4 No R required Recall that fitted values and residuals from

4. (No R required) Recall that fitted values and residuals from the fitted regression line are defined as Using these and equations (1) and (2). show the following equalities hold: Also, give a one-sentence interpretation of what the equalities (3) to (6) mean.

Solution

Y_i = a + bx_i + e_i where the last term represents the error terms.

We have, the sum of errors minimised, so,

(Y_i - y_h )² = e_i² = [ Y_i - ( a + bx_i ) ]²

Partially differentiating with respect to a,

Since, the amount of sum of squares is minimum, the partial derivative must be zero.

Thus,

-2 * [ summation of ( Y_i - ( a + bx_i ) ) = 0

summation of [Y_i - ( a + bx_i ) ] = 0

summation of [ e_i ] = 0   (Thus, 3 is proved)

4)

from 3,

summation of [Y_i - ( a + bx_i ) ] = 0

thus,

summation of [Y_i ] = summation of (a + bx_i ) = summation of y_h

Thus, 4 is proved.

5)

Partially differentiate w.r.t b.

We get,

-2 * [ summation of (Y_i - a - bx_i ) x_i ] = 0

But : (Y_i - a - bx_i ) = e_i

Thus, summation of (e_ix_i ) = 0

Thus, 5 is proved.

6)

Y_h = a + bx_i

Summation of ( Y_h * e_i )

= a * summation of (e_i ) + b * summation of (x_ie_i )

= 0 + 0   (from 3 and 5)

Hope this helps.

Note: here a = Beta₀

b = Beta₁

Ask in case of doubts