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In multiple linear regression, let us say that the F test shows that the model is significant. But the t-tests for beta values does not say that the beta values are non-zero. What can we conclude in such a situation?

Does the fact that the tests for the beta values failed affect the fact that the model is significant?

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You passed the junk regression (F) test but not t tests -- it is the first sign for multicollinearity problem look here for instance stats.stackexchange.com/questions/4099/… – Dmitrij Celov Mar 15 '11 at 15:07
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@chl has nicely answered your question in his reply at stats.stackexchange.com/questions/8237/… : see the second paragraph. – whuber♦ Mar 15 '11 at 15:38
Could you please confirm that you're also the asker for Logic behind the ANOVA F-test in simple linear regression so that we can merge your two accounts. (You will still need to register, see the FAQ.) – chl♦ Mar 15 '11 at 20:39
@CantTell Please register your account and accept some answers. – mbq♦ Jun 15 '11 at 22:10

2 Answers

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One way that this could happen is to put two highly correlated predictors into your model. Because they are highly correlated, each of their coefficients will have relatively wide standard errors. As a result, these coefficients may not be statistically significant separately. Taken together, they may be a strong predictor of the outcome.

The covariance between the coefficient estimates would be pretty negative (increasing the estimate of one of the coefficients leads to a strong decrease in the estimate of the other coefficient because the two are highly linearly related), cancelling out the high variances for the estimators separately. As a result, your model has predictive power and your regressors are important, but their colinearity obscures this fact by creating large (though accurate) standard errors.

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Read the section on tests of joint significance here: http://www.people.fas.harvard.edu/~blackwel/ftests.pdf

Your F-test is of a different hypothesis than the t test, that all beta = 0, rather than some beta = 0.

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