Fractional approaches for the distribution of innovation sequence of INAR(1) processes

Abstract

In this paper, we present a fractional decomposition of the probability generating function of the innovation process of the first-order non-negative integer-valued autoregressive [INAR(1)] process to obtain the corresponding probability mass function. We also provide a comprehensive review of integer-valued time series models, based on the concept of thinning operators with geometric-type marginals. In particular, we develop two fractional approaches to obtain the distribution of innovation processes of the INAR(1) model and show that the distribution of the innovations sequence has geometric-type distribution. These approaches are discussed in detail and illustrated through a few examples.     

Extended Poisson INAR(1) processes with equidispersion,underdispersion and overdispersion

Real count data time series often show the phenomenon of the underdispersion and overdispersion. In this paper, we develop two extensions of the first-order integer-valued autoregressive process with Poisson innovations, based on binomial thinning, for modeling integer-valued time series with equidispersion, underdispersion, and overdispersion. The main properties of the models are derived. The methods of conditional maximum likelihood, Yule–Walker, and conditional least squares are used for estimating the parameters, and their asymptotic properties are established. We also use a test based on our processes for checking if the count time series considered is overdispersed or underdispersed. The proposed models are fitted to time series of the weekly number of syphilis cases and monthly counts of family violence illustrating its capabilities in challenging the overdispersed and underdispersed count data.     

Predicting online gambling self-exclusion: an analysis of the performance of supervised machine learning models

As gambling operators become increasingly sophisticated in their analysis of individual gambling behaviour, this study evaluates the potential for using machine learning techniques to identify individuals who used self-exclusion tools out of a sample of 845 online gamblers, based on analysing trends in their gambling behaviour. Being able to identify other gamblers whose behaviour is similar to those who decided to use self-exclusion tools could, for instance, be used to share responsible gaming messages or other information that aids self-aware gambling and reduces the risk of adverse outcomes. However, operators need to understand how accurate models can be and which techniques work well. The purpose of the article is to identify the most accurate technique out of four highly diverse techniques and to discuss how to deal analytically and practically with a rare event like self-exclusion, which was used by fewer than 1% of gamblers in our data-set. We conclude that balanced training data-sets are necessary for creating effective models and that, on our data-set, the most effective method is the random forest technique which achieves an accuracy improvement of 35 percentage points versus baseline estimates.     

A new approach to modeling positive random variables with repeated measures

In many situations, it is common to have more than one observation per experimental unit, thus generating the experiments with repeated measures. In the modeling of such experiments, it is necessary to consider and model the intra-unit dependency structure. In the literature, there are several proposals to model positive continuous data with repeated measures. In this paper, we propose one more with the generalization of the beta prime regression model. We consider the possibility of dependence between observations of the same unit. Residuals and diagnostic tools also are discussed. To evaluate the finite-sample performance of the estimators, using different correlation matrices and distributions, we conducted a Monte Carlo simulation study. The methodology proposed is illustrated with an analysis of a real data set. Finally, we create an R package for easy access to publicly available the methodology described in this paper.     

Does financial development affect income inequality in the U.S. States?

This paper examines the role of financial development on U.S. state-level income inequality in the 50 states from 1976 to 2011, using fixed-effect estimation. We find robust results whereby financial development linearly increases income inequality for the 50 states. When we divide 50 states into two separate groups of above-average and below-average inequality states than the cross-state average inequality, the effect of financial development on income inequality appears non-linear. When financial development improves, the effect increases at an increasing rate for above-average income-inequality states, whereas an inverted U-shaped relationship exists for below-average income-inequality states. To our knowledge, this paper is the first to examine the role of financial development on U.S. state-level inequality.     

Generalized Tobit models: diagnostics and application in econometrics

The standard Tobit model is constructed under the assumption of a normal distribution and has been widely applied in econometrics. Atypical/extreme data have a harmful effect on the maximum likelihood estimates of the standard Tobit model parameters. Then, we need to count with diagnostic tools to evaluate the effect of extreme data. If they are detected, we must have available a Tobit model that is robust to this type of data. The family of elliptically contoured distributions has the Laplace, logistic, normal and Student-t cases as some of its members. This family has been largely used for providing generalizations of models based on the normal distribution, with excellent practical results. In particular, because the Student-t distribution has an additional parameter, we can adjust the kurtosis of the data, providing robust estimates against extreme data. We propose a methodology based on a generalization of the standard Tobit model with errors following elliptical distributions. Diagnostics in the Tobit model with elliptical errors are developed. We derive residuals and global/local influence methods considering several perturbation schemes. This is important because different diagnostic methods can detect different atypical data. We implement the proposed methodology in an R package. We illustrate the methodology with real-world econometrical data by using the R package, which shows its potential applications. The Tobit model based on the Student-t distribution with a small quantity of degrees of freedom displays an excellent performance reducing the influence of extreme cases in the maximum likelihood estimates in the application presented. It provides new empirical evidence on the capabilities of the Student-t distribution for accommodation of atypical data.     

A class of residuals for outlier identification in zero adjusted regression models

Zero adjusted regression models are used to fit variables that are discrete at zero and continuous at some interval of the positive real numbers. Diagnostic analysis in these models is usually performed using the randomized quantile residual, which is useful for checking the overall adequacy of a zero adjusted regression model. However, it may fail to identify some outliers. In this work, we introduce a class of residuals for outlier identification in zero adjusted regression models. Monte Carlo simulation studies and two applications suggest that one of the residuals of the class introduced here has good properties and detects outliers that are not identified by the randomized quantile residual.