Improved estimation for Poisson INAR(1) models

We consider the first-order Poisson autoregressive model proposed by McKenzie [Some simple models for discrete variate time series. Water Resour Bull. 1985;21:645–650] and Al-Osh and Alzaid [First-order integer valued autoregressive (INAR(1)) process. J Time Ser Anal. 1987;8:261–275], which may be suitable in situations where the time series data are non-negative and integer valued. We derive the second-order bias of the squared difference estimator [Weiß. Process capability analysis for serially dependent processes of Poisson counts. J Stat Comput Simul. 2012;82:383–404] for one of the parameters and show that this bias can be used to define a bias-reduced estimator. The behaviour of a modified conditional least-squares estimator is also studied. Furthermore, we access the asymptotic properties of the estimators here discussed. We present numerical evidence, based upon Monte Carlo simulation studies, showing that the here proposed bias-adjusted estimator outperforms the other estimators in small samples. We also present an application to a real data set.     

Goodness-of-fit tests for binomial AR(1) processes

The binomial AR(1) model describes a nonlinear process with a first-order autoregressive (AR(1)) structure and a binomial marginal distribution. To develop goodness-of-fit tests for the binomial AR(1) model, we investigate the observed marginal distribution of the binomial AR(1) process, and we tackle its autocorrelation structure. Motivated by the family of power-divergence statistics for handling discrete multivariate data, we derive the asymptotic distribution of certain categorized power-divergence statistics for the case of a binomial AR(1) process. Then we consider Bartlett's formula, which is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocorrelations, but which is not applicable when the underlying process is nonlinear. Hence, we derive a novel Bartlett-type formula for the asymptotic distribution of the sample autocorrelations of a binomial AR(1) process, which is then applied to develop tests concerning the autocorrelation structure. Simulation studies are carried out to evaluate the size and power of the proposed tests under diverse alternative process models. Several real examples are used to illustrate our methods and findings.     

A comparison of LS/ML and GMM estimation in a simple AR(1) model

This paper compares least squares (LS)/maximum likelihood (ML) and generalised method of moments (GMM) estimation in a simple. Gaussian autoregressive of order one (AR(1)) model. First, we show that the usual LS/ML estimator is a corner solution to a general minimisation problem that involves two moment conditions, while the new GMM we devise is not. Secondly, we examine asymptotic and finite sample properties of the new GMM estimator in comparison to the usual LS/ML estimator in a simple AR(1) model. For both stable and unstable (unit root) specifications, we show asymptotic equivalence of the distributions of the two estimators. However, in finite samples, the new GMM estimator performs better.     

Extended binomial AR(1) processes with generalized binomial thinning operator

Abstract

In this article, we introduce an extended binomial AR(1) model based on the generalized binomial thinning operator. This operator relaxes the independence assumption of the binomial thinning operator and contains dependent Bernoulli counting series. The new model contains the binomial AR(1) model as a particular case. Some probabilistic and statistical properties are explored. Estimators of the model parameters are derived by conditional maximum likelihood (CML), conditional least squares (CLS) and weighted conditional least squares (WCLS) methods. Some asymptotic properties and numerical results of the estimators are studied. The good performance of the new model is illustrated, among other competitive models in the literature, by an application to the monthly drunken driving counts.     

Binomial AR(1) processes with innovational outliers

Abstract

Binomial integer-valued AR processes have been well studied in the literature, but there is little progress in modeling bounded integer-valued time series with outliers. In this paper, we first review some basic properties of the binomial integer-valued AR(1) process and then we introduce binomial integer-valued AR(1) processes with two classes of innovational outliers. We focus on the joint conditional least squares (CLS) and the joint conditional maximum likelihood (CML) estimates of models’ parameters and the probability of occurrence of the outlier. Their large-sample properties are illustrated by simulation studies. Artificial and real data examples are used to demonstrate good performances of the proposed models.     

Slow-explosive AR(1) processes converging to random walk

Abstract

This article investigates slow-explosive AR(1) processes, which converge to a random walk (RW) process with logarithm rates, to fill the gap between nearly non-stationary AR(1) and moderately deviated AR(1) processes, and derives the asymptotics of the least squares estimator using central limit theorems for (reduced) U-statistic. We successfully establish the smooth link between the nearly non-stationary AR(1) and the moderately deviated AR(1) processes. Some novel results are reported, which include the convergence of the least squares estimator to a biased fractional Brownian motion.     

On Small Sample Properties of the Wald,LR and LM Tests in a Linear Model with AR(1) Errors

When the error terms are autocorrelated, the conventional t-tests for individual regression coefficients mislead us to over-rejection of the null hypothesis. We examine, by Monte Carlo experiments, the small sample properties of the unrestricted estimator of ρ and of the estimator of ρ restricted by the null hypothesis. We compare the small sample properties of the Wald, likelihood ratio and Lagrange multiplier test statistics for individual regression coefficients. It is shown that when the null hypothesis is true, the unrestricted estimator of ρ is biased. It is also shown that the Lagrange multiplier test using the maximum likelihood estimator of ρ performs better than the Wald and likelihood ratio tests.     

Asymptotic Inferences for an AR(1) Model with a Change Point and Possibly Infinite Variance

The basic model in this paper is an AR(1) model with a structural break in the AR parameter β at an unknown time k₀. That is, y_t = β₁y_{t ? 1}I{t ? k₀} + β₂y_{t ? 1}I{t > k₀} + ?_t, t = 1, 2, ???, T, where I{ · } denotes the indicator function. Suppose |β₁| < 1, |β₂| < 1, and {?_t, t ? 1} is a sequence of i.i.d. random variables which are in the domain of attraction of the normal law with zero mean and possibly infinite variance, then the limiting distributions for the least squares estimators of β₁ and β₂ are studied in the present paper, which extend some results in Chong (2001 Chong, T.L. (2001). Structural change in AR(1) models. Econometric Theory 17:87–155.[Crossref], [Web of Science ®] , [Google Scholar]).     

Detection and Estimation of Jump Points in Non parametric Regression Function with AR(1) Noise

In this paper, we propose a method based on wavelet analysis to detect and estimate jump points in non parametric regression function. This method is applied to AR(1) noise process under random design. First, the test statistics are constructed on the empirical wavelet coefficients. Then, under the null hypothesis, the critical values of test statistics are obtained. Under the alternative, the consistency of the test is proved. Afterward, the rate of convergence, the estimators of the number, and locations of change points are given theoretically. Finally, the excellent performance of our method is demonstrated through simulations using artificial and real datasets.     

Error autocorrelation revisited: the AR(1) case

The aim of the paper is to consider the implicit restrictions imposed when adopting an AR(1) error term in the context of the linear regression model. It is shown that these restrictions amount to assuming a largely identical temporal structure for all the variables involved in the specification. Implicit in this is the assumption that these variables are mutually Granger non-causal. The main implication of this result is that in most cases when residual autocorrelation is detected boththe OLS and GLS estimators are biased and inconsistent.     

Monitoring correlated processes with binomial marginals

Few approaches for monitoring autocorrelated attribute data have been proposed in the literature. If the marginal process distribution is binomial, then the binomial AR(1) model as a realistic and well-interpretable process model may be adequate. Based on known and newly derived statistical properties of this model, we shall develop approaches to monitor a binomial AR(1) process, and investigate their performance in a simulation study. A case study demonstrates the applicability of the binomial AR(1) model and of the proposed control charts to problems from statistical process control.     

Empirical likelihood estimation for linear regression models with AR(p) error terms with numerical examples

Linear regression models are useful statistical tools to analyze data sets in different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors. If error terms are correlated the Conditional Maximum Likelihood (CML) estimation method under normality assumption is often used to estimate the parameters of interest. The CML estimation method is required a distributional assumption on error terms. However, in practice, such distributional assumptions on error terms may not be plausible. In this paper, we propose to estimate the parameters of a linear regression model with autoregressive error term using Empirical Likelihood (EL) method, which is a distribution free estimation method. A small simulation study is provided to evaluate the performance of the proposed estimation method over the CML method. The results of the simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.     

Fully observed INAR(1) processes

The innovations of an INAR(1) process (integer-valued autoregressive) are usually assumed to be unobservable. There are, however, situations in practice, where also the innovations can be uncovered, i.e. where we are concerned with a fully observed INAR(1) process. We analyze stochastic properties of such a fully observed INAR(1) process and explore the relation between the INAR(1) model and certain metapopulation models. We show how the additional knowledge about the innovations can be used for parameter estimation, for model diagnostics, and for forecasting. Our findings are illustrated with two real-data examples.     

Higher-order moments,cumulants and spectral densities of the NGINAR(1) process

The motivation for time series with geometric marginal distributions arises from noting that the Poisson distribution is not always suitable for the modeling and analysis of integer-valued time series. The NGINAR(1) process that has been introduced by Risti? et al. (2009) represents a class of such time series. Joint higher-order (factorial) moments and cumulants with some other related statistical measures of the NGINAR(1) process are constructed. Also, the spectral and bispectral density functions of this process are investigated, including their nonparametric estimators, using the multitapering method. A real data example of the nonparametric multitaper spectral estimates is investigated, with a discussion of the results obtained.     

Testing of homogeneity of variance and autocorrelation coefficients of nonlinear mixed models with AR(1) errors based on M-estimation

Homogeneity of between-individual variance and autocorrelation coefficients is one of assumptions in the study of longitudinal data. However, the assumption could be challenging due to the complexity of the dataset. In the paper we propose and analyze nonlinear mixed models with AR(1) errors for longitudinal data, intend to introduce Huber's function in the log-likelihood function and get robust estimation, which may help to reduce the influence of outliers, by Fisher scoring method. Testing of homogeneity of variance among individuals and autocorrelation coefficients on the basis of Huber's M-estimation is studied later in the paper. Simulation studies are carried to assess performance of score test we proposed. Results obtained from plasma concentrations data are reported as an illustrative example.     

First order autoregressive time series with negative binomial and geometric marginals

In this paper we present first order autoregressive (AR(1)) time series with negative binomial and geometric marginals. These processes are the discrete analogues of the gamma and exponential processes introduced by Sim (1990). Many properties of the processes discussed here, such as autocorrelation, regression and joint distributions, are studied.     

Inference for the tail index of a GARCH(1,1) model and an AR(1) model with ARCH(1) errors

For a GARCH(1,1) sequence or an AR(1) model with ARCH(1) errors, one can estimate the tail index by solving an estimating equation with unknown parameters replaced by the quasi maximum likelihood estimation, and a profile empirical likelihood method can be employed to effectively construct a confidence interval for the tail index. However, this requires that the errors of such a model have at least a finite fourth moment. In this article, we show that the finite fourth moment can be relaxed by employing a least absolute deviations estimate for the unknown parameters by noting that the estimating equation for determining the tail index is invariant to a scale transformation of the underlying model.     

A GQL estimation approach for analysing non-stationary over-dispersed BINAR(1) time series

This paper proposes a generalized quasi-likelihood (GQL) function for estimating the vector of regression and over-dispersion effects for the respective series in the bivariate integer-valued autoregressive process of order 1 (BINAR(1)) with Negative Binomial (NB) marginals. The auto-covariance function in the proposed GQL is computed using some ‘robust’ working structures. As for the BINAR(1) process, the inter-relation between the series is induced mainly by the correlated NB innovations that are subject to different levels of over-dispersion. The performance of the GQL approach is tested via some Monte-Carlo simulations under different combination of over-dispersion together with low and high serial- and cross-correlation parameters. The model is also applied to analyse a real-life series of day and night accidents in Mauritius.