Inferences Concerning Coefficients¶

We are most interseted in checking whether a coefficient has an effect or not

\[ \begin{aligned} H_{0}: \theta_{1} = 0 \quad \text{versus} \quad H_{1}: \theta_{1} \neq 0 \end{aligned} \]

We know from above derivations that

\[ \begin{aligned} \frac{\hat{\theta}\_{1} - \theta_{1}}{\sigma^{2} / S_{xx}} \sim \mathcal{N}(0, 1)\newline \frac{SS_{R}}{\sigma^{2}} \sim \chi_{n-2}^{2} \end{aligned} \]

and both the random variables are independent of each other. Hence their division is t-distributed random variable and when \(H_{0}\) is true, \(\theta_{1} = 0\)

\[ \begin{aligned} \frac{\sqrt{S_{xx}}\hat{\theta}\_{1}/\sigma}{\sqrt{\frac{SS_{R}}{\sigma^{2} (n-2)}}} = \hat{\theta}\_{1}\sqrt{\frac{(n-2)S_{xx}}{SS_{R}}} = TS \sim t_{n-2} \end{aligned} \]

We do this since we do not know the exact value of \(\sigma^{2}\) and need to eliminate it with a sample derived version. The hypothesis test at significance level \(\alpha\) simply becomes

\[ \begin{alignat}{4} \text{Reject}\quad &H_{0} \quad\text{if}\quad &\lvert TS \rvert &> &t_{\alpha/2, n-2}\newline \text{Accept}\quad &H_{0} \quad\text{if}\quad &\vert TS \rvert &\leq &t_{\alpha/2, n-2} \end{alignat} \]

which can be converted to a p-value using the \(TS\) and t-distribution. A small p-value will lead to rejection of \(H_{0}\) meaning that the data provides evidence of a relationship between dependent and independent variables.

A confidence interval for \(\theta_{1}\) at \(1-\alpha\) confidence can be obtained as follows

\[ \begin{aligned} P(-t_{\alpha/2, n-2} < (\hat{\theta}\_{1} - \theta_{1})\sqrt{\frac{(n-2)S_{xx}}{SS_{R}}} < t_{\alpha/2, n-2}) = 1-\alpha\newline \text{Confidence Interval is} \quad \bigg(\hat{\theta}\_{1} - t_{\alpha/2, n-2}\sqrt{\frac{SS_{R}}{(n-2)S_{xx}}} < \theta_{1} < \hat{\theta}\_{1} + t_{\alpha/2, n-2}\sqrt{\frac{SS_{R}}{(n-2)S_{xx}}} \bigg) \end{aligned} \]

The hypothesis test for \(\theta_{0}\) can be done in the exact same manner as \(\theta_{1}\) by considering the following test statistic

\[ \begin{aligned} TS = (\hat{\theta}\_{1} - \theta_{1})\sqrt{\frac{n(n-2)S_{xx}}{(\sum_{i=1}^{n} x_{i}^{2})SS_{R}}} \sim t_{n-2} \end{aligned} \]