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This question already has an answer here:

In the context of linear regression, I don't understand why you need to perform an F-test for the H0 that all parameters are zero, instead of just looking at all the t-tests for each parameter.

I understand that they are not the same, but I don't understand why. I sense that it has something to do with the covariance between the ols estimators (because the covariances are not used for the t-tests but all the estimated variance-covariance matrix is used for the F-test).

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marked as duplicate by Community May 27 '15 at 9:18

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The answer in one word: multicollinearity. If two predictors are correlated it might happen that both is insignificant itself (i.e. with $t$-test), but they are jointly significant (with $F$-test). The reason is that multicollinearity will result in the variables mutually increasing each other's standard error, thus giving rise to the insignificance with $t$-test. Nevertheless, together they're significant. The very same works for more than two predictors (or all of the predictors) of course.

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  • $\begingroup$ This looks like the best possible answer $\endgroup$ – Three Diag Feb 2 '17 at 18:26

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