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Title:

Gaussian Maximum Likelihood Estimation For ARMA Models. I. Time Series

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We provide a direct proof for consistency and asymptotic normality of Gaussian maximum likelihood estimators for causal and invertible autoregressive moving-average (ARMA) time series models, which were initially established by Hannan [Journal of Applied Probability (1973) vol. 10, pp. 130-145] via the asymptotic properties of a Whittle's estima...

We provide a direct proof for consistency and asymptotic normality of Gaussian maximum likelihood estimators for causal and invertible autoregressive moving-average (ARMA) time series models, which were initially established by Hannan [Journal of Applied Probability (1973) vol. 10, pp. 130-145] via the asymptotic properties of a Whittle's estimator. This also paves the way to establish similar results for spatial processes presented in the follow-up article by Yao and Brockwell [Bernoulli (2006) in press]. Copyright 2006 The Authors Journal compilation 2006 Blackwell Publishing Ltd. Minimize

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article

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Title:

Strictly stationary solutions of autoregressive moving average equations

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Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining an autoregressive moving average process driven by an independent and identically distributed noise sequence are determined. No moment assumptions on the driving noise sequence are made. Copyright 2010, Oxford University Press.

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining an autoregressive moving average process driven by an independent and identically distributed noise sequence are determined. No moment assumptions on the driving noise sequence are made. Copyright 2010, Oxford University Press. Minimize

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article

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Title:

Autoregressions Generated by the Tent Map

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It is well-known (see e.g. Tong, 1990, Gourieroux, 1997) that if X0 has the uniform distribution function U on [0 � 1], then the sequence of iterates fXn = g(Xn;1)g of the symmetric tent mapg from [0 � 1] onto [0 � 1], is a strictly stationary Markov process with marginal distribution function U. It is also easy to show, using the symmetry of th...

It is well-known (see e.g. Tong, 1990, Gourieroux, 1997) that if X0 has the uniform distribution function U on [0 � 1], then the sequence of iterates fXn = g(Xn;1)g of the symmetric tent mapg from [0 � 1] onto [0 � 1], is a strictly stationary Markov process with marginal distribution function U. It is also easy to show, using the symmetry of the map, that fXng is white noise. In this note we show that if the symmetric tent map is replaced by askewed tent map, then the sequence fXng is a strictly stationary autoregression of order 1 with coe cient = (2=s) ; 1, where s 2 (1 � 1) is the right-derivative ofthetent map at 0. An AR(1) process with uniform marginal distributions and arbitrary coe cient 2 (;1 � 1) can thus be generated by computing the iterates with s =2= ( +1). For the symmetric map s =2and =0. Keywords: Non-linear dynamical system, chaos, nonlinear time series, linear prediction. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-12-23

Source:

http://www-m4.ma.tum.de/Papers/Klueppelberg/chaos.pdf

http://www-m4.ma.tum.de/Papers/Klueppelberg/chaos.pdf Minimize

Document Type:

text

Language:

en

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

High frequency sampling of a continuous-time ARMA process

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Continuous-time autoregressive moving average (CARMA) processes have recently been used widely in the modeling of non-uniformly spaced data and as a tool for dealing with high-frequency data of the form Yn∆, n = 0, 1, 2,., where ∆ is small and positive. Such data occur in many fields of application, particularly in finance and the study of turbu...

Continuous-time autoregressive moving average (CARMA) processes have recently been used widely in the modeling of non-uniformly spaced data and as a tool for dealing with high-frequency data of the form Yn∆, n = 0, 1, 2,., where ∆ is small and positive. Such data occur in many fields of application, particularly in finance and the study of turbulence. This paper is concerned with the characteristics of the process (Yn∆)n∈Z, when ∆ is small and the underlying continuous-time process (Yt)t∈R is a specified CARMA process. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2011-03-30

Source:

http://www-m4.ma.tum.de/Papers/Ferrazzano/bfkcarma110114.pdf

http://www-m4.ma.tum.de/Papers/Ferrazzano/bfkcarma110114.pdf Minimize

Document Type:

text

Language:

en

Subjects:

CARMA process ; high frequency data ; discretely sampled process

CARMA process ; high frequency data ; discretely sampled process Minimize

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Lévy-driven and fractionally integrated ARMA processes with continuous time parameter

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The de nition and properties of Levy-driven CARMA (continuous-time ARMA) processes are reviewed. Gaussian CARMA processes are special cases in which the driving Levy process is Brownian motion. The use of more general Levy processes permits the speci cation of CARMA processes with a wide variety ofmarginal distributions which may be asymmetric a...

The de nition and properties of Levy-driven CARMA (continuous-time ARMA) processes are reviewed. Gaussian CARMA processes are special cases in which the driving Levy process is Brownian motion. The use of more general Levy processes permits the speci cation of CARMA processes with a wide variety ofmarginal distributions which may be asymmetric and heavier tailed than Gaussian. Non-negative CARMA processes are of special interest, partly because of the introduction by Barndor-Nielsen and Shephard (2001) of non-negativeLevy-driven Ornstein-Uhlenbeck processes as models for stochastic volatility. Replacing the Ornstein-Uhlenbeck process byaLevy-driven CARMA process with non-negative kernel permits the modelling of non-negative, heavy-tailed processes with a considerably larger range of autocovariance functions than is possible in the Ornstein-Uhlenbeck framework. We also de ne a class of zero-mean fractionally integrated Levy-driven CARMA processes, obtained by convoluting the CARMA kernel with a kernel corresponding to Riemann-Liouville fractional integration, and derive explicit expressions for the kernel and autocovariance functions of these processes. They are long-memory in the sense that their kernel and autocovariance functions decay asymptotically at hyperbolic rates depending on the order of fractional integration. In order to introduce long-memory into non-negative Levy-driven CARMA processes we replace the fractional integration kernel with a closely related absolutely integrable kernel. This gives a class of stationary non-negative continuous-time Levy-driven processes whose autocovariance functions at lag h also converge to zero at asymptotically hyperbolic rates. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-12-23

Source:

http://www-m4.ma.tum.de/pers/brockwell/ficarma.pdf

http://www-m4.ma.tum.de/pers/brockwell/ficarma.pdf Minimize

Document Type:

text

Language:

en

Subjects:

continuous-time ARMA process ; Levy process ; stochastic volatility ; long memory ; fractional

continuous-time ARMA process ; Levy process ; stochastic volatility ; long memory ; fractional Minimize

DDC:

519 Probabilities & applied mathematics *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Estimation for Non-negative Levy-driven Ornstein-Uhlenbeck Processes

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Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a non-decreasing Levy process constitute a very general class of stationary, non-negative continuous-time processes. In nancial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Levy process, was intro...

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a non-decreasing Levy process constitute a very general class of stationary, non-negative continuous-time processes. In nancial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Levy process, was introduced by Barndor-Nielsen and Shephard (2001) as a model for stochastic volatility toallow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes we take advantage of the non-negativity of the increments of the driving Levy process to study the properties of a highly e cient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0�h�::: �Nh.We also show howto reconstruct the background driving Levy process from a continuously observed realization of the process and use this result to estimate the increments of the Levy process itself when h is small. Asymptotic properties of the coe cient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck process. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-12-23

Source:

http://www-m4.ma.tum.de/Papers/brockwell/LevyOU.pdf

http://www-m4.ma.tum.de/Papers/brockwell/LevyOU.pdf Minimize

Document Type:

text

Language:

en

DDC:

519 Probabilities & applied mathematics *(computed)*

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Title:

Strictly stationary solutions of autoregressive moving average equations

Author:

Description:

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining an autoregressive moving average process driven by an independent and identically distributed noise sequence are determined. No moment assumptions on the driving noise sequence are made.

Necessary and sufficient conditions for the existence of a strictly stationary solution of the equations defining an autoregressive moving average process driven by an independent and identically distributed noise sequence are determined. No moment assumptions on the driving noise sequence are made. Minimize

Publisher:

Oxford University Press

Year of Publication:

2010-09-01 00:00:00.0

Document Type:

TEXT

Language:

en

Subjects:

Miscellanea

Miscellanea Minimize

Rights:

Copyright (C) 2010, Biometrika Trust

Copyright (C) 2010, Biometrika Trust Minimize

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Title:

Parametric estimation of the driving L\'evy process of multivariate CARMA processes from discrete observations

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We consider the parametric estimation of the driving L\'evy process of a multivariate continuous-time autoregressive moving average (MCARMA) process, which is observed on the discrete time grid $(0,h,2h,.)$. Beginning with a new state space representation, we develop a method to recover the driving L\'evy process exactly from a continuous record...

We consider the parametric estimation of the driving L\'evy process of a multivariate continuous-time autoregressive moving average (MCARMA) process, which is observed on the discrete time grid $(0,h,2h,.)$. Beginning with a new state space representation, we develop a method to recover the driving L\'evy process exactly from a continuous record of the observed MCARMA process. We use tools from numerical analysis and the theory of infinitely divisible distributions to extend this result to allow for the approximate recovery of unit increments of the driving L\'evy process from discrete-time observations of the MCARMA process. We show that, if the sampling interval $h=h_N$ is chosen dependent on $N$, the length of the observation horizon, such that $N h_N$ converges to zero as $N$ tends to infinity, then any suitable generalized method of moments estimator based on this reconstructed sample of unit increments has the same asymptotic distribution as the one based on the true increments, and is, in particular, asymptotically normally distributed. ; Comment: 38 pages, four figures; to appear in Journal of Multivariate Analysis Minimize

Year of Publication:

2012-09-05

Document Type:

text

Subjects:

Mathematics - Probability ; Mathematics - Statistics Theory ; 62F10 ; 60G51 ; 60F05 (Primary) 60E07 ; 60G10 (Secondary)

Mathematics - Probability ; Mathematics - Statistics Theory ; 62F10 ; 60G51 ; 60F05 (Primary) 60E07 ; 60G10 (Secondary) Minimize

DDC:

519 Probabilities & applied mathematics *(computed)*

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Title:

Representations of continuous-time ARMA processes

Description:

Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein-Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-dr...

Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein-Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-driven continuous-time ARMA processes obtained by a simple modification of the kernel of the continuous-time ARMA process. Asymptotic properties of the kernel and of the autocovariance function are derived. Minimize

Publisher:

Applied Probability Trust

Year of Publication:

2004-02

Document Type:

Text

Language:

en

Subjects:

Continuous-time ARMA process ; Lévy process ; stochastic volatility ; long memory ; fractional integration ; 60G10 ; 62P05

Continuous-time ARMA process ; Lévy process ; stochastic volatility ; long memory ; fractional integration ; 60G10 ; 62P05 Minimize

Rights:

Copyright 2004 Applied Probability Trust

Copyright 2004 Applied Probability Trust Minimize

Relations:

0021-9002 ; 1475-6072

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Title:

Strictly stationary solutions of multivariate ARMA equations with i.i.d. noise

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We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of multivariate ARMA equations with independent and identically distributed noise. For general ARMA$(p,q)$ equations these conditions are expressed in terms of the characteristic polynomials of the defining equations and moments of the driving noise ...

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of multivariate ARMA equations with independent and identically distributed noise. For general ARMA$(p,q)$ equations these conditions are expressed in terms of the characteristic polynomials of the defining equations and moments of the driving noise sequence, while for $p=1$ an additional characterization is obtained in terms of the Jordan canonical decomposition of the autoregressive matrix, the moving average coefficient matrices and the noise sequence. No a priori assumptions are made on either the driving noise sequence or the coefficient matrices. Minimize

Year of Publication:

2011-05-17

Document Type:

text

Subjects:

Mathematics - Statistics Theory ; Mathematics - Probability ; 60G10

Mathematics - Statistics Theory ; Mathematics - Probability ; 60G10 Minimize

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