The autoregressive moving average model including exogenous covariates, ARMAX(p,q), extends the ARMA(p,q) model by including the linear effect that one or more exogenous series has on the stationary response series yt. The general form of the ARMAX(p,q) model is
You can use this model to check if a set of exogenous variables has an effect on a linear time series. For example, suppose you want to measure how the previous week’s average price of oil, xt, affects this week’s United States exchange rate yt. The exchange rate and the price of oil are time series, so an ARMAX model can be appropriate to study their relationships.
ARMAX models have the same stationarity requirements as ARMA models. Specifically, the response series is stable if the roots of the homogeneous characteristic equation of lie outside of the unit circle according to Wold’s Decomposition .
If the response series yt is not
stable, then you can difference it to form a stationary ARIMA model. Do
this by specifying the degrees of integration
Toolbox™ enforces stability of the AR polynomial. When you specify an
AR model using
arima, the software displays an error if
you enter coefficients that do not correspond to a stable polynomial.
estimate imposes stationarity constraints
The software differences the response series
before including the exogenous covariates if you
specify the degree of integration
D. In other words, the
exogenous covariates enter a model with a stationary
response. Therefore, the
You should assess whether the predictor series
xt are stationary.
Difference all predictor series that are not stationary with
diff during the data preprocessing stage.
is nonstationary, then a test for the significance of
β can produce a false negative. The practical
interpretation of β changes if you difference the
The software uses maximum likelihood estimation for conditional mean models such as ARIMAX models. You can specify either a Gaussian or Student’s t for the distribution of the innovations.
You can include seasonal components in an ARIMAX model (see Multiplicative ARIMA Model) which creates a SARIMAX(p,D,q)(ps,Ds,qs)s model. Assuming that the response series yt is stationary, the model has the form
where Φ(L) and Θ(L)
are the seasonal lag polynomials. If
yt is not stationary, then you
can specify degrees of nonseasonal or seasonal integration using
arima. If you specify
≥ 0, then the software applies degree one seasonal differencing
(Ds = 1) to the response.
Otherwise, Ds = 0. The software
includes the exogenous covariates after it differences the response.
The software treats the exogenous covariates as fixed during estimation and inference.
 Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.
 Wold, H. A Study in the Analysis of Stationary Time Series. Uppsala, Sweden: Almqvist & Wiksell, 1938.