We develop new methods for conducting a finite sample, likelihood-based analysis of the multinomial probit model. Using a variant of the Gibbs sampler, an algorithm is developed to draw from the exact posterior of the multinomial probit model with correlated errors. This approach avoids direct evaluation of the likelihood and, thus, avoids the problems associated with calculating choice probabilities which affect both the standard likelihood and method of simulated moments approaches. Both simulated and actual consumer panel data are used to fit six-dimensional choice models. We also develop methods for analyzing random coefficient and multiperiod probit models.

Statistical inference in long-horizon event studies has been hampered by the fact that abnormal returns are neither normally distributed nor independent. This study presents a new approach to inference that overcomes these difficulties and dominates other popular testing methods. I illustrate the use of the methodology by examining the long-horizon returns of initial public offerings ~IPOs!. I find that the Fama and French ~1993! three-factor model is inconsistent with the observed long-horizon price performance of these IPOs, whereas a characteristic-based model cannot be rejected. RECENT EMPIRICAL STUDIES IN FINANCE document systematic long-run abnormal price reactions subsequent to numerous corporate activities. 1 Since these results imply that stock prices react with a long delay to publicly available information, they appear to be at odds with the Efficient Markets Hypothesis ~EMH!. Long-run event studies, however, are subject to serious statistical difficulties

We consider the problem of identifying differentially expressed genes under different conditions using cDNA microarrays. Standard statistical methods cannot be used because typically there are thousands of genes and few replicates. Because of the many steps involved in the experimental process, from hybridization to image analysis, cDNA microarray data often contain outliers. For example, an outlying data value could occur because of scratches or dust on the surface, imperfections in the glass, or imperfections in the array production. We develop a robust Bayesian hierarchical model for testing for differential expression. Outliers are modeled explicitly using a t-distribution. The model includes an exchangeable prior for the variances which allow different variances for the genes but still shrink extreme empirical variances. Our model can be used for testing for differentially expressed genes among multiple samples, and can distinguish between the different possible patterns of differential expression when there are three or more samples. Parameter estimation is carried out using a novel version of Markov Chain Monte Carlo that is appropriate when the model puts mass on subspaces of the full parameter space. The method is illustrated using two publicly available

Factor analysis is a method in multivariate statistical analysis that can help scientists determine which variables to study in a field and their relationships. We extend the Bayesian approach to factor analysis developed in 1989 by Press and Shigemasu (henceforth PS89) and revised in 1997 to model correlated observation vectors, factor score vectors, and factor loadings. Further, we place a prior distribution on the number of factors and obtain posterior estimates. Hitherto,

Consider the conditionally independent hierarchical model (CIHM) where observations y i are independently distributed from f(y i j` i ), the parameters ` i are independently distributed from distributions g(`j), and the hyperparameters are distributed according to a distribution h(). The posterior distribution of all parameters of the CIHM can be efficiently simulated by Monte Carlo Markov Chain (MCMC) algorithms. Although these simulation algorithms have facilitated the application of CIHM's, they generally have not addressed the problem of computing quantities useful in model selection. This paper explores how MCMC simulation algorithms and other related computational algorithms can be used to compute Bayes factors that are useful in criticizing a particular CIHM. In the case where the CIHM models a belief that the parameters are exchangeable or lie on a regression surface, the Bayes factor can measure the consistency of the data with the structural prior belief. Bayes factors can ...

Press and Shigemasu, 1989, proposed a Bayesian factor analysis model. Factor scores, factor loadings, and disturbance variances and covariances were estimated in closed form using a large sample approximation for one of the terms in the posterior distribution. This paper shows that by using Gibbs sampling or Lindley/Smith optimization ap- proaches to estimation instead of the large sample approximation, both of which are possible in this model, we can obtain improved point estimators in small samples.

Solving any inverse problem requires understanding the uncertainties in the data to know what it means to fit the data. We also need methods to incorporate dataindependent prior information to eliminate unreasonable models that fit the data. Both of these issues involve subtle choices that may significantly influence the results of inverse calculations. The specification of prior information is especially controversial. How does one quantify information? What does it mean to know something about a parameter a priori? In this tutorial we discuss Bayesian and frequentist methodologies that can be used to incorporate information into inverse calculations. In particular we show that apparently conservative Bayesian choices, such as representing interval constraints by uniform probabilities (as is commonly done when using genetic algorithms, for example) may lead to artificially small uncertainties. We also describe tools from statistical decision theory that can be used to...

Exploratory data analysis (EDA) and Bayesian inference (or, more generally, complex statistical modeling)|which are generally considered as unrelated statistical paradigms|can be particularly eective in combination. In this paper, we present a Bayesian framework for EDA based on posterior predictive checks. We explain how posterior predictive simulations can be used to create reference distributions for EDA graphs, and how this approach resolves some theoretical problems in Bayesian data analysis. We show how the generalization of Bayesian inference to include replicated data y and replicated parameters follows a long tradition of generalizations in Bayesian theory.

this article we attempt to remedy this, as well as to answer the call of Dempster (1985) for a more formal criterion for judging influence and to develop such a measure justified in the decision-theoretic framework of learning about parameters of interest

Smoothing regularizers for radial basis functions have been studied extensively, but no general smoothing regularizers for projective basis functions (PBFs), such as the widely-used sigmoidal PBFs, have heretofore th been proposed. We derive new classes of algebraically-simple m-order smoothing regularizers for networks N T of projective basis functions f(W, :r) = 5: big [,c v 5 + v/0] + u0, with general transfer functions g[.]. These regularizers are: RG(m,m) = y}u}ll,Jll 2m- GlobalForm RL(m,m) = y}u}ll,Jll 2m LocalForm With appropriate constant factors, these regularizers bound the corresponding mt*-order smoothing integral Of(W,a: ) 2 In the above expressions, {v j} are the projection vectors, W denotes all the network weights {u j, u0, v j, v0}, and (x) is a weighting function (not necessarily the input density) on the D-dimensional input space. The global and local cases are distinguished by different choices of (x).