A multivariate fractional Brownian motion (mfBm) with component-wise Hurst
exponents is used to model and forecast realized volatility. We investigate the
interplay between correlation coefficients and Hurst exponents and propose a
novel estimation method for all model parameters, establishing consistency and
asymptotic normality of the estimators. Additionally, we develop a
time-reversibility test, which is typically not rejected by real volatility
data. When the data-generating process is a time-reversible mfBm, we derive
optimal forecasting formulae and analyze their properties. A key insight is
that an mfBm with different Hurst exponents and non-zero correlations can
reduce forecasting errors compared to a one-dimensional model. Consistent with
optimal forecasting theory, out-of-sample forecasts using the time-reversible
mfBm show improvements over univariate fBm, particularly when the estimated
Hurst exponents differ significantly. Empirical results demonstrate that
mfBm-based forecasts outperform the (vector) HAR model.

Cet article explore les excursions dans le temps et leurs implications.

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