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Part One Signal Extraction and Likelihood Inference for Linear UC Models 1
1. Introduction 3
1 The Linear State Space Form 3
2 Alternative State Space Representations and Extensions 3
3 The Kalman Filter 4
4 Prediction 5
5 Initialisation and Likelihood Inference 7
6 Smoothing Algorithms 9
6.1 Cross-validatory and auxiliary residuals 10
6.2 Smoothing splines and non parametric regression 10
2. Prediction Theory for Autoregressive-Moving Average Processes 14
Peter Burridge and Kenneth F. Wallis
1 Introduction 14
2 Two Leading Examples 16
2.1 Forecasting the ARMA(1,1) process 16
2.2 Extracting an AR(1) signal masked by white noise 20
3 State-Space Methods and Convergence Conditions 23
3.1 The state-space form and the Kalman filter 23
3.2 Conditions for convergence of the covariance sequence 27
4 Forecasting the ARMA(p, q) Process 29
4.1 Setting up the problem 29
4.2 The invertible moving average case 31
4.3 Moving average with roots on the unit circle 32
4.4 Moving average with roots outside the unit circle 33
5 Signal Extraction in Unobserved-Component ARMA Models 34
5.1 Setting up the problem 34
5.2 The stationary case 37
5.3 The non-stationary detectable case 39
5.4 The non-detectable case 41
6 Discussion 42
Appendix 44
Notes 45
References 46
3. Exact Initial Kalman Filtering and Smoothing for Nonstationary Time Series Models 48
Siem Jan Koopman
1 Introduction 48
2 The Exact Initial Kalman Filter 51
2.1 The nonsingular and univariate case 53
2.2 Automatic collapse to Kalman filter 53
3 Exact Initial Smoothing 54
4 Log-Likelihood Function and Score Vector 54
5 Some Examples 56
5.1 Local-level component model 56
5.2 Local linear trend component model 56
5.3 Common-level component model 57
6 Miscellaneous Issues 58
6.1 Computational costs 59
6.2 Missing values 60
6.3 Numerical performance 60
7 Conclusions 61
Appendix 63
References 66
4. Smoothing and Interpolation with the State-Space Model 68
Piet De Jong
1 Introduction 68
2 The State-Space Model, Kalman Filtering, and Smoothing 69
3 A New Smoothing Result 70
3.1 Fixed-interval smoothing 71
3.2 Classic fixed-interval smoothing 71
3.3 Fixed-point smoothing 71
3.4 Fixed-lag smoothing 72
3.5 Covariances between smoothed estimates 72
4 Signal Extraction 72
5 Interpolation 73
6 Diffuse Smoothing 74
Appendix 74
References 75
5. Diagnostic Checking of Unobserved-Components Time Series Models 77
Andrew C. Harvey and Siem Jan Koopman
1 Properties of Residuals in Large Samples 78
1.1 Local level 79
1.2 Local linear trend 80
1.3 Basic structural model 81
Contents ix
2 Finite Samples 83
2.1 Relationship between auxiliary residuals 84
2.2 Algorithm 85
3 Diagnostics 85
3.1 Tests based on skewness and kurtosis 86
3.2 Monte Carlo experiments 88
4 Miscellaneous Issues 89
4.1 Tests for serial correlation 89
4.2 Residuals from the canonical decomposition 90
4.3 Explanatory variables 91
5 Applications 91
5.1 U.S. exports to Latin America 91
5.2 Car drivers killed and seriously injured in Great Britain 92
5.3 Consumption of spirits in the United Kingdom 93
6 Conclusions 97
Appendix 97
References 98
6. Nonparametric Spline Regression with Autoregressive Moving Average Errors 100
Robert Kohn, Craig F. Ansley and Chi-Ming Wong
1 Introduction 100
2 Penalized Least Squares and Signal Extraction 102
3 Parameter Estimation 104
3.1 Maximum likelihood parameter estimation 104
3.2 Parameter estimation by cross-validation 105
4 Unequally Spaced Observations 106
5 Performance of Function Estimators: Simulation Results 107
6 Examples 110
Appendix 112
References 113
Part Two Unobserved Components in Economic Time Series 115
7. Introduction 117
1 Trends and Cycles in Economic Time Series 117
2 The Hodrick–Prescott Filter 119
3 Canonical Decomposition 121
4 Estimation and Seasonal Adjustment in Panel Surveys 123
5 Seasonality in Weekly Data 124
8. Univariate Detrending Methods with Stochastic Trends 126
Mark W. Watson
1 Introduction 126
x Contents
2 The Model 128
3 Estimation Issues 130
4 Univariate Examples 134
4.1 GNP 135
4.2 Disposable income 140
4.3 Non-durable consumption 142
5 Regression Examples 144
6 Concluding Remarks 146
Notes 147
References 148
9. Detrending, Stylized Facts and the Business Cycle 151
A. C. Harvey and A. Jaeger
1 Introduction 151
2 The Trend Plus Cycle Model 152
3 The Hodrick–Prescott Filter 153
4 Macroeconomic Time Series 155
5 Further Issues 160
5.1 Seasonality 160
5.2 ARIMA methodology and smooth trends 161
5.3 Segmented trends 164
5.4 Spurious cross-correlations between detrended series 164
6 Conclusions 167
Notes 168
References 169
10. Stochastic Linear Trends: Models and Estimators 171
Agustín Maravall
1 Introduction: the Concept of a Trend 171
2 The General Statistical Framework 173
3 Some Models for the Trend Component 175
4 A Frequently Encountered Class of Models 178
5 Extensions and Examples 182
6 The MMSE Estimator of the Trend 184
7 Some Implications for Econometric Modeling 191
8 Summary and Conclusions 196
References 197
11. Estimation and Seasonal Adjustment of Population Means Using Data from Repeated Surveys 201
Danny Pfeffermann
1 State-Space Models and the Kalman Filter 202
2 Basic Structural Models for Repeated Surveys 204
Contents xi
2.1 System equations for the components of
the population mean 204
2.2 Observation equations for the survey estimators 206
2.3 A compact model representation 208
2.4 Discussion 209
3 Accounting for Rotation Group Bias 210
4 Estimation and Initialization of the Kalman Filter 211
5 Simulation and Empirical Results 213
5.1 Simulation results 213
5.2 Empirical results using labour force data 218
6 Concluding Remarks 221
References 222
12. The Modeling and Seasonal Adjustment ofWeekly Observations 225
Andrew Harvey, Siem Jan Koopman and Marco Riani
1 The Basic Structural Time Series Model 227
1.1 Trigonometric seasonality 228
1.2 Dummy-variable seasonality 228
1.3 Weekly data 229
2 Periodic Effects 230
2.1 Trigonometric seasonality 230
2.2 Periodic time-varying splines 231
2.3 Intramonthly effects 232
2.4 Leap years 232
3 Moving Festivals: Variable-Dummy Effects 233
4 Statistical Treatment of the Model 233
5 U.K. Money Supply 235
6 Conclusions 242
Appendix A 243
Appendix B 248
References 249
Part Three Testing in Unobserved Components Models 251
13. Introduction 253
1 Stationarity and Unit Roots Tests 253
2 Seasonality 256
3 Multivariate Stationarity and Unit Root Tests 257
4 Common Trends and Co-integration 258
5 Structural Breaks 259
Notes 259
xii Contents
14. Testing for Deterministic Linear Trend in Time Series 260
Jukka Nyblom
1 Introduction 260
2 Test Statistics 261
3 Eigenvalues of Z′WZ 264
4 Asymptotic Distributions and Efficiency 266
5 Asymptotic Moment-Generating Functions 268
6 Conclusions and Extensions 270
References 270
15. Are Seasonal Patterns Constant Over Time? A Test for Seasonal Stability 272
Fabio Canova and Bruce E. Hansen
1 Regression Models with Stationary Seasonality 275
1.1 Regression equation 275
1.2 Modeling deterministic seasonal patterns 276
1.3 Lagged dependent variables 277
1.4 Estimation and covariance matrices 277
2 Testing for Seasonal Unit Roots 278
2.1 The testing problem 278
2.2 The hypothesis test 279
2.3 Joint test for unit roots at all seasonal frequencies 281
2.4 Tests for unit roots at specific seasonal frequencies 282
3 Testing for Nonconstant Seasonal Patterns 283
3.1 The testing problem 283
3.2 Testing for instability in an individual season 283
3.3 Joint test for instability in the seasonal intercepts 284
4 A Monte Carlo Experiment 286
4.1 First seasonal model 288
4.2 Second seasonal model 292
5 Applications 293
5.1 U.S. post World War II macroeconomic series 293
5.2 European industrial production 297
5.3 Monthly stock returns 298
6 Conclusions 299
References 300
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