Signal Extraction: Efficient Estimation, 'Unit Root'-Tests and Early Detection of Turning Points (Lecture Notes in Economics and Mathematical Systems, 547)

✍ Scribed by Marc Wildi

• Publisher: Springer
• Year: 2004
• Tongue: English
• Leaves: 283
• Edition: 2005
• Category: Library

✦ Synopsis

The material contained in this book originated in interrogations about modern practice in time series analysis. • Why do we use models optimized with respect to one-step ahead foreca- ing performances for applications involving multi-step ahead forecasts? • Why do we infer 'long-term' properties (unit-roots) of an unknown process from statistics essentially based on short-term one-step ahead forecasting performances of particular time series models? • Are we able to detect turning-points of trend components earlier than with traditional signal extraction procedures? The link between 'signal extraction' and the first two questions above is not immediate at first sight. Signal extraction problems are often solved by su- ably designed symmetric filters. Towards the boundaries (t = 1 or t = N) of a time series a particular symmetric filter must be approximated by asymm- ric filters. The time series literature proposes an intuitively straightforward solution for solving this problem: • Stretch the observed time series by forecasts generated by a model. • Apply the symmetric filter to the extended time series. This approach is called 'model-based'. Obviously, the forecast-horizon grows with the length of the symmetric filter. Model-identification and estimation of unknown parameters are then related to the above first two questions. One may further ask, if this approximation problem and the way it is solved by model-based approaches are important topics for practical purposes? Consider some 'prominent' estimation problems: • The determination of the seasonally adjusted actual unemployment rate.

✦ Table of Contents

Contents
Part I Theory
1 Introduction
1.1 Overview
1.2 A General Model-Based-Approach
1.3 An Identification Problem
1.4 The Direct Filter Approach
1.5 Summary
2 Model-Based Approaches
2.1 Introduction
2.2 The Beveridge-Nelson Decomposition
2.3 The Canonical Decomposition
2.3.1 An Illustrative Example
2.3.2 The Airline-Model
2.3.3 An Example
2.3.4 The Revision Error Variance
2.3.5 Concluding Remarks
2.4 Structural Components Model
2.5 CENSUS X-12-ARIMA
3 QMP-ZPC Filters
3.1 Filters : Definitions and Concepts
3.2 A Restricted ARMA Filter Class : QMP-filters
3.3 ZPC-Filters
4 The Periodogram
4.1 Spectral Decomposition
4.2 Convolution Theorem
4.3 The Periodogram for Integrated Processes
4.3.1 Integrated Processes of Order One
4.3.2 The Periodogram for I(2)-Processes
5 Direct Filter Approach (DFA)
5.1 Overview
5.2 Consistency (Stationary MA-Processes)
5.3 Consistency (Integrated Processes)
5.4 Conditional Optimization
5.5 Efficiency
5.6 Inference Under ‘Conditional’ Stationarity
5.6.1 The Asymptotic Distribution of the Parameters of the ‘Linearized’ DFA
5.6.2 Spurious Decrease of the Optimization Criterion
5.6.3 Testing for Parameter Constraints
5.7 Inference : Unit-Roots
5.7.1 I(l)-Process
5.7.2 I(2)-Process
5.8 Links Between the DFA and the MBA
6 Finite Sample Problems and Regularity
6.1 Regularity and Overfitting
6.2 Filter Selection Criterion
6.2.1 Overview
6.2.2 The MC-Criterion
6.3 Cross-Validation
6.4 A Singularity-Penalty
6.5 Variable Frequency Sampling
Part II Empirical Results
7 Empirical Comparisons : Mean Square Performance
7.1 General Framework
7.2 A Simulation Study
7.2.1 Airline-Model
7.2.2 ‘Quasi’-Airline Model
7.2.3 Stationary Input Signals
7.2.4 Conclusions
7.3 ‘Real-World’ Time Series
7.3.1 Mean-Square Approximation of the ‘Ideal’ Trend
7.3.2 Mean-Square Approximation of the ‘Canonical Trend’
7.3.3 Mean Square Approximation of the ‘Canonical Seasonal Adjustment’ Filter
8 Empirical Comparisons : Turning Point Detection
8.1 Turning Point Detection for the ‘Ideal’ Trend
8.1.1 Series Linearized by TRAMO
8.1.2 Series Linearized by X-12-ARIMA
8.2 Turning Point Detection for the Canonical Trend
9 Conclusion
A Decompositions of Stochastic Processes
A.I Weakly Stationary Processes of Finite Variance
A.1.1 Spectral Decomposition and Convolution Theorem
A.1.2 The Wold Decomposition
A.2 Non-Stationary Processes
B Stochastic Properties of the Periodogram
B.I Periodogram for Finite Variance Stationary Processes
B.2 Periodogram for Infinite Variance Stationary Processes
B.2.1 Moving Average Processes of Infinite Variance
B.2.2 Autocorrelation Function, Normalized Spectral Density and (Self) Normalized Periodogram
B.3 The Periodogram for Integrated Processes
C A "Least-Squares" Estimate
C.I Asymptotic Distribution of the Parameters
C.2 A Generalized Information Criterion
D Miscellaneous
D.I Initialization of ARMA-Filters
E Non-Linear Processes
References