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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

$${\Delta}^{D}{y}_{t}=c+{\varphi}_{1}{\Delta}^{D}{y}_{t-1}+\dots +{\varphi}_{p}{\Delta}^{D}{y}_{t-p}+{\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1}+\dots +{\theta}_{q}{\epsilon}_{t-q},$$ | (6-13) |

In lag operator notation, $${L}^{i}{y}_{t}={y}_{t-i}$$.
You can write the ARIMA(*p*,*D*,*q*)
model as

$${\varphi}^{*}(L){y}_{t}=\varphi (L){(1-L)}^{D}{y}_{t}=c+\theta (L){\epsilon}_{t}.$$ | (6-14) |

The signs of the coefficients in the AR lag operator polynomial, $$\varphi (L)$$, are opposite to the right side of Equation 6-13. When specifying and interpreting AR coefficients in Econometrics Toolbox, use the form in Equation 6-13.

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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