In a multiple linear regression analysis, (y_i , x_1i , x_2i ), i = 1, ... , n , are statistically independent and satisfy the model (M1) given by:

y = B₀ + B₁x₁ + B₂x₂ + ? ,

where the response variable y is continuous, the regressor vector X = (x₁, x₂)^T has mean vector (?_x1, ? _x2)^T and positive definite variance-covariance matrix ?_X , the random errors ?_i conditional on X_? are statistically independent and normally distributed with mean zero and variance ?² which does not depend on X .

After trying many multiple linear regression analyses based on the model M1 in Problem 1 above, a data analyst obtained the following statistics.

The regression sum of squares SS(B0) by excluding the two regressors x₁, x₂ .

The regression sums of squares SS(B0), SS(B₀, B₁), by excluding the regressor x₂.

The regression sums of squares SS(B0), SS(B₀, B₂), by excluding the regressor x₁.

The regression sums of squares SS(B0), SS(B₀, B₁, B₂); that is, include all the regressors.

a) Discuss with mathematical proof whether the hypothesis that no regressor has a nonzero effect on the response variable can be tested.

b) Discuss with mathematical proof whether SS(B₀, B₁, B₂) = SS(B₀, B₁) + SS(B₀, B₂) .

c) Discuss with mathematical proof whether S?(B₂ | B₀, B₁ ) = SS(B₂ | B₀ ) .

In a multiple linear regression analysis, {ypxli , x21), i = 1, ..., n , are statistically independent and satisfy the model (M1) given by: 3" :30 +31% +32% +5; where the response variable y is continuous, the regressor vector X = (x1, x2]r has mean vector (,uxl, p)' and positive definite variance-covariance matrix EX , the random errors a, conditional on X,- are statistically independent and normally distributed with mean zero and variance 02 which does not depend on X . After trying many multiple linear regression analyses based on the model M1 a data analyst obtained the following statistics. The regression sum of squares 55([30] by excluding the two regressors x1, x2 . The regression sums of squares 55(30),SS(30, {31), by excluding the regressor x2 . The regression sums of squares 55(30),SS(30, 52), by excluding the regressor x1 . The regression sums of squares 55(30),SS(30,51,,82); that is, include all the regressors. a} Discuss with mathematical proof whether the hypothesis that no regressor has a nonzero effect on the response variable can be tested. b) Discuss with mathematical proof whether 55(50, 51,32) : 55(n,,81) + 55(30, 5?) . (1] Discuss with mathematical proof whether 55(32 | 30,31) = 55(32 | ,8\")