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 *y _{t}*. The general form
of the ARMAX(

$${y}_{t}={\displaystyle \sum _{i=1}^{p}{\varphi}_{i}}{y}_{t-i}+{\displaystyle \sum _{k=1}^{r}{\beta}_{k}}{x}_{tk}+{\epsilon}_{t}+{\displaystyle \sum _{j=1}^{q}{\theta}_{j}}{\epsilon}_{t-j},$$ | (5-16) |

and it has the following condensed form in lag operator notation:

$$\varphi (L){y}_{t}=c+{x}_{t}^{\prime}\beta +\theta (L){\epsilon}_{t}.$$ | (5-17) |

In Equation 5-17, the vector $${x}_{t}^{\prime}$$ 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, *x _{t}*,
affects this week's United States exchange rate

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 $$\varphi (L)={L}^{p}-{\varphi}_{1}{L}^{p-1}-{\varphi}_{2}{L}^{p-2}-\mathrm{...}-{\varphi}_{p}{L}^{p}=0$$ lie outside of the unit circle according to Wold's Decomposition [1].If the response series

*y*is not stable, then you can difference it to form a stationary ARIMA model. Do this by specifying the degrees of integration_{t}`D`

. Econometrics 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. Similarly,`estimate`

imposes stationarity constraints during estimation.The software differences the response series

*y*_{t}*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 ARIMAX(*p*,*D*,*q*) model is$$\varphi (L){y}_{t}={c}^{\ast}+{x}_{t}^{\prime}\beta +{\theta}^{\ast}(L){\epsilon}_{t},$$ **(5-18)**where

*c*=^{*}*c*/(1 –*L*)and^{D}*θ*=^{*}(L)*θ(L)*/(1 –*L*). Subsequently, the interpretation of^{D}*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

*x*are stationary. Difference all predictor series that are not stationary with_{t}`diff`

during the data preprocessing stage. If*x*is nonstationary, then a test for the significance of_{t}*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*)(*p*,_{s}*D*,_{s}*q*)_{s}model. Assuming that the response series_{s}*y*is stationary, the model has the form_{t}$$\varphi (L)\Phi (L){y}_{t}=c+{x}_{t}^{\prime}\beta +\theta (L)\Theta (L){\epsilon}_{t},$$

where

*Φ(L)*and*Θ(L)*are the seasonal lag polynomials. If*y*is not stationary, then you can specify degrees of nonseasonal or seasonal integration using_{t}`arima`

. If you specify`Seasonality`

≥ 0, then the software applies degree one seasonal differencing (*D*= 1) to the response. Otherwise,_{s}*D*= 0. The software includes the exogenous covariates after it differences the response._{s}The software treats the exogenous covariates as fixed during estimation and inference.

[1] Wold, H. *A Study in the Analysis
of Stationary Time Series*. Uppsala, Sweden: Almqvist &
Wiksell, 1938.

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