arXiv subjects

M. Karanasos

Publications and source records attributed to M. Karanasos.

3 recordsLinked to original sources

arXiv2021

A unified theory for ARMA models with varying coefficients: One solution fits all

For the large family of ARMA models with variable coefficients (TV-ARMA), either deterministic or stochastic, we provide an explicit and computationally tractable representation based on the general solution of the associated linear difference equation. Analogous representations are established for the fundamental properties of such processes, including the Wold-Cramér decomposition and their covariance structure as well as explicit optimal linear forecasts based on a finite set of past observations. These results are grounded on the principal determinant, that is a banded Hessenbergian representation of a restriction of the Green function involved in the solution of the linear difference equation associated with TV-ARMA models, built up solely of the autoregressive coefficients of the model. The $L_2$ convergence properties of the model are a consequence of the absolute summability of the aforementioned Hessenbergian representation, which is in line with the asymptotic stability and efficiency of such processes. The invertibility of the model is also a consequence of an analogous condition, but now the Green function is built up of the moving average coefficients. The structural asymmetry between constant and deterministically time-varying coefficient models, that is the backward and forward asymptotic efficiency differ in an essential manner, is formally demonstrated. An alternative approach to the Hessenbergian solution representation is described by an equivalent procedure for manipulating time-varying polynomials. The practical significance of the theoretical results in this work is illustrated with an application to U.S. inflation data. The main finding is that inflation persistence increased after 1976, whereas from 1986 onwards the persistence declines and stabilizes to even lower levels than the pre-1976 period.

math.ST↗

arXiv2021

Explicit and compact representations for the Green's function and the Solution of Linear Difference Equations with variable coefficients

Leibniz' combinatorial formula for determinants is modified to establish a condensed and easily handled compact representation for Hessenbergians, referred to here as Leibnizian representation. Alongside, the elements of a fundamental solution set associated with linear difference equations with variable coefficients of order $p$ are explicitly represented by $p$ banded Hessenbergian solutions, built up solely of the variable coefficients. This yields banded Hessenbergian representations for the elements both of the product of companion matrices and of the determinant ratio formula of the one-sided Green's function (Green's function for short). Combining the above results, the elements of the foregoing notions are endowed with compact representations formulated here by Leibnizian and nested sum representations. We show that the elements of the fundamental solution set can be expressed in terms of the first banded Hessenbergian fundamental solution, called principal determinant function. We also show that the Green's function coincides with the principal determinant function, when both functions are restricted to a fairly large domain. These results yield, an explicit and compact representation of the Green's function restriction along with an explicit and compact solution representation of the previously stated type of difference equations in terms of the variable coefficients, the initial conditions and the forcing term. The equivalence of the Green's function solution representation and the well known single determinant solution representation is derived from first principles. Algorithms and automated software are employed to illustrate the main results of this paper.

math.FA↗

arXiv2014

A closed form to the general solution of linear difference equations with variable coefficients

The determinant of a lower Hessenberg matrix (Hessenbergian) is expressed as a sum of signed elementary products indexed by initial segments of nonnegative integers. A closed form alternative to the recurrence expression of Hessenbergians is thus obtained. This result further leads to a closed form of the general solution for regular order linear difference equations with variable coefficients, including equations of N-order and equations of ascending order.

math.FA↗