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
and it has the following condensed form in lag operator notation:
In Equation 5-17, the vector holds the values of the r exogenous, time-varying predictors at time t, with coefficients denoted β.
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
D. Econometrics Toolbox™ enforces
stability of the AR polynomial. When you specify an AR model using
the software displays an error if you enter coefficients that do not
correspond to a stable polynomial. Similarly,
stationarity constraints during estimation.
The software differences the response series yt before including
the exogenous covariates if you specify the degree of integration
In other words, the exogenous covariates enter a model with a stationary
response. Therefore, the ARIMAX(p,D,q)
where c* = c/(1 – L)D and θ*(L) = θ(L)/(1 – L)D. Subsequently, the interpretation of β has changed to the expected effect a unit increase in the predictor has on the difference between current and lagged values of the response (conditional on those lagged values).
You should assess whether the predictor series xt are
stationary. Difference all predictor series that are not stationary
the data preprocessing stage. If xt 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 predictor series.
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
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 software treats the exogenous covariates as fixed during estimation and inference.
 Wold, H. A Study in the Analysis of Stationary Time Series. Uppsala, Sweden: Almqvist & Wiksell, 1938.