经济代写|计量经济学|ECOM40006/ECOM90013 Assignment 1

Problem 1. In this question we will assume that $x \sim \mathrm{N}(\mu, \Sigma)$ is a pevector, as is $\mu,$ and that the $p \times p$ matrix $\Sigma>0$
(a) If $v$ is any fixed $p$ -vector, show that
$$
g=\frac{v^{\prime}(x-\mu)}{\sqrt{v^{\prime} \Sigma v}} \sim N(0,1)
$$
(b) If $v$ is now a random vector independent of $x$ for which $P\left(v^{\prime} \Sigma v=0\right)=0$, show that $g \sim \mathrm{N}(0,1)$ and is independent of $v .$ Why have we assumed $P\left(v^{\prime} \Sigma v=\right.$
0)$=0$ ? Can you think of an equivalent statement of this assumption? marks)
(c) Hence show that if $y=\left[y_{1}, y_{2}, y_{3}\right]^{\prime} \sim \mathrm{N}\left(0, I_{3}\right)$ then
$$
h=\frac{y_{1} e^{y \Delta}+y_{2} \log \left|y_{3}\right|}{\left[e^{2 y_{1}}+\left(\log \left|y_{3}\right|\right)^{2}\right]^{1 / 2}} \sim N(0,1)
$$
marks)

Problem 2.

Suppose that $x \sim \mathrm{N}(\mu, \Sigma),$ where the $p \times p$ matrix $\Sigma>0,$ and that $v$ is a fixed $p$ -vector. If $r_{i},$ the $i$ th element of the vector $r,$ is the correlation between $x_{i}$ and $v^{\prime} x,$ show that $r=(c D)^{-1 / 2} \Sigma v,$ where $c=v^{\prime} \Sigma v$ and $D=\operatorname{diag}(\Sigma) . \quad$ (3 marks)
Bonus question: When does $r=\Sigma v ?$ (1 mark)
Your answers to the Assignment should be submitted via the LMS no later than $4: 30 \mathrm{pm}$, Thursday 1 April.

No late assignments will be accepted but an incomplete exercise is better than nothing.