Bayesian Methods In Finance

This article looks at the usefulness of Bayesian methods in finance. It covers all the major topics in finance. It discusses the predictability of the mean of asset returns, central to finance, as it relates to the efficiency of financial markets. It reviews the economic relevance of predictability and its impact on optimal allocation. It also describes the Markov chain Monte Carlo (MCMC) and particle filtering algorithms that are important in modern Bayesian financial econometrics. MCMC algorithms have resulted in a tremendous growth in the use of stochastic volatility models in financial econometrics. This article also contains some major contributions of Bayesian econometrics to the literature on empirical asset pricing. Many of the other themes in modern Bayesian econometrics, including the use of shrinkage and the interaction between theory and econometrics are discussed. This article ends up with the discussion of a promising recent development in finance: filtering with parameter learning.

Introduction

This article deals with substantial computation component and discusses powerful computers and simulation algorithms that lead to model development where Bayesian econometric methods are predominant. It includes flexible or nonparametric models. It also distinguishes econometrics from statistics by its combination of economic theory with statistics. This article addresses principles, methods, and applications in different parts. It deals with a set of issues namely, the use of computationally intensive posterior simulation algorithms, heterogeneity, and problems caused by proliferation of parameters. The models discussed in this article have increased range and level of complication. They are strongly infused with economic theory and decision-theoretic issues. This article covers a broad range of the methods and models used by Bayesian econometricians in a wide variety of fields.

Flexible and Nonparametric Modeling

This article is divided into two parts. The first part considers flexible parametric models while the latter is nonparametric. It gives applications to regional growth data and semi parametric estimation of binomial proportions. It reviews methods for flexible mean regression, using either basis functions or Gaussian processes. This article also discusses Dirichlet processes and describes various posterior simulation algorithms for Bayesian nonparametric models. Usefulness is shown in empirical illustrations. Various applications as a function of income and as a cost function for electricity distribution are discussed. This article lists some freely available software that can accommodate many of the methods discussed. It provides a detailed discussion of both theory and computation for flexible treatment of distributions or functional forms or both.

Introduction to Simulation and MCMC Methods

The purpose of this article is to provide an overview of Monte Carlo methods for generating variates from a target probability distribution that are based on Markov chains. These methods, called Markov chain Monte Carlo (MCMC) methods, are widely used to summarize complicated posterior distributions in Bayesian statistics and econometrics. This article begins with an intuitive explanation of the ideas and concepts that underlie popular algorithms such as the Metropolis-Hastings algorithm and multi-block algorithm. It provides the concept of a source or proposal density, which is used to supply a randomization step or an acceptance condition to determine if the candidate draw should be accepted. It is important to assess the performance of the sampling algorithm to determine the rate of mixing. Finally, this article offers an extensive discussion of marginal likelihood calculation using posterior simulator output.

Bayesian Aspects of Treatment Choice

This article discusses the Bayesian approach to decision theory. It focuses on the case of an individual deciding between treatments. It deals with the role of information that is available about other individuals through a propensity score. It also shows the reason for absence of propensity score in the likelihood function but its appearance in the prior. A prior distribution leads to a closed-form expression for the decision rule. The parametric model plays the role of a prior distribution that can be dominated by the data. The next section examines the role of the propensity score in a random effects model with normal distributions for the outcomes and the random effects. It takes up the extension to the case of treatment selection based on unobservables. The main aim of this article is to estimate an average treatment effect for a particular covariate cell.

Bayesian Inference for Time Series State Space Models

This article provides a description of time series methods that emphasize modern macroeconomics and finance. It discusses a variety of posterior simulation algorithms and illustrates their use in a range of models. This article introduces the state space framework and explains the main ideas behind filtering, smoothing, and likelihood computation. It also mentions the particle filter as a general approach for estimating state space models and gives a brief discussion of its methods. The particle filter is a very useful tool in the Bayesian analysis of the kinds of complicated nonlinear state space models that are increasingly being used in macroeconomics. It also deals with conditionally Gaussian state space models and non-Gaussian state space models. A discussion of the advantages and disadvantages of each algorithm is provided in this article. This aims to help with the use of these methods in empirical work.

Exchangeability, Representation Theorems, and Subjectivity

This article is concerned with the foundation of statistical inference in the representation theorems. It shows how different assumptions about the joint distribution of the observable data lead to different parametric models defined by prior and likelihood function. Parametric models arise as an implication of the assumptions about observables. The article presents many extensions and offers description of the subjectivist attitude that underlies much of Bayesian econometrics. This subjectivist interpretation is close to probability. This article discusses exchangeability as the foundation for Bayesian econometrics. It serves as the basis for further extensions to incorporate heterogeneity and dependency across observations. It also discusses representation theorems involving random variables more complicated than Bernoulli random variables. They are not true properties of reality but are useful for making inferences regarding future observables.

Bayesian Applications in Marketing

This article describes various discrete choice models of consumers who may be heterogeneous both in terms of their preferences and in their sensitivities to marketing variables such as price. It addresses a distinct set of challenges that are being posed through the use of hierarchical priors. It considers standard statistical approach that generates discreteness by applying a censoring function to underlying continuous latent variables. This approach generates models that can be employed in situations where more descriptive models are required. Nonparametric and flexible parametric models involving Dirichlet processes and other mixtures are also accepted and favored in marketing. This article outlines several utility specifications that incorporate discreteness and other important aspects of consumer decisions. Computational issues are important when dealing with large marketing data sets and this article discusses on how to implement posterior simulation methods in marketing models.

Bayesian Methods in Microeconometrics

This article is of two-fold interest with the goal of providing an overview of the field and aims at discussing the most recent research in the relevant field. It shows how the computational methods and modelling ideas are being used by Bayesian econometricians. It also discusses linear models and presents a review of the normal linear regression model, deriving marginal, conditional, and predictive posterior densities of interest. This article proceeds further to discuss hierarchical linear models and review approaches to handle endogeneity problems. It presents applications and posterior simulation strategies for nonlinear latent variable models and considers the analysis of treatment effects models and multinomial and multivariate probit models. This article briefly reviews basic Bayesian approaches to the analysis of duration data.

Bayesian Macroeconometrics

This article presents the challenges that arise since macroeconomists often work in data-rich environments. It emphasizes multivariate models that can capture the co-movements of macroeconomic time series analysis. It discusses vector autoregressive (VAR) models distinguishing between reduced-form and structural VARs. Reduced-form VARs summarize the autocovariance properties of the data and provide a useful forecasting tool. The article shows how Bayesian methods have been empirically successful in responding to these challenges. It also encounters dynamic stochastic general equilibrium (DSGE) models that potentially differ in their economic implications. With posterior model probabilities, inference and decisions can be based on model averages. This article discusses inference with linearized as well as nonlinear DSGE models and reviews various approaches for evaluating the empirical fit of DSGE models. It concludes with a discussion of model uncertainty and decision-making with multiple models.