# Autoregressive fractionally integrated moving average

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Autoregressive fractionally integrated moving average

In statistics, autoregressive fractionally integrated moving average models are time series models that generalize ARIMA ("autoregressive integrated moving average") models by allowing non-integer values of the differencing parameter and are useful in modeling time series with long memory.

In an ARIMA model, the "integrated" part of the model includes the differencing operator, in terms of the backspace operator "B", as an integer power of (1 − "B"). For example :$\left(1-B\right)^2=1-2B+B^2 ,,$where:$B^2X_t=X_\left\{t-2\right\} ,.$In a "fractional" model, the power is allowed to be fractional, with the meaning of the term identified using the following formal expansion:$\left(1-B\right)^d=1-dB+frac\left\{d\left(d-1\right)\right\}\left\{2\right\}B^2 +cdots ,.$

References

* C. W. J. Granger and R. Joyeux. "An introduction to long-memory time series and fractional differencing", Journal of Time Series Analysis, 1980.
* J. R. M. Hosking. "Fractional differencing", Biometrika 68(1):165-176, 1981.
* P. M. Robinson. "Time Series With Long Memory", Oxford University Press 2003.

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