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As given in the picture, question ask us to transform dependent variable, by premultiplying with a weighting matrix W, (which will change the regression coefficient into C'β) such that the expression (given in question) has a minimum mean squared error.

So, I solved part a) by solving for the brackets of E[(w′y−c′β)(w′y−c′β)′] and differentiating with respect to w, I obtain the expression as

$w′y=[c′ββ′x′(xββ′x′+σ^2.I)^{−1}]y$

My question is What does this espression signify or simply put what did we do/ what are we trying to achieve and why it is not practical to use above estimator? Any help or suggestion will be helpful.

PS: Is it impractical because $σ^2$ is unknown?

Expression can be rewritten as -

$w′y=[c′ββ′x′(E(yy') )^{−1}]y$

Because Xβ = E(Y|X) and Var(y) =$σ^2$I, By the formula

Var(Y) = E(YY') + E(Y|X)E(Y|X)'

Edit: Expressions for estimated β is given by (X'X)$^{-1}$X'Y where, define A= (X'X)$^{-1}$X'. This gives estimated β = AY. So, the expression W'Y be also denoting another expression for estimated β, an expression computed such that the discrepancy between the stylised expression for est_β ie W'Y and the weighted β ie C'β is minimum? Am I thinking on correct lines?

Question

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    $\begingroup$ Hi: I saw some version of this question earlier and still don't understand it. Is $\beta$ a vector and known ? If not, then what is $C^{\prime} \beta$, a new unknown scalar ? $\endgroup$ – mark leeds Sep 8 at 13:56
  • $\begingroup$ @markleeds Hi, $/beta$ is estimated using OLS and since we have transformed Y by a matrix W, this will change the beta coefficients denoted by C'$/beta$. $\endgroup$ – Elina Gilbert Sep 9 at 2:14
  • $\begingroup$ Hi: So, $\beta$ is a known vector and $W'y$ is still an known vector since $W$ is known also ? $C$ is the only unknown ? $\endgroup$ – mark leeds Sep 9 at 3:11
  • $\begingroup$ @markleeds W is not known. We are trying to transform y (through premultiplying by W) in such a way that it minimizes the expression given in part a). C is a weighted vector and is assumed to be given. So, the question is if we have measured beta as a weighted measure then how should we transform Y so that it minimizes the expression. $\endgroup$ – Elina Gilbert Sep 9 at 4:30
  • $\begingroup$ Hi: I'm sorry for the confusion. That clarifies some of it but can you give the dimension of every variable in the model. I'm confused between what's a vector and what's a matrix. Thanks. $\endgroup$ – mark leeds Sep 9 at 19:52

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