# Generalized Least Squares results

So, I've got the next problem:

Let $$Y\sim N_n(X\beta, \sigma^2 V)$$. Prove that, if $$\hat{\beta} = (X^{\prime}V^{-1}X)^{-1}X^{\prime}V^{-1}Y$$ then:

1. $$SSR = (Y-X\hat{\beta})^{\prime}V^{-1}(Y-X\hat{\beta}) \sim \sigma^{2}\chi^{2}_{(n-p)}$$.
2. $$SSR/(n-p)$$ is UMVUE for $$\sigma^{2}$$.
3. If $$\hat{Y} = X\hat{\beta} = PY$$ then $$P$$ is idempotent but not necessarily symmetric.
4. $$\hat{\beta}$$ is BLUE for $$\beta$$.

To note, the exercise didn't tell anything about the matrix $$V$$, I'm guessing $$V$$ is, at least, a semi-positive definite matrix, or even positive-definite since $$\sigma^{2}V$$ is a covariance matrix...

My attempt:

1. Reading Seber's Linear regression analysis I realize there's a theorem that says that if $$Y\sim N_n(\mu, \Sigma)$$ where $$\Sigma$$ is positive-definite, then $$(Y-\mu)^{\prime}\Sigma^{-1}(Y-\mu)\sim \chi^{2}_{n}$$.

Since $$Y-X\hat{\beta}\sim N_n(0,\sigma^{2}V)$$, and $$\Sigma = \sigma^2 V$$ positive-definite then $$SSR = (Y-X\hat{\beta})^{\prime}\Sigma^{-1}(Y-X\hat{\beta})\sim \chi^{2}_{(n)}$$, but the exercise says the distribution is $$\chi^2_{(n-p)}$$, that would be, if I'm not wrong, iff $$\operatorname{rank}(\Sigma)=n-p$$. If that's so, then how can I prove $$\operatorname{rank}(\Sigma)=n-p$$ ?

1. For this, I think the result is trivial once I have proved 1.
2. I'm totally lost at this, for the idempotent property, it's as simple as

$$P = X\hat{\beta} = X(X^\prime V^{-1}X)^{-1}X^{\prime}V^{-1}$$ $$P^{2} = X(X^\prime V^{-1}X)^{-1}X^{\prime}V^{-1} X(X^\prime V^{-1}X)^{-1}X^{\prime}V^{-1} = X(X^\prime V^{-1}X)^{-1}X^{\prime}V^{-1} = P.$$

But for proving that in general, $$P$$ is not symmetric I'm confused, should I give a counter example or something?

$$\mathbb{E}[\hat{\beta}] = \beta \mbox{ and } Var(\hat{\beta}) = \sigma^{2}(X^\prime V^{-1}X)^{-1}$$

Is that it to conclude $$\hat{\beta}$$ is BLUE?

Any help would be appreciated.

• This is what is called generalized least squares and $\hat\beta$ is the GLS estimator. Mar 17, 2020 at 7:32

In order that $$V^{-1}$$ exist is is necessary that $$V$$ have a rank equal to the number of its rows or its columns. In that context, positive-semi-definite entails positive-definite. And the statement $$Y\sim N_n(X\beta, \sigma^2 V)$$ makes sense only if $$V$$ is positive-semi-definite.
It is correct that $$Y-X\beta\sim\operatorname N(0,\sigma^2 V),$$ but it is not correct that $$Y-X\widehat\beta \sim\operatorname N(0,\sigma^2,V).$$ In fact, the variance of $$Y-\widehat\beta X$$ is a singular matrix of rank $$n-p.$$ Think about where $$\widehat\beta$$ comes from.
To prove that $$\text{SSR}$$ is UMVUE, you need to show that $$\text{SSR}$$ admits no unbiased estimators of zero, i.e. there is no function $$f$$ not depending on $$\sigma$$ for which $$\operatorname E(f(\text{SSR}))$$ remains equal to $$0$$ as $$\sigma>0$$ changes.
Proving that $$\widehat\beta$$ is BLUE for $$\beta$$ should not require the assumption of normality, but only the assumptions on the expected value (an $$n\times1$$ column vector) and the variance (an $$n\times n$$ matrix) of $$Y.$$ This is one whose details I've never gone through, as far as I recall. Maybe this one is worth a separate posted question.