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EC9100
1 (Question 1 Continued overleaf)
UNIVERSITY OF WARWICK
May Examinations 2021/22
Quantitative Methods: Econometrics B
Time Allowed: 3 Hours, plus 15 minutes reading.
Read all instructions carefully – and read through the entire paper at least once before you
start entering your answers.
Sections A and B are based on the material covered in Term 1 and Sections C and D are
based on the material covered in Term 2.
There are FOUR sections in this paper. Answer the ONE question in Section A (30 marks),
ONE of TWO questions in Section B (20 marks), the ONE question in Section C (30 marks)
and ONE of TWO questions in Section D (20 marks).
Approved pocket calculators are allowed
You should not submit answers to more than the required number of questions. If you do,
we will mark the questions in the order that they appear, up to the required number of
questions in each section.
SECTION A: Answer the ONE question
1. The following is a set of results from analyses performed by a researcher, using the data
used by Mroz, T. (1987) “The sensitivity of an empirical model of married women’s
hours of work to economic and statistical assumptions. Econometrica, (55:4), pp. 765-
799. The variables used are as follows:
inlf =1 if in lab frce, 1975
hours # of hours worked, 1975
kidslt6 # kids < 6 years kidsge6 # kids 6-18 age woman's age in yrs educ years of schooling wage hourly wage rate faminc family income, 1975 exper actual labour market experience nwifeinc family income excluding the woman’s income lwage log(wage) expersq experience-squared EC9100 2 (Question 1 Continued overleaf) Out of a sample of 753 women, 428 had worked during the year 1975. i.e., 428 women had reported positive number of hours. Standard errors in parentheses. *** p<0.01, ** p<0.05, *p<0.1 L LR test for Rho=0 in [4] & [5] : chi2(1)=0.03 with Prob>chi2=0.958
[1] [2] [3] [4] [5]
VARIABLES Hours lwage
Labour force
Participation lwage
Labour force
Participation
Estimator Tobit 2-step Heckman Heckman 1st step MLE MLE
nwifeinc -8.814** -0.012** -0.012**
(4.459) (0.005) (0.005)
educ 80.645*** 0.109*** 0.131*** 0.108*** 0.131***
(21.583) (0.016) (0.025) (0.015) (0.025)
exper 131.564*** 0.044*** 0.123*** 0.043*** 0.123***
(17.279) (0.016) (0.019) (0.015) (0.019)
expersq -1.864*** -0.001* -0.002*** -0.001** -0.002***
(0.538) (0.000) (0.001) (0.000) (0.001)
age -54.405*** -0.053*** -0.053***
(7.418) (0.008) (0.008)
kidslt6 -894.020*** -0.868*** -0.867***
(111.878) (0.119) (0.119)
kidsge6 -16.218 0.036 0.036
-8.814** (0.043) (0.043)
Inverse Mill’s
Ratio
0.032
(0.134)
/athrho 0.027
(0.147)
/lnsigma -0.410***
(0.034)
Rho 0.027
(0.147)
Sigma 0.663***
(0.023)
Inverse 0.018
Mill’s Ratio (0.100)
Constant 965.307** -0.578* 0.270 -0.553** 0.266
(446.435) (0.305) (0.509) (0.260) (0.509)
Ln likelihood -832.9
R-squared
EC9100
3 (Continued overleaf)
The column numbers indicate the results from various estimators using STATA, as
follows:
[1] Tobit of the hours equation;
[2] & [3] Heckman 2-step method of estimation of the log(wage) equation;
[4] & [5] Maximum likelihood estimation of the same equations as in [2] and [3].
In the following discussions, you may use the general notation x,y,z etc in your
equations, but clearly state which variables are in which vectors of variables.
(a) Discuss the model specifications and estimations used in columns [2] and [3],
carefully stating all the assumptions that are needed to obtain consistent
estimators. Make sure to motivate and explain the two-steps involved. (7 marks)
(b) Compare the results from columns [2] and [3], with that of [4] and [5]. Discuss the
conclusions you would draw. What do you conclude Provide reasons for your
conclusions. (5 marks)
(c) Discuss the advantages and disadvantages of using [2] and [3] vs [4] and [5].
(4 marks)
(d) The researcher used the Tobit model to model hours worked (column [1]) using the
likelihood method to estimate the equation. Comment on whether Tobit is
appropriate for this or not. Write down the model specification for Tobit. How
would you estimate the same model using a two-step method Explain in detail all
the steps involved. (8 marks)
(e) Explain the role of the extra variables in column [3] compared to that of column [2].
Discuss the properties of the estimator of this model when you drop these extra
variables. (6 marks)
SECTION B: Answer ONE question
2.
(a) Using a regression framework and a binary treatment variable, explain the terms:
Average Treatment Effect (ATE) and Average Treatment Effect on the Treated (ATT).
(4 marks)
(b) Assume you have two observations for each individual (pre and post treatment).
Discuss the assumptions required to identify and estimate the ATT using the
framework discussed in (a). (12 marks)
EC9100
4 (Continued overleaf)
(c) How would you proceed with the model specification and estimation if more time
observations were available covering pre and post treatment period for both
groups (4 marks)
3.
(a) Compare and contrast the within-group (WG), and the Generalised Least Squares
(GLS) estimators of the parameters of the following regression:

yit xi
‘
t 1 zi
‘
i it
for i
2 1,..,N and t 1,…,T.
The number of variables in x and z are k1
and k2
respectively. i
is the unobserved
individual specific variable. Conditional on the regressors, it ~ iid(0,
2
) and
independent of i
. All steps involved in the construction of the estimators, the
assumptions needed to obtain a consistent estimator, needs to be discussed.
(10 marks)
(b) Discuss various ways in which you can estimate 2
. Any proposed methods should
be justified in terms of its statistical properties. (10 marks)
Section C: Answer the ONE question
4. In this part, we set the focus on estimating a vector θ ∈ Θ R
p
from its definition by a set
of H moment conditions:
E [ψ (Y, θ)] = 0 (1)
for which the standard assumptions of GMM (including global and local identification) are
fulfilled. GMM estimation is based on an observed sample of i.i.d. data Yi
, i = 1, …, n.
(a) We consider a given non-singular matrix A(θ) , known function of θ, and the rescaled
moment conditions:
E [ψ

(Y, θ)] = 0 (2)
with:
ψ

(Y, θ) = A (θ) ψ (Y, θ)
(i) Show that moment conditions (1) and (2) define the same continuously updated
GMM estimator θ

n. (3 marks)
(ii) Deduce that the two 2step efficient GMM estimators, based respectively on (1) and
on (2), are asymptotically equivalent, albeit possibly different. (3 marks)
(iii) Deduce more generally that the GMM estimators (continuously updated GMM and
2step efficient GMM) defined above remain asymptotically equivalent if we replace
the function A(.) by a consistent estimator An() such that for all θ ∈ Θ:
A (θ) = P limn→∞
An(θ)
(2 marks)
(Question 4 continued overleaf)
5
(b) We consider a partition of the set of moment functions:
ψ (Y, θ) = ” ψ1 (Y, θ)
ψ2 (Y, θ)
#
with: ψ1 (Y, θ) ∈ R
H1
, ψ2 (Y, θ) ∈ R
H2
, H1 + H2 = H.
(i) We want to replace the second set ψ2 (Y, θ) of moment functions by:
ψ2

(Y, θ) = ψ2 (Y, θ) Cov [ψ2 (Y, θ), ψ1 (Y, θ)] {V ar [ψ1 (Y, θ)]}
1
ψ1 (Y, θ)
Use the results of question (a) to assess to what extent GMM estimators are
modified by the replacement of ψ2 (Y, θ) by ψ2

(Y, θ), with ψ1

(Y, θ) = ψ1 (Y, θ).
Give a precise definition of the matricial functions A (θ) and An(θ). (4 marks)
(ii) Show that an efficient GMM estimator based on moment functions ψ

(Y, θ) (or
ψ

n
(Y, θ) = An (θ) ψ (Y, θ)) can be computed with a block-diagonal weighting
matrix:
min
θ∈Θ
nψˉ
n
(θ)
0 Wnψˉ
n
(θ)
with:
ψˉ
n
(θ) = 1
n
nX
i=1
ψ

(Yi
, θ), Wn =
”
W11,n 0
0 W22,n #
with W11,n (resp. W22,n) square matrix of dimension H1 (resp. H2). (4 marks)
(c) We consider a partition of the parameter vector:
θ =

γ
λ
!
with dim (γ) = q, dim (λ) = r, q + r = p. We denote:
θ

n =

γn
λn
!
We assume that the vector ψ1 (Y, θ) does not depend on λ. We denote:
ψ1 (Y, θ) = φ (Y, γ)
(i) Deduce from the previous question that, if H2 = r, we get a GMM estimator of γ
asymptotically equivalent to γn by using only the moment conditions:
E [φ (Y, γ)] = 0
(5 marks)
(Question 4 continued overleaf)
6
(ii) Explain why this result could be expected without any calculation. (3 marks)
(iii) Discuss more generally the case r < H2. (2 marks) (iv) What is going on when ψ2 (Y, θ) does not depend on any unknown parameter Show that it amounts to estimate γ by efficient GMM based on a set of moments with reduced variance as follows: E h φ (Y, γ) i = 0 V ar h φ (Y, γ) i = V ar [φ (Y, γ)] Cov [φ (Y, γ), ψ2 (Y )] [V ar (ψ2 (Y ))] 1 Cov [ψ2 (Y ), φ (Y, γ)] (4 marks) Section D: Answer ONE question 5. (a) In this question, we consider a causal AR(2) model: Xt+1 = ω + α1Xt + α2Xt 1 + ut+1 |z| ≤ 1 = 1 α1z α2z 2 = 0 ut+1 = weak white noise, α1 > 0, α2 > 0
ρX (h), h = 0, 1, 2, .. stands for the auto-correlation function of the process Xt
:
ρX (h) = Cov [Xt
, Xt h]
V ar (Xt)
(i) Show that for all h = 1, 2, … :
ρX (h) = α1ρX (h 1) + α2ρX (h 2)
(3 marks)
(ii) Deduce that:
ρX (2)
ρX (1) =
α
2
1 + α2 (1 α2)
α1
(4 marks)
(Question 5 continued overleaf)
7
(iii) Deduce that:
ρX (2) < ρX (1) α2 (1 α2) < α1 (1 α1) (1 mark) (iv) Deduce that: ρX (2) < ρX (1) _x005f_x000c_ _x000c_ _x005f _x005f_x000c_ α1 1 2 _x000c_ _x000c_ < α2 1 2 (3 marks) Hint: May help to perform the change of variable: αi = 1 2 + zi , i = 1, 2 (b) In this question, we consider a ARCH(2) model defined by: σt 2 = ω + α1ε 2 t + α2ε 2 t 1 = E[ε 2 t+1 |εs, s ≤ t] ω > 0, E[εt+1 |εs, s ≤ t] = 0
(i) Explain why it is convenient to assume:
α1 + α2 < 1 (2 marks) (ii) Show that the process Xt = ε 2 t can be written as an AR(2) process conformable to the model of question 5(a). (1 mark) (iii) Show that the surprising inequality [ρX (2) ≥ ρX (1)] cannot happen if: α1 >
1
2
(4 marks)
(iv) Interpret this result in terms of volatility clustering. (2 marks)
6. (a) We consider a GARCH(p, q), q ≤ p, model:
σt
2 = ω +
p
X
i=1
αiε
2
t+1 i +
q
X
j=1
βjσt
2
j = E[ε
2
t+1 |εs, s ≤ t]
ω > 0, αi ≥ 0, βj ≥ 0,
p
X
i=1
αi +
q
X
j=1
βj < 1 (Question 6 continued overleaf) 8 (i) Show that Xt = ε 2 t can be written as an ARMA(p, q) model: [1 A(L) B(L)] Xt = ω + [1 B(L)] ut A(L) = p X i=1 αiL i , B(L) = q X i=1 βjL j (2 marks) The weak white noise ut is assumed to be such that this ARMA(p, q) is causal and invertible. (ii) Show that: Xt = ω 1 A(1) B(1) + ut + ∞X i=1 ciut i with, for all z ∈ C : [1 A(z) B(z)] ( ∞X i=1 ciz i ) = A(z) Hint: Write: 1 B(L) = [1 A(L) B(L)] + A(L) (4 marks) (iii) Deduce that c1 = α1 and for all i = 1, 2, ... : ci+1 i X j=1 cj (αi+1 j + βi+1 j ) = αi+1 with the convention of notation: i /∈ {1, 2, ..., p} = αi = 0 i /∈ {1, 2, ..., q} = βi = 0 (4 marks) (iv) Show by a mathematical induction argument that for all i = 1, 2, ... ci ≥ α1 [α1 + β1] i 1 (4 marks) (b) We consider the autocovariance function γX(h) of the process Xt = ε 2 t . (Question 6 continued overleaf) 9 (i) Show that, for all h = 1, 2, ..., by denoting c0 = 1 and σ 2 = V ar (ut) : γX(h) = σ 2 ∞X i=0 cici+h > 0
(3 marks)
(ii) Show that, if:
εt+1 = σtηt+1
with two independent processes {σt} and {ηt}, we have for all h = 1, 2, …:
γX(h) = Cov h σt
2
, σt
2
h
i
(3 marks)
(End)
10