autocorrelation as a predicor in multiple regression models

From: Mike Plichta (michael.m.plichta_at_web.de)
Date: 02/23/05 

Date: Wed, 23 Feb 2005 13:36:31 +0000 (UTC)

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 serie 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.

My idea was to choose a 2-Stage-Strategy:
        
1) Calculate the regression model:

        y= a + x1*ß + x2*ß + e
(where a = const.; xi = i-th predictor; ß = beta-weights; e = error)

2) than, take the resulting unstandardized residuals (res) of the
first step

3) adding them (with a time lag of 1) to the regression model:

        y= a + x1*ß + x2*ß + res(t-1) + e
(where res = residuals of step 1 with a time-lag of 1)

to obtain uncorrelated errors (and prewhiten the data).

In several publications i found a (autoregressive) paramter which
should be multiplicated with the residuals and the other parts of the
regression model - i do not know why and how to calculate this
parameter practically (i guess it is just the
autocorrelation-coefficient?!).

Would it be possible to give me a short note, if i am right with my
ideas or if there are mistakes or misunderstandings?

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