Hypothesis testing in the LRM: The t-test

Summary

Setup:

• The LRM with random sampling and GM

\[y_i=β_1+β_2x_i+ε_i i=1,…,n\]

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

\(H_0: β_2=0\)

• For the null hypothesis \(H_0: β_2=0\) we define a test statistic called the “ t -value ” or the “ t -statistic ”

\[t= \frac{b_2}{SE(b_2)}\]

• If the null hypothesis is true then \(t∼t_{n-2}\)
• We decide on a level of significance \(α\) and reject the null if

\[\left| t \right|>t_{α/2,n-2}\]

\(p\) -values

• For the hypothesis above, we define the \(p\) -value of the hypothesis as the level of significance \(α\) such that we are indifferent between rejecting and not rejecting the null,

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

• We will 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 the GM assumptions are satisfied but the errors \(ε_i\) do not follow a normal distribution then, under mild conditions, the \(t\) -statistic will converge to a standard normal distribution as \(n→∞\) . This follows by a result known as the central limit theorem (CLT).
• Therefore, if \(n\) is large, the procedures outlined in this section will be approximately correct even if the errors are not normal.