Solutions to Review Question

by Qiang Gao, updated at Mar 13, 2017

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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