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.