What is the term for a time series regression having more than one predictor?

Question

It's pretty tough to search the Web for info on something when you don't know what words are commonly used to describe it. In this case, I'm wondering what it's called when you include another predictor in a time series.

As an example, say I'm modeling a variable $X$ using AR(3):

$ X_t = \varphi_1 X_{t-1} + \varphi_2 X_{t-2} + \varphi_3 X_{t-3} + \varepsilon_t $

I want my model to include the effects of another variable—say, $Y$—so my model is now described as:

$ X_t = \varphi_1 X_{t-1} + \varphi_2 X_{t-2} + \varphi_3 X_{t-3} + \beta_1 Y_{t-1} + \beta_2 Y_{t-2} + \beta_3 Y_{t-3} + \varepsilon_t $

What would the term (or terms) be that distinguishes the former model from the latter?

Answer 1

ARIMAX (Box-Tiao) is what it is called when you add covariates to Arima models, is is basically arima + X.

http://www.r-bloggers.com/the-arimax-model-muddle/

Also search for Panel data or TSCS: 'Time-series–cross-section (TSCS) data consist of comparable time series data observed on a variety of units'

See: http://as.nyu.edu/docs/IO/2576/beck.pdf or https://stat.ethz.ch/pipermail/r-sig-mixed-models/2010q4/004530.html

Answer 2

This is called a Transfer Function Model. It has also been referred to as a Dynamic Regression Model.

Answer 3

Searching the SAS documentation for ARIMAX (based on Patrick's answer), I found (in "The ARIMA Procedure: Input Variables and Regression with ARMA Errors") that they listed the following terms:

• ARIMAX
• Box-Tiao
• Transfer Function Model
• Dynamic Regression
• Intervention Model
• Interrupted Time Series Model
• Regression Model with ARMA Errors

The term ARIMAX was used almost equally as much as the term Transfer Function Model. Intervention Model and Interrupted Time Series Model appear to refer to a different kind of model than the others.