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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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4 The estimation procedure 39

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Chapter 4 The estimation procedure This chapter addresses the estimation of the MRW parameters using an itera- ted generalized method of moments (GMM) approach. The originality of this estimation procedure consists in a new moments configuration which improves the performance of the GMM estimation in finite samples. Moreover, we in- troduce an effective initialization method for the variance parameter using a preliminary estimate. 4.1 The iterated GMM estimation Let us denote with index 0 the unknown MRW parameters λ20, ln (T0) and ln (σ0), which we want to estimate from a time series Z (k) , k ∈ N of length N . Accordingly θ0 = ( λ20 ln (T0) ln (σ0) )′ will identify the unknown parameter vector. This is assumed to lie in a given set Θ ⊂ [0, 0.5)×R2.1 The GMM estimator θ̂N is the value of θ, which minimizes the deviation of the sample moments from their theoretical counterparts with respect to the quadratic norm: θ̂N = argmin θ∈Θ QN (θ) (4.1) with QN (θ) = ( 1 N N∑ k=1 f (Z (k) , θ)′ ) WN ( 1 N N∑ k=1 f (Z (k) , θ) ) . (4.2) 1 Without loss of generality we will choose Θ compact for the purpose of empirical ap- plications. 40 Chapter 4. The estimation procedure In this connection, the weighting matrix WN is positive semi-definite and is assumed to converge in probability to a positive definite matrix of constants W . The function f captures the moments employed in the estimation (Greene 2003, pp. 536-538). In this study we employ the moment conditions (3.5), (3.6) and (3.7) with lags...

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