ECON 898: ECONOMETRICS III
PROF. LE WANG
Problem Set #1
Part I: Analytical Questions
Question 1. Let {xn}be an i.i.d sequence of random variables with E[xn] = µand V(xn) =
σ2. Let xNbe the sample mean. Derive the asymptotic distribution of ln(xN). Clearly state
any theorems and assumptions you use in answering the question.
Question 2. Suppose XN
p
−→ σand YN
p
−→ Ywhere Y∼N(0, σ2). Derive the limiting
distribution of X−1
NYNand (X−1
NYN)2. Clearly state any theorems and assumptions you use
in answering the question.
Part II: Applied Questions
Do the following exercises in Stata. Turn in your do file, log file, and graphs. Set the seed
at 123456 at the start of each problem.
Question 1. Simulate data from the following distributions. Plot the histograms of each of
the variables. What are the true and estimated means and variances?
(1) Generate random numbers from the uniform distribution 1
b−a(U[0,1]).
(2) Generate a random variable from the Student-t distribution with 5 degrees of freedom,
t5.
(3) Generate a random variable from the χ2distribution with 1 degree of freedom.
(4) Generate a random variable from the Normal distribution with mean 1 and variance
0.5.
(5) Generate a random variable from the Fdistribution with 10 and 5 degrees of freedom.
Question 2. This problem will show you how the central limit theorem works. Show how
the Central Limit Theorem works for the sample mean with the five separate distributions.
Use n= 10,50,and 100 with m= 100,200, and 1000 Monte Carlo replications. Plot
histograms of the sample means for each distribution and each sample size.
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