ECOM40006/ECOM90013 Econometrics 3
Department of Economics
University of Melbourne
Assignment 3 Solutions
Semester 1, 2021
Let Y1, Y2, . . . , Yn denote a simple random sample from a population with probability
density function
f(y) =
{
θyθ−1, 0 < y < 1, θ > 0,
0, otherwise.
1. Show that the sample mean Y is a consistent estimator of θ/(θ + 1). [7 marks]
Hint: First derive the mean of the population and then remember that laws of large
numbers are your friends.
2. Derive a consistent method of moments estimator, θ˜ say, for θ. [1 mark]
3. Specify the log-likelihood function for this sample. [1 mark]
4. Derive the maximum likelihood estimator, θˆ say, for θ and prove that it is, indeed
a maximum likelihood estimator. [3 marks]
5. Derive the Fisher information for the sample. [2 marks]
6. Suppose that someone wishes to test the null hypothesis null hypothesis H0 : θ = 1
against the alternative that H1 : θ 6= 0. State the true population density function
and describe in words the implication for the population when this null hypothesis
is true. [2 marks]
7. Derive likelihood ratio, Lagrange multiplier and Wald tests for the hypotheses of
Question 6. In each case provide the decision rule that you would use in practice
to apply the test, including any critical value(s) you may need. [12 marks]
8. Without appeal to the generic properties of maximum likelihood estimators, prove
that θˆ is consistent for θ. [6 marks]
Your answers to the Assignment should be submitted via the LMS no later
than 4:30pm, Monday 10 May. Your mark for this assignment may contribute
up to 10% towards your final mark in the subject.
No late assignments will be accepted and an incomplete exercise is better than
nothing.
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