ECON61001 Econometric Methods Ekaterina Kazak Computer Tutorial 4, page 1 of 1 University of Manchester
The purpose of this session is to introduce you to the large sample behaviour of OLS estimator and provide an illustration for the Central Limit Theorem.
Weak Law of Large Numbers and Central Limit Theorem
Consider four different distributions
1. Normal distribution u1 ∼ N (3, 1)
2. Student-t distribution u2 ∼ t(5)
3. Chi-squared distribution u3 ∼ χ2(2)
4. Gamma distribution u4 ∼ Γ(1, 0.5)
a) Plot theoretical densities of uj , j = 1, .., 4. Note, Gamma and Chi-squared distributions are defined only on R+.
b) In order to examine whether the WLLN holds, simulate n = 10000 observations from each of the four distributions. Define a function which computes a sample moving average of an n × 1 vector. Plot the moving average evolution over n for each distribution.
c) Simulate a distribution of the sample mean for each uj , j = 1, .., 4. For each mean, compute the stabilizing transformation and plot its histogram together with N (0, 1) density.
d) Does the CLT hold for u5 ∼ t(1)? What about WLLN?
Exercises to solve at home
Illustrate conclusions of Theorem 3.2 and Theorem 3.3 with the help of a simulation study. For this exercise set the number of simulation draws to mc = 10000. Consider a bivariate linear regression model:
yi = 1 − 3xi + ui, (1) where u ∼ N(0,10), x ∼ N(5,16).
a) Simulate the distribution of the estimated OLS coefficients for slope and intercept (over mc simulation draws) for n = 10. For each simulation step save the theoretical variances of the OLS
coefficient, i.e. V βˆX = σu2(X′X)−1.
b) Standardize the coefficients (re-center and scale with the V βˆX ). Plot the histograms of
the stabilizing transformation of estimated slope and intercept next to each other. Use breaks = 50 option for histograms. On top of each histogram draw a theoretical density of the standard normal distribution and Student-t distribution with n − 2 degrees of freedom. Compute the densities over an equidistant grid {-5, -4.8, …, 4.8, 5}. Which distribution best describes the distribution of the estimated OLS coefficients?
c) Repeat b) for a different standardization: Vˆ βˆX , i.e. re-run the simulation and replace σu2 with its estimate σˆ2 = ni=1 uˆ2i . Which distribution is a better fit for this case?
N n−2
d) Increase the sample size to n = 5000. Explain this result using Theorems 3.2 and 3.3.
Compare your results with the homework answers which will be available on Blackboard on Friday.