ECOM40006/ECOM90013 Econometrics 3
Department of Economics
University of Melbourne
Assignment 1 Solutions
Semester 1, 2021
1. In this question we will assume that x ∼ N(µ,Σ) is a p-vector, as is µ, and that the
p× p matrix Σ > 0.
(a) If v is any fixed p-vector, show that
g =
v′(x− µ)√
v′Σv
∼ N(0, 1). (1 mark)
(b) If v is now a random vector independent of x for which P (v′Σv = 0) = 0, show
that g ∼ N(0, 1) and is independent of v. Why have we assumed P (v′Σv =
0) = 0? Can you think of an equivalent statement of this assumption?
(4 marks)
(c) Hence show that if y = [y1, y2, y3]
′ ∼ N(0, I3) then
h =
y1e
y3 + y2 log |y3|
[e2y3 + (log |y3|)2]1/2 ∼ N(0, 1). (2 marks)
2. Suppose that x ∼ N(µ,Σ), where the p × p matrix Σ > 0, and that v is a fixed
p-vector. If ri, the ith element of the vector r, is the correlation between xi and
v′x, show that r = (cD)−1/2Σv, where c = v′Σv and D = diag(Σ). (3 marks)
Bonus question: When does r = Σv? (1 mark)
Your answers to the Assignment should be submitted via the LMS no later
than 4:30pm, Thursday 1 April.
No late assignments will be accepted but an incomplete exercise is better than
nothing.
1

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