We present several Markov chain Monte Carlo simulation methods that have been widely used in recent years in econometrics and statistics. Among these is the Gibbs sampler, which has been of particular interest to econometricians. Although the paper summarizes some of the relevant theoretical literature, its emphasis is on the presentation and explanation of applications to important models that are studied in econometrics. We include a discussion of some implementation issues, the use of the methods in connection with the EM algorithm, and how the methods can be helpful in model specification questions. Many of the applications of these methods are of particular interest to Bayesians, but we also point out ways in which frequentist statisticians may find the techniques useful.

Introduction Linear Gaussian state space models are used extensively, with unknown parameters usually estimated by maximum likelihood: Wecker & Ansley (1983), Harvey (1989). However, many time series and nonparametric regression applications, such as change point problems, outlier detection and switching regression, require the full generality of the conditionally Gaussian model: Harrison & Stevens (1976), Shumway & Stoffer (1991), West & Harrison (1989), Gordon & Smith (1990). The presence of a large number of indicator variables makes it difficult to estimate conditionally Gaussian models using maximum likelihood, and a Bayesian approach using Markov chain Monte Carlo appears more tractable. We propose a new sampler, which is used to estimate an unknown function nonparametrically when there are jumps in the function and outliers in the observations; it is also applied to a time series change point problem previously discussed by Gordon & Smith (1990). For the first example th

rapolation techniques, especially exponential smoothing and exponentially weighted moving average methods ([20, 71]). Developments of smoothing and discounting techniques in stock control and production planning areas led to formalisms in terms of linear, state-space models for time series with time-varying trends and seasonal patterns, and eventually to the associated Bayesian formalism of methods of inference and prediction. From the early 1960s, practical Bayesian forecasting systems in this context involved the combination of formal time series models and historical data analysis together with methods for subjective intervention and forecast monitoring, so that complete forecasting systems, rather than just routine and automatic data analysis and extrapolation, were in use at that time ([19, 22]). Methods developed in those early days are still in use now in some companies in sales forecasting and stock control areas. There have been major developments in models and methods since t

This paper provides a Bayesian analysis of such a model. The main contribution of our paper is that different features of the data--such as the spectral density of the stationary term, the regression parameters, unknown frequencies and missing observations--are combined in a hierarchical Bayesian framework and estimated simultaneously. A Bayesian test to detect the presence of deterministic components in the data is also constructed. Applications of our methods to simulated and real data suggest that they perform well. We place a smoothness prior, similar to that in Wahba (1980), on the logarithm of the spectral density. To make the estimation of the spectral density computationally tractable, Whittle's (1957) approximation to the Gaussian likelihood is used. This results in a nonparametric regression problem with the logarithm of the periodogram as the dependent variable, the logarithm of the spectral density as the unknown regression curve, and observation errors having log chi-squared distributions. By approximating the logarithm of a chi-squared distribution as a mixture of normals, the approximate log likelihood together with the prior for the spectral density can be expressed as a state space model with errors that are mixtures of normals. The computation is carried out efficiently by Markov chain Monte Carlo using the sampling approach in Carter and Kohn (1994). To make the paper easier to read the full model is introduced in a number of steps. Section 2 shows how to estimate the spectral density of a stationary process in the absence of deterministic components. Section 3 extends the estimation to the signal plus noise model with missing observations. Section 4 shows by example how the results in Sections 2 and 3 can be combined to analyze data and studies emp...

The mixture model likelihood function is invariant with respect to permutation of the components of the mixture. If functions of interest are permutation sensitive, as in classification applications, then interpretation of the likelihood function requires valid inequality constraints and a very large sample may be required to resolve ambiguities. If functions of interest are permutation invariant, as in prediction applications, then there are no such problems of interpretation. Contrary to assessments in some recent publications, simple and widely used Markov chain Monte Carlo (MCMC) algorithms with data augmentation reliably recover the entire posterior distribution. 1 1

In the present paper we discuss Bayesian estimation of a very general model class where the distribution of the observations is assumed to depend on a latent mixture or switching variable taking values in a discrete state space. This model class covers e.g. ønite mixture modelling, Markov switching autoregressive modelling and dynamic linear models with switching. Joint Bayesian estimation of all latent variables, model parameters and parameters determining the probability law of the switching variable is carried out by a new Markov Chain Monte Carlo method called permutation sampling. Estimation of switching and mixture models is known to be faced with identiøability problems as switching and mixture are identiøable only up to permutations of the indices of the states. For a Bayesian analysis the posterior has to be constrained in such a way that identiøablity constraints are fulølled. The permutation sampler is designed to sample eOEciently from the constrained posterior, by ørst sam...

We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regression approaches, noisy measurements can produce inaccurate parameter estimates. Another problem is that for chaotic systems the cost functions that have to be minimized to estimate states and parameters are so complex that common optimization routines may fail. We show that the inclusion of information about the time-continuous nature of the underlying trajectories can improve parameter estimation considerably. Two approaches, which take into account both the errors-in-variables problem and the problem of complex cost functions, are described in detail: shooting approaches and recursive estimation techniques. Both are demonstrated on numerical examples.

A Bayesian analysis is given for a state space model with errors that are finite mixtures of normals and with coefficients that can assume a finite number of different values. A sequence of indicator variables determines which components the errors belong to and the values of the coefficients. The computation is carried out using Markov chain Monte Carlo, with the indicator variables generated without conditioning on the states. Previous approaches use the Gibbs sampler to generate the indicator variables conditional on the states. In many problems, however, there is a strong dependence between the indicator variables and the states causing the Gibbs sampler to converge unacceptably slowly, or even not to converge at all. The new sampler is implemented in 0{n) operations, where n is the sample size, permitting an exact Bayesian analysis of problems that previously had no computationally tractable solution. We show empirically that the new sampler can be much more efficient than previous approaches, and illustrate its applicability to robust nonparametric regression with discontinuities and to a time series change point problem.

Package dlm focuses on Bayesian analysis of Dynamic Linear Models (DLMs), also known as linear state space models (see [H, WH]). The package also includes functions for maximum likelihood estimation of the parameters of a DLM and for Kalman filtering. The algorithms used for Kalman filtering, likelihood evaluation, and sampling from the state vectors are based on the singular value decomposition (SVD) of the relevant variance matrices (see [ZL]), which improves numerical stability over other algorithms.