Python implementation of ARFIMA(p,d,q) simulation. Generalizes to ARMA(p,q) when d = 0, and ARIMA(p,d,q) when d = 1.
User provides an array of coefficients for the AR(p) and MA(q) portions of the series as well as the fractional differencing parameter and the required length. A constant may optionally be specified, as well as the standard deviation of the Gaussian innovations, and the number of seasoning samples to be generated before recording the series.
ARFIMA models with 0 < d < 1 are long-memory processes with autocorrelation functions showing hyberbolic decay as opposed to short-memory ARMA processes with autocorrelation functions that decay at a geometric rate. An example ARFIMA() process is shown below, with Figure 1 showing a plot of the series and Figure 2 showing the ACF plot.
Figure 1 - ARFIMA() series
Figure 2 - ARFIMA() autocorrelation function
p_coeffs : array_like
AR(p) coefficients, where len(p_coeffs) <= 10
q_coeffs : array_like
MA(q) coefficients, where len(q_coeffs) <= 10
d : float
fractional differencing parameter, where -1 < d <= 1
slen : int
number of samples in output ARFIMA series, where
10 <= len(series) <= 100000
alpha : float
series constant (default=0)
sigma : float
standard deviation of innovations
numseas : int
number of seasoning samples (default=100), where
0 <= num(seasoning) <= 10000
series : 1d array
random ARFIMA(p,d,q) series of specified length
MA(q) parameters follow the Box-Jenkins convention which uses a difference representation for the MA(q) process which is the opposite of the standard ARIMA MA(q) summation representation. This matches the operation of SAS/farmasim and R/arfimasim. As such, the SAS/farmafit and R/arfima MA(q) estimates match the sign of the specified MA(q) parameters while the statsmodels ARIMA().fit() estimates have opposite the specified MA(q) parameter signs.
SAS Institute Inc (2013). SAS/IML User's Guide. Cary, NC: SAS Institute Inc.
Veenstra, J.Q. (2012). Persistence and Anti-persistence: Theory and Software (Doctoral Dissertation). Western University, Ontario, Canada.
Python 3.6
Numpy 1.13.1
Statsmodels 0.9.0
Pandas 0.20.3
There are no parameters. The program is set up to access test files in the .\results directory. This path can be modified in the source file.
Please see comments in the source file for additional info including referenced output for the test file.