Hypothesis testing, several restrictions – the F-test

Summary

Setup

• The LRM with random sampling and GM

$$y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+ε_i, i=1,…,n$$

• $ε_i$ follows a normal distribution (conditionally on $x_i$ ), $ε_i|x_i∼N\left( 0,σ^2 \right) i=1,…,n$

$H_0: β_2=0, β_3=0,…,β_k=0$

• For the null hypothesis

$$H_0: β_2=0, β_3=0,…,β_k=0$$

• we have $k-1$ restrictions claiming that no explanatory variable has a significant effect on $y$ . We define a test statistic

$$F= \frac{R^2/(k-1)}{\left( 1-R^2 \right)/(n-k)}$$

• called an “ $F$ -statistic”. If the null hypothesis is true then $F∼F_{k-1,n-k}$ .
• We decide on a level of significance $α$ and reject the null if

$$F>F_{α,k-1,n-k}$$

• The $p$ -value for $H_0: β_2=0, β_3=0,…,β_k=0$ is defined as

$$F=F_{p,k-1,n-k}$$

Large $n$

• If $n$ is large, the procedures outlined in this section will be approximately correct even if the errors are not normal.