10. Consider the linear regression model with assumptions (i) to (iii) and (iv*) with V = k? I, + V1 +V2 , where k e R, Vị is symmetric nonnegative definite with C(V1) C C(X), and V2 is symmetric nonnegative definite with C(V2) C C(X)-. (a) Explain why V = (1– 0)I,n + gl,1, is a special case of the above matrix V. (b) Show that ordinary and generalized least squares estimator coincide.

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10. Consider the linear regression model with assumptions (i) to (iii) and (iv*) with
V = kIn + Vị+V2 ,
1
where k e R, Vị is symmetric nonnegative definite with C(V1) C C(X),
and V2 is symmetric nonnegative definite with C(V2) C C(X)-.
(a) Explain why V = (1– e)In+ gl,1, is a special case of the above matrix V.
(b) Show that ordinary and generalized least squares estimator coincide.
Assumptions
(i) X is a non-stochastic n x p matrix with p < n;
(ii) the matrix X has rank p, i.e. X is of full column rank;
(iii) the elements of the n x1 vector y are observable random vectors;
(iv*) the elements of the n x 1 vector e are non-observable random vari-
ables such that E(e) = 0 and Cov(e) = o?V, where V is a known
n x n symmetric positive definite matrix and o? > 0 is an unknown parameter.
Transcribed Image Text:10. Consider the linear regression model with assumptions (i) to (iii) and (iv*) with V = kIn + Vị+V2 , 1 where k e R, Vị is symmetric nonnegative definite with C(V1) C C(X), and V2 is symmetric nonnegative definite with C(V2) C C(X)-. (a) Explain why V = (1– e)In+ gl,1, is a special case of the above matrix V. (b) Show that ordinary and generalized least squares estimator coincide. Assumptions (i) X is a non-stochastic n x p matrix with p < n; (ii) the matrix X has rank p, i.e. X is of full column rank; (iii) the elements of the n x1 vector y are observable random vectors; (iv*) the elements of the n x 1 vector e are non-observable random vari- ables such that E(e) = 0 and Cov(e) = o?V, where V is a known n x n symmetric positive definite matrix and o? > 0 is an unknown parameter.
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