Bivariate binomial autoregressive models

This paper introduces new classes of bivariate time series models being useful to fit count data time series with a finite range of counts. Motivation comes mainly from the comparison of schemes for monitoring tourism demand, stock data, production and environmental processes. All models are based on the bivariate binomial distribution of Type II. First, a new family of bivariate integer-valued GARCH models is proposed. Then, a new bivariate thinning operation is introduced and explained in detail. The new thinning operation has a number of advantages including the fact that marginally it behaves as the usual binomial thinning operation and also that allows for both positive and negative cross-correlations. Based upon this new thinning operation, a bivariate extension of the binomial autoregressive model of order one is introduced. Basic probabilistic and statistical properties of the model are discussed. Parameter estimation and forecasting are also covered. The performance of these models is illustrated through an empirical application to a set of rainy days time series collected from 2000 up to 2010 in the German cities of Bremen and Cuxhaven.publishe

Integer-valued time series and renewal processes

This research proposes a new but simple model for stationary time series of integer counts. Previous work in the area has focused on mixture and thinning methods and links to classical time series autoregressive moving-average difference equations; in contrast, our methods use a renewal process to generate a correlated sequence of Bernoulli trials. By superpositioning independent copies of such processes, stationary series with binomial, Poisson, geometric, or any other discrete marginal distribution can be readily constructed. The model class proposed is parsimonious, non-Markov, and readily generates series with either short or long memory autocovariances. The model can be fitted with linear prediction techniques for stationary series. Estimation of process parameters based on conditional least squares methods is considered. Asymptotic properties of the estimators are derived. The models sometimes have an autoregressive moving-average structure and we consider the AR(1) count process case in detail. Unlike previous methods based on mixture and thinning tactics, series with negative autocorrelations can be produced

Forecasting with two generalized integer-valued autoregressive processes of order one in the mutual random environment

In this article, we consider two univariate random environment integer-valued autoregressive processes driven by the same hidden process. A model of this kind is capable of describing two correlated non-stationary counting time series using its marginal variable parameter values. The properties of the model are presented. Some parameter estimators are described and implemented on the simulated time series. The introduction of this bivariate integer-valued autoregressive model with a random environment is justified at the end of the paper, where its real-life data-fitting performance was checked and compared to some other appropriate models. The forecasting properties of the model are tested on a few data sets, and forecasting errors are discussed through the residual analysis of the components that comprise the model.Peer Reviewe

Multivariate INAR(1) Regression Models Based on the Sarmanov Distribution.

A multivariate INAR(1) regression model based on the Sarmanov distribution is proposed for modelling claim counts from an automobile insurance contract with different types of coverage. The correlation between claims from different coverage types is considered jointly with the serial correlation between the observations of the same policyholder observed over time. Several models based on the multivariate Sarmanov distribution are analyzed. The new models offer some advantages since they have all the advantages of the MINAR(1) regression model but allow for a more flexible dependence structure by using the Sarmanov distribution. Driven by a real panel data set, these models are considered and fitted to the data to discuss their goodness of fit and computational efficiency

Análise estatística de séries de contagem com estrutura periódica

Doutoramento em MatemáticaOs modelos autoregressivos de valores inteiros multivariados (MINAR) desempenham um papel central na análise estatística de séries temporais de contagem. Dentro do razoavelmente grande espectro de modelos MINAR propostos na literatura, muito poucos focam a análise de séries de contagem com estrutura periódica. A análise dos processos de contagem multivariados apresenta muitos desafios que vão desde a especificação do modelo até à estimação de parâmetros. Esta tese tem como objetivo dar uma contribuição nessa direção. Especificamente, o objetivo deste trabalho é duplo: primeiro, introduzimos o processo multivariado periódico de ordem um, PMINAR(1). As propriedades probabilísticas e estatísticas do modelo são estudadas em detalhe. Para superar as dificuldades computacionais decorrentes da utilização do método da máxima verosimilhança introduzimos uma abordagem baseada na verosimilhança composta. O desempenho do método proposto e outros métodos concorrentes na estimação dos parâmetros é comparado através de um estudo de simulação. A previsão também é abordada. Uma aplicação de dados reais relacionados com a análise de fogos é apresentada. Em segundo lugar, propomos dois modelos INAR (univariado e bivariado) com estrutura periódica, S-PINAR(1) e BS-PINAR(1), respetivamente. Ambos os modelos são baseados no operador signed thinning permitindo contagens de valores positivos e negativos. Apresentamos as propriedades probabilísticas básicas e estatísticas dos modelos periódicos. As inovações são modeladas através das distribuições Skellam univariada e bivariada, respetivamente. Para avaliar o desempenho dos estimadores dos mínimos quadrados condicionais e da máxima verosimilhança condicional, foi realizado um estudo de simulação para o modelo S-PINAR(1).Multivariate INteger–valued AutoRegressive (MINAR) processes play a central role in the statistical analysis of integer-valued time series. Within the reasonably large spectrum of MINAR models proposed in the literature, however, only a few focus on the analysis of time series of count data with periodic structure. The analysis of multivariate counting processes presents many challenging problems ranging from model specification to parameter estimation. This thesis aims at giving a contribution towards this direction. Specifically, the purpose of this research is two-fold: first, we introduce the periodic multivariate process of order one (PMINAR(1) in short). The probabilistic and also the statistical properties of the model are studied in detail. To overcome the computational difficulties arising from the use of the maximum likelihood method we introduce a composite likelihood-based approach. The performance of the proposed method and other competitors methods of estimation is compared through a simulation study. Forecasting is also addressed. An application to a real data set related with the analysis of fire activity is presented. Secondly, we propose two INAR (univariate and bivariate) models with periodic structure, S-PINAR(1) and BS-PINAR(1), respectively. Both models are based on the signed thinning operator allowing for positive and negative counts. We examine the basic probabilistic and also the statistical properties of the periodic models. Innovations are modeled by univariate and bivariate Skellam distributions, respectively. To study the performance of the conditional least squares and conditional maximum likelihood estimators, a simulation study is conducted for the S-PINAR(1) model

Integer-Valued Time Series.

This thesis adresses statistical problems in econometrics. The first part contributes statistical methodology for nonnegative integer-valued time series. The second part of this thesis discusses semiparametric estimation in copula models and develops semiparametric lower bounds for a large class of time series models.

Count Time Series and Discrete Renewal Processes

Most data collected over time has some degree of periodicity (i.e. seasonally varying traits). Climate, stock prices, football season, energy consumption, wildlife sightings, and holiday sales all have cyclical patterns. It should come as no surprise that models that incorporate periodicity are paramount in the study of time series. The following work devises time series models for counts (integer-values) that are periodic and stationary. Foundational work is rst done in constructing a stationary periodic discrete renewal process (SPDRP). The dynamics of the SPDRP are mathematically interesting and have many modeling applications, expositions largely unexplored here. This work develops a SPDRP as a generation mechanism to produce a stationary count time series models with many desirable characteristics, including periodicity, negative autocovariances and long-memory. After development of the SPDRP univariate count models are generalized into multiple dimensions. A multivariate renewal process has many interrelated stochastic processes. The resulting multivariate model has all the desirable properties of its univariate kin, but can also have negative autocovariances between marginal components of the series. To our knowledge, this trait is seldom achieved in current multivariate count methods in tandem with long-memory and periodicit