Hypothesis testing, one restriction – the t-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 \(x_i\) ), \(ε_i|x_i∼N\left( 0,σ^2 \right) i=1,…,n\)

\(H_0: β_j=0\)

• For the null hypothesis \(H_0: β_j=0\) where \(j\) is any value between 1 and \(k\) we define a test statistic

\[t_j= \frac{b_j}{SE(b_j)}\]

• \(t_j\) is called “ The \(t\) -value of the variable ” . If the null-hypothesis is true then \(t∼t_{n-k}\)
• We decide on a level of significance \(α\) and reject the null if

\[\left| t_j \right|>t_{α/2,n-k}\]

• When this null-hypothesis is rejected we say that \(b_j\) is “ statistically significant ” and that \(x_j\) is has statistically significant effect on \(y\) . If we do not reject the null we say that \(b_j\) is “ statistically insignificant ” .

\(p\) -values

• The \(p\) -value for \(H_0: β_j=0\) is defined implicitly as

\[\left| t_j \right|=t_{p/2,n-k}\]

• We reject the null hypothesis if \(p<α\) .
• We will not reject the null hypothesis if \(p>α\) .
• “If \(p\) is low the null must go. If \(p\) is high the null will fly”.

Large \(n\)

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