Multiple-index approach to multiple autoregressive time series model

Time series which have more than one time dependent variable require building an appropriate model in which the variables not only have relationships with each other, but also depend on previous values in time. Based on developments for a sufficient dimension reduction, we investigate a new class of multiple time series models without parametric assumptions. First, for the dependent and independent time series, we simply use a univariate time series central subspace to estimate the autoregressive lags of the series. Secondly, we extract the successive directions to estimate the time series central subspace for regressors which include past lags of dependent and independent series in a mutual information multiple-index time series. Lastly, we estimate a multiple time series model for the reduced directions. In this article, we propose a unified estimation method of minimal dimension using an Akaike information criterion, for situations in which the dimension for multiple regressors is unknown. We present an analysis using real data from the housing price index showing that our approach is an alternative for multiple time series modeling. In addition, we check the accuracy for the multiple time series central subspace method using three simulated data sets.     

Modeling exchange rate dynamics: Non-linear dependence and thick tails

This paper illustrates a new approach to the statistical modeling of non-linear dependence and leptokurtosis in exchange rate data. The student's t autoregressive model withdynamic heteroskedasticity (STAR) of spanos (1992) is shown to provide a parsimonious and statistically adequate representation of the probabilistic information in exchange rate data. For the STAR model, volatility predictions are formed via a sequentially updated weighting scheme which uses all the past history of the series. The estimated STAR models are shown to statistically dominate alternative ARCH-type formulations and suggest that volatility predictions are not necessarily as large or as variable as other models indicate.     

On consistent testing for serial correlation in seasonal time series models

The author considers serial correlation testing in seasonal time series models. He proposes a test statistic based on a spectral approach. Many tests of this type rely on kernel-based spectral density estimators that assign larger weights to low order lags than to high ones. Under seasonality, however, large autocorrelations may occur at seasonal lags that classical kernel estimators cannot take into account. The author thus proposes a test statistic that relies on the spectral density estimator of Shin (2004), whose weighting scheme is more adapted to this context. The distribution of his test statistic is derived under the null hypothesis and he studies its behaviour under fixed and local alternatives. He establishes the consistency of the test under a general fixed alternative. He also makes recommendations for the choice of the smoothing parameters. His simulation results suggest that his test is more powerful against seasonality than alternative procedures based on classical weighting schemes. He illustrates his procedure with monthly statistics on employment among young Americans.     

Modeling exchange rate dynamics: Non-linear dependence and thick tails

This paper illustrates a new approach to the statistical modeling of non-linear dependence and leptokurtosis in exchange rate data. The student's t autoregressive model withdynamic heteroskedasticity (STAR) of spanos (1992) is shown to provide a parsimonious and statistically adequate representation of the probabilistic information in exchange rate data. For the STAR model, volatility predictions are formed via a sequentially updated weighting scheme which uses all the past history of the series. The estimated STAR models are shown to statistically dominate alternative ARCH-type formulations and suggest that volatility predictions are not necessarily as large or as variable as other models indicate.     

Single-Index Additive Vector Autoregressive Time Series Models

Abstract.  We study a new class of nonlinear autoregressive models for vector time series, where the current vector depends on single-indexes defined on the past lags and the effects of different lags have an additive form. A sufficient condition is provided for stationarity of such models. We also study estimation of the proposed model using P-splines, hypothesis testing, asymptotics, selection of the order of the autoregression and of the smoothing parameters and nonlinear forecasting. We perform simulation experiments to evaluate our model in various settings. We illustrate our methodology on a climate data set and show that our model provides more accurate yearly forecasts of the El Niño phenomenon, the unusual warming of water in the Pacific Ocean.     

Robust Bayesian analysis of loss reserving data using scale mixtures distributions

It is vital for insurance companies to have appropriate levels of loss reserving to pay outstanding claims and related settlement costs. With many uncertainties and time lags inherently involved in the claims settlement process, loss reserving therefore must be based on estimates. Existing models and methods cannot cope with irregular and extreme claims and hence do not offer an accurate prediction of loss reserving. This paper extends the conventional normal error distribution in loss reserving modeling to a range of heavy-tailed distributions which are expressed by certain scale mixtures forms. This extension enables robust analysis and, in addition, allows an efficient implementation of Bayesian analysis via Markov chain Monte Carlo simulations. Various models for the mean of the sampling distributions, including the log-Analysis of Variance (ANOVA), log-Analysis of Covariance (ANCOVA) and state space models, are considered and the straightforward implementation of scale mixtures distributions is demonstrated using OpenBUGS.     

A monte carlo comparison of tests based on the durbin-watson statistic with other autocorrelation tests in dynamic models

This paper examines the sampling properties of a number of serial correlation tests in dynamic linear models which include one or two lags of the dependent variable. Among the tests considered are the Durbin-Watson (DW) bounds test, modified versions of the DW proposed recently by King and Wu and Inder, Durbin's m test, Inder's point optimal test and a Hausman type test. Sampling designs include models with one or two lags of the dependent variable. The m, Hausman, and Inder's tests have the best performance, while Inder's modified DW test appears to be better than the other DW based tests. Results also suggest that tests are less powerful and more sensitive to design parameters in models with higher dynamics, with the DW-based tests being the most sensitive.     

Dynamic latent trait models with mixed hidden Markov structure for mixed longitudinal outcomes

We propose a general Bayesian joint modeling approach to model mixed longitudinal outcomes from the exponential family for taking into account any differential misclassification that may exist among categorical outcomes. Under this framework, outcomes observed without measurement error are related to latent trait variables through generalized linear mixed effect models. The misclassified outcomes are related to the latent class variables, which represent unobserved real states, using mixed hidden Markov models (MHMMs). In addition to enabling the estimation of parameters in prevalence, transition and misclassification probabilities, MHMMs capture cluster level heterogeneity. A transition modeling structure allows the latent trait and latent class variables to depend on observed predictors at the same time period and also on latent trait and latent class variables at previous time periods for each individual. Simulation studies are conducted to make comparisons with traditional models in order to illustrate the gains from the proposed approach. The new approach is applied to data from the Southern California Children Health Study to jointly model questionnaire-based asthma state and multiple lung function measurements in order to gain better insight about the underlying biological mechanism that governs the inter-relationship between asthma state and lung function development.     

Reply to comments on modeling asset returns with alternative stable distributions and some extensions *

This paper examines the sampling properties of a number of serial correlation tests in dynamic linear models which include one or two lags of the dependent variable. Among the tests considered are the Durbin-Watson (DW) bounds test, modified versions of the DW proposed recently by King and Wu and Inder, Durbin's m test, Inder's point optimal test and a Hausman type test. Sampling designs include models with one or two lags of the dependent variable. The m, Hausman, and Inder's tests have the best performance, while Inder's modified DW test appears to be better than the other DW based tests. Results also suggest that tests are less powerful and more sensitive to design parameters in models with higher dynamics, with the DW-based tests being the most sensitive.     

Longevity and concentration in survival times: the log-scale-location family of failure time models

Evidence suggests that the increasing life expectancy levels at birth witnessed over the past centuries are associated with a decreasing concentration of the survival times. The purpose of this work is to study the relationships that exist between longevity and concentration measures for some regression models for the evolution of survival. In particular, we study a family of survival models that can be used to capture the observed trends in longevity and concentration over time. The parametric family of log-scale-location models is shown to allow for modeling different trends of expected value and concentration of survival times. An extension towards mixture models is also described in order to take into account scenarios where a fraction of the population experiences short term survival. Some results are also presented for such framework. The use of both the log-scale-location family and the mixture model is illustrated through an application to period life tables from the Human Mortality Database.     

Recurrent events and the exploding Cox model

Counting process models have played an important role in survival and event history analysis for more than 30 years. Nevertheless, almost all models that are being used have a very simple structure. Analyzing recurrent events invites the application of more complex models with dynamic covariates. We discuss how to define valid models in such a setting. One has to check carefully that a suggested model is well defined as a stochastic process. We give conditions for this to hold. Some detailed discussion is presented in relation to a Cox type model, where the exponential structure combined with feedback lead to an exploding model. In general, counting process models with dynamic covariates can be formulated to avoid explosions. In particular, models with a linear feedback structure do not explode, making them useful tools in general modeling of recurrent events.     

Bayesian Analysis of Randomized Response Sum Score Variables

The randomized-response (RR) technique is an effective survey method when collecting sensitive information. In this technique, a probability mechanism using randomization devices is commonly involved in answering to sensitive questions. In order to evaluate the survey at the most accurate extend, self-protection (SP) is introduced to describe the responses by participants who give the evasive answer without taking the result of the randomization device into account. In this study, we propose a Bayesian approach to modeling RR sum score variables under SP assumption. RR data from a Dutch survey on non-compliance with social security regulation in 2004 is used to demonstrate the proposed models.     

Bayesian Kernel Mixtures for Counts

Although Bayesian nonparametric mixture models for continuous data are well developed, there is a limited literature on related approaches for count data. A common strategy is to use a mixture of Poissons, which unfortunately is quite restrictive in not accounting for distributions having variance less than the mean. Other approaches include mixing multinomials, which requires finite support, and using a Dirichlet process prior with a Poisson base measure, which does not allow smooth deviations from the Poisson. As a broad class of alternative models, we propose to use nonparametric mixtures of rounded continuous kernels. An efficient Gibbs sampler is developed for posterior computation, and a simulation study is performed to assess performance. Focusing on the rounded Gaussian case, we generalize the modeling framework to account for multivariate count data, joint modeling with continuous and categorical variables, and other complications. The methods are illustrated through applications to a developmental toxicity study and marketing data. This article has supplementary material online.     

Modelling the role of variables in model-based cluster analysis

In the framework of cluster analysis based on Gaussian mixture models, it is usually assumed that all the variables provide information about the clustering of the sample units. Several variable selection procedures are available in order to detect the structure of interest for the clustering when this structure is contained in a variable sub-vector. Currently, in these procedures a variable is assumed to play one of (up to) three roles: (1) informative, (2) uninformative and correlated with some informative variables, (3) uninformative and uncorrelated with any informative variable. A more general approach for modelling the role of a variable is proposed by taking into account the possibility that the variable vector provides information about more than one structure of interest for the clustering. This approach is developed by assuming that such information is given by non-overlapped and possibly correlated sub-vectors of variables; it is also assumed that the model for the variable vector is equal to a product of conditionally independent Gaussian mixture models (one for each variable sub-vector). Details about model identifiability, parameter estimation and model selection are provided. The usefulness and effectiveness of the described methodology are illustrated using simulated and real datasets.     

A Tutorial of Survival Modeling to Capture Covariate Effect

Mixture modeling in general and expectation–maximization in particular are too cumbersome and confusing for applied health researchers. Consequently, the full potential of mixture modeling is not realized. To remedy the deficiency, this tutorial article is prepared. This article addresses important applied problems in survival analysis and handles them in deeper generality than the existing work, especially from the point of view of taking covariates into account. In specific, the article demonstrates the concepts, tools, and inferencial procedure of mixture modeling using head-and-neck cancer data and survival time after heart transplant surgery data.     

Lag Specification in Rational Distributed Lag Structural Models

This article presents methods for the preliminary specification of distributed lags in structural models in the absence of sufficient a priori information. It is argued that an instrumental-variables procedure produces good results as long as the goodness of fit in the instrument equations is satisfactory. When this is not the case, simple OLS specification is preferable to more complex instrumental-variables methods, even though it may produce some spurious parameters. The procedures are illustrated with a simulated two-equation system and a model of annual supply and price of hogs.     

Hierarchical Bayesian Approach to a Multi-Site Hidden Markov Model

Multivariate data with a sequential or temporal structure occur in various fields of study. The hidden Markov model (HMM) provides an attractive framework for modeling long-term persistence in areas of pattern recognition through the extension of independent and identically distributed mixture models. Unlike in typical mixture models, the heterogeneity of data is represented by hidden Markov states. This article extends the HMM to a multi-site or multivariate case by taking a hierarchical Bayesian approach. This extension has many advantages over a single-site HMM. For example, it can provide more information for identifying the structure of the HMM than a single-site analysis. We evaluate the proposed approach by exploiting a spatial correlation that depends on the distance between sites.     

The double chain markov model

In the class of discrete time Markovian processes, two models are widely used, the Markov chain and the hidden Markov model. A major difference between these two models lies in the relation between successive outputs of the observed variable. In a visible Markov chain, these are directly correlated while in hidden models they are not. However, in some situations it is possible to observe both a hidden Markov chain and a direct relation between successive observed outputs. Unfortunately, the use of either a visible or a hidden model implies the suppression of one of these hypothesis. This paper prsents a Markovian model under random environment called the Double Chain Markov Model which takes into account the maijn features of both visible and hidden models. Its main purpose is the modeling of non-homogeneous time-series. It is very flexible and can be estimated with traditional methods. The model is applied on a sequence of wind speeds and it appears to model data more successfully than both the usual Markov chains and hidden Markov models.     

The stochastic topic block model for the clustering of vertices in networks with textual edges

Due to the significant increase of communications between individuals via social media (Facebook, Twitter, Linkedin) or electronic formats (email, web, e-publication) in the past two decades, network analysis has become an unavoidable discipline. Many random graph models have been proposed to extract information from networks based on person-to-person links only, without taking into account information on the contents. This paper introduces the stochastic topic block model, a probabilistic model for networks with textual edges. We address here the problem of discovering meaningful clusters of vertices that are coherent from both the network interactions and the text contents. A classification variational expectation-maximization algorithm is proposed to perform inference. Simulated datasets are considered in order to assess the proposed approach and to highlight its main features. Finally, we demonstrate the effectiveness of our methodology on two real-word datasets: a directed communication network and an undirected co-authorship network.     

Structured priors for multivariate time series

A class of prior distributions for multivariate autoregressive models is presented. This class of priors is built taking into account the latent component structure that characterizes a collection of autoregressive processes. In particular, the state-space representation of a vector autoregressive process leads to the decomposition of each time series in the multivariate process into simple underlying components. These components may have a common structure across the series. A key feature of the proposed priors is that they allow the modeling of such common structure. This approach also takes into account the uncertainty in the number of latent processes, consequently handling model order uncertainty in the multivariate autoregressive framework. Posterior inference is achieved via standard Markov chain Monte Carlo (MCMC) methods. Issues related to inference and exploration of the posterior distribution are discussed. We illustrate the methodology analyzing two data sets: a synthetic data set with quasi-periodic latent structure, and seasonally adjusted US monthly housing data consisting of housing starts and housing sales over the period 1965 to 1974.