In time series analysis,
the family of generalized state-space models is extremely
rich. However, their likelihood functions are
intractable, except in certain special cases, and this
limits the options in analyses. In practice,
a study typically (1) uses some kind of approximation
to the likelihood function, for instance, one obtained
analytically or by making use of the particle filter
or related methods,
(2) adopts a standard Markov chain Monte Carlo
approach to parameter estimation, or (3) sacrifices
goodness-of-fit for numerical convenice by choosing
an approximating model for which the likelihood can
be computed. Each of these approaches has advantages
and disadvantages, but since none of them yields a
consistent estimate of the likelihood,
model selection remains an outstanding problem for
the general family.
This paper addresses this problem by introducing
a recursive estimator of the
log-likelihood for the generalized state-space model, which
is obtained as a kernel density estimator driven by the
iterations of a Markov chain.
The estimator is very simple to compute, and is shown to converge almost surely
to the exact log-likelihood
as the number of iterations of the Markov chain
approaches infinity.
Keywords: generalized, state-space model, non-Gaussian, nonlinear,
likelihood, recursive, kernel density, estimator, Markov chain,
dynamic model