Question

Consider the general linear model ? = ?? + ?. Use matrix algebra to show that ?̂ is an unbiased estimator of ?.

the last ? has bar

Answer 1

The general linear model is:

is the least square estimate of

The least squre estimate is the one which minimizes the residual errors

Let

   (X' is transpose of X)

   ((AB)' = B'A')

Since, we want the least square esimate of , it should satisfy the following:

Consider equation (2)

  

(Taking tanspose and cancelling the factor 2)

(Premultiplying both sides by )

Hence,

We have,

for all

Now, would be an unbiased estimator of if

We have,

(E(Y) = X)

Hence, is an unbiased estimator of . (Proved)