ARFIMA

ARFIMA stands for AutoRegressive Fractionally Integrated Moving Average.

Stata fits ARFIMA models.

ARFIMA concerns long-memory processes. Long-memory processes are stationary processes whose autocorrelation functions decay slowly. The ARFIMA model provides a parsimonious parameterization of long-memory processes that nests the ARMA (autoregressive moving-average) model, which is widely used for short-memory processes. Long-memory processes can be obtained by fractionally integrating short-memory processes. The ARFIMA model does that. The ARFIMA model also generalizes the ARIMA model by allowing for fractional degrees of integration.

Below we analyze yearly data on the widths of the rings of a tree. This one tree survived 5,000 years! In any case, larger widths represent good years for the tree and narrower widths represent harsh years. Below we estimate the parameters of an ARFIMA model with the fractional difference parameter and a constant. With only those parameters, we will account for the long memory of the process.

. webuse campito, clear
(Campito Mnt. tree ring data from 3435BC to 1969AD)

. arfima width, nolog

ARFIMA regression

Sample: -3435 thru 1969                                Number of obs =   5,405
                                                       Wald chi2(1)  = 1864.44
Log likelihood = -18907.279                            Prob > chi2   =  0.0000



       width 
 
 Coefficient  Std. err.      z    P>|z|     [95% conf. interval]

width        
 
 
       _cons 
 
   44.01432   9.174317     4.80   0.000     26.03299    61.99565
 
 
 
ARFIMA       
 
 
           d 
 
   .4468887   .0103497    43.18   0.000     .4266038    .4671737

     /sigma2 
 
   63.92927   1.229754    51.99   0.000       61.519    66.33955

Note: The test of the variance against zero is one sided, and the two-sided
      confidence interval is truncated at zero.

Below we plot the original series and the fractionally differenced component of the predicted series, which reflects the short-memory component of the process: