Approximations and boundary conditions for continuous-time threshold autoregressive processes

Continuous-time threshold autoregressive (CTAR) processes have been developed in the past few years for modelling non-linear time series observed at irregular intervals. Several approximating processes are given here which are useful for simulation and inference. Each of the approximating processes implicitly defines conditions on the thresholds, thus providing greater understanding of the way in which boundary conditions arise.

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On the Existence and Application of Continuous-Time Threshold Autoregressions of Order Two

Advances in Applied Probability ◽

10.1017/s0001867800027853 ◽

1997 ◽

Vol 29 (01) ◽

pp. 205-227 ◽

Cited By ~ 1

Author(s):

P. J. Brockwell ◽

R. J. Williams

Keyword(s):

Differential Equation ◽

Stochastic Differential Equation ◽

Continuous Time ◽

Linear Time ◽

Autoregressive Process ◽

Initial State ◽

Share Prices ◽

Threshold Autoregressive ◽

Random Time Change ◽

Time Threshold

A continuous-time threshold autoregressive process of order two (CTAR(2)) is constructed as the first component of the unique (in law) weak solution of a stochastic differential equation. The Cameron–Martin–Girsanov formula and a random time-change are used to overcome the difficulties associated with possible discontinuities and degeneracies in the coefficients of the stochastic differential equation. A sequence of approximating processes that are well-suited to numerical calculations is shown to converge in distribution to a solution of this equation, provided the initial state vector has finite second moments. The approximating sequence is used to fit a CTAR(2) model to percentage relative daily changes in the Australian All Ordinaries Index of share prices by maximization of the ‘Gaussian likelihood'. The advantages of non-linear relative to linear time series models are briefly discussed and illustrated by means of the forecasting performance of the model fitted to the All Ordinaries Index.

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On the Existence and Application of Continuous-Time Threshold Autoregressions of Order Two

Advances in Applied Probability ◽

10.2307/1427867 ◽

1997 ◽

Vol 29 (1) ◽

pp. 205-227 ◽

Cited By ~ 4

Author(s):

P. J. Brockwell ◽

R. J. Williams

Keyword(s):

Differential Equation ◽

Stochastic Differential Equation ◽

Continuous Time ◽

Linear Time ◽

Autoregressive Process ◽

Initial State ◽

Share Prices ◽

Threshold Autoregressive ◽

Random Time Change ◽

Time Threshold

A continuous-time threshold autoregressive process of order two (CTAR(2)) is constructed as the first component of the unique (in law) weak solution of a stochastic differential equation. The Cameron–Martin–Girsanov formula and a random time-change are used to overcome the difficulties associated with possible discontinuities and degeneracies in the coefficients of the stochastic differential equation. A sequence of approximating processes that are well-suited to numerical calculations is shown to converge in distribution to a solution of this equation, provided the initial state vector has finite second moments. The approximating sequence is used to fit a CTAR(2) model to percentage relative daily changes in the Australian All Ordinaries Index of share prices by maximization of the ‘Gaussian likelihood'. The advantages of non-linear relative to linear time series models are briefly discussed and illustrated by means of the forecasting performance of the model fitted to the All Ordinaries Index.

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Nonlinear Time Series Analysis with R

10.1093/oso/9780198782933.001.0001 ◽

2018 ◽

Author(s):

Ray Huffaker ◽

Marco Bittelli ◽

Rodolfo Rosa

Keyword(s):

Time Series ◽

Time Series Analysis ◽

Particle Physics ◽

Linear Time ◽

Nonlinear Time Series ◽

Nonlinear Time Series Analysis ◽

Series Analysis ◽

Linear Behavior ◽

Non Linear ◽

Scientific Principles

In the process of data analysis, the investigator is often facing highly-volatile and random-appearing observed data. A vast body of literature shows that the assumption of underlying stochastic processes was not necessarily representing the nature of the processes under investigation and, when other tools were used, deterministic features emerged. Non Linear Time Series Analysis (NLTS) allows researchers to test whether observed volatility conceals systematic non linear behavior, and to rigorously characterize governing dynamics. Behavioral patterns detected by non linear time series analysis, along with scientific principles and other expert information, guide the specification of mechanistic models that serve to explain real-world behavior rather than merely reproducing it. Often there is a misconception regarding the complexity of the level of mathematics needed to understand and utilize the tools of NLTS (for instance Chaos theory). However, mathematics used in NLTS is much simpler than many other subjects of science, such as mathematical topology, relativity or particle physics. For this reason, the tools of NLTS have been confined and utilized mostly in the fields of mathematics and physics. However, many natural phenomena investigated I many fields have been revealing deterministic non linear structures. In this book we aim at presenting the theory and the empirical of NLTS to a broader audience, to make this very powerful area of science available to many scientific areas. This book targets students and professionals in physics, engineering, biology, agriculture, economy and social sciences as a textbook in Nonlinear Time Series Analysis (NLTS) using the R computer language.

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