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 formβˆδˆ 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 answer (2 marks) 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
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