I am not working on Hayashi, econometrics page 21, Influential analysis part.
Let's consider the classiacal regression model. Then, accroding to the book, the following holds: $$b^{(i)}-b=-(\frac{1}{1-p_i})(X^tX)^{-1}x_ie_i$$ where
$b^{(i)}$ is a $k\times 1$ vector and the OLS estimate when the $i^{th}$ observation is omitted.
$b=(X^tX)^{-1}X^ty$ is a $k\times 1$ vector and the OLS estimate of all observation.
$X=\begin{bmatrix} & {x_1}^t \\ & {x_2}^t \\ & ...& \\ &{x_n}^t& \end{bmatrix}$ is the regressor $n\times k$ matrix of rank $k$, where $x_i, 1\le i\le n$ is a real $k-$vector. (presuming $n\le k$)
$e_i=y_i-{x_i}^tb, 1\le i \le n$ is the OLS residual of $i^{th}$ observation.
$p_i$ is the $i^{th}$ diagonal element of the projectin matrix $P$ onto $image(X)$, i.e., $P=X(X^tX)^{-1}X^t$.
I tried to solve it through a direct calculation but soon realized that it is nearly impossible to do so due to the existence of the inverse term. Is there any good trick? Or any reference to look for the proof of the result? I'd really appreciate any help. Thanks!