Econometric Methods, ECO341A, Semester I, 2015-16

Homework II (60 points)

Instructor: M.A. Rahman

Deadline: 2:50 pm, September 23, 2015.

Please read the instructions carefully and follow them while writing answers.

• Solutions to homework should be written in A4 size loose sheets. If you are not comfortable

writing on white sheets, please ask for biology paper in Tarun Book Store.

• Questions should be answered in order as they appear in the homework. Every new question

should begin in a new page. Please number all the pages of your homework solution.

• Please leave a margin of one inch from top and one inch from left. Staple the sheets on the

top-left.

• Matlab assignments (if any) and written answers should be together and in order.

1. (8 points) Matlab Exercise. Suppose Yi ∼ Bernoulli(p) for i = 1, 2, · · · , n, and the following

20 values were generated: 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0 from the distribution. Use

the Newton-Raphson procedure to maximize the log-likelihood function and find the MLE of the

parameter p.

2. (12 points) Matlab Exercise. Let Yi ∼ N(θ, σ2) for i = 1, 2, · · · , n, and suppose 30 values

presented in Q2.xlsx were generated from the distribution. Use the Newton-Raphson procedure

to maximize the log-likelihood function and find the MLE of (θ, σ2).

3. (6 + 6 = 12 points) Let X1,X2, · · · be independent and identically distributed random

variables with mean µ, variance σ2 and µ4 = E(X1 − µ)4 = α4σ4 with α4 > 1. Then show the

following,

(a) s2np−→ σ2

(b)√n(

s2n − σ2) d−→ N

(

0, (α4 − 1)σ4)

.

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Note that α4 is the kurtosis and s2n is the unbiased estimator of σ2. (Hint: Define s2mle =∑n

i=1(Xi−

X̄)2/n and show s2n converges in probability to s2mle ). Unlike lecture notes, you will have to show

that variance of s2mle goes to zero.

4. (8 points) Let X1,X2, · · · be a sequence of independent and identically distributed random

variables each uniform on [0,1]. For the geometric mean,

Gn = (X1X2 · · ·Xn)1/n,

show that Gnp−→ c for some constant c. Find c.

5. (4 + 6 = 10 points) Let X1,X2, · · · ,Xm be a random sample from a N(µ, σ2) distribution,

and independently let Y1, Y2, · · · , Yn be a random sample of size n from a N(µ, λσ2) distribution

with λ > 0 unknown.

(a) Find the MLE of λ if µ and σ2 are known.

(b) Suppose, µ and σ2 are unknown. Find the MLE’s of µ, σ2 and λ.

6. (1 + 3 + 3 + 3 = 10 points) Consider a centered Laplace distribution that has the following

pdf,

f(x|β) = 1

2βexp(−|x|/β),

where the shape parameter β > 0. Based on the above pdf, answer the following.

(a) Find E(X).

(b) Find the method of moments estimator for β using the first moment. What do you observe?

(c) Find the method of moments estimator for β using the second moments.

(d) Find the maximum likelihood estimator of β. Is it same as the method of moments estimator?

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