57 research outputs found
A recursive online algorithm for the estimation of time-varying ARCH parameters
In this paper we propose a recursive online algorithm for estimating the
parameters of a time-varying ARCH process. The estimation is done by updating
the estimator at time point with observations about the time point to
yield an estimator of the parameter at time point . The sampling properties
of this estimator are studied in a non-stationary context -- in particular,
asymptotic normality and an expression for the bias due to non-stationarity are
established. By running two recursive online algorithms in parallel with
different step sizes and taking a linear combination of the estimators, the
rate of convergence can be improved for parameter curves from H\"{o}lder
classes of order between 1 and 2.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5009 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Statistical inference for time-varying ARCH processes
In this paper the class of ARCH models is generalized to the
nonstationary class of ARCH models with time-varying coefficients.
For fixed time points, a stationary approximation is given leading to the
notation ``locally stationary ARCH process.'' The asymptotic
properties of weighted quasi-likelihood estimators of time-varying ARCH
processes () are studied, including asymptotic normality. In
particular, the extra bias due to nonstationarity of the process is
investigated. Moreover, a Taylor expansion of the nonstationary ARCH process in
terms of stationary processes is given and it is proved that the time-varying
ARCH process can be written as a time-varying Volterra series.Comment: Published at http://dx.doi.org/10.1214/009053606000000227 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions
Hawkes (1971) introduced a powerful multivariate point process model of
mutually exciting processes to explain causal structure in data. In this paper
it is shown that the Granger causality structure of such processes is fully
encoded in the corresponding link functions of the model. A new nonparametric
estimator of the link functions based on a time-discretized version of the
point process is introduced by using an infinite order autoregression.
Consistency of the new estimator is derived. The estimator is applied to
simulated data and to neural spike train data from the spinal dorsal horn of a
rat.Comment: 20 pages, 4 figure
Local inference for locally stationary time series based on the empirical spectral measure
The time varying empirical spectral measure plays a major role in the treatment of inference problems for locally stationary processes. The properties of the empirical spectral measure and related statistics are studied - both when its index function is fixed or dependent on the sample size. In particular we prove a general central limit theorem. Several applications and examples are given including semiparametric Whittle estimation, local least squares estimation and spectral density estimation
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