MULTISCALE ANALYSIS OF HIGH FREQUENCY EXCHANGE RATE TIME SERIES by SANDRA ABOSEDE ABIOLA OGUNYA A thesis presented for the degree of Doctor of Philosophy of the University of London and the Diploma of Imperial College Department of Mathematics Imperial College London 180 Queens Gate London SW7 2BZ United Kingdom =yý °--ý, 'y .ý 2007 Declaration I certify that this thesis, and the research to which it refers, are the product of my own work, and that any ideas or quotations from the work of other people, published or otherwise, are fully acknowledged in accordance with the standard referencing practices of the discipline. ý Signed: 4z2ý, Copyright Copyright in text of this thesis rests with the Author. Copies (by any process) either in full, or of extracts, may be made only in accordance with instructions given by the Author and lodged in the doctorate thesis archive of the college central library. Details may be obtained from the Librarian. This page must form part of any such copies made. Further copies (by any process) of copies made in accordance with such instructions may not be made without the permission (in writing) of the Author. The ownership of any intellectual property rights which may be described in this thesis is vested in Imperial College, subject to any prior agreement to the contrary, and may not be made available for use by third parties without the written permission of the University, which will prescribe the terms and conditions of any such agreement. Further information on the conditions under which disclosures and exploitation may take place is available from the Imperial College registry. 3 Abstract The aim of this thesis is to introduce a new methodology for the modelling and analysis of high frequency seasonal time series. We propose a general modelling methodology that involves the usage of the class of cyclostationary processes and multiscale analysis methods. In particular, we concentrate on a (much studied) data set consisting of five minute Deutsche Mark-US Dollar exchange rate returns. A cyclostationary process has a well defined period. Such a process can be represented as stationary across periods and non-stationary within periods. Wavelet analysis is an ideal multiscale tool for modelling this non-stationary behaviour as the discrete wavelet transform is able to decompose a time series into components, representing structure associated with both time and scale. In the first part of this thesis, we detect cyclostationarity in the data set and look at the use of wavelet analysis in the modelling of seasonal time series. Secondly, we de- velop general linear and non-stationary models for cyclostationary seasonal time series. Finally, we provide a general multiscale algorithm that attempts to capture and simu- late the correlation and distributional properties of high frequency seasonal time series. This thesis, therefore develops methodology for the modelling and analysis of seasonal processes using multiscale methods. 4 Acknowledgements Firstly, I would like to thank my supervisors Dr. S. Olhede and Prof. A. T. Walden for their excellent encouragement and generosity of time. Thanks to members of the statistics department, in particular to Dr. E. McCoy and Dr. N. Adams for their helpful comments on my transfer report. I would also like to thank the Engineering and Physical Sciences Research Council for their financial support by means of a Research Studentship. Finally, but not least, I am eternally grateful to my family and friends for supporting me throughout my PhD and I am especially indebted to Harry MacDonald. J Contents Declaration 2 Copyright 3 Abstract 4 Acknowledgements 5 List of Figures 10 List of Tables 13 List of Notation 14 1 Introduction 16 1.1 Outline Thesis 16 of ................................ 2 Time Series Analysis of Financial and Econometric Data 18 2.1 Time Series Analysis 18 .............................. 2.2 Financial Econometric Time Series 19 and ................... 2.3 Empirical Foreign Exchange (FX) Data 21 ................... 2.4 Empirical Analysis Features FX Data 25 and of ................ 2.5 Summary 30 .................................... 3 Mathematical Modelling and Inference for Time Series 31 3.1 Stochastic Process Models 32 ........................... 3.2 Stationary Stochastic Process Models 34 .................... 3.3 Spectral Analysis 35 ................................ 6 CONTENTS 7 3.4 Estimation Weakly Stationary Processes 36 of ................. 3.4.1 Estimation the 37 of mean ........................ 3.4.2 Estimation the 38 of autocovariance sequence .............. 3.4.3 The Periodogram 39 ............................ 3.4.4 Tapering 40 ................................ 3.4.5 A Simple Multitaper Spectral Estimate of the Spectral Density Function 41 ................................. 3.5 Wavelet Analysis 43 ................................ 3.6 Summary 45 .................................... 4 Literature Review 46 Econometric Financial Time Series Models 46 4.1 and ............... 4.1.1 Parametric Models 46 ........................... 4.1.2 Non-parametric Model 48 ......................... Modelling High Frequency Financial Time Series 50 4.2 .............. 4.2.1 Modelling High Frequency FX Returns 51 ................ 4.2.2 The Modelling of High Frequency Financial and Econometric Data 53 4.3 Modelling Non-stationary Time Series 54 .................... Periodically Correlated Cyclostationary Models 54 4.3.1 or ......... 4.3.2 Locally Stationary Models 55 ....................... 4.4 Wavelets Financial Time Series 56 and ...................... Wavelets Multiscale for Seasonal Time Series 56 4.5 and methods ......... 5 Loe ve Spectral Analysis 60 5.1 Cyclostationarity 60 ................................ Spectral Theory Harmonizable Processes 62 5.2 of ................. 5.2.1 Spectral Representation Theory 62 .................... Spectral Representation Stationary Processes 64 5.2.2 of ........... Representation Periodically Correlated Processes 65 5.2.3 Spectral of ... 5.3 Hurd Gerr's Procedure 66 and .......................... 5.3.1 Estimation the Loeve Spectrum 67 of .................. 5.3.2 Locve Spectral Coherence 68 ....................... Simplifying Loeve Spectral Representations 69 5.3.3 Interpreting and ... CONTENTS 8 5.4 Inference for Cyclostationary Time Series 70 .................. 5.4.1 Spectral Density Estimates Rn 72 of ................... 5.4.2 Detecting Cyclostionarity in Rn 75 .................... 5.5 Summary 81 .................................... 6 Wavelet Decomposition of a Cyclostationary Process 85 6.1 DWT Decomposition Seasonal Time Series 85 of a ............... 6.1.1 Partial DWT 92 .............................. 6.1.2 Neccessary Sufficient Condition for Cyclostationarity 93 and ..... 6.1.3 Practical Considerations 94 ........................ 6.2 Li Hinich's Model 95 and ............................. 6.2.1 Persistent Waveforms 95 ......................... 6.2.2 A Seasonal Time Series Model 97 .................... 6.2.3 Wavelet Decomposition the FX 99 of returns series .......... 6.3 Summary 100 .................................... 7 Modelling Periodic Correlation 103 7.1 Uniform Modulation 104 .............................. 7.1.1 Evolutionary Spectrum of Uniformly Modulated Stationary Pro- 105 cesses .................................. 7.1.2 Evolutionary Spectra for Uniformly Modulated Cyclostationary Processes 107 ................................ 7.2 Modelling Cyclostationary Time Series 112 .................... 7.2.1 Motivation 112 ............................... 7.2.2 General Model 113 ............................. 7.2.3 Uniform Periodic Modulation Stationary Processes (Class A) 115 of .. 7.2.4 Uniform Periodic Modulation of Cyclostationary Processes (Class 116 .................... 7.2.5 Special Case of Class A and Class B................. 119 7.2.6 Harmonic Series Representation PC Process 120 of ........... 7.3 Statis tical Properties the Class A C Models 122 of and ............. 7.3.1 Statistical Properties the Class A Model 123 of ............. 7.3.2 Statistical Properties the Class C Model 125 of ............. CONTENTS 9 7.4 Statistical Properties of the Wavelet Coefficients obtained from the Cy- Models 128 clostationary .............................. 7.4.1 Wavelet Covariance Properties of Class A.............. 130 7.4.2 Wavelet Correlation Matrices Simulated Time Series 135 of some ... 7.4.3 Distribution the { W. t }M of sample of rvs . i. 140 7.5 Summary 143 .................................... 8 Modelling Returns using Wavelets and Cyclostationarity 145 8.1 Motivation 145 .................................. 8.2 Modelling High Frequency Financial Returns 147 ............... 8.2.1 Detecting Jumps 147 ........................... 8.2.2 Missing Data 160 ............................. 8.2.3 The Model 167 .............................. 8.2.4 Estimating the Time-Varying Standard Deviation the Model 168 of . 8.2.5 Reducing the Dimensionality the Model 170 of ............. 8.3 Multiscale Simulation High Frequency Log Absolute Returns 171 of ..... 8.3.1 Matching the Correlation Distributional Properties 173 and ..... 8.3.2 Simulation Algorithm for the log 185 absolute returns series ..... 8.4 Conclusion 187 .................................. 9 Conclusions 191 9.1 Original Contributions Findings 191 and ..................... 9.2 Recommendations For Further Research 193 ................... Bibliography 194 List of Figures 2.1 DM-$ five 26 minute returns rn .......................... 2.2 Average and standard deviation of rn for each