An inversion formula for a matrix polynomial about a (unit) root

Faliva, Mario; Zoia, Maria Grazia

doi:10.1080/03081081003685936

Cited by 4 publications

(1 citation statement)

References 5 publications

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“…Several authors have investigated the problem of the inversion of a matrix polynomial about a pole (see e.g., Avrachenkov et al (2001), Faliva and Zoia (2002), Faliva and Zoia (2003), Faliva and Zoia (2009), Faliva and Zoia (2011), Langenhop (1971), Franchi and Paruolo (2019b)) ever since.…”

Section: Introductionmentioning

confidence: 99%

Cointegrated Solutions of Unit-Root VARs: An Extended Representation Theorem

Faliva¹,

Zoia²

2021

Preprint

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This paper establishes an extended representation theorem for unit-root VARs. A specific algebraic technique is devised to recover stationarity from the solution of the model in the form of a cointegrating transformation. Closed forms of the results of interest are derived for integrated processes up to the 4-th order. An extension to higher-order processes turns out to be within the reach on an induction argument.

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Section: Introductionmentioning

confidence: 99%

Cointegrated Solutions of Unit-Root VARs: An Extended Representation Theorem

Faliva¹,

Zoia²

2021

Preprint

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Representation of I(1) and I(2) Autoregressive Hilbertian Processes

Beare

Seo

2019

Econom. Theory

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We develop versions of the Granger-Johansen representation theorems for I(1) and I(2) vector autoregressive processes that apply to processes taking values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series. An I(1) or I(2) solution to an autoregressive law of motion is obtained when the inverse of the autoregressive operator pencil has a pole of first or second order at one. We obtain a range of necessary and sufficient conditions for such a pole to be of first or second order. Cointegrating and attractor subspaces are characterized in terms of the behavior of the autoregressive operator pencil in a neighborhood of one.

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Cointegration and Representation of Cointegrated Autoregressive Processes in Banach Spaces

Seo

2022

Econom. Theory

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We extend the notion of cointegration for time series taking values in a potentially infinite dimensional Banach space. Examples of such time series include stochastic processes in $C[0,1]$ equipped with the supremum distance and those in a finite dimensional vector space equipped with a non-Euclidean distance. We then develop versions of the Granger–Johansen representation theorems for I(1) and I(2) autoregressive (AR) processes taking values in such a space. To achieve this goal, we first note that an AR(p) law of motion can be characterized by a linear operator pencil (an operator-valued map with certain properties) via the companion form representation, and then study the spectral properties of a linear operator pencil to obtain a necessary and sufficient condition for a given AR(p) law of motion to admit I(1) or I(2) solutions. These operator-theoretic results form a fundamental basis for our representation theorems. Furthermore, it is shown that our operator-theoretic approach is in fact a closely related extension of the conventional approach taken in a Euclidean space setting. Our theoretical results may be especially relevant in a recently growing literature on functional time series analysis in Banach spaces.

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An inversion formula for a matrix polynomial about a (unit) root

Cited by 4 publications

References 5 publications

Cointegrated Solutions of Unit-Root VARs: An Extended Representation Theorem

Cointegrated Solutions of Unit-Root VARs: An Extended Representation Theorem

Representation of I(1) and I(2) Autoregressive Hilbertian Processes

Cointegration and Representation of Cointegrated Autoregressive Processes in Banach Spaces

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