Abstract
Non-Gaussian state-space models arise in several applications, and within this framework the binary time series setting provides a relevant example. However, unlike for 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 key functionals of these distributions are, in principle, available, but their calculation requires the evaluation of multivariate Gaussian cumulative distribution functions. Leveraging sun properties, we address this issue via novel Monte Carlo methods based on independent samples from the smoothing distribution, that can easily be adapted to the filtering and predictive case, thus improving state-of-the-art approximate and sequential Monte Carlo inference in small-to-moderate dimensional studies. Novel sequential Monte Carlo procedures that exploit the sun properties are also developed to deal with online inference in high dimensions. Performance gains over competitors are outlined in a financial application.
Highlights
Keywords Dynamic probit model · Kalman filter · Particle filter · State-space model · sun Despite the availability of several alternative approaches for dynamic inference and prediction of binary time series (MacDonald and Zucchini 1997), state-space models are a source of constant interest due to their flexibility in accommodating a variety of representations and dependence structures via an interpretable formulation (West and Harrison 2006; Petris et al 2009; Durbin and Koopman 2012)
Motivated by this consideration and by the additive structure of the sun filtering distribution, we further develop in Sect. 4.2.2 a partially collapsed sequential Monte Carlo procedure which recursively samples via lookahead methods (Lin et al 2013) only the multivariate truncated normal term in the sun additive representation, while keeping the Gaussian component exact
The partially collapsed lookahead particle filter for sampling recursively from p(θ t | y1:t ) requires to store and update, for each particle trajectory, the sufficient statistics at−k|t−k−1 and Pt−k|t−k−1 via Kalman filter recursions applied to the model (4)–(5), with every zt replaced by the particles generated under the lookahead routine
Summary
Despite the availability of several alternative approaches for dynamic inference and prediction of binary time series (MacDonald and Zucchini 1997), state-space models are a source of constant interest due to their flexibility in accommodating a variety of representations and dependence structures via an interpretable formulation (West and Harrison 2006; Petris et al 2009; Durbin and Koopman 2012). Data Science and Analytics, Bocconi University, Milan, Italy to provide closed-form expressions for the filtering, predictive and smoothing distributions in the general multivariate dynamic probit model p(yt | θ t ) = Φm (Bt Ft θ t ; Bt Vt Bt ),. The quantities Ft , Vt , Gt , Wt , a0 and P0 denote, instead, known matrices controlling the location, scale and dependence structure in the state-space model (1)–(2).
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