The "Rough" HAR Model

2026-09-18

AI Summary

本文通过在AR和HAR模型中加入MA(1)成分，提出了离散时间近似方法来模拟连续时间下的波动率粗糙模型，并证明了新模型在预测资产波动率方面优于传统模型。

Abstract

This paper proposes discrete-time approximations to rough continuous-time models of realized variance (RV). The leading rough models can be viewed as autoregressive processes driven by fractional Gaussian noise. We show that the Wold representation of this noise concentrates its dependence at the first lag when the Hurst parameter is below one half. Augmenting the autoregressive (AR) and heterogeneous autoregressive (HAR) models with a first-order moving-average (MA(1)) component therefore approximates the roughness, and the MA coefficient maps almost linearly into the Hurst parameter. We refer to these extensions as the "rough" AR and "rough" HAR models. Estimating them on the log RV of ten ETFs, we find negative MA coefficients for every asset, and the implied Hurst parameters align closely with the estimates from the continuous-time models. In the HAR literature, the negative MA(1) component is a significant feature that has been largely overlooked. In out-of-sample comparisons, the "rough" models outperform their classical counterparts for nearly every asset and horizon, with the largest gains at short horizons, and their accuracy is comparable to that of the rough continuous-time models but much easier to estimate by standard off-the-shelf software.

Problem

• Rough Models
• Realized Variance
• Hurst Parameter
• Fractional Gaussian Noise
• Moving-Average Component

Innovation

• Rough Models
• Hurst Parameter
• MA(1) Component
• Discrete-Time Approximations
• Realized Variance