Missing data and forecasting in multivariate time series: An application of the common components dynamic linear model

Summary The paper deals with missing data and forecasting problems in multivariate time series making use of the Common Components Dynamic Linear Model (DLMCC), presented in Quintana (1985), and West and Harrison (1989). Some results are presented and discussed: exploiting the correlation between series, estimated by the DLMCC, the paper shows as it is possible to update state vector posterior distributions for the unobserved series. This is realized on the base of the updating of the observed series state vectors, for which the usual Kalman filter equations can be applied. An application concerning some Italian private consumption series provides an example of the model capabilities.     

A simple hidden markov model for bayesian modeling with time dependent data

Consider a set of real valued observations collected over time. We pro¬pose a simple hidden Markow model for these realizations in which the the predicted distribution of the next future observation given the past is easily computed. The hidden or unobservable set of parameters is assumed to have a Markov structure of a special type. The model is quite flexible and can be used to incorporate different types of prior information in straightforward and sensible ways.     

A bivariate integer-valued long-memory model for high-frequency financial count data

We propose a bivariate integer-valued fractional integrated (BINFIMA) model to account for the long-memory property and apply the model to high-frequency stock transaction data. The BINFIMA model allows for both positive and negative correlations between the counts. The unconditional and conditional first- and second-order moments are given. The model is capable of capturing the covariance between and within intra-day time series of high-frequency transaction data due to macroeconomic news and news related to a specific stock. Empirically, it is found that Ericsson B has mean recursive process while AstraZeneca has long-memory property.     

A factor model approach for the joint segmentation with between‐series correlation

We consider the detection of changes in the mean of a set of time series. The breakpoints are allowed to be series specific, and the series are assumed to be correlated. The correlation between the series is supposed to be constant along time but is allowed to take an arbitrary form. We show that such a dependence structure can be encoded in a factor model. Thanks to this representation, the inference of the breakpoints can be achieved via dynamic programming, which remains one the most efficient algorithms. We propose a model selection procedure to determine both the number of breakpoints and the number of factors. This proposed method is implemented in the FASeg R package, which is available on the CRAN. We demonstrate the performances of our procedure through simulation experiments and present an application to geodesic data.     

A statistic to check model adequacy in time series

A simple statistic is suggested to examine if the assumptions on variances in a fitted time series model is valid or not. The properties of the statistic are discussed and examples are considered.     

Forecasting time series with missing data using Holt's model

This paper deals with the prediction of time series with missing data using an alternative formulation for Holt's model with additive errors. This formulation simplifies both the calculus of maximum likelihood estimators of all the unknowns in the model and the calculus of point forecasts. In the presence of missing data, the EM algorithm is used to obtain maximum likelihood estimates and point forecasts. Based on this application we propose a leave-one-out algorithm for the data transformation selection problem which allows us to analyse Holt's model with multiplicative errors. Some numerical results show the performance of these procedures for obtaining robust forecasts.     

A refined parameter estimating approach for HIV dynamic model

HIV dynamic models, a set of ordinary differential equations (ODEs), have provided new understanding of the pathogenesis of HIV infection and the treatment effects of antiviral therapies. However, to estimate parameters for ODEs is very challenging due to the complexity of this nonlinear system. In this article, we propose a comprehensive procedure to deal with this issue. In the proposed procedure, a series of cutting-edge statistical methods and techniques are employed, including nonparametric mixed-effects smoothing-based methods for ODE models and stochastic approximation expectation–maximization (EM) approach for mixed-effects ODE models. A simulation study is performed to validate the proposed approach. An application example from a real HIV clinical trial study is used to illustrate the usefulness of the proposed method.     

A geometric time series model with a new dependent Bernoulli counting series

ABSTRACT

New generalized binomial thinning operator with dependent counting series is introduced. An integer valued time series model with geometric marginals based on this thinning operator is constructed. Main features of the process are analyzed and determined. Estimation of the parameters are presented and some asymptotic properties of the obtained estimators are discussed. Behavior of the estimators is described through the numerical results. Also, model is applied on the real data set and compared to some relevant INAR(1) models.     

A dynamic multiple-variety choice adaption model

The choice of a product on one purchase occasion by one consumer could be multiple varieties and influenced by past usage experience of this product. To mimic the real situation, this article proposes a new dynamic multiple-variety choice (DMC) model which incorporates quantitative and qualitative dynamics into an additive utility function. This model exhibits three major features of consumer purchase behavior: more than one variety purchased, learning behavior from use experience, and forgetting with the passage of time. All these are achieved by combining a simultaneous demand model with Bayesian learning theory embedded in an exponential function. The model is tested and validated using Hong Kong television viewing data. Empirical results show that including Bayesian learning in a multiple-choice model significantly improves model performance and prediction accuracy, and consideration of the effect of forgetting when studying learning behavior renders the Bayesian learning model much more accurate in practical application.     

Power-law scaling behavior analysis of financial time series model by voter interacting dynamic system

We investigate the power-law scaling behaviors of returns for a financial price process which is developed by the voter interacting dynamic system in comparison with the real financial market index (Shanghai Composite Index). The voter system is a continuous time Markov process, which originally represents a voter's attitude on a particular topic, that is, voters reconsider their opinions at times distributed according to independent exponential random variables. In this paper, the detrended fluctuation analysis method is employed to explore the long range power-law correlations of return time series for different values of parameters in the financial model. The findings show no indication or very weak long-range power-law correlations for the simulated returns but strong long-range dependence for the absolute returns. The multiplier distribution is studied to demonstrate directly the existence of scale invariance in the actual data of the Shanghai Stock Exchange and the simulation data of the model by comparison. Moreover, the Zipf analysis is applied to investigate the statistical behaviors of frequency functions and the distributions of the returns. By a comparative study, the simulation data for our constructed price model exhibits very similar behaviors to the real stock index, this indicates somewhat rationality of our model to the market application.     

A dynamic hurdle model for zero-inflated count data

Excess zeros are encountered in many empirical count data applications. We provide a new explanation of extra zeros, related to the underlying stochastic process that generates events. The process has two rates: a lower rate until the first event and a higher one thereafter. We derive the corresponding distribution of the number of events during a fixed period and extend it to account for observed and unobserved heterogeneity. An application to the socioeconomic determinants of the individual number of doctor visits in Germany illustrates the usefulness of the new approach.     

Inverse Lindley power series distributions: a new compounding family and regression model with censored data

This paper introduces a new class of distributions by compounding the inverse Lindley distribution and power series distributions which is called compound inverse Lindley power series (CILPS) distributions. An important feature of this distribution is that the lifetime of the component associated with a particular risk is not observable, rather only the minimum lifetime value among all risks is observable. Further, these distributions exhibit an unimodal failure rate. Various properties of the distribution are derived. Besides, two special models of the new family are investigated. The model parameters of the two sub-models of the new family are obtained by the methods of maximum likelihood, least square, weighted least square and maximum product of spacing and compared them using the Monte Carlo simulation study. Besides, the log compound inverse Lindley regression model for censored data is proposed. Three real data sets are analyzed to illustrate the flexibility and importance of the proposed models.     

A semiparametric approach for modelling multivariate nonlinear time series

In this article, a semiparametric time‐varying nonlinear vector autoregressive (NVAR) model is proposed to model nonlinear vector time series data. We consider a combination of parametric and nonparametric estimation approaches to estimate the NVAR function for both independent and dependent errors. We use the multivariate Taylor series expansion of the link function up to the second order which has a parametric framework as a representation of the nonlinear vector regression function. After the unknown parameters are estimated by the maximum likelihood estimation procedure, the obtained NVAR function is adjusted by a nonparametric diagonal matrix, where the proposed adjusted matrix is estimated by the nonparametric kernel estimator. The asymptotic consistency properties of the proposed estimators are established. Simulation studies are conducted to evaluate the performance of the proposed semiparametric method. A real data example on short‐run interest rates and long‐run interest rates of United States Treasury securities is analyzed to demonstrate the application of the proposed approach. The Canadian Journal of Statistics 47: 668–687; 2019 © 2019 Statistical Society of Canada     

A bayesian approach to dynamic tobit models

This paper develops a posterior simulation method for a dynamic Tobit model. The major obstacle rooted in such a problem lies in high dimensional integrals, induced by dependence among censored observations, in the likelihood function. The primary contribution of this study is to develop a practical and efficient sampling scheme for the conditional posterior distributions of the censored (i.e., unobserved) data, so that the Gibbs sampler with the data augmentation algorithm is successfully applied. The substantial differences between this approach and some existing methods are highlighted. The proposed simulation method is investigated by means of a Monte Carlo study and applied to a regression model of Japanese exports of passenger cars to the U.S. subject to a non-tariff trade barrier.     

A bayesian approach to dynamic tobit models

This paper develops a posterior simulation method for a dynamic Tobit model. The major obstacle rooted in such a problem lies in high dimensional integrals, induced by dependence among censored observations, in the likelihood function. The primary contribution of this study is to develop a practical and efficient sampling scheme for the conditional posterior distributions of the censored (i.e., unobserved) data, so that the Gibbs sampler with the data augmentation algorithm is successfully applied. The substantial differences between this approach and some existing methods are highlighted. The proposed simulation method is investigated by means of a Monte Carlo study and applied to a regression model of Japanese exports of passenger cars to the U.S. subject to a non-tariff trade barrier.     

Bootstrapping a time series model: some empirical results

The bootstrap is a methodology for estimating standard errors. The idea is to use a Monte Carlo simulation experiment based on a nonparametric estimate of the error distribution. The main objective of this article is to demonstrate the use of the bootstrap to attach standard errors to coefficient estimates in a second-order autoregressive model fitted by least squares and maximum likelihood estimation. Additionally, a comparison of the bootstrap and the conventional methodology is made. As it turns out, the conventional asymptotic formulae (both the least squares and maximum likelihood estimates) for estimating standard errors appear to overestimate the true standard errors. But there are two problems:i. The first two observations y₁ and y₂ have been fixed, and ii. The residuals have not been inflated. After these two factors are considered in the trial and bootstrap experiment, both the conventional maximum likelihood and bootstrap estimates of the standard errors appear to be performing quite well.     

A dynamic model for double‐bounded time series with chaotic‐driven conditional averages

In this work, we introduce a class of dynamic models for time series taking values on the unit interval. The proposed model follows a generalized linear model approach where the random component, conditioned on the past information, follows a beta distribution, while the conditional mean specification may include covariates and also an extra additive term given by the iteration of a map that can present chaotic behavior. The resulting model is very flexible and its systematic component can accommodate short‐ and long‐range dependence, periodic behavior, laminar phases, etc. We derive easily verifiable conditions for the stationarity of the proposed model, as well as conditions for the law of large numbers and a Birkhoff‐type theorem to hold. A Monte Carlo simulation study is performed to assess the finite sample behavior of the partial maximum likelihood approach for parameter estimation in the proposed model. Finally, an application to the proportion of stored hydroelectrical energy in Southern Brazil is presented.     

A geometric time-series model with an alternative dependent Bernoulli counting series

In this article, we first introduce an alternative way for construction of the generalized binomial thinning operator with dependent counting series. Some properties of this thinning operator are derived and discussed. Then, by using this thinning operator, we introduce an integer-valued time-series model with geometric marginals. Some conditional and unconditional properties of this model are derived and discussed. Some estimation methods are considered and for some of them, asymptotic properties of the obtained estimates are derived. Performances of the estimates are discussed through some simulations. Finally, a real data example is considered and the goodness-of-fit of this model is compared with the models based on the binomial, negative binomial, and dependent binomial thinning operators.     

Time series analysis of categorical data using auto-odds ratio function

In this paper, we consider the auto-odds ratio function (AORF) as a measure of serial association for a stationary time series process of categorical data at two different time points. Numerical measures such as the autocorrelation function (ACF) have no meaningful interpretation, unless the time series data are numerical. Instead, we use the AORF as a measure of association to study the serial dependency of the categorical time series for both ordinal and nominal categories. Biswas and Song [Discrete-valued ARMA processes. Stat Probab Lett. 2009;79(17):1884–1889] provided some results on this measure for Pegram's operator-based AR(1) process with binary responses. Here, we extend this measure to more general set-ups, i.e. for AR(p) and MA(q) processes and for a general number of categories. We discuss how this method can effectively be used in parameter estimation and model selection. Following Weiß [Empirical measures of signed serial dependence in categorical time series. J Stat Comput Simul. 2011;81(4):411–429], we derive the large sample distribution of the estimator of the AORF under independent and identically distributed (iid) set-up. Some simulation results and two categorical data examples (one is ordinal and other nominal) are presented to illustrate the proposed method.     

A wavelet-based time-varying autoregressive model for non-stationary and irregular time series

In this work we propose an autoregressive model with parameters varying in time applied to irregularly spaced non-stationary time series. We expand all the functional parameters in a wavelet basis and estimate the coefficients by least squares after truncation at a suitable resolution level. We also present some simulations in order to evaluate both the estimation method and the model behavior on finite samples. Applications to silicates and nitrites irregularly observed data are provided as well.