Bayesian modelling of skewness and kurtosis with two-piece scale and shape distributions

We formalise and generalise the definition of the family of univariate double two--piece distributions, obtained by using a density--based transformation of unimodal symmetric continuous distributions with a shape parameter. The resulting distributions contain five interpretable parameters that control the mode, as well as the scale and shape in each direction. Four-parameter subfamilies of this class of distributions that capture different types of asymmetry are discussed. We propose interpretable scale and location-invariant benchmark priors and derive conditions for the propriety of the corresponding posterior distribution. The prior structures used allow for meaningful comparisons through Bayes factors within flexible families of distributions. These distributions are applied to data from finance, internet traffic and medicine, comparing them with appropriate competitors

Jointness in Bayesian variable selection with applications to growth regression

The authors present a measure of jointness to explore dependence among regressors in the context of Bayesian model selection. The jointness measure they propose equals the posterior odds ratio between those models that include a set of variables and the models that only include proper subsets. They show its application in cross-country growth regressions using two data-sets from the model-averaging growth literature.Statistical&Mathematical Sciences,Climate Change,Educational Technology and Distance Education,Economic Theory&Research,Achieving Shared Growth

On the effect of prior assumptions in Bayesian model averaging with applications to growth regression

This paper examines the problem of variable selection in linear regression models. Bayesian model averaging has become an important tool in empirical settings with large numbers of potential regressors and relatively limited numbers of observations. The paper analyzes the effect of a variety of prior assumptions on the inference concerning model size, posterior inclusion probabilities of regressors, and predictive performance. The analysis illustrates these issues in the context of cross-country growth regressions using three datasets with 41 to 67 potential drivers of growth and 72 to 93 observations. The results favor particular prior structures for use in this and related contexts.Educational Technology and Distance Education,Geographical Information Systems,Statistical&Mathematical Sciences,Science Education,Scientific Research&Science Parks

Non-Gaussian dynamic Bayesian modelling for panel data

A first order autoregressive non-Gaussian model for analysing panel data is proposed. The main feature is that the model is able to accommodate fat tails and also skewness, thus allowing for outliers and asymmetries. The modelling approach is to gain sufficient flexibility, without sacrificing interpretability and computational ease. The model incorporates individual effects and we pay specific attention to the elicitation of the prior. As the prior structure chosen is not proper, we derive conditions for the existence of the posterior. By considering a model with individual dynamic parameters we are also able to formally test whether the dynamic behaviour is common to all units in the panel. The methodology is illustrated with two applications involving earnings data and one on growth of countries.autoregressive modelling; growth convergence; individual effects; labour earnings; prior elicitation; posterior existence; skewed distributions

On Bayesian nonparametric modelling of two correlated distributions

In this paper, we consider the problem of modelling a pair of related distributions using Bayesian nonparametric methods. A representation of the distributions as weighted sums of distributions is derived through normalisation. This allows us to define several classes of nonparametric priors. The properties of these distributions are explored and efficient Markov chain Monte Carlo methods are developed. The methodology is illustrated on simulated data and an example concerning hospital efficiency measurement

Model-based Clustering of non-Gaussian Panel Data

In this paper we propose a model-based method to cluster units within a panel. The underlying model is autoregressive and non-Gaussian, allowing for both skewness and fat tails, and the units are clustered according to their dynamic behaviour and equilibrium level. Inference is addressed from a Bayesian perspective and model comparison is conducted using the formal tool of Bayes factors. Particular attention is paid to prior elicitation and posterior propriety. We suggest priors that require little subjective input from the user and possess hierarchical structures that enhance the robustness of the inference. Two examples illustrate the methodology: one analyses economic growth of OECD countries and the second one investigates employment growth of Spanish manufacturing firmsautoregressive modelling; employment growth; GDP growth convergence; hierarchical prior; model comparison; posterior propriety; skewness

Adaptive MC^3 and Gibbs algorithms for Bayesian Model Averaging in Linear Regression Models

The MC$^3$ (Madigan and York, 1995) and Gibbs (George and McCulloch, 1997) samplers are the most widely implemented algorithms for Bayesian Model Averaging (BMA) in linear regression models. These samplers draw a variable at random in each iteration using uniform selection probabilities and then propose to update that variable. This may be computationally inefficient if the number of variables is large and many variables are redundant. In this work, we introduce adaptive versions of these samplers that retain their simplicity in implementation and reduce the selection probabilities of the many redundant variables. The improvements in efficiency for the adaptive samplers are illustrated in real and simulated datasets

On Describing Multivariate Skewness: A Directional Approach

Most multivariate measures of skewness in the literature measure the overall skewness of a distribution. While these measures are perfectly adequate for testing the hypothesis of distributional symmetry, their relevance for describing skewed distributions is less obvious. In this article, we consider the problem of characterising the skewness of multivariate distributions. We define directional skewness as the skewness along a direction and analyse parametric classes of skewed distributions using measures based on directional skewness. The analysis brings further insight into the classes, allowing for a more informed selection of particular classes for particular applications. In the context of Bayesian linear regression under skewed error we use the concept of directional skewness twice. First in the elicitation of a prior on the parameters of the error distribution, and then in the analysis of the skewness of the posterior distribution of the regression residuals.Bayesian methods, Multivariate distribution, Multivariate regression, Prior elicitation, Skewness.