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I have an F-statistic, $F(4,10)$, my constant and 4 Coefficients $\beta_2 , \beta_3 , \beta_4$ and $\beta_5$

I already know that the (in this case) 10 reflects the number of obsverations. But what exactly does the 4 tell me? Is it the number of coefficients to be tested? Also what does the outcome - $F(4,10) = 9.59$ tell me?

Thank you in advance.

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With $F$-test in multiple linear regression, you test the $H_0$ that a "trivial" model $$y = \beta_1 + u$$ (i.e. dependent variable as a function of intercept and random error only) is equally good in predicting variability in the dependent variable as your "main" model $$y = \beta_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 + \beta_5 x_5 + u$$.

Basically, you test the significance of the model as a whole. In $F(k, (n-k-1))$, parameter $k$ is the number of parameters restricted (4), and $n-k-1$ is the degrees of freedom in multiple linear regression - i.e. you are estimating your model with 15 observations only, which is far too low for 5 parameter-estimation.

Also, the critical value for $F(4,10)=2.6$ (at 5% sig. level), so you are fine (reject $H_0$ in favour of significant model). However, let me repeat that estimating 5 coeffs using 15 observations is generally not a good idea.

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    $\begingroup$ You'd largely be correct in the social sciences about 5 parameters with 15 observations. But if the data are from a field with high measurement precision and highly controlled cases so low random variation, the methods are fine. $\endgroup$ – Heteroskedastic Jim Nov 8 '18 at 12:48

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