ARFIMA Python

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

The Math: ARFIMA

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.

The Math: FIGARCH

Traditional GARCH models assume that volatility shocks decay exponentially. FIGARCH introduces a fractional difference parameter to the variance equation to model hyperbolic decay, capturing long-memory volatility clustering. The conditional variance solves:

$$\sigma_t^2 = \omega^* + \left[ 1 - \frac{\phi(L)}{\beta(L)} (1-L)^{d_v} \right] \epsilon_t^2$$

Where:

• $\sigma_t^2$ is the conditional variance.
• $\omega^*$ is the adjusted baseline variance constant.
• $d_v$ is the fractional volatility parameter ($0 \le d_v \le 1$).
• $\phi(L)$ and $\beta(L)$ are the ARCH and GARCH polynomials.
• $\epsilon_t^2$ represents the squared residuals extracted from the mean equation.

Key Features

• Initialization: Uses the Geweke and Porter-Hudak (GPH) estimator to seed fractional parameters ($d$ and $d_v$) for optimization.
• Fractional Differencing: Computes fractional integration and variance convolutions using Fast Fourier Transform (FFT).
• Model Selection: Evaluates lag structures using bounded optimization and standard information criteria (AIC, BIC, HQIC, AICc).
• Inference: Extracts standard errors, t-statistics, and p-values from the inverse Hessian matrix.
• Forecasting: Calculates in-sample fitted values and out-of-sample predictions for both the mean and variance.

Installation

pip install arfima

Changelog

[1.1.0] - 2026-09-13 :

Added Added a FIGARCH class to model fractionally integrated GARCH processes. Added Guweke and Porter Hudak(GPH) estimator to generate a better seed fractional difference parameter for both ARFIMA and FIGARCH.

• v1.0.5: Fix import
• v1.0.4: Fix import
• v1.0.3: Fix import
• 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.