416 research outputs found
Multivariate COGARCH(1,1) processes
Multivariate processes are introduced as a
continuous-time models for multidimensional heteroskedastic observations. Our
model is driven by a single multivariate L\'{e}vy process and the latent
time-varying covariance matrix is directly specified as a stochastic process in
the positive semidefinite matrices. After defining the process, we analyze its probabilistic properties. We show a
sufficient condition for the existence of a stationary distribution for the
stochastic covariance matrix process and present criteria ensuring the
finiteness of moments. Under certain natural assumptions on the moments of the
driving L\'{e}vy process, explicit expressions for the first and second-order
moments and (asymptotic) second-order stationarity of the covariance matrix
process are obtained. Furthermore, we study the stationarity and second-order
structure of the increments of the multivariate
process and their "squares".Comment: Published in at http://dx.doi.org/10.3150/09-BEJ196 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes
The class of multivariate L\'{e}vy-driven autoregressive moving average
(MCARMA) processes, the continuous-time analogs of the classical vector ARMA
processes, is shown to be equivalent to the class of continuous-time state
space models. The linear innovations of the weak ARMA process arising from
sampling an MCARMA process at an equidistant grid are proved to be
exponentially completely regular (-mixing) under a mild continuity
assumption on the driving L\'{e}vy process. It is verified that this continuity
assumption is satisfied in most practically relevant situations, including the
case where the driving L\'{e}vy process has a non-singular Gaussian component,
is compound Poisson with an absolutely continuous jump size distribution or has
an infinite L\'{e}vy measure admitting a density around zero.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ329 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Quasi maximum likelihood estimation for strongly mixing state space models and multivariate L\'evy-driven CARMA processes
We consider quasi maximum likelihood (QML) estimation for general
non-Gaussian discrete-ime linear state space models and equidistantly observed
multivariate L\'evy-driven continuoustime autoregressive moving average
(MCARMA) processes. In the discrete-time setting, we prove strong consistency
and asymptotic normality of the QML estimator under standard moment assumptions
and a strong-mixing condition on the output process of the state space model.
In the second part of the paper, we investigate probabilistic and analytical
properties of equidistantly sampled continuous-time state space models and
apply our results from the discrete-time setting to derive the asymptotic
properties of the QML estimator of discretely recorded MCARMA processes. Under
natural identifiability conditions, the estimators are again consistent and
asymptotically normally distributed for any sampling frequency. We also
demonstrate the practical applicability of our method through a simulation
study and a data example from econometrics
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