Contents
Preface xi
1 Introduction 1
1.1 Basic Description 1
1.2 Simple Descriptive Techniques 5
1.2.1 Trends 5
1.2.2 Seasonal Cycles 8
1.3 Transformations 9
1.4 Example 9
1.5 Conclusions 13
1.6 Exercises 13
2 Probability Models 15
2.1 Introduction 15
2.2 Stochastic Processes 15
2.3 Examples 17
2.4 Sample Correlation Function 18
2.5 Exercises 20
3 Autoregressive Moving Average Models 23
3.1 Introduction 23
3.2 Moving Average Models 23
3.3 Autoregressive Models 25
vii
viii CONTENTS
5.5.1 Duality between Causality
and Stationarity 26
3.3.2 Asymptotic Stationarity 28
3.3.3 Causality Theorem 28
3.3.4 Covariance Structure of AR Models 29
3.4 ARMA Models 32
3.5 ARIMA Models 33
3.6 Seasonal ARIMA 35
3.7 Exercises 36
4 Estimation in the Time Domain 39
4.1 Introduction 39
4.2 Moment Estimators 39
4.3 Autoregressive Models 40
4.4 Moving Average Models 42
4.5 ARMA Models 43
4.6 Maximum Likelihood Estimates 44
4.7 Partial ACF 47
4.8 Order Selections 49
4.9 Residual Analysis 53
4.10 Model Building 53
4.11 Exercises 54
5 Examples in SPLUS 59
5.1 Introduction 59
5.2 Example 1 59
5.3 Example 2 62
5.4 Exercises 68
6 Forecasting 69
6.1 Introduction 69
6.2 Simple Forecasts 70
6.3 Box and Jenkins Approach 71
6.4 Treasury Bill Example 73
6.5 Recursions 77
6.6 Exercises 77
7 Spectral Analysis 79
7.1 Introduction 79
7.2 Spectral Representation Theorems 79
7.3 Periodogram 83
7.4 Smoothing of Periodogram 85
7.5 Conclusions 89
7.6 Exercises 89
CONTENTS ix
8 Nonstationarity 93
8.1 Introduction 93
8.2 Nonstationarity in Variance 93
8.3 Nonstationarity in Mean: Random Walk
with Drift 94
8.4 Unit Root Test 96
8.5 Simulations 98
8.6 Exercises 99
9 Heteroskedasticity 101
9.1 Introduction 101
9.2 ARCH 102
9.3 GARCH 105
9.4 Estimation and Testing for ARCH 107
9.5 Example of Foreign Exchange Rates 109
9.6 Exercises 116
10 Multivariate Time Series 117
10.1 Introduction 117
10.2 Estimation of m and G 121
10.3 Multivariate ARM A Processes 121
10.3.1 Causality and Invertibility 122
10.3.2 Identifiability 123
10.4 Vector AR Models 124
10.5 Example of Inferences for VAR 127
10.6 Exercises 135
11 State Space Models 137
11.1 Introduction 137
11.2 State Space Representation 137
11.3 Kalman Recursions 140
11.4 Stochastic Volatility Models 142
11.5 Example of Kalman Filtering of Term Structure 144
11.6 Exercises 150
12 Multivariate GARCH 153
12.1 Introduction 153
12.2 General Model 154
12.2.1 Diagonal Form 155
12.2.2 Alternative Matrix Form 156
12.3 Quadratic Form 156
12.3.1 Single-Factor GARCH(1,1) 156
12.3.2 Constant-Correlation Model 157
x CONTENTS
12.4 Example of Foreign Exchange Rates 157
12.4.1 The Data 158
12.4.2 Multivariate GARCH in SPLUS 158
12.4.3 Prediction 166
12.4-4 Predicting Portfolio Conditional
Standard Deviations 167
12.4.5 BEKK Model 168
12.4-6 Vector-Diagonal Models 169
12.4.7 ARM A in Conditional Mean 170
12.5 Conclusions 171
12.6 Exercises 171
13 Cointegrations and Common Trends 173
13.1 Introduction 173
13.2 Definitions and Examples 174
13.3 Error Correction Form 177
13.4 Granger's Representation Theorem 179
13.5 Structure of Cointegrated Systems 183
13.6 Statistical Inference for Cointegrated Systems 184
13.6.1 Canonical Correlations 184
13.6.2 Inference and Testing 186
13.7 Example of Spot Index and Futures 188
13.8 Conclusions 193
13.9 Exercises 193
References 195
Index 201