1. In a typical multiple linear regression model where x1 and x2 are non-random

regressors, the expected value of the response variable y given x1 and x2 is denoted

by E(y | 2,, X2). Build a multiple linear regression model for E (y | *,, *2) such that the

value of E(y | x1, X2) may change as the value of x2 changes but the change in the

value of E(y | X1, X2) may differ in the value of x1 . How can such a potential difference

be tested and estimated statistically?

2. For any multiple linear regression model, the total sum of squares can be decomposed

into the sum of squares contributed solely by the predictor vector and the sum of

squares contributed solely by the residual vector. To assess the importance of a set of

the regressors, they can be taken jointly by regressing the response variable on this set

of regressors, either including or excluding other regressors.

a) Discuss, using mathematical proof in vector or matrix expressions, how this set of

regressors can be assessed by use of the sums of squares of the residual vectors.

Will including or excluding regressors change the conclusion in a)? Why or why not?

Provide proof for your assertions.