Particle filters and Bayesian inference in financial econometrics

In this paper we review sequential Monte Carlo (SMC) methods, or particle filters (PF), with special emphasis on its potential applications in financial time series analysis and econometrics. We start with the well-known normal dynamic linear model, also known as the normal linear state space model, for which sequential state learning is available in closed form via standard Kalman filter and Kalman smoother recursions. Particle filters are then introduced as a set of Monte Carlo schemes that enable Kalman-type recursions when normality or linearity or both are abandoned. The seminal bootstrap filter (BF) of Gordon, Salmond and Smith (1993) is used to introduce the SMC jargon, potentials and limitations. We also review the literature on parameter learning, an area that started to attract much attention from the particle filter community in recent years. We give particular attention to the Liu-West filter (2001), Storvik filter (2002) and particle learning (PL) of Carvalho, Johannes, Lopes and Polson (2010). We argue that the BF and the auxiliary particle filter (APF) of Pitt and Shephard (1999) define two fundamentally distinct directions within the particle filter literature. We also show that the distinction is more pronounced with parameter learning and argue that PL, which follows the APF direction, is an attractive extension. One of our contributions is to sort out the research from BF to APF (during the 1990s), from APF to now (the 2000s) and from Liu-West filter to Storvik filter to PL. To this end, we provide code in R for all the examples of the paper. Readers are invited to find their own way into this dynamic and active research arena.