1.1.5.md

Solution to Review Question

by Qiang Gao, updated at Mar 13, 2017

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Chapter 1 Finite-Sample Properties of OLS

Section 1 The Classical Linear Regression Model

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Review Question 1.1.5 (Multicollinearity for the simple regression model)

Show that Assumption 1.3 for the simple regression model is that the nonconstant regressor ($$ x_{i2} $$) is really nonconstant (i.e. $$ x_{i2} \neq x_{j2} $$ for some pairs of $$ (i, j) $$, $$ i \neq j $$, with probability one).

Solution

The simple regression model is

$$ \mathbf{y} = \beta_1 \cdot \mathbf{1} + \beta_2 \mathbf{x}_2 + \boldsymbol{\varepsilon}. $$

Assumption 1.3 requires that $$ { \mathbf{1}, \mathbf{x}2 } $$ are linearly independent with probability one. This means $$ \mathbf{x}2 $$ is not proportional to $$ \mathbf{1} $$ with probability one, i.e., $$ x{i2} \neq x{j2} $$ for some pairs of $$ (i, j) $$, $$ i \neq j $$, with probability one.

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