22
Dec 18

Application: distribution of sigma squared estimator




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}.

 

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