ARIMA Model Including Exogenous Covariates

ARIMAX(p,D,q) Model

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(p,q) model is

$$y_t={\displaystyle \sum _{i=1}^p{\varphi }_i}{y}_{t-i}+{\displaystyle \sum _{k=1}^r{\beta }_k}{x}_{tk}+{\epsilon }_t+{\displaystyle \sum _{j=1}^q{\theta }_j}{\epsilon }_{t-j},$$

(1)

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

$$\varphi (L)y_t=c+x_t^{\prime }\beta +\theta (L){\epsilon }_t.$$

(2)

In Equation 2, 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 y_t. The exchange rate and the price of oil are time series, so an ARMAX model can be appropriate to study their relationships.

Conventions and Extensions of the ARIMAX Model

• 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_t 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 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

  $$\varphi (L)y_t=c^{\ast }+x_t^{\prime }\beta +{\theta }^{\ast }(L){\epsilon }_t,$$

  (3)

  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 x_t are stationary. Difference all predictor series that are not stationary with diff during the data preprocessing stage. If x_t 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)(p_s,D_s,q_s)_s model. Assuming that the response series y_t is stationary, the model has the form

  $$\varphi (L)\Phi (L)y_t=c+x_t^{\prime }\beta +\theta (L)\Theta (L){\epsilon }_t,$$

  where Φ(L) and Θ(L) are the seasonal lag polynomials. If y_t is not stationary, then you can specify degrees of nonseasonal or seasonal integration using arima. If you specify Seasonality ≥ 0, then the software applies degree one seasonal differencing (D_s = 1) to the response. Otherwise, D_s = 0. The software includes the exogenous covariates after it differences the response.

• The software treats the exogenous covariates as fixed during estimation and inference.

References

[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.