Corsi (2009) A Simple Approximate Long-Memory Model of Realized Volatility

Summary

Corsi proposes an additive cascade of volatility components defined over different time horizons, which reduces to a simple autoregressive-type model in realized volatility: the Heterogeneous Autoregressive model of Realized Volatility (HAR-RV). The model regresses daily realized volatility on its own past daily, weekly, and monthly realized-volatility averages. Despite its simplicity and the absence of any true long-memory mechanism, simulations show it reproduces the main empirical features of financial returns — apparent long memory, fat tails, and self-similarity — parsimoniously, and it delivers remarkably good out-of-sample volatility forecasts.

Key Claims

• Heterogeneous Market Hypothesis motivation. Market participants act over different time horizons (short-term traders, medium-term investors, long-term institutions). Each cares about, and generates, volatility at its own horizon, producing a volatility cascade in which lower-frequency (longer-horizon) volatility influences higher-frequency (shorter-horizon) volatility but not vice versa.
• The HAR-RV model. The cascade yields a linear model for (log) realized volatility using components realized over daily, weekly, and monthly horizons: $$RV^{(d)}_{t+1} = c + \beta_d\,RV^{(d)}_t + \beta_w\,RV^{(w)}_t + \beta_m\,RV^{(m)}_t + \varepsilon_{t+1},$$ where $RV^{(w)}_t = \tfrac{1}{5}\sum_{i=0}^{4}RV^{(d)}_{t-i}$ and $RV^{(m)}_t = \tfrac{1}{22}\sum_{i=0}^{21}RV^{(d)}_{t-i}$ are weekly and monthly averages. It is a constrained AR(22) estimable by ordinary least squares.
• Apparent, not true, long memory. HAR-RV is a short-memory model, yet the sum of a few components at geometrically spaced horizons mimics the slowly (hyperbolically) decaying autocorrelation function of realized volatility — reproducing the "long memory" without fractional integration.
• Reproduces stylized facts. Simulations show the model generates the observed long-range dependence, fat-tailed return distributions, and self-similarity/scaling of financial returns.
• Strong forecasting performance. Empirically, HAR-RV forecasts realized volatility at least as well as fractionally integrated (ARFIMA) long-memory models across daily, weekly, and monthly horizons, while being far simpler to estimate and interpret.

Concepts Introduced or Extended

• Heterogeneous Autoregressive (HAR) Model
• Realized Volatility
• Long Memory and Fractional Integration

Entities Mentioned

• Fulvio Corsi

Quotes

"This volatility cascade leads to a simple AR-type model in the realized volatility with the feature of considering different volatility components realized over different time horizons and thus termed Heterogeneous Autoregressive model of Realized Volatility (HAR-RV)."

"In spite of the simplicity of its structure and the absence of true long-memory properties, simulation results show that the HAR-RV model successfully achieves the purpose of reproducing the main empirical features of financial returns."

My Take

HAR-RV is the model everyone actually uses. Its trick is almost embarrassingly simple — a three-term OLS regression of realized volatility on its daily, weekly, and monthly averages — but that is exactly why it won: it delivers the slowly decaying autocorrelations of realized volatility that had motivated the whole fractionally integrated literature, without the estimation headaches of ARFIMA, and it forecasts just as well. The economic story (a cascade of heterogeneous horizons) gives the restriction a rationale, but even taken as a pure approximation the model is the natural companion to the realized-volatility measurement papers (ABDL, BNS). It also composes cleanly with jump separation: replacing or augmenting the RV components with bipower variation and the jump part gives the HAR-RV-CJ family used to forecast continuous and jump volatility separately.