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The F-statistic formula is:

$$ F=\frac{(TSS-RSS)/p}{RSS/(n-p-1)} $$

Where $TSS$ is total sum of squares and is equal to $\sum_{i=1}^n(y_i-\bar{y})^2$ and $RSS$ is the residual sum of squares and is equal to $\sum_{i=1}^2(y_i-\hat{y}_i)^2$.

$p$ is the number of predictors in the regression and $n$ is the sample size.

This F-statistic is testing the null hypothesis that:

$$ H_0:\beta_1=\beta_2=\dots=\beta_p=0 $$

Sometimes we want to test that a particular subset of $q$ coefficients are 0. The book I am learning from states the following. The null hypothesis is:

$$ H_0:\beta_{p-q+1} = \beta_{p-q+2} = \beta_p = 0 $$

where for convenience the variables chosen for omission are at the end of the list.

We fit a second model that uses all the variables except those last $q$. The residual sum of squares for that model is referred to as $RSS_0$. The appropriate F-statistic is then:

$$ F=\frac{(RSS_0-RSS)/q}{RSS/(n-p-1)} $$

1) What exactly do they mean by "the variables chosen for omission are at the end of the list"? I don't understand how $\beta_{p-q+1}$ and $\beta_{p-q+2}$ specify which coefficients to remove (i.e. the $q$ coefficients).

2) What happens to $TSS$ in the numerator of the second F-statistic equation?

3) How does this report the partial effect of adding a variable to a model?

4) Why does the F-test of a model that leaves out coefficient $j$ while leaving all others in (i.e. $q=1$) exactly equal the $\text{p-value}$ and $\text{t-statistic}^2$ of coefficient $j$ in a model that leaves all coefficients in?

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1) As said in the text, for the convenience (of writing). They can be any $\beta$s that you want test.

2) You can think in this way: for the first model, you have $TSS - RSS$; for the second model, you have $TSS - RSS_0$. So the deference is $RSS - RSS_0$

3) Because $RSS - RSS_0$ is the contribution of $q$ variables after other $p-q$ variables already in the model.

4) A) When $q=1$, both the F test described here and t-test you described test the same null hypothesis $H_0: \beta_j = 0$. B) Also you can check the relationship between t distribution and F distribution, you will find under certain conditions, $t^2 = F$. C) Can prove that square of t statistics = F statistics under this situation and you may find it from internet.

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