Bayesian Inferences and Forecasts With Multiple Autoregressive Moving Average Models

The main objective of this paper is to develop convenient Bayesian techniques for estimation and forecasting which can be used to analyze multiple (multivariate) autoregressive moving average processes. Based on the conditional likelihood function and the least squares estimates of the residuals, the marginal posterior distribution of the coefficients of the model is approximated by a matrix t distribution, the marginal posterior distribution of the precision matrix is approximated by a Wishart distribution, and the predictive distribution is approximated by a multivariate t distribution. Some numerical examples are given to demonstrate the idea of using the proposed techniques to analyze different types of multiple ARMA models.     

Hidden Markov Mixture Autoregressive Models: Stability and Moments

This article introduces a parsimonious structure for mixture of autoregressive models, where the weighting coefficients are determined through latent random variables, as functions of all past observations. These latent variables follow a Markov model. We propose a dynamic programming algorithm for forecasting, which reduces the volume of calculations. We also derive limiting behavior of unconditional first moment of the process and an appropriate upper bound for the limiting value of the variance. Further more, we show convergence and stability of the second moment. Finally, we illustrate the efficacy of the proposed model by simulation.     

Monitoring Variance Change in Infinite Order Moving Average Processes and Nonstationary Autoregressive Processes

This article considers the sequential monitoring problem of variance change in stationary and non stationary time series. We suggest a CUSUM of squares procedure to detect variance change in infinite order moving average processes, and a residual CUSUM of squares procedure to detect variance change in non stationary autoregressive processes. Moreover, we introduce a bandwidth parameter to improve the monitoring power when change point does not occur at the early stage of monitoring. It is shown that both procedures have the same null distribution. The procedures are illustrated via a simulation study and an investigation of daily Mexico/US exchange rates.     

Bayesian Analysis of Two-Regime Threshold Autoregressive Moving Average Model with Exogenous Inputs

We consider Bayesian analysis of threshold autoregressive moving average model with exogenous inputs (TARMAX). In order to obtain the desired marginal posterior distributions of all parameters including the threshold value of the two-regime TARMAX model, we use two different Markov chain Monte Carlo (MCMC) methods to apply Gibbs sampler with Metropolis-Hastings algorithm. The first one is used to obtain iterative least squares estimates of the parameters. The second one includes two MCMC stages for estimate the desired marginal posterior distributions and the parameters. Simulation experiments and a real data example show support to our approaches.     

On-Line Learning for the Infinite Hidden Markov Model

We develop a sequential Monte Carlo algorithm for the infinite hidden Markov model (iHMM) that allows us to perform on-line inferences on both system states and structural (static) parameters. The algorithm described here provides a natural alternative to Markov chain Monte Carlo samplers previously developed for the iHMM, and is particularly helpful in applications where data is collected sequentially and model parameters need to be continuously updated. We illustrate our approach in the context of both a simulation study and a financial application.     

Inference on Filtered and Smoothed Probabilities in Markov-Switching Autoregressive Models

ABSTRACT

We derive a statistical theory that provides useful asymptotic approximations to the distributions of the single inferences of filtered and smoothed probabilities, derived from time series characterized by Markov-switching dynamics. We show that the uncertainty in these probabilities diminishes when the states are separated, the variance of the shocks is low, and the time series or the regimes are persistent. As empirical illustrations of our approach, we analyze the U.S. GDP growth rates and the U.S. real interest rates. For both models, we illustrate the usefulness of the confidence intervals when identifying the business cycle phases and the interest rate regimes.     

A nonparametric Bayesian model for inference in related longitudinal studies

Summary.  We discuss a method for combining different but related longitudinal studies to improve predictive precision. The motivation is to borrow strength across clinical studies in which the same measurements are collected at different frequencies. Key features of the data are heterogeneous populations and an unbalanced design across three studies of interest. The first two studies are phase I studies with very detailed observations on a relatively small number of patients. The third study is a large phase III study with over 1500 enrolled patients, but with relatively few measurements on each patient. Patients receive different doses of several drugs in the studies, with the phase III study containing significantly less toxic treatments. Thus, the main challenges for the analysis are to accommodate heterogeneous population distributions and to formalize borrowing strength across the studies and across the various treatment levels. We describe a hierarchical extension over suitable semiparametric longitudinal data models to achieve the inferential goal. A nonparametric random-effects model accommodates the heterogeneity of the population of patients. A hierarchical extension allows borrowing strength across different studies and different levels of treatment by introducing dependence across these nonparametric random-effects distributions. Dependence is introduced by building an analysis of variance (ANOVA) like structure over the random-effects distributions for different studies and treatment combinations. Model structure and parameter interpretation are similar to standard ANOVA models. Instead of the unknown normal means as in standard ANOVA models, however, the basic objects of inference are random distributions, namely the unknown population distributions under each study. The analysis is based on a mixture of Dirichlet processes model as the underlying semiparametric model.     

Non parametric Bayesian analysis of the two-sample problem with censoring

Testing for differences between two groups is a fundamental problem in statistics, and due to developments in Bayesian non parametrics and semiparametrics there has been renewed interest in approaches to this problem. Here we describe a new approach to developing such tests and introduce a class of such tests that take advantage of developments in Bayesian non parametric computing. This class of tests uses the connection between the Dirichlet process (DP) prior and the Wilcoxon rank sum test but extends this idea to the DP mixture prior. Here tests are developed that have appropriate frequentist sampling procedures for large samples but have the potential to outperform the usual frequentist tests. Extensions to interval and right censoring are considered and an application to a high-dimensional data set obtained from an RNA-Seq investigation demonstrates the practical utility of the method.     

A Class of Markov Chain Models for Average Run Length Computations for Autocorrelated Processes

The average run length (ARL) of conventional control charts is typically computed assuming temporal independence. However, this assumption is frequently violated in practical applications. Alternative ARL computations have often been conducted via time consuming and yet not necessarily very accurate simulations. In this article, we develop a class of Markov chain models for evaluating the run length performance of traditional control charts for autocorrelated processes. We show extensions from the univariate AR(1) model to the general multivariate VARMA(p, q) time series. The results of the proposed method are highly comparable to those of simulations and with significantly less computational overhead.     

A Bayesian Analysis of Autoregressive Models with Exogenous Variables and Power-Transformed and Threshold GARCH Errors

Consider a class of autoregressive models with exogenous variables and power transformed and threshold GARCH (ARX-PTTGARCH) errors, which is a natural generalization of the standard and special GARCH model. We propose a Bayesian method to show that combining Gibbs sampler and Metropolis-Hastings algorithm to give a Bayesian analysis can be applied to estimate parameters of ARX-PTTGARCH models with success.     

Bayesian Identification of Moving Average Models

This study approaches the Bayesian identification of moving average processes using an approximate likelihood function and a normal gamma prior density. The marginal posterior probability mass function of the model order is developed in a convenient form. Then one may investigate the posterior probabilities over the grid of the order and choose the order with the highest probability to solve the identification problem. A comprehensive simulation study is carried out to demonstrate the performance of the proposed procedure and check its adequacy in handling the identification problem. In addition, the proposed Bayesian procedure is compared with some non Bayesian automatic techniques and another Bayesian technique. The numerical results support the adequacy of using the proposed procedure in solving the identification problem of moving average processes.     

Approximately Median-Unbiased Estimation of Autoregressive Models

A hedonic model of automobile prices that takes gasoline costs into account is developed and used to examine whether gasoline price increases (especially those related to the 1973 and 1979 oil shocks) changed consumer evaluations of the relative qualities of used cars in the U.S. during 1970–1981. We test the null hypothesis that the characteristics' coefficients remained constant over time. It is rejected if gasoline costs are excluded from the model but not if they are included. Alternative approaches are developed to show that the gasoline price increases alone can explain much of the observed changes in the coefficients.     

A Sequential Point Process Model and Bayesian Inference for Spatial Point Patterns with Linear Structures

Abstract. We introduce a flexible spatial point process model for spatial point patterns exhibiting linear structures, without incorporating a latent line process. The model is given by an underlying sequential point process model. Under this model, the points can be of one of three types: a ‘background point’ an ‘independent cluster point’ or a ‘dependent cluster point’. The background and independent cluster points are thought to exhibit ‘complete spatial randomness’, whereas the dependent cluster points are likely to occur close to previous cluster points. We demonstrate the flexibility of the model for producing point patterns with linear structures and propose to use the model as the likelihood in a Bayesian setting when analysing a spatial point pattern exhibiting linear structures. We illustrate this methodology by analysing two spatial point pattern datasets (locations of bronze age graves in Denmark and locations of mountain tops in Spain).     

Bayesian inference for rare errors in populations with unequal unit sizes

We describe a Bayesian model for a scenario in which the population of errors contains many 0s and there is a known covariate. This kind of structure typically occurs in auditing, and we use auditing as the driving application of the method. Our model is based on a categorization of the error population together with a Bayesian nonparametric method of modelling errors within some of the categories. Inference is through simulation. We conclude with an example based on a data set provided by the UK's National Audit Office.     

Statistical Inference for Partially Hidden Markov Models

ABSTRACT

In this article we introduce a new missing data model, based on a standard parametric Hidden Markov Model (HMM), for which information on the latent Markov chain is given since this one reaches a fixed state (and until it leaves this state). We study, under mild conditions, the consistency and asymptotic normality of the maximum likelihood estimator. We point out also that the underlying Markov chain does not need to be ergodic, and that identifiability of the model is not tractable in a simple way (unlike standard HMMs), but can be studied using various technical arguments.     

Moments of Mixture Periodic Autoregressive Models

This article deals with the study of some properties of a mixture periodically correlated autoregressive (MPAR _S ) time series model, which extends the mixture time invariant parameter autoregressive (MAR) model, that has recently received a considerable interest from many economic time series analysts, to mixture periodic parameter autoregressive model. The aim behind this extension is to make the model able to capture, in addition to all features captured by the classical MAR model, the periodicity feature exhibited by the autocovariance structure of many encountered financial and environmental time series with eventual multimodal distributions. Our main contribution here is obtaining of the second moment periodically stationary condition for a MPAR _S (K; 2,…, 2) model, furthermore the closed-form of the second moment is obtained.     

Choosing Prior Hyperparameters: With Applications to Time-Varying Parameter Models

Time-varying parameter models with stochastic volatility are widely used to study macroeconomic and financial data. These models are almost exclusively estimated using Bayesian methods. A common practice is to focus on prior distributions that themselves depend on relatively few hyperparameters such as the scaling factor for the prior covariance matrix of the residuals governing time variation in the parameters. The choice of these hyperparameters is crucial because their influence is sizeable for standard sample sizes. In this article, we treat the hyperparameters as part of a hierarchical model and propose a fast, tractable, easy-to-implement, and fully Bayesian approach to estimate those hyperparameters jointly with all other parameters in the model. We show via Monte Carlo simulations that, in this class of models, our approach can drastically improve on using fixed hyperparameters previously proposed in the literature. Supplementary materials for this article are available online.     

Conditional Correlation Models of Autoregressive Conditional Heteroscedasticity With Nonstationary GARCH Equations

In this article, we investigate the effects of careful modeling the long-run dynamics of the volatilities of stock market returns on the conditional correlation structure. To this end, we allow the individual unconditional variances in conditional correlation generalized autoregressive conditional heteroscedasticity (CC-GARCH) models to change smoothly over time by incorporating a nonstationary component in the variance equations such as the spline-GARCH model and the time-varying (TV)-GARCH model. The variance equations combine the long-run and the short-run dynamic behavior of the volatilities. The structure of the conditional correlation matrix is assumed to be either time independent or to vary over time. We apply our model to pairs of seven daily stock returns belonging to the S&P 500 composite index and traded at the New York Stock Exchange. The results suggest that accounting for deterministic changes in the unconditional variances improves the fit of the multivariate CC-GARCH models to the data. The effect of careful specification of the variance equations on the estimated correlations is variable: in some cases rather small, in others more discernible. We also show empirically that the CC-GARCH models with time-varying unconditional variances using the TV-GARCH model outperform the other models under study in terms of out-of-sample forecasting performance. In addition, we find that portfolio volatility-timing strategies based on time-varying unconditional variances often outperform the unmodeled long-run variances strategy out-of-sample. As a by-product, we generalize news impact surfaces to the situation in which both the GARCH equations and the conditional correlations contain a deterministic component that is a function of time.     

Modeling the density of US yield curve using Bayesian semiparametric dynamic Nelson-Siegel model

Abstract

This paper proposes the Bayesian semiparametric dynamic Nelson-Siegel model for estimating the density of bond yields. Specifically, we model the distribution of the yield curve factors according to an infinite Markov mixture (iMM). The model allows for time variation in the mean and covariance matrix of factors in a discrete manner, as opposed to continuous changes in these parameters such as the Time Varying Parameter (TVP) models. Estimating the number of regimes using the iMM structure endogenously leads to an adaptive process that can generate newly emerging regimes over time in response to changing economic conditions in addition to existing regimes. The potential of the proposed framework is examined using US bond yields data. The semiparametric structure of the factors can handle various forms of non-normalities including fat tails and nonlinear dependence between factors using a unified approach by generating new clusters capturing these specific characteristics. We document that modeling parameter changes in a discrete manner increases the model fit as well as forecasting performance at both short and long horizons relative to models with fixed parameters as well as the TVP model with continuous parameter changes. This is mainly due to fact that the discrete changes in parameters suit the typical low frequency monthly bond yields data characteristics better.     

On Mixture Periodic Vector Autoregressive Models

This article deals with the study of some properties of a mixture periodically correlated n-variate vector autoregressive (MPVAR) time series model, which extends the mixture time invariant parameter n-vector autoregressive (MVAR) model that has been recently studied by Fong et al. (2007 Fong, P.W., Li, W.K., Yau, C.W., Wong, C.S. (2007). On a mixture vector autoregressive model. The Canadian Journal of Statistics 35:135–150.[Crossref], [Web of Science ®] , [Google Scholar]). Our main contributions here are, on the one side, the obtaining of the second moment periodically stationary condition for a n-variate MPVAR_S(n; K; 2, …, 2) model; furthermore, the closed-form of the second moment is obtained and, on the other side, the estimation, via the Expectation-Maximization (EM) algorithm, of the coefficient matrices and the error variance matrix.