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A Closed-Form Filter for Binary Time Series
Non-Gaussian state-space models arise in several applications. Within this
framework, the binary time series setting is a source of constant interest due
to its relevance in many studies. However, unlike Gaussian state-space models,
where filtering, predictive and smoothing distributions are available in
closed-form, binary state-space models require approximations or sequential
Monte Carlo strategies for inference and prediction. This is due to the
apparent absence of conjugacy between the Gaussian states and the likelihood
induced by the observation equation for the binary data. In this article we
prove that the filtering, predictive and smoothing distributions in dynamic
probit models with Gaussian state variables are, in fact, available and belong
to a class of unified skew-normals (SUN) whose parameters can be updated
recursively in time via analytical expressions. Also the functionals of these
distributions depend on known functions, but their calculation requires
intractable numerical integration. Leveraging the SUN properties, we address
this point via new Monte Carlo methods based on independent and identically
distributed samples from the smoothing distribution, which can naturally be
adapted to the filtering and predictive case, thereby improving
state-of-the-art approximate or sequential Monte Carlo inference in
small-to-moderate dimensional studies. A scalable and optimal particle filter
which exploits the SUN properties is also developed to deal with online
inference in high dimensions. Performance gains over competitors are outlined
in a real-data financial application
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