# Proof that $F$-statistic of dropping single coefficient is square of $z$-score in linear regression

Show that the $F$ statistic $\left(F=\frac{(\text{RSS}_1-\text{RSS}_0)/(p_1-p_0)}{\text{RSS}_1/(N-p_1-1)}\right)$ for dropping a single coefficient from a model is equal to the square of the corresponding $z$-score $z_j=\frac{\hat{\beta}_j}{\hat{\sigma}\sqrt{v_j}}$. Here, $\text{RSS}_{1,2}$ denotes the residual sum of squares (the squared errors you make by using the linear model). $v_j$ is the $j$th diagonal element of $(X^{T}X)^{-1}$ and $p_{1,2}$ are the numbers of parameters in the corresponding regression models.

This is exercise 3.1 of the book "Elements of Statistical Learning". Here the link to the book for further reference: http://www-stat.stanford.edu/~tibs/ElemStatLearn/

I have tried simply plugging things into the $F$-fraction but i can't simplify the diffenece of the residual sum of squares because i dont know if the parameters $\beta$ of the $2$ models are equal in general. Further more i have no idea how tho get the $j$th diagonal element of a matrix into this formula! I think I lack some deeper understanding on my part. Generally i am not so bad in statistics, especially linear models but somehow i still havent found my way into this book.

It seems that this excercise needs some more reasoning than simple algebraic calculations.

Help is appreciated

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