Durham E-Theses Essays on the nonlinear and nonstochastic nature of stock market data Vorlow, Constantine Euripides How to cite: Vorlow, Constantine Euripides (2002) Essays on the nonlinear and nonstochastic nature of stock market data, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/4154/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 The copyright of this thesis rests with the author. No quotation from it should be published without his prior written consent and information derived from it should be acknowledged. THE UNIVERSITY OF DURHAM SCHOOL OF ECONOMICS, FINANCE AND BUSINESS Essays on the Nonlinear and Nonstochastic Nature of Stock Market Data Constantine Euripides Vorlow Submitted for the Qualification of Ph.D. in Finance September 2002 1 8 JUN 2003 Supervisor: Prof. Antonios Antoniou External Examiners: Prof. Lucio Sarno and Dr. Turalay Kenc © Copyright 2002 by Constantine Euripides Vorlow All Rights Reserved ii CONTENTS LIST OF TABLES vii LIST OF FIGURES ix ACKNOWLEDGMENT xv ABSTRACT xvi 1. Introduction 1 1.1 Chaos and Markets 1 1.2 Information and noise 3 1.3 Complexity and Chaos 4 1.4 Mandelbrot, chaos and finance 6 1.5 About this thesis 8 1.5.1 Software and data 10 1.6 Concluding remarks 11 2. Literature review 12 2.1 Introduction 12 2.2 The basics 12 2.2.1 The early literature 14 2.2.1.1 The EMH in developed and developing markets ... 15 2.3 Bubbles, fads, anomalies and overreaction 16 2.3.1 Asset-pricing models and "anomalies" 20 2.4 A new approach: chaotic determinism 21 2.5 Wavelets in finance 26 2.6 Conclusions 29 3. Statistical processes for asset prices 32 3.1 Introduction 32 3.2 Statistical properties of asset returns 33 3.3 Martingales and Random Walks 35 3.4 Self-similarity, Affinity, Scale Invariance and Fractals 38 3.5 Multifractal and 1// processes 40 iii 3.6 Levy processes 43 3.7 Long memory processes 43 3.8 Conclusions 44 4. Dynamical systems 46 4.1 Introduction 46 4.2 Dynamical systems and Chaos 47 4.2.0.1 Definition of Chaos and stylised facts 51 4.2.0.2 Chaotic attractors 53 4.2.1 Ergodic Theory 55 4.3 Phase Space Reconstruction 58 4.3.1 Invariant measures 59 4.3.1.1 Euclidian Dimension d and Similarity dimension Ds 59 4.3.1.2 Hausdorff Dimension DH 60 4.3.1.3 Capacity (Kolmogorov) Dimension D0 and Box-counting dimension 61 4.3.1.4 Information Dimension D\ 61 4.3.1.5 Correlation dimension D2 63 4.3.1.6 Kolmogorov Entropy 64 4.3.1.7 Lyapunov exponents 65 4.3.2 How to reconstruct the phase space 68 4.3.2.1 Time-delay embedding 70 4.3.3 Phase-space reconstruction: some practical issues 71 4.3.3.1 Average Mutual Information 72 4.3.3.2 False Nearest Neighbours 73 4.3.3.3 Other approaches 74 4.4 Conclusions 75 5. Recurrence Analysis 77 5.1 Introduction 77 5.1.1 Close Returns plots 78 5.2 RPs and phase-space reconstruction 79 5.2.1 RPs and correlation dimension 82 5.3 Recurrence Plots 83 5.4 Recurrence plots and Market Efficiency 88 5.4.1 "Zooming in" 94 iv 5.4.2 Recurrence Quantification Analysis 97 5.5 Conclusions 101 6. Surrogate Data Analysis 103 6.1 Introduction 103 6.2 Previous research 104 6.3 Why use Surrogate Data? 107 6.4 The general SDA hypothesis testing framework 110 6.4.1 SDA Hypothesis and Statistics Ill 6.4.1.1 Types of null Hypothesis 112 6.4.1.2 The SDA discriminating statistics 114 6.5 Types of Surrogates 116 6.5.1 Simple shuffled surrogate data 117 6.5.2 Fourier Transform based surrogate data 117 6.5.3 Amplitude Adjusted Fourier transformed surrogate data . .119 6.5.4 Corrected AAFT surrogates 120 6.5.5 ARM A based surrogates 120 6.5.6 Pseudo-periodic surrogates 122 6.6 SDA applications 123 6.6.1 Simulations 124 6.6.2 SDA with financial time series 127 6.6.2.1 Theiler's "crinkle" statistic 131 6.6.2.2 Time reversibility 134 6.7 Conclusions 137 7. Wavelet Theory 139 7.1 Introduction 139 7.1.1 The basic framework 140 7.2 The Continuous and Discrete Wavelet Transforms 146 7.3 Definitions 148 7.3.1 A brief history of wavelets 151 7.4 Wavelets vs Spectral analysis 152 7.4.1 Fourier Series and transforms 153 7.5 Criteria and Properties of Wavelet Functions 161 7.6 Multiresolution Analysis 164 v 7.7 Wavelet Regression 166 7.8 Wavelet Shrinkage-Thresholding 169 7.8.1 Thresholding functions and rules 171 7.9 A simple wavelet: The Haar 174 7.10 The Maximal Overlap Discrete Wavelet Transform 176 7.11 Conclusions 178 8. Time-frequency analysis 180 8.1 Introduction 180 8.1.1 Previous research 181 8.1.2 Datasets and their descriptive statistics 182 8.2 A time-frequency approach: the CWT scalograms 185 8.3 The DWT of the FTSE 198 8.3.1 Multiresolution Analysis 199 8.3.2 Waveshrinking the FTSE 205 8.4 Nonlinear Determinism 211 8.4.1 Exploring dynamics via lag and phase plots 218 8.4.2 Searching for nonlinear dependence 226 8.4.3 Recurrence Quantification Analysis 229 8.5 Conclusions 235 9. Conclusions 237 9.1 What can we deduce from our results 237 9.2 Chaos, Self-organised Criticality and the future 242 9.2.1 Self organised criticality 244 9.2.2 Investigating the causes of chaos and the future of research . 246 9.3 Conclusion 249 vi LIST OF TABLES 5.1 Time Delays and Embedding Dimensions for the 6 time series 90 5.2 FTSE returns RQA results from RP in 5.10 (b) 98 6.1 The first 20 lags of the autocorrelation function of a simulated AR(1) process and the bias of the surrogate data ACF statistic. 50 phase- amplitude adjusted FT surrogate sequences were generated 129 6.2 Theiler's "Crinkle" statistic SDA results on returns sequences. 50 ph• ase randomised (FT) amplitude adjusted surrogates ("polished") where generated for each test. The test is one (right) sided t-student based which means that for a 5% significance level, 19 surrogates would suf• fice. We can reject the null for all variables at a = 2% except for US (NYSE at 6% and Dow-Jones at 10%) and Japan (at 4%). Therefore, evidence of nonlinearity is extremely strong in almost all of the returns sequences 133 6.3 Time reversibility SDA results on the levels of the each index and the corresponding returns sequences. 50 phase randomised (FT) amplitude adjusted surrogates ("polished") where generated for each test. The results here do not support strongly the rejection of the null as in table (6.2). Only the US markets and the Greek can support the rejection of the null for a low significance level 136 8.1 Descriptive statistics. Jarque-Bera p-values within parenthesis 184 8.2 Autocorrelation function coefficients with their corresponding Q-statistics and their probability values 185 8.3 Partial autocorrelation function coefficients with their corresponding Q-statistics and probability values 185 8.4 The 3 subsamples used in figures 8.9-8.11 189 8.5 Dates and positions of th 19th largest oscillations in the FTSE series as these are identified by the 19 largest DWT wavelet coefficients 189 8.6 Augmented Dickey-Fuller unit root test results with MacKinnon critical values for rejection of hypothesis of a unit root. It is obvious that even for 1% statistical significance level the Waveshrink residuals are a stationary process 206 vii 8.7 The parameters of the Waveshrink denoising process on the FTSE re• turns using the d6 wavelet 206 8.8 The descriptive statistics of the Waveshrink obtained processes 207 8.9 The first 20 values of the autocorrelation (ACF) and partial autocorre- lation(PACF) functions for the Waveshrink fitted values and residuals with the corresponding Q-statistic values 212 8.10 Kolmogorov-Smirnov (K-S) test of composite normality 212 8.11 Jarque-Bera (J-B) test of normality. The test is distributed as a x2 distribution with 2 degrees of freedom test. The critical value for a = 5% is 5.991 212 8.12 First 15 values of the Average Mutual Information (AMI) functions (see figure 8.24) of the actual FTSE returns, Waveshrink fits and Waveshrink residual sequences. Bold numbers indicate the 1st local minimums. An interesting finding is that both the returns and residual sequences exh• ibit a local minimum as well where the 1st minimum of the Waveshrink fit is (r = 12) 214 8.13 FNN algorithm results for the Waveshrink residuals 215 8.14 FNN algorithm results for the FTSE returns 216 8.15 FNN algorithm results for the Waveshrink fitted values 217 8.16 Maximal Lyapunov Exponent 227 8.17 Correlation dimension d for Waveshrink fit.