Wildi Signal Extraction

Efficient Estimation, 'Unit Root'-Tests and Early Detection of Turning Points
1. Auflage 2005
ISBN: 978-3-540-26916-8
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 547, 279 Seiten, eBook

Reihe: Lecture Notes in Economics and Mathematical Systems

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.

Zielgruppe

Research

Autoren/Hrsg.

Wildi, Marc

Inhaltsverzeichnis

Theory.- Model-Based Approaches.- QMP-ZPC Filters.- The Periodogram.- Direct Filter Approach (DFA).- Finite Sample Problems and Regularity.- Empirical Results.- Empirical Comparisons : Mean Square Performance.- Empirical Comparisons : Turning Point Detection.- Conclusion.

Leseproben

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.