Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

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(

$${y}_{t}={\displaystyle \sum _{i=1}^{p}{\varphi}_{i}}{y}_{t-i}+{\displaystyle \sum _{k=1}^{r}{\beta}_{k}}{x}_{tk}+{\epsilon}_{t}+{\displaystyle \sum _{j=1}^{q}{\theta}_{j}}{\epsilon}_{t-j},$$ | (6-16) |

$$\varphi (L){y}_{t}=c+{x}_{t}^{\prime}\beta +\theta (L){\epsilon}_{t}.$$ | (6-17) |

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 [2].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

where$$\varphi (L){y}_{t}={c}^{\ast}+{x}_{t}^{\prime}\beta +{\theta}^{\ast}(L){\epsilon}_{t},$$ **(6-18)***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}where$$\varphi (L)\Phi (L){y}_{t}=c+{x}_{t}^{\prime}\beta +\theta (L)\Theta (L){\epsilon}_{t},$$

*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] 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.

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

Was this topic helpful?