ARFIMA Python

A lightweight, fast Python library for modeling long-memory time series using AutoRegressive Fractionally Integrated Moving Average (ARFIMA) models.

The Math

Traditional ARIMA models require integer differencing, which can erase long-term trends. ARFIMA allows for fractional differencing to capture "long-memory" processes. The model solves the general equation:

$$\phi(L)(1-L)^d y_t = \theta(L)w_t$$

Where:

• $L$ is the lag operator.
• $d$ is the fractional integration parameter ($$-0.5 < d < 0.5$$).
• $\phi$ and $\theta$ are the Autoregressive (AR) and Moving Average (MA) polynomials.
• $w_t$ represents the white noise.

Key Features

• Fast Fractional Differencing: Uses Fast Fourier Transform (FFT) for fractional integration and differencing.
• Automatic Grid Search: Automatically finds the optimal lag structure using information criteria (AIC, BIC, HQIC, or AICc).
• Statistical Inference: Calculates the inverse Hessian matrix to provide standard errors, t-statistics, and exact p-values.
• Forecasting: Built-in methods for both in-sample historical reconstruction and out-of-sample predictions.

Installation

pip install arfima

Changelog

• v1.0.2: Minor bug fixes and docstring improvements.
• v1.0.1: Fixed inverted AR/MA polynomials in forecast generation.
• v1.0.0: Initial release.