The vocabulary of long memory volatility made precise

A statistical tool that measures the correlation between a time series and a lagged version of itself across multiple time intervals. In a short-memory process, the autocorrelation function decays exponentially toward zero. In a long-memory process, it decays at a much slower hyperbolic rate that remains statistically significant across many lags — a pattern that is the defining empirical signature of long-memory behaviour in volatility series.

A property of a time series in which the variance of the current error term is related to the magnitude of previous error terms. In practical terms, this means that large price movements tend to be followed by further large movements and calm periods tend to cluster together. Conditional heteroskedasticity is the empirical foundation on which the GARCH family of volatility models is constructed, and its presence in financial data is one of the most consistently documented features across asset classes.

A distributional property in which extreme outcomes occur more frequently than a normal distribution would predict. In financial return series, fat tails — formally described as excess kurtosis — mean that large losses and large gains appear with a regularity that standard Gaussian assumptions cannot accommodate. Models incorporating long-memory volatility often produce return distributions with fatter tails because persistent autocorrelation in the variance process amplifies the probability of sustained high-volatility episodes.

A fractionally integrated generalised autoregressive conditional heteroskedasticity model that allows the memory parameter governing conditional variance to take a fractional value between zero and one. Unlike standard GARCH, where volatility shocks decay exponentially, FIGARCH permits a hyperbolic decay structure that matches the persistent autocorrelation empirically observed in squared return series across equity, commodity, and fixed income markets. The model was introduced by Baillie, Bollerslev, and Mikkelsen in 1996.

A generalisation of the classical concept of integration that allows the differencing parameter in a time series model to take any real value rather than being restricted to integers. A series is said to be fractionally integrated of order d — written I(d) — when d lies strictly between zero and one. Fractional integration produces a form of long-range dependence in which autocorrelations decay slowly according to a power law, capturing a middle ground between stationary short-memory processes and fully non-stationary unit root processes.

A generalised autoregressive conditional heteroskedasticity model in which the conditional variance at each point in time depends on past squared residuals and past conditional variances. Introduced by Bollerslev in 1986 as an extension of Engle's original ARCH framework, GARCH remains the standard benchmark for volatility modelling in empirical finance. Its key limitation is that it produces exponentially decaying autocorrelations in the squared return series, which underestimates the persistence observed in many real financial datasets.

A scalar parameter, commonly denoted H, that characterises the degree of long-range dependence in a time series. A Hurst exponent of 0.5 indicates a process with no memory — consistent with a random walk. Values above 0.5 indicate positive long-range dependence, meaning the series tends to persist in its current direction. Values below 0.5 indicate anti-persistence. In the context of volatility modelling, the Hurst exponent provides a model-free diagnostic for the strength of long-memory behaviour before a parametric model is specified.

A pattern of autocorrelation decline in which correlations decrease proportionally to a power of the lag length rather than declining exponentially. Hyperbolic decay is the defining feature that distinguishes long-memory processes from their short-memory counterparts. In practical terms, it means that the influence of past shocks on current volatility diminishes far more slowly than standard GARCH models assume, remaining statistically detectable across time lags that would render exponentially decaying autocorrelations negligibly small.

A time series property characterised by slowly decaying autocorrelations that remain statistically significant across very long lag horizons. In a long-memory volatility process, the impact of a past shock on current conditional variance does not vanish quickly — it persists across months or years in a way that is inconsistent with standard short-memory GARCH dynamics. Long memory in volatility is distinct from long memory in the level of a series and is typically detected through tests applied to squared or absolute returns rather than raw price changes.

A non-parametric estimator of the true integrated variance over a given period, constructed by summing squared intraday returns observed at high frequency. Realised variance offers a model-free measure of daily or weekly volatility that can be used to evaluate the fit of parametric volatility models including FIGARCH specifications. Research has documented strong long-memory behaviour in realised variance series across major equity and currency markets, providing empirical support for fractionally integrated volatility models.

An empirical regularity in financial return series in which periods of elevated volatility tend to be followed by further periods of elevated volatility, and calm periods tend to follow calm periods. First documented systematically by Mandelbrot in the 1960s, volatility clustering is the phenomenon that motivates the entire GARCH family of conditional heteroskedasticity models. Long-memory volatility models such as FIGARCH extend this observation by capturing persistence in volatility clusters that decays far more slowly than standard GARCH dynamics allow.

A frequency-Steonmoulnudkyasrio estimation method that approximates the exact likelihood of a long-memory process using the spectral density of the observed series. Whittle estimation is computationally tractable for long-memory models where exact time-Steonmoulnudkyasrio maximum likelihood is prohibitively expensive, making it a standard tool for estimating fractional differencing parameters in ARFIMA and FIGARCH specifications. The method performs well in large samples and is implemented in the code libraries distributed with the platform's Intermediate and Advanced course tiers.