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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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6 Statistical testing procedures 57

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Chapter 6 Statistical testing procedures In Chapter 4 we introduced a new iterated GMM estimator for the MRW, with an enhanced moments function and an efficient initialization procedure for the variance parameter. This estimator is robust to variations in the starting values for λ2 and ln (T ) and allows for the estimation of the parameters intermittency coefficient λ2 and logarithmic standard deviation ln (σ) with good performance in finite samples. Yet the logarithmic decorrelation scale ln (T ) can only be reliably estimated from very large datasets with the sample sizeN much greater than T (N ≫ T ). In this chapter we discuss some statistical testing procedures for the MRW model. Traditional hypothesis tests for the GMM framework can naturally be applied to the MRW, e.g. tests about the values of the model parameters using the Wald statistic. These are outlined in the following section. A particular interest is accorded to the intermittency coefficient λ2, which gives information about the multifractal property of the data-generating pro- cess. In this connection it is interesting to investigate whether the process is multifractal, i.e. whether λ2 is greater than zero. 6.1 The Wald test This section outlines the testing methodology for hypotheses about the values of the model parameters. To simplify matters we will use in the following the example of the parameter λ2, whereas similar tests can be constructed for ln (T ) and ln (σ). For a a positive real number, we test the hypotheses H0 : λ 2 = a vs. H1 : λ2 = a 58 Chapter 6. Statistical...

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