## ARIMA Model

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

 ${\Delta }^{D}{y}_{t}=c+{\varphi }_{1}{\Delta }^{D}{y}_{t-1}+\dots +{\varphi }_{p}{\Delta }^{D}{y}_{t-p}+{\epsilon }_{t}+{\theta }_{1}{\epsilon }_{t-1}+\dots +{\theta }_{q}{\epsilon }_{t-q},$ (1)
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

 ${\varphi }^{*}\left(L\right){y}_{t}=\varphi \left(L\right){\left(1-L\right)}^{D}{y}_{t}=c+\theta \left(L\right){\epsilon }_{t}.$ (2)
Here, ${\varphi }^{*}\left(L\right)$ is an unstable AR operator polynomial with exactly D unit roots. You can factor this polynomial as $\varphi \left(L\right){\left(1-L\right)}^{D},$ where$\varphi \left(L\right)=\left(1-{\varphi }_{1}L-\dots -{\varphi }_{p}{L}^{p}\right)$ is a stable degree p AR lag operator polynomial (with all roots lying outside the unit circle). Similarly, $\theta \left(L\right)=\left(1+{\theta }_{1}L+\dots +{\theta }_{q}{L}^{q}\right)$ 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, $\varphi \left(L\right)$, are opposite to the right side of Equation 1. When specifying and interpreting AR coefficients in Econometrics Toolbox, use the form in Equation 1.

### 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).

## 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.