FIGARCHpdq

FIGARCH(p,d,q) stands for Fractionally Integrated GARCH with orders p, d, q. It is a class of econometric models for conditional variance that extends the standard GARCH framework by allowing long memory in volatility through a fractional integration parameter d. FIGARCH models are used to capture persistent, slowly decaying effects of shocks on volatility observed in financial time series.

The general specification uses a fractional differencing operator (1 − L)^d to induce long-range dependence. A common form is

phi(L) h_t = omega + [1 − theta(L)] (1 − L)^d epsilon_t^2,

where phi(L) = 1 − phi1 L − … − phip L^p, theta(L) = 1 − theta1 L − … − thetaq L^q, and (1 − L)^d is the fractional differencing operator defined by a binomial expansion with parameter d. Here h_t denotes the conditional variance of the return r_t, and epsilon_t^2 is the squared innovation, often expressed as z_t^2 h_t with z_t being a standardized shock (E[z_t^2] = 1). The parameter d governs the memory of volatility: 0 ≤ d < 1, with d = 0 reducing FIGARCH to a standard GARCH(p,q) and larger d implying longer memory.

Estimation and interpretation: FIGARCH models are typically estimated by maximum likelihood under an assumed distribution for the innovations (e.g., normal or Student-t). The estimation requires constrained nonlinear optimization to enforce positivity of h_t and bounds on d (often 0 ≤ d < 0.5 for practicality). The model provides a flexible framework for forecasting volatility and assessing risk, especially when empirical volatility shows slow decay of shocks.

Variants and related models include FIEGARCH and HYGARCH, which address related long-memory or leverage effects. FIGARCH(p,d,q) remains a widely used tool in financial econometrics for modeling persistent volatility.