scholarly journals Stationary Threshold Vector Autoregressive Models

2018 ◽  
Vol 11 (3) ◽  
pp. 45 ◽  
Author(s):  
Galyna Grynkiv ◽  
Lars Stentoft
Keyword(s):  
Steady State ◽  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Threshold Models ◽  
Bernoulli Distribution ◽  
Vector Autoregressive Model ◽  
Economic Data ◽  
Vector Autoregressive Models ◽  
Vector Autoregressive ◽  
Necessary And Sufficient

This paper examines the steady state properties of the Threshold Vector Autoregressive model. Assuming that the trigger variable is exogenous and the regime process follows a Bernoulli distribution, necessary and sufficient conditions for the existence of stationary distribution are derived. A situation related to so-called “locally explosive models”, where the stationary distribution exists though the model is explosive in one regime, is analysed. Simulations show that locally explosive models can generate some of the key properties of financial and economic data. They also show that assessing the stationarity of threshold models based on simulations might well lead to wrong conclusions.

Approximating the stationary distribution of an infinite stochastic matrix

1991 ◽  
Vol 28 (1) ◽  
pp. 96-103 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Stationary Distribution ◽  
Numerical Approximation ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Finite System ◽  
Balance Equations ◽  
The Given ◽  
Steady State Balance

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.

A REPRESENTATION THEORY FOR A CLASS OF VECTOR AUTOREGRESSIVE MODELS FOR FRACTIONAL PROCESSES

2008 ◽  
Vol 24 (3) ◽  
pp. 651-676 ◽  
Author(s):  
SØren Johansen
Keyword(s):  
Representation Theory ◽  
Linear Combination ◽  
Unit Root ◽  
Autoregressive Model ◽  
Autoregressive Models ◽  
Vector Autoregressive Model ◽  
Order Zero ◽  
Vector Autoregressive Models ◽  
Vector Autoregressive

Based on an idea of Granger (1986, Oxford Bulletin of Economics and Statistics 48, 213–228), we analyze a new vector autoregressive model defined from the fractional lag operator 1 − (1 − L)d. We first derive conditions in terms of the coefficients for the model to generate processes that are fractional of order zero. We then show that if there is a unit root, the model generates a fractional process Xt of order d, d > 0, for which there are vectors β so that β‼Xt is fractional of order d − b, 0 < b ≤ d. We find a representation of the solution that demonstrates the fractional properties. Finally we suggest a model that allows for a polynomial fractional vector, that is, the process Xt is fractional of order d, β‼Xt is fractional of order d − b, and a linear combination of β‼Xt and ΔbXt is fractional of order d − 2b. The representations and conditions are analogous to the well-known conditions for I(0), I(1), and I(2) variables.

scholarly journals Multiple cycles and the Bautin bifurcation in the Goodwin model of a class struggle

2013 ◽  
Vol 18 (3) ◽  
pp. 265-274 ◽  
Author(s):  
Giovanni Bella
Keyword(s):  
Steady State ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Class Struggle ◽  
Necessary And Sufficient Conditions ◽  
Goodwin Model ◽  
Multiple Cycles ◽  
Bautin Bifurcation ◽  
Necessary And Sufficient

The aim of this paper is to present the necessary and sufficient conditions for the emergence of a generalized Hopf (i.e., Bautin) bifurcation in the Goodwin’s model of a class struggle, and determine the parameter regions where multiple attracting and repelling limit cycles around the steady state may coexist.

scholarly journals REPRESENTATION OF I(1) AND I(2) AUTOREGRESSIVE HILBERTIAN PROCESSES

2019 ◽  
Vol 36 (5) ◽  
pp. 773-802 ◽  
Author(s):  
Brendan K. Beare ◽  
Won-Ki Seo
Keyword(s):  
Separable Hilbert Space ◽  
Sufficient Conditions ◽  
General Setting ◽  
Operator Pencil ◽  
Second Order ◽  
Vector Autoregressive ◽  
Functional Time Series ◽  
Vector Autoregressive Processes ◽  
Necessary And Sufficient ◽  
Autoregressive Hilbertian Processes

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.

scholarly journals STRUCTURE-REVERSIBILITY OF A TWO-DIMENSIONAL REFLECTING RANDOM WALK AND ITS APPLICATION TO QUEUEING NETWORK

2014 ◽  
Vol 29 (1) ◽  
pp. 1-25 ◽  
Author(s):  
Masahiro Kobayashi ◽  
Masakiyo Miyazawa ◽  
Hiroshi Shimizu
Keyword(s):  
Random Walk ◽  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Queueing Network ◽  
Necessary And Sufficient Conditions ◽  
Negative Integer ◽  
Product Form ◽  
Two Dimensional ◽  
Necessary And Sufficient

We consider a two-dimensional reflecting random walk on the non-negative integer quadrant. It is assumed that this reflecting random walk has skip-free transitions. We are concerned with its time-reversed process assuming that the stationary distribution exists. In general, the time-reversed process may not be a reflecting random walk. In this paper, we derive necessary and sufficient conditions for the time-reversed process also to be a reflecting random walk. These conditions are different from but closely related to the product form of the stationary distribution.

A Theorem on the Exact Nonsimilar Steady-State Motions of a Nonlinear Oscillator

1992 ◽  
Vol 59 (2) ◽  
pp. 418-424 ◽  
Author(s):  
A. F. Vakakis ◽  
T. K. Caughey
Keyword(s):  
Steady State ◽  
Nonlinear Oscillator ◽  
General Theorem ◽  
Sufficient Conditions ◽  
Steady Motion ◽  
Strongly Nonlinear ◽  
Forcing Functions ◽  
Steady Motions ◽  
Necessary And Sufficient ◽  
Strongly Nonlinear Oscillator

In this work the steady-state motions of a nonlinear, discrete, undamped oscillator are examined. This is achieved by using the notion of exact steady state, i.e., a motion where all coordinates of the system oscillate equiperiodically, with a period equal to that of the excitation. Special forcing functions that are periodic but not necessarily harmonic are applied to the system, and its steady response is approximately computed by an asymptotic methodology. For a system with cubic nonlinearity, a general theorem is given on the necessary and sufficient conditions that a excitation should satisfy in order to lead to an exact steady motion. As a result of this theorem, a whole class of admissible periodic functions capable of producing steady motions is identified (in contrast to the linear case, where the only excitation leading to a steady-state motion is the harmonic one). An analytic expression for the modal curve describing the steady motion of the system in the configuration space is derived and numerical simulations of the steady-state motions of a strongly nonlinear oscillator excited by two different forcing functions are presented.

Approximating the stationary distribution of an infinite stochastic matrix

1991 ◽  
Vol 28 (01) ◽  
pp. 96-103 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Stationary Distribution ◽  
Numerical Approximation ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Finite System ◽  
Balance Equations ◽  
The Given ◽  
Steady State Balance

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.

scholarly journals Persistence and Stability for a Class of Forced Positive Nonlinear Delay-Differential Systems

2021 ◽  
Vol 174 (1) ◽  
Author(s):  
D. Franco ◽  
C. Guiver ◽  
H. Logemann
Keyword(s):  
Steady State ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Breeding Programmes ◽  
Delay Differential Systems ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Nonlinear Delay ◽  
Stability Concept

AbstractPersistence and stability properties are considered for a class of forced positive nonlinear delay-differential systems which arise in mathematical ecology and other applied contexts. The inclusion of forcing incorporates the effects of control actions (such as harvesting or breeding programmes in an ecological setting), disturbances induced by seasonal or environmental variation, or migration. We provide necessary and sufficient conditions under which the states of these models are semi-globally persistent, uniformly with respect to the initial conditions and forcing terms. Under mild assumptions, the model under consideration naturally admits two steady states (equilibria) when unforced: the origin and a unique non-zero steady state. We present sufficient conditions for the non-zero steady state to be stable in a sense which is reminiscent of input-to-state stability, a stability notion for forced systems developed in control theory. In the absence of forcing, our input-to-sate stability concept is identical to semi-global exponential stability.

Some Reconsiderations of Simon's City-Size Distribution Model

1977 ◽  
Vol 9 (9) ◽  
pp. 1043-1053 ◽  
Author(s):  
A Okabe
Keyword(s):  
Steady State ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
City Size ◽  
Distribution Model ◽  
Type Model ◽  
Size Distributions ◽  
City Size Distribution ◽  
Necessary And Sufficient ◽  
Over Time

In conjunction with the empirical findings that the form of city-size distributions is stable over time, this paper reexamines Simon's (1955) model and provides a better understanding of that model. First, a Simon-type model is proposed which is a generalization of Simon's model. Second, Simon's model is reexamined with respect to the ‘steady state’. Third, in the context of the Simon-type model, the necessary and sufficient conditions for the ‘steady state’ with the Yule (1924) city-size distributions are investigated. Last, the necessary and sufficient conditions for the ‘asymptotically steady state’ are obtained.

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