I see this expression in a lot of articles about time-series regression, and I am not sure what it means.

"Autoregression" means estimating a vector from its past values, right? Since we are estimating the data itself, and not the error, how can the error be autoregressive?

I can think of two answers :

  1. It is referring to the temporally-nonstationary nature of the system, i.e. a regressor we compute for, let's say, y(1:100) might not work for y(101:200)
  2. It is referring to accumulated error.

Are one of these correct? Or does it mean something else?

Thanks for any help!


A regression model with autoregressive error can be written as follows:

$y_t=\beta_0+\beta_1 x_1+\beta_2 x_2+...+\beta_k x_k+n_t$

where $n_t = \phi_1 n_{t-1}+\phi_2 n_{t-2}+...+\phi_p n_{t-p}+\varepsilon_t$.

But of course, the $n_t$ are not observable. You could also think of it as a way of specifying a particular covariance structure on $n$ (and thereby on the conditional $y$). For example, if the autoregressive model is of order 1, then the $i,j$ element of $\text{Var}(n)$ is $\sigma^2\phi_1^{|i-j|}$.

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  • $\begingroup$ Thank you so much for your answer ! But I don't understand one point. We can model $n_t$ as in your second equation for any given time-series vector and a regression model on it, right? So what makes an error autoregressive and the other not? Is there a property that the $\phi$'s should hold, or a test? $\endgroup$ – jeff Sep 19 '15 at 13:42
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    $\begingroup$ If you fit that model, it's a model with an autoregressive error term. Normally, however, we restrict the parameters such that the AR model is stationary. Fitting a model doesn't imply it's suitable, however. $\endgroup$ – Glen_b Sep 19 '15 at 14:13

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