16.5 Hypothesis Testing on Parameters

The $t$ tests are used to conduct hypothesis tests on the regression coefficients obtained in simple linear regression.

$$H_0: \beta_1 = 0$$ $$H_1: \beta_1 \neq 0$$

The test statistic used for this test is:

$$t_{stat} = \dfrac{\hat{\beta_1}}{se(\hat{\beta_1})}$$

where $\beta^1$ is the least square estimate of $\beta_1$, and $se(\hat{\beta_1})$ is its standard error. The value of $se(\hat{\beta_1})$can be calculated as follows:

$$se(\hat{\beta}_1)= \sqrt{\frac{\frac{\displaystyle \sum_{i=1}^n e_i^2}{n-2}}{\displaystyle \sum_{i=1}^n (x_i-\bar{x})^2}}$$

The test statistic, $T_0$, follows a $t$ distribution with $(n−2)$ degrees of freedom, where $n$ is the total number of observations. The null hypothesis, $H_0$, is not rejected if the calculated value of the test statistic $(t_{stat})$ is such that:

$$-t_{\alpha/2,n-2}\lt T_0\lt t_{\alpha/2,n-2}$$

where $t_{\alpha/2,n-2}$ and $-t_{\alpha/2,n-2}$ are the critical values for the two-sided hypothesis. $t_{\alpha/2,n-2}$ is the percentile of the $t$ distribution corresponding to a cumulative probability of $(1−\alpha/2)$ and $\alpha$ is the significance level.

The test indicates if the fitted regression model is of value in explaining variations in the observations or if you are trying to impose a regression model when no true relationship exists between $x$ and $y$. Failure to reject $H_0: \beta_1 = 0$ implies that no linear relationship exists between $x$ and $y$.