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Fractional difference of 2 is the 2nd derivative or 2nd difference. note: applying fractional differencing changes the units of the problem. If we started with Prices then take fractional differences, we no longer are in Price units. determining the order of differencing to make a time series stationary may be an iterative, exploratory process.
If the time-series is suspected to exhibit long-range dependence, then the d parameter may be allowed to have non-integer values in an autoregressive fractionally integrated moving average model, which is also called a Fractional ARIMA (FARIMA or ARFIMA) model.
Seasonal ARIMA (SARIMA or periodic ARMA) models periodic variation. Autoregressive fractionally integrated moving average (ARFIMA, or Fractional ARIMA, FARIMA) model time-series that exhibits long memory. Multiscale AR (MAR) is indexed by the nodes of a tree instead of integers.
Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which ...
X-13ARIMA-SEATS, successor to X-12-ARIMA and X-11, is a set of statistical methods for seasonal adjustment and other descriptive analysis of time series data that are implemented in the U.S. Census Bureau's software package. [3]
Long-range and short-range dependent processes are characterised by their autocovariance functions.. In short-range dependent processes, the coupling between values at different times decreases rapidly as the time difference increases.
Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, [3] which have a more complicated stochastic structure. Contrary to the AR model, the finite MA model is always stationary.
However in ARIMA, implicitly (from the binomial expansion) the strength of past values goes weaker further into the past. With the introduction of the fractional d exponent, the strength of the past histories can be taken into account for a better model result.