Smoothing Hazard Functions and Time-Varying Effects in Discrete Duration and Competing Risks Models

State space or dynamic approaches to discrete or grouped duration data with competing risks or multiple terminating events allow simultaneous modelling and smooth estimation of hazard functions and time-varying effects in a flexible way. Full Bayesian or posterior mean estimation, using numerical integration techniques or Monte Carlo methods, can become computationally rather demanding or even infeasible for higher dimensions and larger data sets. Therefore, based on previous work on filtering and smoothing for multicategorical time series and longitudinal data, our approach uses posterior mode estimation. Thus we have to maximize posterior densities or, equivalently, a penalized likelihood, which enforces smoothness of hazard functions and time-varying effects by a roughness penalty. Dropping the Bayesian smoothness prior and adopting a nonparametric viewpoint, one might also start directly from maximizing this penalized likelihood. We show how Fisher scoring smoothing iterations can be carried out efficiently by iteratively applying linear Kalman filtering and smoothing to a working model. This algorithm can be combined with an EM-type procedure to estimate unknown smoothing- or hyperparameters. The methods are applied to a larger set of unemployment duration data with one and, in a further analysis, multiple terminating events from the German socio-economic panel GSOEP

Measuring inflation persistence: a structural time series approach

Time series estimates of inflation persistence incur an upward bias if shifts in the inflation target of the central bank remain unaccounted for. Using a structural time series approach we measure different sorts of inflation persistence allowing for an unobserved timevarying inflation target. Unobserved components are identified using Kalman filtering and smoothing techniques. Posterior densities of the model parameters and the unobserved components are obtained in a Bayesian framework based on importance sampling. We find that inflation persistence, expressed by the halflife of a shock, can range from 1 quarter in case of a costpush shock to several years for a shock to longrun inflation expectations or the output gap.Inflation persistence, inflation target, Kalman filter, Bayesian analysis.

Dynamic detection of change points in long time series

We consider the problem of detecting change points (structural changes) in long sequences of data, whether in a sequential fashion or not, and without assuming prior knowledge of the number of these change points. We reformulate this problem as the Bayesian filtering and smoothing of a non standard state-space model. Towards this goal, we build a hybrid algorithm that relies on particle filtering and MCMC ideas. The approach is illustrated by a GARCH change point model

Inverse Problems and Data Assimilation

These notes are designed with the aim of providing a clear and concise introduction to the subjects of Inverse Problems and Data Assimilation, and their inter-relations, together with citations to some relevant literature in this area. The first half of the notes is dedicated to studying the Bayesian framework for inverse problems. Techniques such as importance sampling and Markov Chain Monte Carlo (MCMC) methods are introduced; these methods have the desirable property that in the limit of an infinite number of samples they reproduce the full posterior distribution. Since it is often computationally intensive to implement these methods, especially in high dimensional problems, approximate techniques such as approximating the posterior by a Dirac or a Gaussian distribution are discussed. The second half of the notes cover data assimilation. This refers to a particular class of inverse problems in which the unknown parameter is the initial condition of a dynamical system, and in the stochastic dynamics case the subsequent states of the system, and the data comprises partial and noisy observations of that (possibly stochastic) dynamical system. We will also demonstrate that methods developed in data assimilation may be employed to study generic inverse problems, by introducing an artificial time to generate a sequence of probability measures interpolating from the prior to the posterior