22
Dec 18

## Application: distribution of sigma squared estimator

For the formulation of multiple regression and classical conditions on its elements see Application: estimating sigma squared. There we proved unbiasedness of the OLS estimator of $\sigma^2.$ Here we do more: we characterize its distribution and obtain unbiasedness as a corollary.

### Preliminaries

We need a summary of what we know about the residual $r=y-\hat{y}$ and the projector $Q=I-P$ where $P=X^T(X^TX)^{-1}X^T:$

(1) $\Vert r\Vert^2=e^TQe.$

$P$ has $k$ unities and $n-k$ zeros on the diagonal of its diagonal representation, where $k$ is the number of regressors. With $Q$ it's the opposite: it has $n-k$ unities and $k$ zeros on the diagonal of its diagonal representation. We can always assume that the unities come first, so in the diagonal representation

(2) $Q=UDU^{-1}$

the matrix $U$ is orthogonal and $D$ can be written as

(3) $D=\left(\begin{array}{cc}I_{n-k}&0\\0&0\end{array}\right)$

where $I_{n-k}$ is an identity matrix and the zeros are zero matrices of compatible dimensions.

### Characterization of the distribution of $s^2$

Exercise 1. Suppose the error vector $e$ is normal: $e\sim N(0,\sigma^2I).$ Prove that the vector $\delta =U^{-1}e/\sigma$ is standard normal.

Proof. By the properties of orthogonal matrices

$Var(\delta)=E\delta\delta^T=U^{-1}Eee^TU/\sigma^2=U^{-1}U=I.$

This, together with the equation $E\delta =0$, proves that $\delta$ is standard normal.

Exercise 2. Prove that $\Vert r\Vert^2/\sigma^2$ is distributed as $\chi _{n-k}^2.$

Proof. From (1) and (2) we have

$\Vert r\Vert^2/\sigma^2=e^TUDU^{-1}e/\sigma^2=(U^{-1}e)^TD(U^{-1}e)/\sigma^2=\delta^TD\delta.$

Now (3) shows that $\Vert r\Vert^2/\sigma^2=\sum_{i=1}^{n-k}\delta_i^2$ which is the definition of $\chi _{n-k}^2.$

Exercise 3. Find the mean and variance of $s^2=\Vert r\Vert^2/(n-k)=\sigma^2\chi _{n-k}^2/(n-k).$

Solution. From Exercise 2 we obtain the result proved earlier in a different way:

$Es^2=\sigma^2E\chi _{n-k}^2/(n-k)=\sigma^2.$

Further, using the variance of a standard normal

$Var(s^2)=\frac{\sigma^4}{(n-k)^2}\sum_{i=1}^{n-k}Var(\delta_i^2)=\frac{2\sigma^4}{n-k}.$