Often called the global test. Test the model for significancy using an $F$ test

Motivation

To test if the model is useful in explaining the dependent variable, we can use t-test on each $\beta_i$ of the model.

However, this method is generally not recommended, because it for each t-test, the probability of getting a Type I error increases.

To properly test the utility of a multiple linear regression model, we will need to do a global test (test that encompasses all the $\beta$ parameters)

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Warning

A rejection in the $H_0$ of the test means that the model is “statistically useful”. However, this does not mean the “best”.

Instead, there might be other models which is more reliable.

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Assumptions

Assumptions for the error component

Hypotheses

• $H_0 : \beta_1 = \beta_2 = \dots = \beta_p =0$
• $H_1 :$ Atleast one $\beta_i \neq 0$

Test statistic

$$ \begin{align*} F &= \frac{(SS_{yy} - SSE)/p}{SSE/(n-(p+1))}\\ &= \frac{MS(\text{Model})} {MSE} \end{align*} $$

Critical region