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Advanced Microeconometrics
Semester 1, 2020
Online, non-invigilated
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120 minutes; you are allowed to use more time if you need.
This exam is weighted at 50% of your total mark for this course. This is an open book exam – all materials permitted.
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Semester One Final Examination, 2020 ECON7320 Advanced Microeconometrics
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Semester One Final Examination, 2020 ECON7320 Advanced Microeconometrics
Answer ALL questions.
Show all your working. Explain your answers. Marks are as indicated (Total marks: 100)
Question 1: Maximum Likelihood [25 marks]
Consider the random variable Y with Y ∼ Poisson(λ) such that Prob[Y = y] = λye−λ
and E[Y ] = VAR[Y ] = λ. Suppose that we have an independent random sample of size N .
(a) Write down the likelihood function and log likelihood function. (3 marks)
(b) Obtain the maximum likelihood estimator of λ. (7 marks)
(c) Now suppose that there is a single covariate x, and
Y |X = x ∼ Poisson(λ0 + λ1x).
You obtain the STATA output in Figure 1 overleaf for the full model and the model restricting λ1 = 0 respectively. Perform a likelihood ratio test of
H0 :λ1 =0 against a two sided alternative at the 0.05 level.
(d) Perform a Wald test of
H0 :λ1 =0 against a two sided alternative at the 0.05 level.
(e) Explain which additional information you would need to perform a Wald test of
H0 :λ0 =λ1 =0
against a two sided alternative and how you would implement the test at the 0.05
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Semester One Final Examination, 2020 ECON7320 Advanced Microeconometrics
FIGURE 1. STATA output for Question 1
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Semester One Final Examination, 2020 ECON7320 Advanced Microeconometrics
Question 2: General Method of Moments [25 marks]
Consider the linear model:
y=Xβ+u, E[u|Z]=0
where y is a N × 1 vector of outcomes, X is a N × K matrix of regressors, Z is a N × L matrix of instrumental variables, u is a N × 1 vector of unobserved error terms and β is a K × 1 vector of parameters. Assume throughout that L ≥ K and Z′X has rank K.
(a) Given a L × L symmetric, positive definite, full rank weight matrix WN , the GMM estimator minimizes the objective function
1 ′ ′ 1 ′ NZu WN NZu .
Explain how this objective function relates to the method of moments, and the drawback
of method of moments when L > K. (b) Show that the GMM estimator is
(d) For which Z and WN does the GMM estimator equal the OLS estimator? (3 marks)
(e) For which WN does the GMM estimator equal the two-stage least squares estimator?
(f) If VAR[u|Z] = σ2IN and N = 50, would you favour the two-stage least squares or optimal two-step GMM estimator? Justify your answer. (3 marks)
(g) If VAR[u|Z] = Ω ̸= σ2IN and N = 10,000, would you favour the two-stage least squares or optimal two-step GMM estimator? Justify your answer. (3 marks)
βGMM =(X′ZWNZ′X)−1X′ZWNZ′y.
(c) If L = K, explain why any WN leads to the same βGMM.
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Semester One Final Examination, 2020 ECON7320 Advanced Microeconometrics
Question 3: Structural Modelling [25 marks]
Consider the production function for a monopolist making widgets:
and the demand function:
Q = ALαV P = Q−1/2
where Q is output, L is labour, P is the unit price, A is a constant and V is firm heterogeneity, which is unobserved by the researcher but observed by the firm. The wage per unit of labour is W , for which firms are price takers (i.e. wages are exogenously determined by local labour market conditions, and are independent of V ). The parameter of interest is α, which is known to satisfy 0 < α < 2.
(a) Show that the monopoly profit is given by A1/2Lα/2V 1/2 − W L.
(b) Show that the profit maximizing choice of labour is
W β (α/2)A1/2V 1/2 ,
β=1. α/2−1
(c) Your colleague has a sample of widget manufacturers of size N = 1, 024, and presents you with the STATA output in Figure 2 overleaf. Your colleague is puzzled, because it is well known that widget manufacture exhibits decreasing returns to scale. Comment on your colleague’s findings. (7 marks)
(d) Advise your colleague on how to improve his estimate of α. State any assumptions that you make. (5 marks)
(e) Your colleague informs you that they would also like to construct a 0.95 confidence
interval for
Provide detailed instructions on how this can be achieved.
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β=1. α/2−1
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Semester One Final Examination, 2020 ECON7320 Advanced Microeconometrics
FIGURE 2. STATA output for Question 3
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Semester One Final Examination, 2020 ECON7320 Advanced Microeconometrics
Question 4: Non-parametric Methods [25 marks]
Consider an independent random sample of size N drawn from the continuous population probability density function f(y).
(a) Is the function
K(u)= 2 1(−1