Linear Models and Time-Series Analysis

A comprehensive and timely edition on an emerging new trend in time series

Linear Models and Time-Series Analysis: Regression, ANOVA, ARMA and GARCH sets a strong foundation, in terms of distribution theory, for the linear model (regression and ANOVA), univariate time series analysis (ARMAX and GARCH), and some multivariate models associated primarily with modeling financial asset returns (copula-based structures and the discrete mixed normal and Laplace). It builds on the author's previous book, Fundamental Statistical Inference: A Computational Approach, which introduced the major concepts of statistical inference. Attention is explicitly paid to application and numeric computation, with examples of Matlab code throughout. The code offers a framework for discussion and illustration of numerics, and shows the mapping from theory to computation.

The topic of time series analysis is on firm footing, with numerous textbooks and research journals dedicated to it. With respect to the subject/technology, many chapters in Linear Models and Time-Series Analysis cover firmly entrenched topics (regression and ARMA). Several others are dedicated to very modern methods, as used in empirical finance, asset pricing, risk management, and portfolio optimization, in order to address the severe change in performance of many pension funds, and changes in how fund managers work.
* Covers traditional time series analysis with new guidelines
* Provides access to cutting edge topics that are at the forefront of financial econometrics and industry
* Includes latest developments and topics such as financial returns data, notably also in a multivariate context
* Written by a leading expert in time series analysis
* Extensively classroom tested
* Includes a tutorial on SAS
* Supplemented with a companion website containing numerous Matlab programs
* Solutions to most exercises are provided in the book

Linear Models and Time-Series Analysis: Regression, ANOVA, ARMA and GARCH is suitable for advanced masters students in statistics and quantitative finance, as well as doctoral students in economics and finance. It is also useful for quantitative financial practitioners in large financial institutions and smaller finance outlets.

Preface xiii
Part I Linear Models: Regression and ANOVA 1
1 The Linear Model 3
1. 1 Regression, Correlation, and Causality 3
1. 2 Ordinary and Generalized Least Squares 7
1. 2. 1 Ordinary Least Squares Estimation 7
1. 2. 2 Further Aspects of Regression and OLS 8
1. 2. 3 Generalized Least Squares 12
1. 3 The Geometric Approach to Least Squares 17
1. 3. 1 Projection 17
1. 3. 2 Implementation 22
1. 4 Linear Parameter Restrictions 26
1. 4. 1 Formulation and Estimation 27
1. 4. 2 Estimability and Identi ability 30
1. 4. 3 Moments and the Restricted GLS Estimator 32
1. 4. 4 Testing With h = 0 34
1. 4. 5 Testing With Nonzero h 37
1. 4. 6 Examples 37
1. 4. 7 Con dence Intervals 42
1. 5 Alternative Residual Calculation 47
1. 6 Further Topics 51
1. 7 Problems 56
1. A Appendix: Derivation of the BLUS Residual Vector 60
1. B Appendix: The Recursive Residuals 64
1. C Appendix: Solutions 66
2 Fixed E ects ANOVA Models 77
2. 1 Introduction: Fixed, Random, and Mixed E ects Models 77
2. 2 Two Sample t-Tests for Di erences in Means 78
2. 3 The Two Sample t-Test with Ignored Block E ects 84
2. 4 One-Way ANOVA with Fixed E ects 87
2. 4. 1 The Model 87
2. 4. 2 Estimation and Testing 88
2. 4. 3 Determination of Sample Size 91
2. 4. 4 The ANOVA Table 93
2. 4. 5 Computing Con dence Intervals 97
2. 4. 6 A Word on Model Assumptions 103
2. 5 Two-Way Balanced Fixed E ects ANOVA 107
2. 5. 1 The Model and Use of the Interaction Terms 107
2. 5. 2 Sums of Squares Decomposition without Interaction 108
2. 5. 3 Sums of Squares Decomposition with Interaction 113
2. 5. 4 Example and Codes 117
3 Introduction to Random and Mixed E ects Models 127
3. 1 One-Factor Balanced Random E ects Model 128
3. 1. 1 Model and Maximum Likelihood Estimation 128
3. 1. 2 Distribution Theory and ANOVA Table 131
3. 1. 3 Point Estimation, Interval Estimation, and Signi cance Testing 137
3. 1. 4 Satterthwaite's Method 139
3. 1. 5 Use of SAS 142
3. 1. 6 Approximate Inference in the Unbalanced Case 143
3. 1. 6. 1 Point Estimation in the Unbalanced Case 144
3. 1. 6. 2 Interval Estimation in the Unbalanced Case 150
3. 2 Crossed Random E ects Models 152
3. 2. 1 Two Factors 154
3. 2. 1. 1 With Interaction Term 154
3. 2. 1. 2 Without Interaction Term 157
3. 2. 2 Three Factors 157
3. 3 Nested Random E ects Models 162
3. 3. 1 Two Factors 162
3. 3. 1. 1 Both E ects Random: Model and Parameter Estimation 162
3. 3. 1. 2 Both E ects Random: Exact and Approximate Con dence Intervals 167
3. 3. 1. 3 Mixed Model Case 170
3. 3. 2 Three Factors 174
3. 3. 2. 1 All E ects Random 174
3. 3. 2. 2 Mixed: Classes Fixed 176
3. 3. 2. 3 Mixed: Classes and Subclasses Fixed 177
3. 4 Problems 177
3. A Appendix: Solutions 178
Part II Time-Series Analysis: ARMAX Processes 185
4 The AR(1) Model 187
4. 1 Moments and Stationarity 188
4. 2 Order of Integration and Long-Run Variance 195
4. 3 Least Squares and ML Estimation 196
4. 3. 1 OLS Estimator of a 196
4. 3. 2 Likelihood Derivation I 196
4. 3. 3 Likelihood Derivation II 198
4. 3. 4 Likelihood Derivation III 198
4. 3. 5 Asymptotic Distribution 199
4. 4 Forecasting 200
4. 5 Small Sample Distribution of the OLS and ML Point Estimators 204
4. 6 Alternative Point Estimators of a 208
4. 6. 1 Use of the Jackknife for Bias Reduction 208
4. 6. 2 Use of the Bootstrap for Bias Reduction 209
4. 6. 3 Median-Unbiased Estimator 211
4. 6. 4 Mean-Bias Adjusted Estimator 211
4. 6. 5 Mode-Adjusted Estimator 212
4. 6. 6 Comparison 213
4. 7 Con dence Intervals for a 215
4. 8 Problems 219
5 Regression Extensions: AR(1) Errors and Time-varying Parameters 223
5. 1 The AR(1) Regression Model and the Likelihood 223
5. 2 OLS Point and Interval Estimation of a 225
5. 3 Testing a = 0 in the ARX(1) Model  229
5. 3. 1 Use of Con dence Intervals 229
5. 3. 2 The Durbin-Watson Test 229
5. 3. 3 Other Tests for First-order Autocorrelation 231
5. 3. 4 Further Details on the Durbin-Watson Test 236
5. 3. 4. 1 The Bounds Test, and Critique of Use of p-Values 236
5. 3. 4. 2 Limiting Power as a ± 1 239
5. 4 Bias-Adjusted Point Estimation 243
5. 5 Unit Root Testing in the ARX(1) Model 246
5. 5. 1 Null is a = 1 248
5. 5. 2 Null is a < 1 256