This paper surveys the fundamental principles of subjective Bayesian inference in econometrics and the implementation of those principles using posterior simulation methods. The emphasis is on the combination of models and the development of predictive distributions. Moving beyond conditioning on a fixed number of completely specified models, the paper introduces subjective Bayesian tools for formal comparison of these models with as yet incompletely specified models. The paper then shows how posterior simulators can facilitate communication between investigators (for example, econometricians) on the one hand and remote clients (for example, decision makers) on the other, enabling clients to vary the prior distributions and functions of interest employed by investigators. A theme of the paper is the practicality of subjective Bayesian methods. To this end, the paper describes publicly available software for Bayesian inference, model development, and communication and provides illustrations using two simple econometric models. *This paper was originally prepared for the Australasian meetings of the Econometric Society in Melbourne, Australia,

This paper is concerned with the Bayesian estimation of non-linear stochastic differential equations when only discrete observations are available. The estimation is carried out using a tuned MCMC method, in particular a blocked Metropolis-Hastings algorithm, by introducing auxiliary points and using the Euler-Maruyama discretisation scheme. Techniques for computing the likelihood function, the marginal likelihood and diagnostic measures (all based on the MCMC output) are presented. Examples using simulated and real data are presented and discussed in detail.

This paper discusses and documents the algorithms of SsfPack 2.2. SsfPack is a suite of C routines for carrying out computations involving the statistical analysis of univariate and multivariate models in state space form. The emphasis is on documenting the link we have made to the Ox computing environment. SsfPack allows for a full range of different state space forms: from a simple time-invariant model to a complicated time-varying model. Functions can be used which put standard models such as ARMA and cubic spline models in state space form. Basic functions are available for ltering, moment smoothing and simulation smoothing. Ready-to-use functions are provided for standard tasks such as likelihood evaluation, forecasting and signal extraction. We show that SsfPack can be easily used for implementing, tting and analysing Gaussian models relevant to many areas of econometrics and statistics. Some Gaussian illustrations are given.

This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a time-varying correlation. Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for short-dated options.

We investigate the issue of model uncertainty in cross-country growth regressions using Bayesian Model Averaging (BMA). We find that the posterior probability is spread widely among many models, suggesting the superiority of BMA over choosing any single model. Out-of-sample predictive results support this claim. In contrast to Levine and Renelt (1992), our results broadly support the more ‘optimistic ’ conclusion of Salai-Martin (1997b), namely that some variables are important regressors for explaining cross-country growth patterns. However, care should be taken in the methodology employed. The approach proposed here is firmly grounded in statistical theory and immediately leads to posterior and predictive inference. Copyright © 2001 John Wiley & Sons, Ltd. 1.

Abstract: The paper investigates the common dynamic properties of business cycle fluctuations across countries, regions, and the world. We employ a Bayesian dynamic latent factor model to estimate common components in the main macroeconomic aggregates (output, consumption and investment) in a sixty-country sample covering seven regions of the world. In particular, we simultaneously estimate (i) a dynamic factor common to all aggregates, regions, and countries (the world factor); (ii) a set of 7 regional dynamic factors common across aggregates within a region; (iii) 60 country factors to capture dynamic comovement across aggregates within each country; and (iv) a component for each aggregate that captures idiosyncratic dynamics. We decompose the volatility in each aggregate into the fraction due to the world, region, country, and idiosyncratic components. The results indicate that the world factor is an important source of volatility for aggregates in most countries, providing evidence for a world business cycle. We find that the region-specific factor plays only a minor role in explaining fluctuations in economic activity. While the world and regional factors together account for a larger share of fluctuations in output than in consumption, the country-specific and idiosyncratic components play much larger roles in explaining investment dynamics. We also explore how the three aggregates in each country relate to the world, region and country factors, and document similarities and differences across regions, countries and aggregates. We link the empirical results to the economic structures of the countries in the sample.

A new Bayesian method is proposed for the analysis of discretely sampled diffusion processes. The method, which is termed high frequency augmentation (HFA), is a simple numerical method that is applicable to a wide variety of univariate or multivariate diffusion and jump-diffusion processes. It is furthermore useful when observations are irregularly observed, when one or more elements of the multivariate process are latent, or when microstructure effects add error to the observed data. The Markov chain-Monte Carlo-based procedure can be used to attain the posterior distributions of the parameters of the drift and diffusion functions as well as the posteriors of missing or latent data. Several examples are explored. First, posteriors of the parameters of a geometric Brownian motion are attained using HFA and compared with those obtained using standard analytical methods in a short Monte Carlo study. Second, a stochastic volatility model is estimated on a sample of S&P500 returns, a problem for which posteriors are analytically intractable. Third, it is shown how the method can be used to estimate an interest rate process using data that suffer from severe rounding. Finally, extension of the method to jump-diffusions is described and applied to the analysis of the U.S dollar/German mark exchange rate.

This paper designs and implements a Bayesian dynamic latent factor model for a vector of data describing the Iowa economy. Posterior distributions of parameters and the latent factor are analyzed by Markov Chain Monte Carlo methods, and coincident and leading indicators are computed by using posterior mean values of current and predictive distributions for the latent factor. JEL Codes: C11, C32, E32. Keywords: Markov chain, Monte Carlo, index model, latent dynamic factor Running Head: Bayesian Leading Indicators * Manuscript received September 1996; revised December 1997. 1 This paper was initially prepared for presentation at the July 1996 NBER/NSF Seminar on Forecasting and Empirical Methods in Macroeconomics. We thank Francis Diebold, Robert Engle, John Geweke, Beth Ingram, Thomas Sargent, Christopher Sims, James Stock, Ruey Tsay, Mark Watson, and three anonymous referees for helpful comments. Sid Chib graciously supplied us with GAUSS code for posterior analysis of regression mo...

Using a new Bayesian method for the analysis of diffusion processes, this article finds that the nonlinear drift in interest rates found in a number of previous studies can be confirmed only under prior distributions that are best described as informative. The assumption of stationarity, which is common in the literature, represents a nontrivial prior belief about the shape of the drift function. This belief and the use of ``flat'' priors contribute strongly to the finding of nonlinear mean reversion. Implementation of an approximate Jeffreys prior results in virtually no evidence for mean reversion in interest rates unless stationarity is assumed. Finally, the article documents that nonlinear drift is primarily a feature of daily rather than monthly data, and that these data contain a transitory element that is not reflected in the volatility of longermaturity yields.

I argue for thinking of program evaluation as a decision problem. There are two steps. First, a counselor determines which program (treatment or control) each individual joins, based for example on maximizing the probability of employment or expected earnings. Second, the policymaker decides whether: to assign all individuals to treatment or to control, or to allow the counselor to choose. This framework has two advantages. Individualized assignment rules (known as profiling) can raise the average impact, improving cost effectiveness by exploiting treatment-impact heterogeneity. Second, it accounts systematically for inequality and uncertainty, and the policymaker’s attitude toward these, in the evaluation.