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In a simple linear regression
$SSR = \sum(\hat{y} - \bar{y})^2$
$SSXY = \sum(x - \bar{x})(y-\bar{y})$
$SSX = \sum(x - \bar{x})^2$
$SSR$ can be computed by dividing $SSXY^2$ by $SSX$. Namely, $SSR = SSXY^2/SSX$
Could someone illustrate mathematically how this is obtained? Thanks!!
From the answers at Is there a way to use the covariance matrix to find coefficients for multiple regression? we know the least squares equations for the estimated coefficient $\widehat\beta$ are
$$SS_X \widehat\beta = SS_{XY}$$
where the predicted values are given by
$$\widehat{y}_i=x_i\widehat{\beta}.$$
Squaring both sides and summing, recognizing that $\Sigma_i x_i^2=SS_X$, yields
$$SS_R=SS_X \widehat\beta^2.$$
Plugging in the solution $\widehat\beta^2 = \left(SS_{XY} / SS_X\right)^2$ from the first equation turns this into
$$SS_R=SS_X \left(SS_{XY} / SS_X\right)^2 =SS_{XY}^2/SS_X.$$
R, so I basically did y_hat %*% y_hat)
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