Quantitative Methods in High-Frequency Financial Econometrics: Modeling Univariate and Multivariate Time Series Zur Erlangung des akademischen Grades eines Doktors der Wirtschaftswissenschaften (Dr. rer. pol.) von der Fakult¨atf¨ur Wirtschaftswissenschaften der Universit¨atKarlsruhe (TH) genehmigte DISSERTATION von M. Sc. Wei Sun Tag der m¨undlichen Pr¨ufung:30.11.2007 Referent: Prof. Dr. Svetlozar T. Rachev Korreferent: Prof. Dr. Andreas Geyer-Schulz Karlsruhe (2007) 2 Contents Acknowledgments 15 Preface 17 1 Introduction 19 2 Mining High-Frequency Financial Data 23 2.1 High-Frequency Financial Data . 25 2.2 Information Extraction and Knowledge Discovery . 27 2.2.1 Informative Data . 28 2.2.2 Data Quality . 31 2.2.3 Aggregation . 31 2.2.4 Data Cleaning . 33 2.2.5 Data Snooping . 34 2.2.6 Pattern Recognition . 35 2.3 Computational Data Mining . 37 2.3.1 Cluster Analysis . 37 2.3.2 K-Nearest-Neighbour Method . 41 2.3.3 Neural Networks . 42 2.3.4 Wavelet Analysis . 46 2.3.5 Other Methods . 47 2.4 Statistical Data Mining . 49 2.4.1 Robustness . 49 2.4.2 Visualization . 50 2.4.3 Standard Models . 50 3 4 CONTENTS 2.4.4 Nonparametric Methods . 51 2.5 Evaluation of Data Mining Methods . 52 2.5.1 Criteria Based on Statistical Goodness-of-fit Techniques . 53 2.5.2 Criteria Based on Score Functions . 55 2.5.3 Criteria Based on Loss Functions . 56 2.5.4 Criteria Based on Bayesian Methods . 57 2.5.5 Criteria Based on Computational Methods . 58 3 High-Frequency Financial Econometrics 61 3.1 Mechanisms in Economic Settings . 62 3.2 Formation of Market Price . 63 3.3 Transparency of the Market . 63 3.4 Liquidity of the Market . 64 3.5 Volatility of the Market . 65 3.6 Pattern Recognition and Stylized Facts . 68 3.6.1 Random Durations . 70 3.6.2 Distributional Properties of Returns . 70 3.6.3 Autocorrelation . 70 3.6.4 Seasonality . 71 3.6.5 Clustering . 71 3.6.6 Long-range Dependence . 72 4 Long Range Dependence and Fractal Processes 73 4.1 Estimation and Detection of LRD in the Time Domain . 73 4.1.1 The Rescaled Adjusted Range Approach . 73 4.1.2 ARFIMA Model . 75 4.1.3 Variance-Type Method . 76 4.1.4 Absolute Moments Method . 77 4.2 Estimation and Detection of LRD in the Frequency Domain . 78 4.2.1 Periodogram Method . 78 4.2.2 Whittle-Type Methods . 78 4.3 Econometric Modeling of LRD . 80 CONTENTS 5 4.3.1 GARCH-Type Extension . 80 4.3.2 Stochastic Volatility Type Extension . 81 4.3.3 Unit Root Type Extension . 81 4.3.4 Regime Switching Type Extension . 82 4.4 Fractal Processes and Long-Range Dependence . 82 4.4.1 Specification of the Fractal Processes . 82 4.4.2 Estimation of Fractal Processes . 84 4.4.3 Simulation of Fractal Processes . 87 4.4.4 Implications of Fractal Processes . 88 5 Modeling Univariate High-Frequency Time Series I 91 5.1 Introduction . 91 5.2 Specification of the self-similar processes . 93 5.2.1 Fractional Gaussian noise . 94 5.2.2 Fractional stable noise . 94 5.3 Empirical analysis . 95 5.3.1 Data and Methodology . 96 5.3.2 Preliminary Test . 97 5.3.3 Results . 99 5.4 Conclusions . 100 6 Modeling Univariate High-Frequency Time Series II 109 6.1 Introduction . 109 6.2 Point processes in modeling durations . 112 6.3 Empirical study . 115 6.3.1 The data . 116 6.3.2 The methodology of finding the best model . 116 6.4 Results . 117 6.4.1 Preliminary Tests . 117 6.4.2 Goodness of fit test . 120 6.5 Conclusions . 122 7 Modeling Multivariate High-Frequency Time Series I 133 6 CONTENTS 7.1 Introduction . 133 7.2 Unconditional copulas and tail dependence . 135 7.2.1 Definition of unconditional copulas and tail dependence . 135 7.2.2 Test of tail dependence . 137 7.3 Data and empirical methodology . 139 7.3.1 Data . 139 7.3.2 Empirical methodology . 141 7.4 Analysis of the marginal distribution . 142 7.4.1 The self-similarity parameter . 143 7.4.2 Specification of the self-similar processses . 143 7.4.3 Estimation of the self-similarity parameter . 144 7.4.4 The parameters of a stable Non-Gaussian distribution . 146 7.5 Simulating the co-movement of international equity markets . 147 7.5.1 Simulation of the marginal distribution . 147 7.5.2 Simulation of the multi-dimensional copulas . 148 7.6 Empirical results . 149 7.7 Conclusion . 152 8 Modeling Multivariate High-Frequency Time Series II 165 8.1 Introduction . 165 8.2 Skewed Student’s t Copula . 168 8.2.1 Multivariate skewed Student’s t distribution . 168 8.2.2 Simulation Algorithm . 170 8.3 L´evy Processes with Specifications . 171 8.3.1 L´evyprocessses . 171 8.3.2 L´evyStable Distribution . 172 8.3.3 Fractional Brownian Motion . 173 8.3.4 L´evyStable Motion . 173 8.4 Data and empirical methodology . 175 8.4.1 Data . 175 8.4.2 Empirical methodology . 176 8.4.3 Empirical results . 178 CONTENTS 7 8.5 Conclusions . ..