# Econ122B UCI Derivations Worksheet

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Economics 122B
Applied Econometrics II
Fall 2020
Homework 2
Due by 4pm on 5/14/2020
Derivations
Problem 1: Given the model y = X? + ? with E(?|X) = 0 and V ar(?|X) = ?,
please derive the expected value and covariance matrix of the estimator
?? GLS = (X 0 ??1 X)?1 X 0 ??1 y.
(1)
If you are given X and y only, is this estimator feasible? Discuss what you may need
to do, listing the steps in the process explicitly.
Problem 2: Given the model y = X? + ? with E(?|X) = 0 and V ar(?|X) = ?,
please derive the expected value and covariance matrix of the OLS estimator. Is
OLS biased in this setting? Is it efficient? If you are unwilling to place additional
assumptions about the form of heteroskedasticity, how can you estimate the standard
errors of the OLS estimator? Please discuss.
This material is in your class notes. However, your understanding of the
material is essential, so do not copy blindly  make sure you understand
the steps and the main ideas.
Application
Problem 3: Generate a covariate matrix X of dimension n × k, where n = 250 and
k = 2. The first column in X should be a column of ones and the other column can
be generated from a normal distribution with variance 10. Let ? = (0.5, 1)0 and let
? 2 = 1. Simulate y from the model y = X? + ?, where ? ? N (0, ? 2 In ). Estimate ?
by ordinary least squares and produce the forecast point estimates y?f for Xf = X.
Also produce the standard errors of your forecast and plot your point and interval
estimates of your forecast in a graph. [Hint: see your class notes.] Provide details on
For this exercise, you can use any software you are comfortable with, but
program the steps yourself, following the theoretical derivations presented
in class.