EC9A3
UNIVERSITY OF WARWICK
May Examinations 2018/19
Advanced Econometric Theory
Time Allowed: 3 Hours
Answer ALL questions in each section.
Use a separate answer booklet for each section.
Read carefully the instructions on the answer book provided and make sure that the particulars
required are entered on each answer book.
Section A: Answer ALL questions
1.
(a) Define Convergence Almost Surely.
(3 marks)
(b) Consider the linear regression model,
yi =
x0
β
i
0 +
i, where
xi is an
L-dimensional
vector of endogenous covariates, i.e.,
E[
xii] 6= 0. Suppose that there is a set of
instruments
z
K×1
i ∈ R
that is available to the researcher to achieve identification of
β0.
State and explain the two conditions that the set of instruments
zi needs to satisfy to
identify
β0.
(5 marks)
2. In order to study the effect of education on wages in a certain population, an i.i.d. sample of
n individuals are chosen and, for each individual, the following variables are recorded:
Xi = number of years of education of individual
i.
Yi = wage in dollars of individual
i.
h
Assume that
V ar(
X)
> 0,
V ar(
Y )
> 0, and that
E (
X −
E(
X))2 (
Y −
E(
Y ))2i
< ∞.
The researcher is interested in conducting large sample inference about the correlation
coefficient between wages and years of education:
cov(
X, Y )
ρX,Y = q
q
V ar(
X)
V ar(
Y )
(Question 2 continued overleaf)
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EC9A3
In particular, she is interested in testing a hypothesis about the sign of
ρX,Y . To conduct this
test, the researcher construct a statistic based on the sample covariance, given by
n
cov(
X, Y ) =
n−1 X
X
Y
d
i − ¯
Xn
i − ¯
Yn
i=1
where ¯
Xn and ¯
Yn are the sample averages of {
Xi}
n
and {
Y
, respectively. Complete
i=1
i}
n
i=1
the following points.
(a) Derive the asymptotic distribution of
cov(
X, Y ).
d
Hint: In order to have a non-degerate distribution, the statistic should be properly
normalized. You may find useful deriving the asymptotic distribution of:
n
n−1 X (
Xi −
E(
X)) (
Yi −
E(
Y ))
i=1
(properly normalized).
(8 marks)
(b) Provide a consistent estimator of the asymptotic
variance of
cov(
X, Y ). To answer this
d
question you do not need to formally prove consistency, just sketch the arguments and
indicate which results are being used.
(5 marks)
(c) Prove that
ρX,Y ≤ 0 if and only if
cov(
X, Y ) ≤ 0.
(2 marks)
(d) Using the finding in previous points, propose a procedure to conduct the following
hypothesis test:
H0 :
ρX,Y ≤ 0
(1)
H1 :
ρX,Y > 0
For the answer to be valid, the proposed test (i) should be consistent, (ii) have an
asymptotic size of
α = 5%. In particular, you should:
(i) Describe the test procedure.
(5 marks)
(ii) Sketch an argument to show that the proposed test is consistent.
(4 marks)
(iii) Sketch an argument to show that the proposed test has an asymptotic size of
α = 5%.
(4 marks)
(e) Suppose that the null hypothesis of the test in Equation (1) is rejected. Is this sufficient
evidence to conclude that an increase in education will cause an increase in expected
wages? Justify your answer.
(2 marks)
(Continued overleaf)
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EC9A3
3. Consider the following sampling process. We select a number
n of individuals from a
population, and for each individual, we collect
Y1
,i and
Y2
,i. We assume that:
"
#
"
#
"
#!
Y
µ
σ2
0
{(
Y
1
,i
i
1
,i, Y2
,i)}
n
are independent and
∼
N
,
i=1
Y2
,i
µi
0
σ2
where the unknown parameters are ({
µ
n
i}
n
, σ2) ∈ Θ
×
i=1
n = R
R++. As the notation
indicates, the parameter space Θ
n depends on the sample size. Our objective in this problem
is to estimate
σ2. We are not interested in the parameters {
µi}
n , i.e., from our perspective
i=1
{
µi}
n
is a vector of nuisance parameters.
i=1
(a) Explain why {(
Y1
,i, Y2
,i)}
n
is NOT necessarily an i.i.d. sequence of random variables.
i=1
(2 marks)
(b) Consider the following MLE estimator of ({
µi}
n , σ2)
i=1
Y
P
n
(
Y
ˆ
µ
1
,i +
Y2
,i
i=1
1
,i −
Y2
,i)2
i =
,
ˆ
σ2 =
,
2
n
Prove the following results.
Hint: For the following points, you may find useful that
Y1
i ⊥
Y2
i when (
Y1
,i, Y2
,i) is
distributed bivariate normal and
cov(
Y1
,i, Y2
,i) = 0.
(i) Show that the MLE estimator of
µi is inconsistent.
(5 marks)
(ii) Show that the MLE estimator of
σ2 is inconsistent.
(5 marks)
Remark: The situation in this excercise is known as incidental parameters problem. The size
of the vector of nuisance parameters {
µi}
n
grows with the sample size n (i.e., they are
i=1
incidental) and this negatively affects the properties of the parameter of interest.
Section B: Answer ALL questions
4. Consider the following model for an observed binary variable
yit such that:
0
Pr(
yit = 1|
yi(
t−1)
, yi(
t−2)
, .., yi1
, xiT , ..., xi1) = Λ(
x β +
α
it
i)
(
i = 1
, ..., N ;
t = 1
, .., T )
where Λ is the logistic cumulative distribution function (cdf) [Λ(
z) = exp(
z)
/[1 + exp(
z)]]
.
(Question 4 continued overleaf)
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EC9A3
(a) Derive the likelihood function for this model and explain how you can obtain a
consistent estimator of
β, without making any assumptions about the form of the
distribution of
αi|
xi1
, xi2
, ..., xiT . All necessary assumptions need to be stated and linked
to the derivations.
(4 marks)
(b) Verify that the above method discussed in (a) works for the case of
T = 2
. (4 marks)
(c) Compare and contrast the method with the method that relies on the assumption
regarding the distribution of
αi|
xi1
, xi2
, ..., xiT . (5 marks)
5. Consider the following dynamic probit model:
0
Pr(
yit = 1|
yi(
t−1)
, yi(
t−2)
, .., yi1) = Φ(
γyi(
t−1) +
x β +
α
it
i)
(
i = 1
, ..., N ;
t = 1
, .., T )
where Φ is the standard normal cdf.
(a) What are the ‘initial conditions’ problems in these type of models?
(3 marks)
(b) Explain how you would estimate the parameters of the above model addressing the
initial conditions problem. Discussion of one method is enough. However, all the
necessary assumptions need to be discussed to motivate the method.
(9 marks)
Section C: Answer ALL questions
6. Provide a statement of the Wold Representation Theorem and discuss its meaning and
limitations.
(a) Sketch out a proof of the Theorem for univariate stochastic processes.
(6 marks)
Consider a purely nondeterministic, stationary VARMA(p,q) process
Yt = (
y0
y0 )0
1
,t
2
,t
!
!
!
!
Φ11(
L) Φ12(
L)
y1
,t
Ψ
u
=
11(
L)
Ψ12(
L)
1
,t
.
(2)
Φ21(
L) Φ22(
L)
y2
,t
Ψ21(
L) Ψ22(
L)
u2
,t
(b) Assume
Yt to be covariance-stationary. Does
y1
,t have an independent ARMA
representation (not involving
y2
,t)? Provide a proof of your statement.
(6 marks)
(Continued overleaf)
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EC9A3
7. In a VAR (possibly with infinite lag order) involving current and past values of a stochastic
vector
Yt, the innovations are assumed to be linear combinations of the ‘true shocks’ of the
system and, moreover, the spaces spanned by the innovations and the true ‘true shocks’ are
assumed to coincide, i.e.
νt = Θ
εt
(3)
where Θ is non singular,
νt are the VAR residuals and
εt are the ‘true shocks’.
(a) Show that necessary condition for Equation 3 to hold is that the number of variables in
the VAR equal the number of shocks.
(3 marks)
(b) Show that condition in Equation 3 is equivalent to requiring that
εt can be linearly
determined from current and lagged values of
Yt:
εt =
P roj(
εt|
Yt, Yt−1
, Yt−2
, . . . )
.
(5 marks)
Recall that
νt =
P roj(
Yt|
Yt−1
, Yt−2
, . . . ).
(c) How does this condition relate to the condition of invertibility of the moving average?
(5 marks)
(End)
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