Abstract
The construction of an importance density for partially non‐Gaussian state space models is crucial when simulation methods are used for likelihood evaluation, signal extraction, and forecasting. The method of efficient importance sampling is successful in this respect, but we show that it can be implemented in a computationally more efficient manner using standard Kalman filter and smoothing methods. Efficient importance sampling is generally applicable for a wide range of models, but it is typically a custom‐built procedure. For the class of partially non‐Gaussian state space models, we present a general method for efficient importance sampling. Our novel method makes the efficient importance sampling methodology more accessible because it does not require the computation of a (possibly) complicated density kernel that needs to be tracked for each time period. The new method is illustrated for a stochastic volatility model with a Student's t distribution.
Highlights
For the modelling of an observed time series y1, ... , yn, we consider a parametric model that we formulate conditionally on a dynamic latent factor or a time-varying parameter vector αt, for time index t = 1, ... , n
We show that our modified efficient importance sampling (EIS) (MEIS) method can lead to some considerable reductions in computing time
We have presented a new modification of the EIS method for the analysis of partially non-Gaussian state space models that include a wide range of time series models of interest
Summary
When (a) the observation density for yt conditional on αt is Gaussian, (b) the relation between yt and αt is linear, and (c) the dynamic model for αt is linear and Gaussian, our time series modelling framework reduces to the linear Gaussian state space model as discussed and reviewed in, for example, Harvey (1989) and Durbin and Koopman (2012, Part I) In this framework, we can rely on the celebrated Kalman filter and its related smoothing method for the signal extraction of αt, the evaluation of the likelihood function for a specific value of ψ, and the forecasting of yt.
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