Proove that c The squared correlation r2 can be interpreted

Proove that:

(c) The squared correlation r^2 can be interpreted as \'\'the fraction of the variability in y explained by the regression,\'\' that is,

Solution

Var (y^) = Summation (i=1....n) (y_i^{^} - y-bar)

Here y^ is nothing but the estimated y and y-bar is the mean of the regressand; so any deviation of the y value from the mean of the regressand is attributable to the regressors X_is

(y-y^)² however is the residual sum of squares or squared Error deviation attributable to the existence of the stochastic variables in the regression model

thus, (y- y^)² = explained variation² + unexplained variation² due to error term= total variation

therefore,

r^{2 =} explained sum of squares = ESS= summation (i=1....n)(y^ - y-bar)²

   Total sum of squares = TSS= Summation (i=1...n)( y-y-bar)²