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I was running a regression with 1000 observations and 7 explanatory variables (with a constant term included as a "variable"). So, the population model initially stated was:

$Y_{i} = \beta_{1} + \beta_{2}X_{2i} + ....+ \beta_{6}X_{6i} + u_{i}$

So the sample model is:

$\hat{Y_{i}} = b_{1} + b_{2}X_{2i} + ....+ b_{6}X_{6i}$

(Estimated via OLS, meaning that I used the Gauss-Markov assumptions: https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem )

After performing an individually significant test over $\beta_{6}$ i got the result that I can not reject the Null Hypothesis of $\beta_{6} = 0$.

But, if I make a F-test, of joint significance, I get that I can reject the Null Hypothesis of $\beta_{6} = 0$.

What's the intuition? (and maybe the mathematics behind)

How can it be that one test contradicts the other?

Thanks!

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    $\begingroup$ Perhaps you are misinterpreting the F test. The usual F test in linear regression assesses the overall usefulness of the model which just means that at least 1 of the predictors has a significant $\beta$ and not necessarily $\beta_6$. $\endgroup$ – Michael R. Chernick Jun 14 '17 at 19:45
  • $\begingroup$ @Michael Although the thrust of that comment is correct, how you expressed it is not, because it is possible for all coefficients to be insignificant (individually) while the overall regression is nevertheless significant. See stats.stackexchange.com/questions/3549. $\endgroup$ – whuber Jun 14 '17 at 23:04
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    $\begingroup$ @whuber. You make a good point. I guess I have given a situation where the two tests could lead to seemingly conflicting results. But the cumulative effect of all six parameters could make the model significant even when none are statistically significant individually. $\endgroup$ – Michael R. Chernick Jun 14 '17 at 23:11
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    $\begingroup$ @Michael, very well put: that nicely summarizes this thread and the duplicate. $\endgroup$ – whuber Jun 15 '17 at 1:46
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No, you cannot reject the null nypothesis $\beta_6=0$ using the F-test. Consider a situation in which the coefficients are correlated. The F-test takes account of this correlation while an individual t-test does not. Hence, you cannot infer from the F-test that $\beta_6=0$. If you can reject the F-test null hypothesis then you only know that $(\beta_1,\dots,\beta_6)\neq(\underbrace{0,\dots,0}_{6\times})$. But it may still be the case that $$(\beta_1,\beta_2,\beta_3,\beta_4,\beta_5,\beta_6) = (0,0,0,-123,45,0)$$ or $$(\beta_1,\beta_2,\beta_3,\beta_4,\beta_5,\beta_6) = (0,0,0,0,0,.4)$$ (i.e. $\beta_6=0.4\neq0$) -- and that eventhough you rejected the null hypothesis of the F-test!

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