CONTENTS
DEDICATION v
ACKNOWLEDGMENTS
PREFACE xxi
xix
INTRODUCTION
 1 Fourier versus Wavelet Analysis
.2 Seasonality Filtering
,3 Denoising
.4 Identification of Structural Breaks
,5 Scaling
.6 Aggregate Heterogeneity and Timescales
.7 Multiscale Cross-Correlation
.8 Outline

2
LINEAR FILTERS
2.1Introduction
2.2 Filters in Time Domain
2.2.1Infinite Impulse Response (IIR) Filters
2.2.2Noncausal Finite Impulse Response (FIR) Filters
2.2.3Causal FIR Filters
2.3 Filters in the Frequency Domain
2.3.1Frequency Response
2.3.2Low-Pass and High-Pass Filters
2.4 Filters in Practice
2.4.1 The EWMA and Volatility Estimation
2.4.2 The Hodrick-Prescott Filter
2.4.3 The Baxter-King (BK) Filter
2.4.4 Filters in Technical Analysis of Financial Markets
3 OPTIMUM LINEAR ESTIMATION
3. I Introduction
3.2 The Wiener Filter and Estimation
3.2.1 Example: Real Wage Estimation
3.2.2 Signal-to-Noise Ratio
3.2.3 Comments on Wiener Filtering and Estimation
3.2.4 Pitfalls of the Sample Autocorrelation and Cross-
Correlation
3.3 Recursive Filtering and the Kalman Filter
3.3.1 Recursive Mean Estimation
3.3.2 The Kalman Filter and Estimation
3.4 Prediction with the Kalman Filter
3.4.1 Convergence of Kalman Gain
3.4.2 Example: Adaptive Expectations
3.5 Vector Kalman Filter Estimation
3.5.1 Time-Varying Coefficients in a Regression
3.5.2 An Autoregressive Model
3.5.3 A Simple Vector Autoregression
3.5.4 Vector Kalman Filter Prediction
3.5.5 Summary: The Kalman Filter
3.6 Applications
3.6.1 Scalar Kalman Filter Simulation
3.6.2 Vector Kalman Filter Simulation
3.6.3 Bayesian Vector Autoregression
3.6.4 Time-Varying Beta Estimation
CONTENTS ix
4
DISCRETE WAVELET TRANSFORMS
4. I Introduction
4.1.1 Fourier Transform
4.1.2 Short-Time Fourier Transform
4.1.3 Wavelet Transform
4.2 Properties of the Wavelet Transform
4.2.1 Continuous Wavelet Functions
4.2.2 Continuous versus Discrete Wavelet Transform
4.3 Discrete Wavelet Filters
4.3.1 Haar Wavelets
4.3.2 Daubechies Wavelets
4.3.3 Minimum Bandwidth Discrete-Time Wavelets
4.4 The Discrete Wavelet Transform
4.4.1 Implementation of the DWT: Pyramid Algorithm
4.4.2 Partial Discrete Wavelet Transform
4.4.3 Multiresolution Analysis
4.4.4 Analysis of Variance
4.4.5 Example: IBM Returns
4.4.6 Example: An Overlapping Generations Model
4.5 The Maximal Overlap Discrete Wavelet Transform
4.5. I Definition
4.5.2 Multiresolution Analysis
4.5.3 Analysis of Variance
4.5.4 Example: IBM Stock Prices
4.5.5 Example: AR(1) with Seasonalities
4.6 Practical Issues in Implementation
4.6.1 Selecting a Wavelet Basis
4.6.2 Nondyadic Length Time Series
4.6.3 Boundary Conditions
4.7 Applications
4.7.1 Filtering FX Intraday Seasonalities
4.7.2 Causality and Cointegration in Economics
4.7.3 Money Growth and Inflation

5
WAVELETS
AND STATIONARY PROCESSES
5. I Introduction
5.2 Wavelets and Long-Memory Processes
5.2.1 Fractional Difference Processes
5.2.2 The DWT of Fractional Difference Processes
CONTENTS
$.2.3 Simulation of Fractional Difference Processes 167
5.2.4 Ordinary Least-Squares Estimation of Fractional Difference
Processes 170
5.2.5 Approximate Maximum Likelihood Estimation of Fractional
Difference Processes 172
5,2.6 Example: IBM Stock Prices 174
Generalizations of the DWT and MODWT 176
5.3.1 The Discrete Wavelet Packet Transform 176
5.3,2 The Maximal Overlap Discrete Wavelet Packet Transform 179
5.3.3 Example: Mexican Money Supply 180
Wavelets and Seasonal Long Memory 183
5.4.1 Seasonal Persistent Processes 183
5.4.2 Simulation of Seasonal Persistent Processes 185
5.4.3 Basis Selection Procedures 188
5.4.4 Ordinary Least-Squares Estimation of Seasonal Persistent
Processes 189
5.4.5 Approximate Maximum Likelihood Estimation of Seasonal
Persistent Processes 192
Applications 194
5.5.1 Mexican Money Supply 194
5.5.2 Japanese GNR Seasonality, and Trends 196
5.5.3 U.S. Unemployment, Consumer Price Index, and Tourism
Revenues 198
6
WAVELET DENOISING
6. I Introduction
6.2 Nonlinear Denoising via Thresholding
6.2.1 Hard Thresholding
6.2.2 Soft Thresholding
6.2.3 Other Thresholding Rules
6.3 Threshold Selection
6.3.1 Universal Thresholding
6.3.2 Minimax Estimation
6.3.3 Stein's Unbiased Risk Estimate
6.3.4 Hypothesis Testing
6.3.5 Bayesian Methodology
6.3.6 Cross-Validation
6.4 Implementing Wavelet Denoising
6.4.1 Standard Denoising
6.4.2 Translation-Invariant Denoising
6.5 Applications
CONTENTS
6.5.1 IBM Stock Prices
6.5.2 IBM Stock Returns
6.5.3 IBM Volatility
6.5.4 Outlier Testing

7
WAVELETS FOR VARIANCE-COVARIANCE
ESTIMATION
7. I Introduction
7.2 The Wavelet Variance
7.2. I Estimating the Wavelet Variance
7.2.2 Confidence Intervals for the Wavelet Variance
7.2.3 Example: Simulated AR(1) Model
7.2.4 Example: IBM Stock Prices
7.3 Testing Homogeneity of Variance
7,3. I Locating a Variance Change
7.3.2 Example: IBM Stock Prices
7,3.3 Extension: Multiple Variance Changes
7.4 The Wavelet Covariance and Cross-Covariance
7.4. I Estimation
7.4.2 Confidence Intervals
7.4.3 Example: Monthly Foreign Exchange Rates
7.5 The Wavelet Correlation and Cross-Correlation
7.5. I Estimation
7.5.2 Confidence Intervals
7.5.3 Example: Monthly Foreign Exchange Rates
7.6 Applications
7.6.1 Scaling Laws in FX Markets
7.6.2 Multiscale Beta Estimation
7.7 Univariate and Bivariate Spectrum Analysis
7.7.1 Univariate Spectrum Analysis
7.7.2 Univariate Spectrum Estimation
7.7.3 Equivalent Degrees of Freedom for a Spectral Estimator
7.7.4 Bivariate Spectrum Analysis
7.7.5 Bivariate Spectrum Estimation 

8
ARTIFICIAL NEURAL NETWORKS
8. I Introduction
8.2 Activation Functions
8.2.1 Deterministic Activation Functions
8.3 Eeedforward Networks
8.3.1 Example: IBM Stock Price and Volatility Prediction
8.4 Recurrent Networks
8.4.1 Output-Recurrent Model
8.4.2 Hidden-Recurrent Model
8.4.3 Output-Hidden Recurrent Model
8.5 Network Selection
8.5.1 Information Theoretic Criteria
8.5.2 Cross-Validation
8.5.3 Bayesian Regularization
8.5.4 Early Stopping
8.5.5 Bagging
8.6 Adaptivity
8.7 Estimation of Recurrent Networks
8.7.1 Extended Kalman Filter for Recurrent Networks
8.7.2 Multistream Training for Recurrent Networks
8.7.3 An Example: IBM Volatility Prediction
8.8 Applications of Neural Network Models
8.8.1 Option Pricing
8.8.2 Filtering, Adaptation, and Predictability in Foreign
Exchange Markets
NOTATIONS 315
BIBLIOGRAPHY 323
INDEX 349