In this paper we discuss how to compare various (possibly misspecified) density forecast models using the Kullback-Leibler Information Criterion (KLIC) of a candidate density forecast model with respect to thetruedensity. TheKLIC-differential between a pair of competing models is the (predictive) log-likelihood ratio (LR) between the two models. Even though the true density is unknown, using the LR statistic amounts to comparing models with the KLIC as a loss function and thus enables us to assess which density forecast model can approximate the true density more closely. We also discuss how this KLIC is related to the KLIC based on the probability integral transform (PIT) in the framework of Diebold et al. (1998). While they are asymptotically equivalent, the PIT-based KLIC is best suited for evaluating the adequacy of each density forecast model and the original KLIC is best suited for comparing competing models. In an empirical study with the S&P500 and NASDAQ daily return series, we find strong evidence for rejecting the Normal-GARCH benchmark model, in favor of the models that can capture skewness in the conditional distribution and asymmetry and long-memory in the conditional variance.

In this paper we propose a testing procedure for comparing the predictive abilities of possibly misspecified density forecast models. We use the minimum Kullback-Leibler Information Criterion (KLIC) divergence measure to define the distance between the candidate density forecast model and the true model. We use the fact that the inverse-normal transform of the probability integral transforms (PIT) should be IID standard normal as discussed in Berkowitz (2001). To compare the performance of density forecast models in the tails, we use the censored likelihood functions to compute the tail minimum KLIC. The reality check test of White (2000) is then constructed using our distance measure as a loss function. To highlight the merits of our approach, we use the daily S&P500 and NASDAQ return series to conduct an empirical density forecast comparison exercise. A large set of distributions, including some recently proposed flexible distributions to accommodate higher moments, and the ARCH-family volatility specifications are studied. Our empirical findings lend further support of fat-tailedness and skewness of return distributions. In addition, the choice of conditional distribution specification appears to be a much more dominant factor in determining the quality of density forecasts than the choice of volatility specification.

This paper considers a Bayesian analysis of the linear regression model under independent sampling from general scale mixtures of Normals. Using a common reference prior, we investigate the validity of Bayesian inference and the existence of posterior moments of the regression and scale parameters. We find that whereas existence of the posterior distribution does not depend on the choice of the design matrix or the mixing distribution, both of them can crucially intervene in the existence of posterior moments. We identify some useful characteristics that allow for an easy verification of the existence of a wide range of moments. In addition, we provide full characterizations under sampling from finite mixtures of Normals, Pearson VII or certain Modulated Normal distributions. For empirical applications, a numerical implementation based on the Gibbs sampler is recommended.

100-8630 JAPANNOTE: IMES Discussion Paper Series is circulated in order to stimulate discussion and comments. Views expressed in Discussion Paper Series are those of authors and do not necessarily reflect those of the Bank of Japan or the Institute for Monetary and Economic Studies. IMES Discussion Paper Series 2002-E-14

Data heterogeneity appears when the sample comes from at least two different populations. We analyze three types of situations. In the first and simplest case the majority of the data come from a central model and a few isolated observations come from a contaminating distribution. The data from the contaminating distribution are called outliers and they have been studied in depth in the statistical literature. In the second case we still have a central model but the heterogeneous data may appear in clusters of outliers which mask each other. This is the multiple outlier problem which is much more difficult to handle and it has been analyzed and understood in the last few years. The few Bayesian contributions to this problem are presented. In the third case we do not have a central model but instead different groups of data have been generated by different models. For multivariate normal this problem has been analyzed by mixture models under the name of cluster analysis, but a challenging area of research is to develop a general methodology for applying this multiple model approach to other statistical problems. Heterogeneity implies in general an increase in the uncertainty of predictions, and we present in this paper a procedure to measure this effect.

In this paper we model Value-at-Risk (VaR) for daily stock index returns using a collection of parametric models of the ARCH family based on the skewed Student distribution. We show that models that rely on a symmetric density distribution for the error term underperform with respect to skewed density models when the left and right tails of the distribution of returns must be modelled. Thus, VaR for traders having both long and short positions is not adequately modelled using usual Normal or Student distributions. We suggest using an APARCH model based on the skewed Student distribution to fully take into account the fat left and right tails of the returns distribution. This allows for an adequate modelling of large returns defined on long and short trading positions. The performances of all models are assessed on daily data for the CAC40, DAX, NASDAQ, NIKKEI and SMI stock indexes. We also compute the expected short-fall and the average multiple of tail event to risk measure for the new model. Keywords: Value-at-Risk, Expected short-fall, Skewed Student distribution, APARCH, short trading JEL classification: C52, C53, G15 1 Department of Quantitative Economics, Maastricht University and Center for Operations Research and Econometrics, UCL; email: giot@core.ucl.ac.be or p.giot@ke.unimaas.nl 2 Departement des Sciences Economiques, Universite de Liege. This research was done when S. Laurent was visiting the Department of Quantitative Economics at Maastricht University. email: S.Laurent@ulg.ac.be 3 Corresponding author While remaining responsible for any errors in this paper, the authors would like to thank Luc Bauwens, Jon Danielsson, Philippe Lambert and Jean-Pierre Urbain for useful remarks and suggestions. 1

Introduction In scientic elds such as petroleum engineering, civil engineering, geography, geology, statistical spatial prediction is an important problem. Usually data from these scientic elds are thought as a sample from a realization of a random eld. The prevalent technique for statistical spatial prediction is Kriging (Cressie, 1993). Most of theories related to spatial prediction assume that the data are generated from a Gaussian random eld. However, non-Gaussian characteristics, such as nonnegative continuous variables with skewed distribution, often with a heavy right or left tail, appear in many data sets from scientic elds. We need ways to model those kinds of data sets. A common way to model this type of data is to assume that the random eld of interest is the result of an unknown nonlinear transformation of a Gaussian random eld. Trans-Gaussian kriging is the kriging variant used for prediction in transformed Gaussian random elds, where the normalizing tr

The use of a more general distribution function is often justified by treating the proposed distribution function as an alternative hypothesis. The suitability of such a distribution is then determined by the outcome of a nested test on a standard distributional assumption, which is typically a restricted version of the proposed distribution.

We study the beha viour of the informa tionma trix (IM) test when ma ximum likelihoodestima torsa re repla ced with robust estima tors. The la tter ma yunma sk outliersa nd hence improve the power of the test. We investiga te in deta il the loca la symptotic power of the IM test in the norma l model, for va rious estima torsa nd undera ra nge of lo ca#a lterna#jS es. These loca# a#FjAzS9PP es include conta#O#P4jS9 neighbourhoods, Student's t (with degrees of freedoma#domSP hing infinity) , skewness,a nda tilted norma l. Simula tion studies for fixed a# terna# ives confirm tha t in ma# y ca#P# the use of robust estima# ors substa ntia lly increa ses the power of the IM test. JEL cla# sifica##A0 : C12, C15 Key-words: informa tion ma trix test, robustness Corresponding author. Address: K.U.Leuven, Department of Economics, Naamsestraat 69, 3000 Leuven, BelgP m. Tel. +32 16 326652. Fax +32 16 326796. Email: dirk.hoorelbeke@econ.kuleuven.ac.be Introducti White (1982) introduced the informa tion ma trix (IM) test a# a# omnibus test for misspecifica#j#0 ofa pa ra metric model. The test exploits the wellknown property tha t, a# the model, the sum of the Hessi a# of the loglikelihood a# d the outer product of the score h a# zero expecta#OqS . So if, a# pa ra meter estima#Sj , the sa mplea ver a#S of the sum of the Hessi a# a# d the outer product of the score di#ers significa ntly from zero, this is evidence a#OAO st the model. The IM test is typica#P# implemented using ma ximum likelihood (ML) estima tes of thepa ra meters. In this pa per we explore the potentia# of repla##F g the ML estima#Sj with robust estima#SO s. Specifica#O tention is given to the e#ect on power, conjecturing tha tunma sking outliers will lea# to a# incr ea# ed power of the IM test. In mostca# es consider...

In this article we present new statistical methodology for the analysis of repeated measures of spatially correlated growth data. Our motivating application, a ten year study of height growth in a plantation of even-aged white spruce, presents several challenges for statistical analysis. Here, the growth measurements arise from an asymmetric distribution, with heavy tails, and thus standard longitudinal regression models based on a Gaussian error structure are not appropriate. We seek more flexibility for modeling both skewness and fat tails, and achieve this within the class of skew-elliptical distributions. Within this framework, robust space-time regression models are formulated using random effect growth curves, with coefficients arising from an underlying multivariate spatial process. Computational difficulties arise when data are collected at a large number of locations, and we consider two approaches for spatial modeling in the large data context. Both approaches are compared within the context of our application, and inference is conducted in a Bayesian framework, with implementation based on hybrid Monte Carlo.