What does error being autoregressive mean?

Question

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!

Answer 1

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 $\sigma^2\phi_1^{|i-j|}$.