What is the reason for not including an intercept term in AR and ARMA models?

Question

In econometric literature it is usually argued that in case of estimating an equation, an intercept term must be always included regardless of its statistical importance because removing the constant cause other regression parameters be biased (for example see Brooks, introductory econometrics for finance). Now my question is given this reasoning, why in models like AR (autoregressive regression) and ARMA (Autoregressive–moving-average model) which deal with time series data, an intercept term is not usually included?

Answer 1

ARMA models can easily be formulated with an intercept term, and these are in common use. For example an AR($1$) model with an intercept term can be written as:

$$\begin{equation} \begin{aligned} X_t &= \mu + \phi (X_{t-1} - \mu) + \varepsilon_t \\[6pt] &= (1-\phi) \mu + \phi X_{t-1} + \varepsilon_t. \end{aligned} \end{equation}$$

In the special case where $\mu = 0$ this simplifies to the formula you have probably seen:

$$X_t = \phi X_{t-1} + \varepsilon_t.$$

For pedagogical purposes, it is common for books and lecture notes on time-series models to omit the intercept term because it does not really add anything of substance to understanding the model form (it is just a shift in location). It is relatively simple to add the intercept term into the model by defining a zero-mean model (no intercept term) for $\{ \tilde{X}_t \}$ and defining $\{ X_t \}$ with $X_t = \mu + \tilde{X}$.

Answer 2

To begin with in arima models the constant is mandatory if d=0 i.e.no differencing is in play. If d<>=0 then the constant is optional. If d<>=0 and a constant is in the model there is a steady state constant reflecting a "slope" or growth reflecting growth as compared to deterministic growth via time/counting numbers related predictor variables in an armaX model e.g. X=1,2,3,,,t.

See Constant in arima model whether to include or exclude? for a related discussion.

The reason a constant should always be included in a regression model is that if it is omitted then the prediction equation is forced to go through the point 0,0 i.e. the origin.