Understanding error term in AR model

Question

In AR model, the value at a time $\tau$ is modeled as linear regression of past values and an additional error term ($\epsilon_{\tau}$) at time $\tau$. In this what is the error term?

Answer 1

An autoregression of order $p$, AR($p$), is $$ x_t = c + \varphi_1 x_{t-1} + \dots + \varphi_p x_{t-p} + \varepsilon_t. $$ Its conditional mean, conditioning on information up to time $t-1$, $I_{t-1}$, is $$ \mathbb{E}(x_t|I_{t-1}) = c + \varphi_1 x_{t-1} + \dots + \varphi_p x_{t-p}. $$ Hence, $$ \varepsilon_t=x_t-\mathbb{E}(x_t|I_{t-1}). $$ Therefore, one way of looking at the error term in an AR($p$) model is that it is the difference between $x_t$ and its conditional mean.

(I have replaced your time index $\tau$ with a more common $t$.)