## 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) |

*D*th 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

$${\varphi}^{*}(L){y}_{t}=\varphi (L){(1-L)}^{D}{y}_{t}=c+\theta (L){\epsilon}_{t}.$$ | (2) |

*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 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.