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The autoregressive integrated moving average (ARIMA) process
generates nonstationary series that are integrated of order * D*,
denoted

A series that you can model as a stationary ARMA(* p*,

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

where $${\Delta}^{D}{y}_{t}$$ denotes a * 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*,

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

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

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.

)
process [1]. Econometrics Toolbox fits
and forecasts ARIMA(q,p,D)
processes directly, so you do not need to difference data before modeling
(or backtransform forecasts).q |

[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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