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I am reading some lecture notes, and in it, in order to justify fitting an autoregressive correlation structure, the professor plots an autocorrelation plot of the actual data.

But, should he not plot the autocorrelation plot of the residuals after fitting some model, and then if that plot showcases autocorrelation, then should he add autoregressive correlation?

If he only plots autocorrelation of the data, then how does he know that a standard regression model would not manage to account for that correlation?

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When you fit a model with an autoregressive correlation the structure, the underlying assumption is (in some shape or form), that $$ Y_{ij} = f(X) + R_{ij} $$ ie. that your response variable is realized from some stochastic variable which has mean depending on some predictors, $f(X)$, and that there is some stochastics process (for example, a Brownian motion), $\{R_{ij} \}$, specifying the entire covariance structure. If you want to fit a model such as this, you should have some argument, that you believe this to be the data-generating process. One way (and a very good way) of making this argument is to calculate some residuals (for which the residuals of a normal linear model with mean structure $f(X)$ are a good bet) and plot the empirical correlation between these - in time series analysis, you would often calculate and plot them for/against difference in the observed times. If these exhibit some pattern, that you recognize as an autoregressive process, then you should in fact construct your model as such. If not, then you should perhaps remain in the linear normal model (or whatever else might seem reasonable in the given setting).

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  • $\begingroup$ Fine answer, one question though: why would you make the choice for the model to use dependent on the results of some analyses? If you have good reasons to assume the data-generating process is influenced by autocorrelation, shouldn't the model reflect that? $\endgroup$
    – IWS
    May 27, 2017 at 7:06

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