Some computational aspects of Gaussian CARMA modelling

Tómasson, Helgí

doi:10.1007/s11222-013-9438-9

Cited by 21 publications

(16 citation statements)

References 30 publications

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“…As shown in the introduction, the ctarma package developed by [30] contains several routines for the simulation and the estimation of a Gaussian CARMA(p,q) model using both frequency and time-domain approaches. Since in this paper we focus on the state-space representation of a CARMA(p,q) model, we conduct our comparison considering only the time-domain approach and refer to [29] for a complete and detailed explanation of the frequency-domain approach for the simulation and the estimation of a Gaussian CARMA(p,q) model. Our comparison is based on two exercises.…”

Section: Ctarma Packagementioning

confidence: 99%

“…Based on our knowledge, the R packages available on CRAN deal only with CARMA(p,q) models driven by a standard Brownian Motion [30] or Gaussian CAR(p) models [33]. For example the ctarma package developed by [29] is an useful package for the simulation and the estimation of a CARMA(p,q) model driven by a Brownian Motion. Another package for continuous Autoregressive model is the cts developed by [33] which deals with a modified version of the CAR(p) model named CZAR(p) by [32].…”

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Implementation of Lévy CARMA model in Yuima package

Iacus

Mercuri

2015

Comput Stat

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The paper shows how to use the R package yuima available on CRAN for the simulation and the estimation of a general Lévy Continuous Autoregressive Moving Average (CARMA) model. The flexibility of the package is due to the fact that the user is allowed to choose several parametric Lévy distribution for the increments. Some numerical examples are given in order to explain the main classes and the corresponding methods implemented in yuima package for the CARMA model. of the Deutsche Mark/US Dollar daily exchange rate. Moreover [10] proposed the fractionally integrated CARMA model in order to capture the long range dependence usually observed in financial time series.The interest on the CARMA model is manifold since it can be used to model directly some given time series but it is also a main block for the construction of a more general process like the COGARCH(p,q) as in [5].The aim of this work is to develop in the yuima package a complete computational scheme for the simulation and the estimation of a general Lévy CARMA model. Based on our knowledge, the R packages available on CRAN deal only with CARMA(p,q) models driven by a standard Brownian Motion [30] or Gaussian CAR(p) models [33]. For example the ctarma package developed by [29] is an useful package for the simulation and the estimation of a CARMA(p,q) model driven by a Brownian Motion. Another package for continuous Autoregressive model is the cts developed by [33] which deals with a modified version of the CAR(p) model named CZAR(p) by [32]. Since the CAR(p) model is a special case of a CARMA(p,q), the ctarma package is a valid benchmark for the functions implemented in the yuima package and for this reason a direct comparison is given in this paper where a Gaussian CARMA(p,q) model is considered. Moreover, in the yuima package, once the estimation of the coefficients is done, it is possible to recover the underlying Lévy process from the observed data using the methodology in [7] and extended to the multivariate CARMA(p,q) by [8]. In this way we are able to simulate trajectories of a CARMA model without an explicit assumption on the distribution at time one of the underlying Lévy process. The outline of this paper is the following. In Sect. 2 we review the main results about the CARMA(p,q) process. In particular we focus the attention on the condition for the existence of the second order stationary solution of the CARMA process. In Sect. 3 we explain the estimation procedure implemented in the yuima package if the data are observed in equally space-time intervals. In Sect. 4 we describe the main classes and corresponding methods available in the yuima package for a CARMA model. We show how to use them for simulation and estimation of a Gaussian CARMA model and we conduct a comparison with the methods availables in the ctarma package. In Sect. 5 we present some numerical examples about the simulation and the estimation of Lévy CARMA models.

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Section: Ctarma Packagementioning

confidence: 99%

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confidence: 99%

Implementation of Lévy CARMA model in Yuima package

Iacus

Mercuri

2015

Comput Stat

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“…Generation of this Gaussian CARMA term is outside the scope of this paper. (We refer the reader to (Tómasson 2013) and references therein.) 2.…”

Section: Overview Of Resultsmentioning

confidence: 99%

Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes

Kawai

2015

Methodol Comput Appl Probab

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We address the issue of sample path simulation of Lévy-driven continuous-time autoregressive moving average (CARMA) processes. Approximate discrete-time simulation schemes are constructed along with quantifiable error analysis for stable, second-order and non-negative CARMA processes, based upon the so-called series representation of infinitely divisible laws and associated Lévy processes. We prove that under suitable conditions, the simulation scheme can be improved in terms of second-order structure, finite dimensional laws as well as sample path properties. The simulation procedure is often quite simple and allows one to conduct super-sampling without running the algorithm once again. The computational complexity of the proposed scheme is not affected much by the sampling scheme, such as sampling frequency and irregular spacing. Numerical results are presented throughout to illustrate the effectiveness of the proposed simulation scheme.

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“…Stochastic processes and autoregressive processes are covered in many textbooks in a quite unified parametrization (e.g., [23][24][25][26][27][28][29][30]). In contrast to this, various parametrizations of Autoregressive Moving Average (ARMA) processes exist in stochastic theory [31][32][33][34][35]. A general connection of these topics, i.e., stochastic processes, covariance functions, and the collocation theory in the context of geodetic usage is given by [5].…”

Section: Introductionmentioning

confidence: 99%

A Generic Approach to Covariance Function Estimation Using ARMA-Models

et al. 2020

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Covariance function modeling is an essential part of stochastic methodology. Many processes in geodetic applications have rather complex, often oscillating covariance functions, where it is difficult to find corresponding analytical functions for modeling. This paper aims to give the methodological foundations for an advanced covariance modeling and elaborates a set of generic base functions which can be used for flexible covariance modeling. In particular, we provide a straightforward procedure and guidelines for a generic approach to the fitting of oscillating covariance functions to an empirical sequence of covariances. The underlying methodology is developed based on the well known properties of autoregressive processes in time series. The surprising simplicity of the proposed covariance model is that it corresponds to a finite sum of covariance functions of second-order Gauss-Markov (SOGM) processes. Furthermore, the great benefit is that the method is automated to a great extent and directly results in the appropriate model. A manual decision for a set of components is not required. Notably, the numerical method can be easily extended to ARMA-processes, which results in the same linear system of equations. Although the underlying mathematical methodology is extensively complex, the results can be obtained from a simple and straightforward numerical method.

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Some computational aspects of Gaussian CARMA modelling

Cited by 21 publications

References 30 publications

Implementation of Lévy CARMA model in Yuima package

Implementation of Lévy CARMA model in Yuima package

Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes

A Generic Approach to Covariance Function Estimation Using ARMA-Models

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