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Statistical Inference in Multifractal Random Walk Models for Financial Time Series

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Cristina Sattarhoff

The dynamics of financial returns varies with the return period, from high-frequency data to daily, quarterly or annual data. Multifractal Random Walk models can capture the statistical relation between returns and return periods, thus facilitating a more accurate representation of real price changes. This book provides a generalized method of moments estimation technique for the model parameters with enhanced performance in finite samples, and a novel testing procedure for multifractality. The resource-efficient computer-based manipulation of large datasets is a typical challenge in finance. In this connection, this book also proposes a new algorithm for the computation of heteroscedasticity and autocorrelation consistent (HAC) covariance matrix estimators that can cope with large datasets.

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List of symbols 11

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List of symbols bN bandwidth C (c) circulant matrix with the first column c dB (u) White Noise process in continuous time d mean distance d2 mean square distance f moments function f∗ the moments function in Bacry et al. (2008a, 2008b) F matrix of sample moments F (j : k) matrix containing rows j till k of F Ft public information at time t F ∗ extended matrix of sample moments, employed in the HAC algorithm introduced in this thesis F ∗j the jth column in matrix F ∗ h lag of the autocorrelation/autocovariance function i index j index k index k the number of volatility components in the MSM model ℓ sampling interval L variable employed in the HAC estimation algorithms mµ an MRW average employed in the multifractality test 12 List of symbols mσ an MRW average employed in the multifractality test M multifactality test statistic M (q, τ) empirical qth order absolute moment of returns with period τ N cascade step; sample size Pt price at time t q moment order; number of moment conditions employed in the GMM estimation Q set containing the finite moment orders q QN (θ) objective function in the GMM estimation rt returns between t− 1 and t in the classic financial volatility models r (dt,N) returns in dt at cascade step N r (t) returns between t− 1 and t in MV models r (t, ℓ) returns between t− 1 and t with sampling interval ℓ in MV models S asymptotic covariance matrix of the normalized sample mean...

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