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EC9100
1 (Continued overleaf)
UNIVERSITY OF WARWICK
May Examinations 2022/23
Quantitative Methods: Econometrics B
Time Allowed: 3 Hours, plus 15 minutes reading time
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. 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).
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 ALL questions
1. (a) The two tables given below, are taken from Angrist & Pischke ‘Mostly Harmless
Econometrics’, and refer to the study by Krueger, Alan B. (1999): “Experimental
estimates of education production functions”, The Quarterly Journal of Economics, 114,
497-532.
Notes: Adapted from Krueger (1999), Table 1. The table shows the means of variables by treatment
status. The P-Values in the last column is for the F-test of equality of variable means across all three
groups. All variables except attrition are for the first year a student is observed. The free-lunch variable
is the fraction receiving free-lunch. The percentile score is the average percentile score on three
Stanford Achievement Tests. The attrition rate is the proportion lost to follow-up before completing
third grade.
2 (Continued Overleaf)
Table 2.2.2: Comparison of treatment and control characteristics
In the Tennessee STAR experiment – Regression results
Notes: Adapted from Krueger (1999), Table 5. The dependent variable is the Stanford Achievement
Test percentile score. Robust standard errors that allow for correlated errors within classes are shown
in parentheses. The sample size is 5,681.
Concentrating on the kindergarten study results presented in Tables 2.2.1 and 2.2.2, answer
only ONE question from the list of 4 given below:
i. Provide a broad context to set the scene of the investigation and write down the main
equation that is estimated. For example, you might want to consider the following
points among others. What was the research question and why is this important in
economics What is the role of the dummy variables in the above specification.
or
ii. Discuss the experimental protocol in detail. You should discuss how the experiment
was set up and how the randomisation was carried out.
or
iii. Discuss the estimated results in the above two tables.
or
iv. Discuss any limitations of this experimental design.
(20 marks)
(b) Discuss the difference-in-difference estimator of a treatment effect. Make sure that
you list all the assumptions necessary to identify the coefficient on the treatment
dummy as the “average-treatment-effect-on-the-treated” (ATT) and obtain an
unbiased/consistent estimator using OLS. (10 marks)
3 (Continued Overleaf)
SECTION B: Answer ONE question
2. A researcher has access to a cross-sectional survey of a group of individuals – both
employed and unemployed. The survey collected the usual demographic variables as
well as the number of hours the individual usually worked in a week and the weekly
wages for those employed people. The researcher would like to model the following
outcomes: weekly hours worked (answer in part (a) below) and log wages (answer in
part (b) below).
Set up the appropriate econometric specifications and discuss the estimation of these
models using a two-step procedure, stating clearly all the assumptions that are
necessary for your estimator to be consistent. You will have to discuss all identification
issues related to these estimations.
Hint
You may assume that for a jointly normally distributed random variables and u,
[

] ～ {[
0
0
],[

2

2
]},
the conditional means are:
and where, and are the pdf and cdf of a
standard normal variable, and .
(a) Weekly worked hours (10 marks):
(b) Log wages (10 marks):
3. A researcher is interested in estimating the effect of schooling on log wages using
observations on individuals over time. The schooling variable is the years of schooling
and is time-invariant. In the absence of a suitable instrument, the researcher proceeds
with the estimation using Generalised Least Squares (GLS) method.
(a) Discuss in detail the GLS estimator, stating all the assumptions that are needed for
this estimator to be consistent. (12 marks)
(b) Is the proposed method suitable to answer the research question Discuss your
reasons as to why it is or it is not, a suitable method (Hint: consider all the
assumptions that you have discussed in (a)). If there is something the researcher
can do to solve this problem (if there is one), discuss how the researcher should
proceed. (8 marks)

( | ] E c
= ( | ) u
u
E u c

=
( ) /[1 ( )] c c =
EC9100
Section C: Answer the ONE question
4. We consider a GARCH(1,1) model:
εt =
q htut
ht = ω + αε2
t 1 + βht 1
ω > 0, α ≥ 0, β ≥ 0, α + β < 1 where (ut) is a strong white noise with: E (ut) = 0, E _x005f_x0010_ u 2 t _x005f = 1, E u 4 t = κ (a) Show that if E (ε 4 t ) < ∞, we have the ARCH (∞) representation: ht = ω 1 β + α ∞X i=0 β i ε 2 t 1 i (6 marks) (b) In order to assess how much restrictive is the condition E (ε 4 t ) < ∞, we compute: h 2 t = ω 2 + α 2 ε 4 t 1 + β 2h 2 t 1 + 2ωαε2 t 1 + 2ωβht 1 + 2αβht 1ε 2 t 1 Show that if E [h 2 t ] < ∞: h 1 κα2 β 2 2αβi E h h 2 t i = ω 2 + 2ω (α + β) ω 1 α β (8 marks) (c) Deduce that: E ε 4 t < ∞ (α + β) 2 + (κ 1) α 2 < 1 (5 marks) (Continued overleaf) 4 EC9100 (d) Explain why this condition is very restrictive when volatility is highly persistent. (6 marks) (e) Show that when this condition is fulfilled: E ε 4 t = κω2 1 + α + β (1 α β) (1 κα2 β 2 2αβ) (5 marks) (Continued overleaf) 5 EC9100 Section D: Answer ONE question 5. The model of interest consists of the following three equations (for i = 1, ..., n): log (w1i) = x 01,iβ1 + η1,i log (w0i) = x 00,iβ0 + η0,i qi = z 0iπ + εi The first two equations are wage equations for the union and non-union sector respectively (see explanations below). Note that the case x1,i = x0.i and β1 = β0 except for the intercepts α1, α0, is not precluded The third equation is a sectoral choice equation. qi is an unobserved variable representing the utility gain from union membership. Its dichotomous realization Di is observed union status. If qi > 0, Di
is set equal to one, meaning that the i-th worker is a union member and her
observed wage wi
is w1i
. If qi < 0, conversely, Di is set equal to zero, meaning that worker i is employed in the non-union sector and that wi = w0i . We assume that (η1,i, η0,i, εi) are independent of the regressors (x1i , x0,i, zi). (a) i) Show that the observed salary can be written: log (wi) = Dix 01,iβ1 + (1 Di) x 00,iβ0 + ui (1) with: ui = Diη1,i + (1 Di) η0,i (1 mark) ii) We assume that the couple (εi , η1,i) and the couple (εi , η0,i) follow the same joint probability distribution. Explain (a rigorous proof is not required) why we expect that: E (ui) = 0 (3 marks) (Continued overleaf) 6 EC9100 iii) Explain why we cannot estimate the parameters of interest by OLS in equation (1). (3 marks) (b) We define: Pi (zi) = Pr[Di = 1 |zi)] i) Explain why x1iPi (zi) and x0i [1 Pi(zi)] are valid instruments to instrument the variables Dix1i and [1 Di ] x0i respectively in equation (1). (3 marks) ii) Explain why we may expect that x1iPi (zi) and x0i [1 Pi(zi)] are relevant instruments. (3 marks) iii) How would you use these instruments in practice (2 marks) (c) We consider in this question the case where x1,i = x0.i and β1 = β0 except for the intercepts α1, α0. i) Show that the wage gap, that is the gain from moving a unionized person i with attributes xi and zi from the nonunionized to the unionized sector is: E[log (w1i) log (w0i)|Di = 1, xi , zi ] = (α1 α0) + E[η1,i η0,i |Di = 1, zi ] (2 marks) ii) Discuss the possibility to get a consistent estimator of the wage gap by using the instrumental variable estimator defined above. (3 marks) (Continued overleaf) 7 EC9100 6. (a) We consider the following system of demand/supply equations for t = 1, ..., T: Qt = α + βPt + εt Qt = γ + θPt + ut where Qt and Pt are observable, where α, β, γ, and θ are fixed unknown parameters, and where εt and ut are serially and contemporaneously independent random variables with zero means and variances σε 2 and σ 2 u . (i) Show that: Pt = α + εt γ ut θ β Qt = [α + εt ]θ [γ + ut ]β θ β (2 marks) (ii) Deduce that: V ar (Pt) = σε 2 + σ 2 u (θ β) 2 V ar (Qt) = θ 2σε 2 + β 2σ 2 u (θ β) 2 Cov (Pt , Qt) = θσε 2 + βσ2 u (θ β) 2 (2 marks) (iii) Show that for given estimators θ, β of (θ, β), the first and third equations above allow to compute a moment estimator ( σ 2 ε , σ 2 u ) of (σε 2 , σ2 u ) by computing: " σ 2 ε σ 2 u # = " 1 1 θ β # 1 " s 2 p spq # θ β 2 where s 2 p (resp. spq) stands for the sample variance of Pt (resp. sample covariance between Pt and Qt). (2 marks) (Continued overleaf) 8 EC9100 (iv) We obviously deduce (calculation not requested) that: σε 2 = _x0010_ βs2 p spq θ β _x0011_ = s 2 p _x0010_ β b θ β σ 2 u = θs2 p + spq _x0010_ θ β = s 2 p θ + b _x0011_ _x0010_ θ β where: b = spq s 2 p Interpret the estimator b. (2 marks) (v) We complete the computation of the moment estimators by plugging in σε 2 and σ 2 u in the above formula for V ar (Qt) (that is replaced by its sample counterpart s 2 q ). Show that we get: θ β s 2 q s 2 p = θ 2 b β _x0011_ + β 2 θ b This equation can be rewritten (a proof is not requested): θ b β b = r 2 pq 1 s 2 q s 2 p where r 2 pq is the squared sample correlation between Pt and Qt . (2 marks) (vi) Deduce from above formulas that any estimate of β is possible, and any estimate of θ; but given one, there is a unique moment estimate of the other. Note that b is always between min β, θ and max β, θ . Explain why these results are conformable to the analysis developed in the lectures. (4 marks) (Continued overleaf) 9 EC9100 (b) We now consider a more general model in which an observable variable xt may affect both the quantity supplied and the quantity demanded. More precisely, the model is altered to allow α and γ to be functions of xt : Pt = α0 + α1xt + βPt + εt Qt = γ0 + γ1xt + θPt + ut From formulas of question 1a), this leaves the conditional variance matrix of (Pt , Qt) given xt unchanged but alters the means to: E[Pt |xt ] = α0 γ0 + (α1 γ1)xt θ β E[Qt |xt ] = α0θ γ0β + [α1θ γ1β] xt θ β i) Explain why the OLS estimators dp and dq of the slope coefficients of regression of Pt and Qt on the variable xt allow us to compute consistent estimators of α1 and γ1 by solving the two equations: dp = α1 γ1 θ β dq = α1θ γ1β θ β These equations can be solved for ( α1, γ1) in terms of θ, β : α1 = dq βdp γ1 = dq θdp (3 marks) ii) Deduce that if α1 is known to be zero: β = dq dp Explain this result in terms of IV (Instrumental Variables) estimation. (3 marks) 10 (End)