Question

Weighted least squares is a modification of standard regression analysis that may be used for a set of data when the assumption of variance homogeneity does not hold. (Assume the responses are independent.) If the ith response is an average of mi equally variable observations, then Var(Vi) ynx1-Xnxp ßpx1 + En x1, where E(c) _ 0, Cov(c)-σ2V, and In this case, we have the model 1112 0 The fixed and known positive definite matrix Vnxn has rank n. The weighted least squares estimator of ß is given by (a) Derive the expectation of ßw, E[3W1 (b) Derive the covariance matrix of Bw, Cov(Bw) (c) Find the exact distribution of Bw. If it is necessary to make any reasonable further assumptions in order to find the distribution of Bw, provide them

Answer 1

$(a)~E(\hat{\boldsymbol{\beta}}_{W})=({\bf X}^\prime{\bf V}^{-1}{\bf X})^{-1}{\bf X}^\prime{\bf V}^{-1}E({\bf y})=({\bf X}^\prime{\bf V}^{-1}{\bf X})^{-1}{\bf X}^\prime{\bf V}^{-1}{\bf X}\boldsymbol{\beta}=\boldsymbol{\beta}\\ (b)~Cov(\hat{\boldsymbol{\beta}}_{W})=({\bf X}^\prime{\bf V}^{-1}{\bf X})^{-1}{\bf X}^\prime{\bf V}^{-1}Cov({\bf y}){\bf V}^{-1}{\bf X}({\bf X}^\prime{\bf V}^{-1}{\bf X})^{-1}\\ =({\bf X}^\prime{\bf V}^{-1}{\bf X})^{-1}{\bf X}^\prime{\bf V}^{-1}{\bf V}{\bf V}^{-1}{\bf X}({\bf X}^\prime{\bf V}^{-1}{\bf X})^{-1}\sigma^2;\\ ~since~Cov({\bf y})=Cov(\boldsymbol{\epsilon })=\sigma^2{\bf V}\\ =({\bf X}^\prime{\bf V}^{-1}{\bf X})^{-1}\sigma^2\\ (c)~Assume,~\boldsymbol{\epsilon }\sim N_n({\bf 0},~{\bf V}\sigma^2).\\ Then~{\bf y}\sim N_n({\bf X}\boldsymbol{\beta},~{\bf V}\sigma^2)~provided~{\bf X}~is~non-stochastic.\\ hence~\hat{\boldsymbol{\beta}}_W\sim N_n(\boldsymbol{\beta},~({\bf X}^\prime{\bf V}^{-1}{\bf X})^{-1}\sigma^2)$