The autoregressive integrated moving average (ARIMA) process generates nonstationary series that are integrated of order D, denoted I(D). A nonstationary I(D) process is one that can be made stationary by taking D differences. Such processes are often called difference-stationary or unit root processes.
A series that you can model as a stationary ARMA(p,q) process after being differenced D times is denoted by ARIMA(p,D,q). The form of the ARIMA(p,D,q) model in Econometrics Toolbox™ is
$${\Delta}^{D}{y}_{t}=c+{\varphi}_{1}{\Delta}^{D}{y}_{t-1}+\dots +{\varphi}_{p}{\Delta}^{D}{y}_{t-p}+{\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1}+\dots +{\theta}_{q}{\epsilon}_{t-q},$$ | (5-13) |
where $${\Delta}^{D}{y}_{t}$$ denotes a Dth differenced time series, and $${\epsilon}_{t}$$ is an uncorrelated innovation process with mean zero.
In lag operator notation, $${L}^{i}{y}_{t}={y}_{t-i}$$. You can write the ARIMA(p,D,q) model as
$${\varphi}^{*}(L){y}_{t}=\varphi (L){(1-L)}^{D}{y}_{t}=c+\theta (L){\epsilon}_{t}.$$ | (5-14) |
Here, $${\varphi}^{*}(L)$$ is an unstable AR operator polynomial with exactly D unit roots. You can factor this polynomial as $$\varphi (L){(1-L)}^{D},$$ where$$\varphi (L)=(1-{\varphi}_{1}L-\dots -{\varphi}_{p}{L}^{p})$$ is a stable degree p AR lag operator polynomial (with all roots lying outside the unit circle). Similarly, $$\theta (L)=(1+{\theta}_{1}L+\dots +{\theta}_{q}{L}^{q})$$ is an invertible degree q MA lag operator polynomial (with all roots lying outside the unit circle).
The signs of the coefficients in the AR lag operator polynomial, $$\varphi (L)$$, are opposite to the right side of Equation 5-13. When specifying and interpreting AR coefficients in Econometrics Toolbox, use the form in Equation 5-13.
Note: In the original Box-Jenkins methodology, you difference an integrated series until it is stationary before modeling. Then, you model the differenced series as a stationary ARMA(p,q) process [1]. Econometrics Toolbox fits and forecasts ARIMA(p,D,q) processes directly, so you do not need to difference data before modeling (or backtransform forecasts). |
[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.