[SOLVED] EMET 4314/8014 Advanced Econometrics I 2022

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.