ARFIMA
ARFIMA stands for AutoRegressive Fractionally Integrated Moving Average.
Stata’s new arfima command 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.
Below we plot the original series and the fractionally differenced component
of the predicted series, which reflects the short-memory component of the
process:
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