Question

Consider the simple linear regression model and let e = y −y_hat, i = 1,...,n be the least-squares residuals, where y_hat = β_hat + β_hat * x the fitted values.

(a) Find the expected value of the residuals, E(ei).

(b) Find the variance of the fitted values, V ar(y_hat ). (Hint: Remember that y_bar i and β1_hat are uncorrelated.)

Answer 1

a)

Yhat = b0 + b1*X
Yhat = YBar + R*(Sy/Sx)*(X - XBar)

And, the Error (E) terms are expressed as:

E=(Y - Yhat)

SumE=Sum(Y - Yhat)

SumE=Sum(Y - YBar + R*(Sy/Sx)*(X - XBar) )

SumE=SumY - SumYBar + R*(Sy/Sx)*Sum(X - XBar)

SumE=N*YBar - N*YBar + 0 = 0,

because the summation of a constant N times is N times that constant (N*YBar) and Sum(X - XBar) = 0.

Thus, as indicated above, if the sum of the error terms are zero, so will the mean.

b)

var(yi^) = var(b0^) + xi^2

yi^ - ybar = b0^ + b1^ (xi - xbar)

var(y^) = var(b0^) + (xi-xbar)^2 var(b1^)

=