Signal Extraction (eBook)

Foreword 7
Contents 9
Part I Theory 12
1 Introduction 14
1.1 Overview 14
1.2 A General Model- Based- Approach 18
1.3 An Identification Problem 21
1.4 The Direct Filter Approach 23
1.5 Summary 25
2 Model-Based Approaches 28
2.1 Introduction 28
2.2 The Beveridge-Nelson Decomposition 29
2.3 The Canonical Decomposition 30
2.3.1 An Illustrative Example 31
2.3.2 The Airline-Model 35
2.3.3 An Example 40
2.3.4 The Revision Error Variance 44
2.3.5 Concluding Remarks 46
2.4 Structural Components Model 47
2.5 CENSUS X-12-ARIMA 50
3 QMP-ZPC Filters 56
3.1 Filters : Definitions and Concepts 56
3.2 A Restricted ARMA Filter Class : QMP-filters 62
3.3 ZPC-Filters 65
4 The Periodogram 76
4.1 Spectral Decomposition 76
4.2 Convolution Theorem 80
4.3 The Periodogram for Integrated Processes 87
5 Direct Filter Approach (DFA) 102
5.1 Overview 103
5.2 Consistency (Stationary MA-Processes) 105
5.3 Consistency (Integrated Processes) 113
5.4 Conditional Optimization 123
5.5 Efficiency 126
5.6 Inference Under 'Conditional' Stationarity 131
5.6.1 The Asymptotic Distribution of the Parameters of the 'Linearized' DFA 132
5.6.2 Spurious Decrease of the Optimization Criterion 138
5.6.3 Testing for Parameter Constraints 140
5.7 Inference : Unit-Roots 140
5.7.1 I(l)-Process 141
5.7.2 I(2)-Process 154
5.8 Links Between the DFA and the MBA 156
6 Finite Sample Problems and Regularity 158
6.1 Regularity and Overfitting 159
6.2 Filter Selection Criterion 162
6.2.1 Overview 162
6.2.2 The MC-Criterion 163
6.3 Cross-Validation 165
6.4 A Singularity-Penalty 166
6.5 Variable Frequency Sampling 170
Part II Empirical Results 176
7 Empirical Comparisons: Mean Square Performance 178
7.1 General Framework 178
7.2 A Simulation Study 180
7.2.1 Airline-Model 181
7.2.2 'Quasi'-Airline Model 187
7.2.3 Stationary Input Signals 190
7.2.4 Conclusions 193
7.3 'Real-World' Time Series 197
7.3.1 Mean-Square Approximation of the 'Ideal' Trend 200
7.3.2 Mean-Square Approximation of the 'Canonical Trend' 213
7.3.3 Mean Square Approximation of the 'Canonical Seasonal Adjustment' Filter 217
8 Empirical Comparisons: Turning Point Detection 224
8.1 Turning Point Detection for the 'Ideal' Trend 225
8.1.1 Series Linearized by TRAMO 226
8.1.2 Series Linearized by X-12-ARIMA 230
8.2 Turning Point Detection for the Canonical Trend 233
9 Conclusion 236
A Decompositions of Stochastic Processes 240
A.1 Weakly Stationary Processes of Finite Variance 240
A.1.1 Spectral Decomposition and Convolution Theorem 240
A. 1.2 The Wold Decomposition 242
A.2 Non-Stationary Processes 244
B Stochastic Properties of the Periodogram 246
B.I Periodogram for Finite Variance Stationary Processes 246
B.2 Periodogram for Infinite Variance Stationary Processes 254
B.2.1 Moving Average Processes of Infinite Variance 254
B.2.2 Autocorrelation Function, Normalized Spectral Density and (Self) Normalized Periodogram 255
B.3 The Periodogram for Integrated Processes 257
C A 'Least-Squares' Estimate 266
C.I Asymptotic Distribution of the Parameters 266
C.2 A Generalized Information Criterion 277
D Miscellaneous 280
D.1 Initialization of ARMA-Filters 280
E Non-Linear Processes 282
References 286

1.5 Summary (p.14-16)

For economic time series, interesting signals are often seasonally adjusted components or trends, see chapter 2 (recall that component definitions depend on strong a priori assumptions, see section 1.3). An efficient and general signal estimation method is needed for these important applications because economic time series are characterized by randomness (the DGP is not deterministic) and complex dynamics. Moreover, 'typical' users are often interested in signal estimates for time points near the upper boundary t = N1. Consequently, filters are heavily asymmetric so that efficient estimation methods are required. A new method, the DFA, is presented here. The book is organized as follows:

• In chapter 2, model-based approaches are presented. The aim is not to provide an exhaustive list of existent methods but to describe established procedures which are implemented in 'widely used' software packages. The objective is to compare the DFA to established MBA.

• The main concepts needed for the description of filters in the frequency domain (such as transfer functions, amplitude functions or phase functions) are proposed in chapter 3. A new filter class (ZPC-filters) is derived whose characteristics 'match' the signal estimation problem.

• For the DFA, an eminent role is awarded to the periodogram (or to statistics directly related to the periodogram). It 'collects' and transforms the information of the sample XI,...,XN into a form suitable for the signal estimation problem. Therefore, properties of the periodogram and technical details related to the DFA are analyzed in chapter 4. In particular, the statistic is analyzed for integrated processes. Stochastic properties of squared periodogram ordinates are analyzed in the appendix. Both kind of results are omitted in the 'traditional' time series literature and are needed here for proving theoretical results in chapter 5. An explorative instrument for assessing possible 'unit-root misspecification' of the filter design for the DFA is proposed also.

• The main theoretical results for the DFA are reported in chapter 5: the consistency, the efficiency, the generalization to non-stationary integrated input processes, the generalized conditional optimization (resulting in asymmetric filters with smaller time delays) and the asymptotic distribution of the estimated filter parameters (which enables hypothesis testing). In particular, a generalized unit-root test is proposed which is designed for the signal estimation problem.

• In order to prove the results in chapter 5, regularity assumptions are needed. One of these assumptions is directly related to finite sample issues (overfitting problem). Therefore, the overfitting problem is analyzed in chapter 6. Overparameterization and overfitting are distinguished and new procedures are proposed for 'tackling' their various aspects. An estimation of the order of the asymmetric filter is presented (which avoids more specifically overparameterization), founding on the asymptotic distribution of the parameter estimates. The proposed method does not rely on 'traditional' information criteria, because the DGP of Xt is not of immediate concern. However, it is shown in the appendix that 'traditional' information criteria (like AIC for example) may be considered as special cases of the proposed method. Also, new procedures ensuring regularity of the DFA solution are proposed which solve specific overfitting problems. The key idea behind these new methods is to modify the original optimization criterion such that overfitting becomes 'measurable'. It is felt that these ideas may be useful also when modelling the DGP for the MBA.

• Empirical results which are based on the simulation of artificial processes (1(2), 1(1) and stationary processes) and on a 'real-world' time series are presented in chapter 7. The DFA is compared with the MBA with respect to mean square performances. It is shown that the DFA performs as well as maximum likelihood estimates for artificial times series. If the DGP is unknown, as is the case for the 'real-world' time series, the DFA outperforms two established MBA, namely TRAMO/SEATS and CENSUS X-12-ARIMA (see chapter 2 for a definition). The increased performance is achieved with respect to various signal definitions (two different trend signals and a particular seasonal adjustment) both 'in' and 'out of sample'. It is also suggested that statistics relying on the one-step ahead forecasts, like 'traditional' unit-root tests (augmented Dickey-Fuller and Phillips- Perron tests) or diagnostic tests (like for example Ljung-Box tests) may be misleading for the signal estimation problem if the true DGP is unknown. Instead, specific instruments derived in chapters 4, 5 and 6 are used for determining the optimal filter design for the DFA. These instruments, which are based on estimated filter errors (rather than one-step ahead forecasting errors of the model), indicate smaller integration orders for the analyzed time series (1(1)- instead of I(2)-processes as 'proposed' by the majority of the unit-root tests). A possible explanation for these differences may be seen in the fact that filter errors implicitly account for one- and multi-step ahead forecasts simultaneously. A further analysis of the revision errors (filter approximation errors) suggests that the I(2)-hypothesis should be rejected indeed.