of-sample individual as in (d) (education= 16,experience= 10).
Construct an 90% forecast interval for their wage (5 points). Hint: To obtain the forecast
interval for the wage, apply the exponential function to both endpoints.
MLE: Tobit Regression (25 points)
Consider the Tobin (1958) regression model:
Y
∗
=X
T
β+ewithe|X∼N(0,σ
2
),(2)
Y= max{Y
∗
,0}.(3)
Tobin (1958) used this model to study household consumptionYof durable goods. He observed
that in survey data,Yis zero for a positive fraction of households. He proposed treating the
observedYas acensoredrealization from a latent continuous variableY
∗
(like “willingness to
pay”); that is,Y=Y
∗
whenY
∗
>0 andY= 0 whenY
∗
≤0. Here the observed variableYis
censoredY
∗
from below at zero. After all, negative pay is infeasible. This model is known as
Tobit regression,censored regression, orType I Tobit model.
Sincee|X∼N(0,σ
2
) is assumed in (2), Tobit model (2)–(3) is parametric and its unknown
parameters (β,σ) can be estimated using the maximum likelihood method. By definition, it is
easy to write out the distribution function ofYconditional onX:
P(Y≤y|X=x) =P(Y
∗
≤0|X=x)
1[y≤0]
·P(Y
∗
≤y|X=x)
1[y>0]
=P(x
T
β+e≤0)
1[y≤0]
·P(x
T
β+e≤y)
1[y>0]
= Φ(−x
T
β/σ)
1[y≤0]
·Φ((y−x
T
β)/σ)
1[y>0]
,(4)
where1[·] is the indicator function
1
and Φ(·) is the CDF ofN(0,1). Taking derivative of (4)
with respect toyyields the likelihood function:
f
Y|X
(y|x) = Φ(−x
T
β/σ)
1[y≤0]
·[σ
−1
φ((y−x
T
β)/σ)]
1[y>0]
,(5)
whereφ(·) is the PDF ofN(0,1).
Use theCHJ2004dataset (see the data description file). The variablestinkindandincome
are household transfers received in-kind and household income, respectively. Divide both vari-
ables by 1000 to standardize.
(a) Estimate a linear regression oftinkindonincomeandincome
2
(5 points).
(b) Calculate the percentage of censored observations (tinkind= 0) (3 points). Estimate a
linear regression oftinkindonincomeandincome
2
by omitting the censored observations
(5 points).
(c) Estimate a Tobit regression oftinkindonincomeandincome
2
(6 points). Explain the
differences between your results in (a)–(c) (6 points). Hint: You can use thetobit()
function in theAERpackage. Alternatively, you can also use (5) to code up the maximum
likelihood estimation and inference by yourself and check if you can replicate the results
returned bytobit(). But this is optional.
1
1[A] = 1 if eventAoccurs and1[A] = 0, otherwise.
2
2SLS and GMM (25 points)
In an influential paper, David Card (1995) suggested that if a potential student lives close to a
college, this reduces the cost of attendance and raises the likelihood that the student will attend
college. However, college proximity does not directly affect a student’s skills or abilities, so it
should not affect their wage. These considerations suggest that college proximity can be a valid
IV for education in a wage regression.
Use theCard1995dataset to replicate the baseline analysis conducted in Card (1995). Con-
sider the following model:
log(wage) =β
0
+β
1
education+β
2
experience+β
3
experience
2
/100
+β
4
black+β
5
south+β
6
smsa+u,(6)
whereeducationis years of schooling andexperience=age−education−6. For all the
questions below, only use data for 1976. See the data description file for variable definitions.
(a) Estimate model (6) by OLS and 2SLS (usingnearc4as the instrument foreducation)
(8 points).
(b) Estimate model (6) by 2SLS using instruments{nearc4a,nearc4b}(3 points). What is
the impact on the structural estimate of model (6) (2 points)?
(c) Are the instruments used in (b) strong or weak? Test it (4 points).
(d) Use the 2SLS regression in (b) to test the exogeneity ofeducation(4 points).
(e) Use the 2SLS regression in (b) to test the exogeneity of instruments (4 points).
(f) (Optional) Re-estimate model (6) using the same set of instruments as in (a) by efficient
GMM. Do the results change meaningfully? Hint: Use thegmm()function in thegmm
package. This question has no points; it is here just because we don’t have room for it in
Tutorial 4.
Theoretical Questions (20 points)
1. Prove that the regression errors of LPM must be heteroskedastic (5 points).
2. Prove thatE[Y|X] = arg min
g
E[(Y−g(X))
2
]; i.e.,E[Y|X] has the smallestmean squared
error(MSE) among all functions ofX. Hint: Consider (Y−g(X))
2
= [(Y−E[Y|X]) +
(E[Y|X]−g(X))]
2
and apply the law of iterated expectation (LIE) (5 points).
3. Consider the following simple linear regression model:
y
i
=βx
i
+ε
i
, i= 1,...,n,(7)
where{x
i
,y
i
}
n
i=1
are i.i.d.,E[ε
i
|x
i
] = 0, and V[ε
i
|x
i
] =σ
2
. Supposex
i
>1 andE[x
2
i
]<∞
for alli= 1,...,n. We have the following three estimators ofβ.
ˆ
β
A
=
n
X
i=1
x
i
y
i
/
n
X
i=1
x
2
i
,
ˆ
β
B
=
n
X
i=1
y
i
/
n
X
i=1
x
i
,
ˆ
β
C
=
1
n
n
X
i=1
y
i
x
i
.
3
(a) Show that
ˆ
β
A
,
ˆ
β
B
and
ˆ
β
C
are all unbiased (6 points). Hint: Use (7) and LIE.
(b) Among
ˆ
β
A
,
ˆ
β
B
and
ˆ
β
C
, which one has the smallest variance? Why? (2 points) Hint:
Review the Gauss-Markov Theorem.
2
(c) Show that
ˆ
β
C
is consistent (2 points). Hint: Apply WLLN and LIE.
2
When we say an estimator is linear, we mean the estimator is a linear function ofy.
4
