1.4.1.md

Solution to Review Question

by Qiang Gao, updated at May 10, 2017

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Chapter 1 Finite-Sample Properties of OLS

Section 4 Hypothesis Testing under Normality

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Review Question 1.4.1 (Conditionial vs. unconditional distribution)

(a) Do we know from Assumptions 1.1—1.5 that the marginal (unconditional) distribution of $$ \mathbf{b} $$ is normal?

(b) Are the statistics $$ z_k $$, $$ t_k $$, and $$ F $$ distributed independently of $$ \mathbf{X} $$?

Solution

(a) Under Assumptions 1.1—1.5,

$$ \mathbf{b} \mid \mathbf{X} \sim N( \boldsymbol{\beta}, \sigma^2 \cdot ( \mathbf{X}' \mathbf{X} )^{-1} ). \tag{1.4.2} $$

Because the variance of $$ \mathbf{b} $$ depends on $$ \mathbf{X} $$, when the marginal distribution of $$ \mathbf{X} $$ is unknown, the marginal distribution of $$ \mathbf{b} $$ is also unknown, so it is not necessarily normal.

(b) Under Assumptions 1.1—1.5 and the null hypothesis,

$$ \begin{align} z_k \mid \mathbf{X} & \sim N(0, 1), \tag{1.4.3} \\ t_k \mid \mathbf{X} & \sim t(n - K), \tag{1.4.5} \\ F \mid \mathbf{X} & \sim F(# \mathbf{r}, n - K). \tag{1.4.9} \end{align} $$

These conditional distributions does not depend on the value of $$ \mathbf{X} $$, so their marginal distributions are independent of $$ \mathbf{X} $$.

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Copyright ©2017 by Qiang Gao