Optimal Linear Filtering, Smoothing and Trend Extraction for Processes with Unit Roots and Cointegration

Thomakos, Dimitrios D.

doi:10.2139/ssrn.1113331

Cited by 10 publications

(15 citation statements)

References 29 publications

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“…Here, of course, the signal is a random walk, and unlike the signal in Example I, which is a finite dimensional stationary but singular process, the random walk signal is infinite dimensional and non‐stationary, an ultimately unpredictable process of orthogonal increments. See Thomakos (, b) for a number of results concerning the application of SSA to the random walk process.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

Signal Identification in Singular Spectrum Analysis

Khan

Poskitt

2016

Aus NZ J of Statistics

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This paper provides an information theoretic analysis of the signal identification problem in singular spectrum analysis. We present a signal-plus-noise model based on the Karhunen-Loève expansion and use this model to motivate the construction of a minimum description length criterion that can be employed to identify the dimension (rank) of the signal component. We show that under very general regularity conditions the criterion will identify the true signal dimension with probability one as the sample size increases. A by-product of this analysis is a procedure for selecting a window length consistent with the Whitney embedding theorem. The upshot is a modeling strategy that results in a specification that yields a signal-noise reconstruction that minimises mean squared reconstruction error. Empirical results obtained using simulated and real world data series indicate that theoretical properties presented in the paper are reflected in observed behaviour, even in relatively small samples, and that the minimum description length modeling strategy provides the practitioner with an effective addition to the SSA tool box.

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Section: Numerical Illustrationsmentioning

confidence: 99%

Signal Identification in Singular Spectrum Analysis

Khan

Poskitt

2016

Aus NZ J of Statistics

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“…business cycles) considerations and was related to the extraction of components of certain frequencies. Thomakos [22,24] has a line of research where he derives several new results of SSA-based analysis under the assumptions of time series that have a unit root and cointegration. His work expands that of Phillips [25][26][27] on smoothing and trend extraction for unit root processes.…”

Section: Literature Reviewmentioning

confidence: 99%

A review on singular spectrum analysis for economic and financial time series

Hassani

Thomakos

2010

Statistics and Its Interface

150

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In recent years Singular Spectrum Analysis (SSA), a relatively novel but powerful technique in time series analysis, has been developed and applied to many practical problems across different fields. In this paper we review recent developments in the theoretical and methodological aspects of the SSA from the perspective of analyzing and forecasting economic and financial time series, and also represent some new results. In particular, we (a) show what are the implications of SSA for the, frequently invoked, unit root hypothesis of economic and financial times series; (b) introduce two new versions of SSA, based on the minimum variance estimator and based on perturbation theory; (c) discuss the concept of causality in the context of SSA; and (d) provide a variety of simulation results and real world applications, along with comparisons with other existing methodologies.

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“…The RT‐SSA reconstructions will therefore be close to that based on the corresponding population ensemble model as given in and , and this will ultimately be manifest in a signal component

s^{d} false(t false) \leftrightarrow scriptH false({boldY}_{scriptS} false) = scriptH false({bold1}_{m} {bold1}_{m}^{⊤} boldY false) + o false(1 false)

. An explicit representation of the series corresponding to

scriptH false({bold1}_{m} {bold1}_{m}^{⊤} boldY false)

as an m point smoother is presented in Thomakos, (, section 3.1). Hence we find that, despite the values of YY ⊤ for each series being not too dissimilar from each other, and close to their common theoretical limiting value, the first step RT‐SSA reconstructions accurately reproduce the different singular components of each series.…”

Section: Numerical Illustrations IImentioning

confidence: 64%

“…The first component is a random walk, of course, a non‐stationary process of orthogonal increments, and in the structural times series literature the process in is referred to as a local level model. See Thomakos () for a number of results concerning the application of SSA to random walks.…”

Section: Numerical Illustrations Imentioning

confidence: 99%

On Singular Spectrum Analysis And Stepwise Time Series Reconstruction

Poskitt

2019

Journal Time Series Analysis

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This article provides a detailed statistical analysis of a new approach to singular spectrum analysis (SSA). It examines SSA constructed using re-scaled trajectories (RT-SSA) and presents a theoretical analysis of RT-SSA under very general conditions concerning the structure of the observed series. The spectral features of population ensemble models implicit in the large sample properties of RT-SSA are investigated, motivating a new time series modelling methodology based on a stepwise application of RT-SSA. The operation of the theoretical results is illustrated via numerical examples involving trend stationary and difference stationary processes, and a random walk with drift. An analysis of the S&P 500 index also serves as a vehicle to demonstrate the practical impact of the stepwise RT-SSA processing methodology.Note that diagonal averaging is a minimum norm linear idempotent operator in the sense that for any matrix A the Hankel matrix (A) minimizes the Frobenius norm of the approximation error, that is, ‖A−(A)‖ ≤ ‖A−H‖ for all conformable Hankel matrices H, and (A ± H) = (A) ± (H) = (A) ± H. After applying diagonal averaging the resulting Hankel matrices S and E = X − S give the signal-noise decomposition X = S + E of the trajectory matrix, and x(t) = s(t) + e(t), t = 1, … , N, yields the associated SSA signal-noise reconstruction of the original time series. The decomposition constructed via (3) and (4) thus defines a signal-plus-noise specification for x(t) with a k(m + 1) element parameter vector ( 1 , … , k , u ′ 1 , … , u ′ k ) ′ (a combined functional-structural model in the terminology of Kendal and Stuart, 1979, chapter 29).Experimental and empirical evidence indicating that SSA performs well relative to more conventional signal extraction techniques has lead to SSA being employed in various discipline areas including, geophysics, the analysis of electro-encephalograms and DNA micro-arrays, applications to economic and financial time series, and in wileyonlinelibrary.com/journal/jtsa 69 image processing, see for example the references in Jolliffe (2002, chapters 12.1 and 12.2), the contributions to the special edition of Statistics and It's Interface (Volume 3, Number 3, 2010), and for a comparative study of SSA and different techniques of trend evaluation in seasonally adjusted series Alexandrov et al. (2012). Despite this broad range of application, however, to this author's knowledge detailed probabilistic and statistical analyses of the theoretical underpinnings of SSA per se, divorced from any substantive practical question or other contextual information, are lacking in the time series analysis literature. The purpose of this article is to address this lacuna.It is well known from Wold's theorem that a stochastic process x(t) can be represented in a unique way as x(t) = y(t) + z(t) where y(t) and z(t) are subordinate to x(t), mutually uncorrelated, and y(t) is singular while z(t) is regular. 1 The spectral distribution of x(t) equals the sum of that of y(t) and z(t), and the Lebesgue deco...

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Optimal Linear Filtering, Smoothing and Trend Extraction for Processes with Unit Roots and Cointegration

Cited by 10 publications

References 29 publications

Signal Identification in Singular Spectrum Analysis

Signal Identification in Singular Spectrum Analysis

A review on singular spectrum analysis for economic and financial time series

On Singular Spectrum Analysis And Stepwise Time Series Reconstruction

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