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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 abbreviations 9 List of symbols 11 List of figures 15 List of tables 17 1 Introduction 19 2 The concept of multifractal volatility 23 2.1 The Efficient Markets Hypothesis . . . . . . . . . . . . . . . . . 23 2.2 Stylized facts of financial time series . . . . . . . . . . . . . . . . 24 2.3 Classic models of financial volatility . . . . . . . . . . . . . . . . 25 2.4 Multifractal volatility models . . . . . . . . . . . . . . . . . . . 27 3 The Multifractal Random Walk model 31 4 The estimation procedure 39 4.1 The iterated GMM estimation . . . . . . . . . . . . . . . . . . . 39 4.2 The HAC covariance matrix estimation . . . . . . . . . . . . . . 41 4.3 The numerical minimization procedure . . . . . . . . . . . . . . 43 4.4 The initialization method for the parameter ln (σ) . . . . . . . . 45 5 Finite sample behaviour of the estimation procedure 47 5.1 The Monte Carlo simulation study . . . . . . . . . . . . . . . . 47 5.2 The estimation performance . . . . . . . . . . . . . . . . . . . . 48 5.3 The performance of the initialization method for ln (σ) . . . . . 51 5.4 The robustness of the estimation procedure . . . . . . . . . . . . 52 6 Statistical testing procedures 57 6.1 The Wald test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 8 Contents 6.2 The multifractality test . . . . . . . . . . . . . . . . . . . . . . . 59 7 Empirical study 63 8 The efficient computation of HAC estimators 67 8.1 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 8.2 Computational gains . . . . . . . . . . . . . . . . . . . . . . . . 69 9 Conclusion 73 A Empirical study of the stylized facts of financial time series 77 B QQ plots 83 C The asymptotic distribution of the statistic M 87 D MATLAB code 93 References 97

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