(OLS) How to get sample variance-covariance matrix of residuals?

**derksheng** · 07-16-2012 07:00 AM

Consider the OLS:

(1) y = a + b*x + e

Where e is the error term. I understand that the population variance-covariance matrix for the residuals is ee', but for the sample var-covar matrix do I need to do any d.f. adjustments?

**Dason** · 07-16-2012 08:42 AM

Are you assuming the covariance matrix looks like $\sigma^2 I_{nxn}$ (ie the observations have constant variance and are independent)?

If so and if you only have an intercept and a slope and you're using the residuals as your stand in for the errors (so instead of $e_i$ you use $r_i = y_i - (a + b*x_i)$ ) it is typical to use $\hat{\sigma}^2 = r'r/(n-2) = \frac{1}{n-2}\sum_{i=1}^nr_i^2$

Also note that to get the population covariance matrix you actually need E[ee'].

**derksheng** · 07-16-2012 09:14 AM

Very informative post. Thanks for that.

I'm actually working with heteroscedasticity and serial correlation.

**Dason** · 07-16-2012 09:25 AM

Are there a few parameters in the proposed covariance matrix? It's not really possible to just estimate the covariance matrix in general without more information.

**derksheng** · 07-16-2012 09:54 AM

I wasn't aware that it wasn't straight forward. I have the cross-sectional (1) $\gamma_i = F \beta + \epsilon$ . Both gamma and epsilon are stochastic.

F is a matrix of factors (the first column is 1, for the intercept).

Using "Portfolio Weighted Least Squares" from Chandra and Balachandran (1992)*, we have $\beta = w^T \gamma$ and $Cov(\beta) = w^T S w$ , where S is the estimated variance-covariance matrix computed from (1) using $\epsilon$ (but the authors don't provide details). $w^T = (F^T D^{-1} F)^{-1} F^T D^{-1}$ .
$D$ is the diagonal of $S$

So I'll need the var-covar of (1) to estimate this regression.

*More Powerful Portfolio Approaches to Regressing Abnormal Returns on Firm-SpecificVariables for Cross-Sectional
http://www.jstor.org/discover/10.230...id=56310633793

**Dason** · 07-16-2012 09:57 AM

I'm not on campus right now and can't access that article otherwise I'd take a look at it.

**derksheng** · 07-16-2012 10:02 AM

Oh thanks for your help.

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