Fractional integration (or fractional differencing) was devised independently by Granger and Joyeux (1980) and Hosking (1981) to provide a way to model a process with “long-memory”—defined as having autocorrelations which damp off at a slow enough rate that they sum to infinity. Standard arima models can only approximate this type of behavior, as a non-integrated ARMA model has autocorrelations that are exponentially damped, and are thus summable, and any integrated model has autocorrelations which don’t damp at all.

 

Fractional integration is modelled with the help of the specialized lag function \((1 - L)^d \) where \(d\) is non-integer. This can be expanded in the time domain using a generalized binomial expansion. In fact, in RATS, the coefficient on lag \(k\) can be obtained as %BINOMIAL(d,k)*(-1)^k—this gives rise to an infinite one-sided lag polynomial.

 

There has been only a modest amount of empirical work done in econometrics using fractional integration, particularly when compared with the amount done using unit root statistics. Part of this is clearly due to the fact that a first difference is easy to compute and has a relatively simple economic interpretation, while a fractional difference can’t even be computed exactly in the time domain—the filter always has to be truncated. It is also true that fractional integration is another entrant in the set of models proposed to explain long term behavior of data sets which in many cases simply aren’t long enough to separate one clearly from the others.

 

Fractional integration is a natural candidate for attack in the frequency domain since the lag function  can be computed easily there. For instance, Geweke and Porter-Hudak (1983) proposed a method for estimating \(d\) which involves analyzing the behavior of the periodogram of the data for low frequencies. The procedure @GPH implements this. A more recent refinement of this by Andrews and Guggenberger (2003) is available as @AGFRACTD.

 

If you just need to fractionally difference (or integrate) a series with a known value of \(d\), you can use the DIFFERENCE instruction with the option FRACTION (you might also need the DIFFERENCE option as well). The frequency domain computations described here are used by DIFFERENCE. Because this is an infinite lag polynomial, the calculation is subject to wraparound effect. DIFFERENCE has a PADDING option if you need to pad out with a non-zero value which is sometimes appropriate. For instance, Baillie, Bollerslev and Mikkelson(1996) estimate a FIGARCH model, which requires calculation of a fractionally differenced squared residual. Since the fractional difference is applied to a squared series, a zero value for padding would be inappropriate. Instead, it uses an unconditional estimate of the squared residual:

 

  diff(fraction=d,pad=uupresample) uu / uufilter

 

As a general rule, if you run across a paper which uses fractional differencing or integration as part of the calculation (that is, doing something more than just estimating \(d\)) and it does not provide any details for how the infinite filter was handled, you should probably look for a better, more carefully done, example.

 

The example FRACTINT.RPF estimates an ARFIMA model (AutoRegressive, Fractionally Integrated, Moving Average) model using frequency domain techniques. This is done using the instruction MAXIMIZE applied to a criterion function which is defined frequency by frequency. This function (the "Whittle Likelihood"), which can be used more broadly than just for ARFIMA models, looks similar to those used in time domain likelihoods. However, where you would usually see a squared residual divided by a variance, it uses the ratio between the periodogram of the data and the spectral density implied by the model and its parameters. Fox and Taqqu (1986) work through the theory of this method of estimation. It’s possible to concentrate out the innovation variance, leaving a simpler function than the one shown here, but we’ve found that leaving the function in its original form works better.

 

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