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Consider the following multiple linear regression model $$Y_i = \beta_0 + \beta_1X_{i1} + \dotsc + \beta_kX_{ik} + \varepsilon_i$$ where $\varepsilon_i$ is assumed to be normaly distrubuted with mean zero and the variance is $\sigma_{\varepsilon}$ for all $i$. Further $X_{ij}$ is not categorical for all $j = 1, \dotsc, k$.

Let's say that we want to perform an omnibus F-test. From what I've learnt, we test $$H_0: \beta_0 + \varepsilon_i$$ against $$H_1: \beta_0 + \beta_1X_{i1} + \dotsc + \beta_kX_{ik} + \varepsilon_i.$$ Assume that all of the estimated betas are not zero and we reject the omnibus null hypothesis. Would you conclude that not all betas are zero or all betas are not zero?

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The test is looking to test for significance "as a whole". An alternative (and in my opinion, better) way to represent these hypotheses is:

\begin{align} H_0 :\ &\ \beta_1 = \beta_2 = \ldots = \beta_k = 0, \text{for all } i \in \{1, 2, \ldots, k\}\\ H_1 :\ &\ \beta_i \neq 0,\text{for some } i\in \{1, 2, \ldots, k\}. \end{align}

That is, $H_1$ states that at least one of the $\beta_i$ ($i>0$) is non-zero. It could be just one of them, or it could be all of them - this test does not distinguish between one, all, or somewhere in between. It does not tell us which $\beta_i$ are driving significance.

We exclude $\beta_0$ from these hypotheses since testing $\beta_0 = 0$ would be testing if the model intercept is zero. This is a bad test to do since a change of units (e.g. Celsius to Kelvin) could change the statistical significance of $\beta_0$.

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  • $\begingroup$ So, even if the estimated slopes are not close to zero, we still conclude that not all $\beta_i$'s are zero because of the reason above? $\endgroup$
    – Vicky
    Commented Mar 12, 2021 at 9:10
  • $\begingroup$ Yes - and if they're not close to zero that is what you would want to do $\endgroup$
    – jcken
    Commented Mar 12, 2021 at 9:35

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