ECOM40006/ECOM90013 Econometrics 3
Department of Economics
University of Melbourne
Assignment 2 Solutions
Semester 1, 2021
1. It is common for wage equations to be estimated with some measure of the level of
education as one of the expanatory variables. To allow for a non-linear response,
this variable often enters the equation in both level and squared forms. So, for the
i-th individual the equation may look something like
wagesi = βeduci + δeduc
2
i + x

iθ + ui, i = 1, . . . , n,
where wi denotes the wage of the i-th individual, educi their level of educational
attainment, and xi is a vector of observations on all the other expanators in the
equation (including the intercept). Note that we have said nothing about how any
of these variables are measured. In any event, the postulated model allows for a
quadratic relationshp between wages and education and an obvious question to ask
is where is the turning point in the relationship between wages and education given
the x’s. Elementary mathematics tells us that this occurs where educi = −β/(2δ).
Suppose that the joint asymptotic distribution of [βˆ, δˆ, θˆ′]′ is of the formβˆδˆ
θˆ
 ∼
a
N
βδ
θ
 , n−1
 σ2β σ2βδ Σ′βθσ2βδ σ2δ Σ′δθ
Σβθ Σδθ Σθ
 .
(a) What is the marginal asymptotic distribution of [βˆ, δˆ]? (1 mark)
(b) Find the asymptotic distribution for the turning point of the wage equation
as a function of education. (4 marks)
2. Let Y1, Y2, . . . , Yn denote a simple random sample of size n from a Normal population
with mean µ and variance 1. Consider the first observation Y1 as an estimator for
µ.
(a) Show that Y1 is an unbiased estimator for µ. (1 mark)
(b) Find Pr (|Y1 − µ| ≤ 1). (2 marks)
(c) Based on your answer to 2(b), is Y1 a consistent estimator for µ? Explain your
1
Your answers to the Assignment should be submitted via the LMS no later
than 4:30pm, Thursday 22 April.
No late assignments will be accepted and an incomplete exercise is better than
nothing.
Your mark for this assignment may contribute up to 10% towards your final
mark in the subject.
2 