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Problem set 1 for econometrics iii, a graduate-level economics course taught by prof. Le wang. The problem set includes analytical questions related to the asymptotic distribution of sample means and limiting distributions, as well as applied questions that require using stata to simulate data and estimate parameters. The document also includes hints and instructions for each question.
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PROF. LE WANG
Problem Set #
Part I: Analytical Questions
Question 1. Let {xn} be an i.i.d sequence of random variables with E[xn] = μ and V(xn) = σ^2. Let xN be 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 σ and YN −→p Y where Y ∼ N(0, σ^2 ). Derive the limiting distribution of X N− 1 YN and (X− N 1 YN )^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 (^) b−^1 a (U [0, 1]). (2) Generate a random variable from the Student-t distribution with 5 degrees of freedom, t 5. (3) Generate a random variable from the χ^2 distribution 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 F distribution 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. 1
2 PROF. LE WANG
(1) Standard normal, N (0, 1) (2) Uniform, U [0, 1] (3) Student-t, t 5
Hint: In the class, we fixed the sample size and simulated the sample mean for m = 50, 100 , .. replications.
Question 3.
(1) Following the steps in the class, show that if one regressor is endogenous (i.e. corre- lated with error term), OLS estimator is inconsistent even when the sample size goes to infinity. (2) Simulate a data set from the following model
y = β 1 + β 2 x + u x = z + 0. 5 u u ∼ N (0, 1) z ∼ N (0, 1) where β 1 = 10 and β 2 = 2. (a) Simulate a data set of 150 observations. Estimate the effects of x on y, β 2 , via OLS estimation. Is it consistent? Explain. (b) Simulate a data set of 1000 observations. Estimate the effects of x on y, β 2 , via OLS estimation. Does the result get better? Explain. (c) With the information at hand, how can you consistently estimate the effects of x on y, β 2? Show your work.