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In a regression problem, with $y=X\theta+\epsilon$ and $X$ is an $n$ by $p$ matrix the ‘weighted least squares estimate is the minimizer $\theta^{*}$ of $f(\theta)=\sum_{i=1}^{n}\omega_{i}(y_i-x_i^{'}\theta)^2$ for suitable positive ‘weights’ $\omega_i$, where $x_i^{'}$ are the rows of $X$. It can be shown that if $X$ has full rank, then the minimizer $\theta^{*}=(X'WX)^{-1}X'Wy$ where $W=\textrm{diag}(\omega_1,\omega_2,...,\omega_n)$.

Now suppose that the observations are independent, so that the covariance matrix $\Sigma$ of y is diagonal, but that the diagonal elements: $\sigma_1^{2},\sigma_2^{2},...,\sigma_n^{2}$ are not all equal. We say the observations are ‘heteroscedastic’.

Question 1: Show that the covariance matrix of $\theta'$ is given by: $$\operatorname{cov}(\theta^{*})=(X'WX)^{-1}X^{'}W\Sigma WX(X'WX)^{-1}$$ and that the variance of a linear combination $a\theta^{*}$ is: $$\operatorname{Var}[a'\theta^{*}]=a'(X'WX)^{-1}X^{'}W\Sigma WX(X'WX)^{-1}a$$

Question 2:

Show that, for any such linear combination, $\operatorname{Var}[a'\theta^{*}]$ is minimized by the choice of the weights: $\omega_i=1/\sigma_i^2$(i.e, when $W=\Sigma^{-1}$).

I solved question 1, but I have no idea about question 2. I appreciate any help. Thanks!

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3 Answers 3

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It's early and i'm on the bus...Maybe i'm sleeping, but i tried to solve it:

For $W= S:= \Sigma^{-1}, $

$$a^\top (X^\top SX)^{-1}X^\top S S^{-1} S X(X^\top SX)^{-1} a $$

Where $S^{-1} S = \textrm{diag}(1)$

$$a^\top (X^\top SX)^{-1} X^\top SX(X^\top SX)^{-1}a = a (X^\top SX)^{-1}a .$$

where $(X^\top SX)^{-1} X^\top SX= \textrm{diag}(1)~~~a (X^\top SX)^{-1}a,$ where $(X^\top SX)^{-1}$ is homoscedastic covariance matrix of Theta vector and it is minimal cov matrix for the gauss-markov theorem.

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$\require{cancel}$ $$\operatorname{Cov}(\theta^\ast)=E[{\theta^\ast}{\theta^\ast}']-E[{\theta^\ast}]E[{\theta^\ast}]'$$

$$E[{\theta^\ast}]E[{\theta^\ast}]'=\theta \theta'$$

$$y = X\theta+u$$

$$E[{\theta^\ast}{\theta^\ast}']'=E[((X'WX)^{−1}X'Wy)((X'WX)^{−1}X'Wy)']=\\ E[(X'WX)^{−1}X'Wyy'WX(X'WX)^{−1}]=E[(X'WX)^{−1}X'W(X\theta+u)(\theta'X'+u')WX(X'WX)^{−1}]\\ =E[(X'WX)^{−1}X'W(X\theta\theta'X+u\theta'X+X\theta u'+uu')WX(X'WX)^{−1}]= E[\cancel{(X'WX)^{−1}(X'WX)}\theta\theta'\cancel{(XWX)(X'WX)^{−1}}]+\\ \underbrace{E[(X'WX)^{−1}X'W(u\theta'X)WX(X'WX)^{−1}]}_{=0}+\\ \underbrace{E[(X'WX)^{−1}X'W(X\theta u')WX(X'WX)^{−1}]}_{=0}+\\ E[(X'WX)^{−1}X'W(uu')WX(X'WX)^{−1}]\\ =\theta \theta' +(X'WX)^{−1}X'W\overbrace{\Sigma}^{E[uu']} WX(X'WX)^{−1} $$

Hence

$$\operatorname{Cov}(\theta^\ast)=\underbrace{\cancel{\theta \theta'} +(X'WX)^{−1}X'W\Sigma WX(X'WX)^{−1}}_{E[{\theta^\ast}{\theta^\ast}']}- \underbrace{\cancel{\theta \theta'}}_{E[{\theta^\ast}]E[{\theta^\ast}]'}\\ =(X'WX)^{−1}X'W\Sigma WX(X'WX)^{−1}$$

Then

$$\operatorname{Var}(a'\theta^\ast)=E[a'\theta^\ast{\theta^\ast}'a]-E[a'\theta^\ast]E[{\theta^\ast}'a]\\ =a'E[\theta^\ast{\theta^\ast}]a - a'E[\theta^\ast]E[\theta^\ast]'a\\ = a'\left(E[\theta^\ast{\theta^\ast}] - E[\theta^\ast]E[\theta^\ast]'\right)a\\ =a'\operatorname{Cov}(\theta^\ast)a$$


If you can use the results of the unweighted version, the result can be directly obtained by recognizing a linear transformation of both $X$ and $y$ that generalizes the weighted linear regression estimator.

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There are two methods of showing $\operatorname{Var}(a'\theta^*) \geq \operatorname{Var}(a'\hat{\theta})$, where $\hat{\theta} = (X'\Sigma^{-1}X)^{-1}X'\Sigma^{-1}y$ is the weighted-least squares estimate of $\theta$ with the "theoretical" weights $\Sigma$, and $\theta^* = (X'WX)^{-1}X'Wy$ is an arbitrary weighted-least squares estimate. The first method links the problem to an OLS problem and then applies the Gauss-Markov theorem (as @Danilo attempted but he did not clearly finish the argument). The second method is a brutal-force evaluation of the difference $\operatorname{Var}(a'\theta^*) - \operatorname{Var}(a'\hat{\theta})$.

Method 1

Rewrite the linear model $y = X\theta + \epsilon$ as $y_0 = X_0\theta + u$, where $y_0 = \Sigma^{-1/2}y$, $X_0 = \Sigma^{-1/2}X$, $u = \Sigma^{-1/2}\epsilon$. The latter representation then corresponds to an OLS problem as the error $u$ is homoscedastic in view of $\operatorname{Var}(u) = \Sigma^{-1/2}\Sigma\Sigma^{-1/2} = I_{(n)}$. The Gauss-Markov theorem then applies: since $a'\theta^* = a'(X'WX)^{-1}X'Wy = a'(X'WX)^{-1}X'W\Sigma^{1/2}y_0$ is an unbiased linear estimate of $a'\theta$ (i.e., $E[a'\theta^*] = a'\theta$), it follows that \begin{align} \operatorname{Var}(a'\theta^*) \geq \operatorname{Var}(a'(X_0'X_0)^{-1}X_0'y_0) = \operatorname{Var}(a'\hat{\theta}). \end{align} This completes the proof.

Method 2

Since $\operatorname{Var}(a'\theta^*) - \operatorname{Var}(a'\hat{\theta}) = a'((X'WX)^{-1}X'W\Sigma WX(X'WX)^{-1} - (X'\Sigma^{-1}X)^{-1})a$, if we can show that the matrix $(X'WX)^{-1}X'W\Sigma WX(X'WX)^{-1} - (X'\Sigma^{-1}X)^{-1} \geq 0$ (i.e., the difference is a positive semi-definite matirx), the result then follows. To this end, note that \begin{align} & (X'WX)^{-1}X'W\Sigma WX(X'WX)^{-1} - (X'\Sigma^{-1}X)^{-1} \\ =& (X'WX)^{-1}[X'W\Sigma WX - (X'WX)(X'\Sigma^{-1}X)^{-1}(X'WX)](X'WX)^{-1} \\ =& (X'WX)^{-1}X'W[\Sigma - X(X'\Sigma^{-1}X)^{-1}X']WX(X'WX)^{-1} \\ =& (X'WX)^{-1}X'W\Sigma^{1/2}[I_{(n)} - \Sigma^{-1/2}X(X'\Sigma^{-1}X)^{-1}(\Sigma^{-1/2}X)']\Sigma^{1/2}WX(X'WX)^{-1}, \end{align} hence it suffices to prove $I_{(n)} - \Sigma^{-1/2}X(X'\Sigma^{-1}X)^{-1}(\Sigma^{-1/2}X)' \geq 0$, which follows from the matrix ("hat matrix") $H := \Sigma^{-1/2}X(X'\Sigma^{-1}X)^{-1}(\Sigma^{-1/2}X)'$ is symmetric and idempotent. This completes the proof.

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