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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

where ΔDyt denotes a Dth differenced time series, and εt is an uncorrelated innovation process with mean zero.

In lag operator notation, Liyt=yti. You can write the ARIMA(p,D,q) model as

Here, ϕ*(L) is an unstable AR operator polynomial with exactly D unit roots. You can factor this polynomial as ϕ(L)(1L)D, whereϕ(L)=(1ϕ1LϕpLp) is a stable degree p AR lag operator polynomial (with all roots lying outside the unit circle). Similarly, θ(L)=(1+θ1L++θqLq) 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, ϕ(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.


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.

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