Re: autocorrelation as a predicor in multiple regression models

From: Ray Koopman (koopman_at_sfu.ca)
Date: 02/23/05 

Date: 23 Feb 2005 11:13:07 -0800

Mike Plichta wrote:
> I have got time series data (900 sec with a sampling rate of 0.1;
> resulting in 9000 time points) recorded with a neuroimaging
> technique). Furthermore, there is a stimulation every 15 sec.
> I assume an idealized model of brain response related to the
> stimulations. What i have done is to calculate a multiple regression
> with the idealized model(-variables) as predictors and the time
> series as criterion. The model fits quite good (R=.57) and i was
> very happy until i inspected the residuals. The residuals are
> highly autocorrelated and so the assumptions of the regression model
> are violated and the estimates are biased.

Correlations among the errors will change the covariance matrix of
the estimated weights, but will not bias them. If y = Xb + e, where
E(e) = 0 and E(ee') = vC, where v is a scalar and C is symmetric
positive definite, then the Gauss-Markov estimate of b is (X'WX)^X'Wy,
where ' denotes transposition, ^ denotes inversion, and W = C^.