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I am looking at this model, which is used when the residuals of your typical least squares regression model is serially correlated. https://online.stat.psu.edu/stat510/lesson/8/8.1. I think on google, it is also called regression plus time series errors.

It seems like the resulting coefficients of the regression + arima model is pretty much the same as just your original regression model. As a result, is there a reason to even take this extra step to rebuild your model jointly with an ARIMA model for the residuals - since your coefficients are pretty much the same so I'd imagine prediction and inference to be the same as well?

Thanks

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A regular linear model requires independent observations, non-independent observation will have an effect on inference (SE, p-values, ...), not really on coefficients. For that reason you use an ARIMA on the residuals, to correct inference.

Edit: to expand a little on not really on coefficients, there may be some differences in the coefficients depending on whether you include an intercept in the model and how the ARIMA model is estimated, which is usually done with maximum likelihood and CSS (Conditional Sum of Squares) compared to a linear model which uses OLS; in this case as your sample size increases the differences should get smaller.

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  • $\begingroup$ The statement regarding no effect on coefficients is not quite correct. See "How can I handle autocorrelated residuals?" and "Correcting for autocorrelation in simple linear regressions in R". $\endgroup$ – Richard Hardy May 19 '20 at 7:28
  • $\begingroup$ @RichardHardy I can't find what you are referring to regarding the coefficients. Could you summarize? $\endgroup$ – user2974951 May 19 '20 at 7:48
  • $\begingroup$ Point estimates of model coefficients change if ARMA structure is introduced in the residuals. They are not the same as when assuming residuals being i.i.d. I think Francis X. Diebold makes it pretty clear; perhaps you would benefit from reading his entire post (posts) on the topic, not just the excerpts that I present. $\endgroup$ – Richard Hardy May 19 '20 at 9:52

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