Advanced Econometrics I
(EMET 4314/8014)
First Semester Final Examination– June, 2022
Beginning of Exam Questions
1. [1 mark total] Write the following statement by hand:
I hereby declare
◦ to uphold the principles of academic integrity, as defined in the University Academic Misconduct Rules;
◦ that your work in the final exam in no part involves copying, cheating, collusion, fabrication, plagiarism or recycling.
2. [20 marks total]
Consider the scalar model Yi = β0+ β 1Xi1+ ei where ei|Xi1 ~ N(0,1).
You have available a random sample (Xi1, Yi), i = 1, . . . , N.
Let β0and β 1be the OLS estimators obtained from a regression of Yi on a constant and Xi1 .
(a) [2 marks] State β 1 in terms of sample moments ofthe data (that is, sample
means, variances, and covariances). No derivation, just state the result.
(b) [2 marks] Derive Var (β 1|Xi1).
You observe an additional variable, Xi2 . Denote by 0 and 1 the OLS estimators from a regression of Xi1 on a constant and Xi2 . Define X-i1 := π-0+ π-1Xi2 .
Let θ0and θ 1be the OLS estimators obtained from a regression of Yi on a constant and Xi1 .
(c) [4 marks] Derive θ 1 in terms of sample moments of the data.
(d) [2 marks] Derive Var (θ 1|Xi1, Xi2).
(e) [5 marks] Prove or disprove: θ = β 1+ op (1).
(f) [5 marks] Which estimator do you prefer: β 1 or θ 1? Why?
3. [20 marks total - 5 marks each]
Are the following statements true or false? Provide a complete explanation. Use mathematical derivations where necessary.
(Note: you will not receive any credit without providing a correct explanation.)
(a) Let the discrete random variable have the following distribution:
P(Y = 1) = π1, P(Y = 2) = π2, P(Y = 3) = π3,
where π1 ∈ (0,1), π2 ∈ (0,1), π3 ∈ (0,1), and π1 + π2 + π3 = 1.
In a random sample of size N you observe N1 realizations for which Y = 1, N2 realizations for which Y = 2, N3 realizations for which Y = 3, so that N1 + N2+ N3 = N.
Then the maximum likelihood estimate of π1 is N1/N.
(b) Let X be a Bernoulli random variable, that is, X = 1 with probability π and X = 0 with probability 1 — π where π ∈ (0,1). Let Y be another random variable (not Bernoulli distributed) and assume that Cov(X, Y) ≠ 0.
Then Cov(X, XY) = E(Y) + (1 — π) · Cov(X, Y).
(c) Let the random variable Z be such that E(Z) = 3 and E(Z2 ) = 13. Then a lower bound for P(—2 < Z < 8) is given by 21/25.
(d) The Monte Carlo simulation of the simple schooling model from week 7, as summarized by the Julia code and corresponding output below, illustrates that the OLS estimator is a consistent estimator for the return to schooling.
JULIA CODE
1 using Distributions , Random, Plots
2
3 function schooling__sample (b2 , n ;
4 p=13.2 , b1 =4.7 , b3=0,
5 su =0.175 , sa =7.2)
6 u = rand (Normal (0 , s q rt (su ) ) , n)
7 a = rand (Normal (0 , s q rt ( sa ) ) , n)
8 S = p .+ a
9 Y = b1 .+ b2∗S .+ b3∗a .+ u
10 return S , Y
11 end
12
13 rep = 100000
14 b2 = Array{ Floa t 6 4 } (undef , rep )
15 for r in 1 : rep
16 n = 1000
17 x , y = schooling__sample (0 .075 , n)
18 b1_tmp , b2_tmp= [ones (n , 1) x ]\y
19 b2 [ r ] = b2_tmp
20 end
21 histogram (b2 , normed = false )
OUTPUT
4. [20 marks total] Consider the model
Yi = µ(Xi, θ) + ei, where ei|Xi ~ N(0, σ2e).
The variables Yi and ei are scalars and dim(Xi) = K × 1 and dim(θ) = L × 1 where K ≠ L. The functional form. of µ is considered known but is left unspecified here.
You have available a random sample (Xi, Yi), i = 1, . . . , N, to estimate the un- known parameters θ and the scalar σ2e.
(a) [3 marks] Derive the conditional log likelihood function L(θ, σ2e).
(b) [3 marks] Derive the score function.
(c) [3 marks] Derive the expected value of the score conditional on Xi.
(d) [2 marks] Determine the MLE of σ2e as a function of θ ML (the MLE of θ).
(e) [3 marks] Derive the Hessian matrix as the derivative of the score.
(f) [6 marks] Show that the information equality holds here.