Conditional Maximum Likelihood Method Research Articles

Threshold integer-valued autoregressive model with serially dependent innovation

ABSTRACT Nonlinearity in count time series is commonly encountered in practice. To better explain the nonlinear phenomena in count time series, this article introduces a new threshold INAR(1) model using the idea of serially dependent innovation. The proposed model contains some existing INAR(1) models as special cases and establishes a connection between the threshold INAR(1) model and the threshold INARCH(1) model. The basic probabilistic and statistical properties of the new model are investigated. Model parameters, including the threshold variable, are estimated by the conditional least squares, modified quasi-likelihood and conditional maximum likelihood methods. Asymptotic properties and numerical results of the estimates are also studied. An application to the monthly trading volume of stock B in Shanghai Stock Exchange is conducted to show the practicability of the proposed model.

Periodic INAR(1) model with Bell innovations distribution

Abstract In this paper, we introduce a class of periodic integer-valued autoregressive BL-PINAR(1) models with Bell innovations distribution based on the binomial thinning operator. The basic probabilistic and statistical properties of this class are studied. Indeed, the first and the second moment periodically stationary conditions are established. The closed forms of these moments are, under the obtained conditions, derived. Furthermore, the periodic autocovariance structure is also considered while providing the closed form of the periodic autocorrelation function. The conditional least squares (CLS), Yule–Walker (YW), weighted conditional least squares (WCLS), and conditional maximum likelihood (CML) methods are applied to estimate the underlying parameters. The asymptotic properties of the CLS and the YW estimators are obtained. The performances of these methods are compared through a simulation study. An application on a real data set is provided.

Bivariate first-order random coefficient integer-valued autoregressive processes based on modified negative binomial operator

In this paper, a new bivariate random coefficient integer-valued autoregressive process based on modified negative binomial operator with dependent innovations is proposed. Basic probabilistic and statistical properties of this model are derived. To estimate unknown parameters, Yule-Walker, conditional least squares and conditional maximum likelihood methods are considered and evaluated by Monte Carlo simulations. Asymptotic properties of the estimators are derived. Moreover, coherent forecasting and possible extension of the proposed model is provided. Finally, the proposed model is applied to the monthly crime datasets and compared with other models.

Bivariate Random Coefficient Integer-Valued Autoregressive Model Based on a ρ-Thinning Operator

While overdispersion is a common phenomenon in univariate count time series data, its exploration within bivariate contexts remains limited. To fill this gap, we propose a bivariate integer-valued autoregressive model. The model leverages a modified binomial thinning operator with a dispersion parameter ρ and integrates random coefficients. This approach combines characteristics from both binomial and negative binomial thinning operators, thereby offering a flexible framework capable of generating counting series exhibiting equidispersion, overdispersion, or underdispersion. Notably, our model includes two distinct classes of first-order bivariate geometric integer-valued autoregressive models: one class employs binomial thinning (BVGINAR(1)), and the other adopts negative binomial thinning (BVNGINAR(1)). We establish the stationarity and ergodicity of the model and estimate its parameters using a combination of the Yule–Walker (YW) and conditional maximum likelihood (CML) methods. Furthermore, Monte Carlo simulation experiments are conducted to evaluate the finite sample performances of the proposed estimators across various parameter configurations, and the Anderson-Darling (AD) test is employed to assess the asymptotic normality of the estimators under large sample sizes. Ultimately, we highlight the practical applicability of the examined model by analyzing two real-world datasets on crime counts in New South Wales (NSW) and comparing its performance with other popular overdispersed BINAR(1) models.

The Circumstance-Driven Bivariate Integer-Valued Autoregressive Model.

The novel circumstance-driven bivariate integer-valued autoregressive (CuBINAR) model for non-stationary count time series is proposed. The non-stationarity of the bivariate count process is defined by a joint categorical sequence, which expresses the current state of the process. Additional cross-dependence can be generated via cross-dependent innovations. The model can also be equipped with a marginal bivariate Poisson distribution to make it suitable for low-count time series. Important stochastic properties of the new model are derived. The Yule-Walker and conditional maximum likelihood method are adopted to estimate the unknown parameters. The consistency of these estimators is established, and their finite-sample performance is investigated by a simulation study. The scope and application of the model are illustrated by a real-world data example on sales counts, where a soap product in different stores with a common circumstance factor is investigated.

A Time-Varying Mixture Integer-Valued Threshold Autoregressive Process Driven by Explanatory Variables.

In this paper, a time-varying first-order mixture integer-valued threshold autoregressive process driven by explanatory variables is introduced. The basic probabilistic and statistical properties of this model are studied in depth. We proceed to derive estimators using the conditional least squares (CLS) and conditional maximum likelihood (CML) methods, while also establishing the asymptotic properties of the CLS estimator. Furthermore, we employed the CLS and CML score functions to infer the threshold parameter. Additionally, three test statistics to detect the existence of the piecewise structure and explanatory variables were utilized. To support our findings, we conducted simulation studies and applied our model to two applications concerning the daily stock trading volumes of VOW.

Asymmetric beta-binomial GARCH models for time series with bounded support

In this paper, we introduce a new class of asymmetric beta-binomial generalized autoregressive conditional heteroscedastic (GARCH) models for bounded integer-valued time series, which can capture the asymmetric impact of positive and negative observations. We study the stationarity conditions of the process and derive the moment and covariance functions. Furthermore, we estimate the unknown parameters using the conditional maximum likelihood (CML) method. The asymptotic properties of the estimators are discussed, as well as their finite-sample performance. Finally, we illustrate the model to real time series data in the field of meteorology.

Risk Of Eviction And Women’s Technology Adoption: Some Evidence From Southwestern Nigeria

ABSTRACT: In the presence of tenure insecurity that many women face in developing societies because of patriarchal norms, farmers may not adopt farm technologies because there is no assurance of reaping the benefits. This article examines how the risk of eviction affects women’s fertilizer use and adoption of early maturing seed varieties. For the empirical analysis, I employ individual-level data that was collected from 180 women rice farmers in southwestern Nigeria with a multistage sampling technique and analyse the data with both econometric methods that address possible endogeneity issues and those that do not. I analyse the relationship between the risk of eviction and fertilizer use with the Two Stage Least Squares (2SLS) and Ordinary Least Squares (OLS) methods and analyse the relationship between the risk of eviction and the adoption of early-maturing seed varieties with the Two Stage Conditional Maximum Likelihood (2SCML) and probit methods. I find women who are cultivating family-allocated, husband-allocated and purchased plots to be significantly less likely to have fears of eviction than those who are cultivating fixed-rent plots, even as the risk of eviction have different effects on fertilizer use and on the adoption of early-maturing rice varieties respectively. While it makes the women spend less on fertilizer, it makes them more likely to adopt early maturing seed varieties. These would suggest that the women are responding to the unfavourable production environment from tenure insecurity with lower investments in soil fertility and shortened time horizons towards minimizing their losses in the event of eviction. The results also suggest that the women may invest in soil fertility if the operating environment becomes more favourable with better tenure. The policy implications of the findings are thus evident. Women are evidently not averse to adopting farm technologies by default; to thus improve adoption rates of farm technologies, policymakers should adapt technologies to local socioeconomic conditions, while exploring how to make tenancy contracts and the social norms that underpin customary tenure systems more amenable to granting women tenure security. These would help protect land resources from degradation, make agricultural incomes sustainable, and facilitate socio-economic development.

A new threshold INAR(1) model based on modified negative binomial operator with random coefficient

In this paper, a new threshold INAR(1) model based on modified negative binomial operator with random coefficient is proposed. Basic probabilistic and statistical properties of this process are established. Then the conditional least squares (CLS) and the conditional maximum likelihood (CML) methods are applied to estimate the model parameters when the threshold value is known or not. The asymptotic properties of the CLS-estimator and the CML-estimator are also been discussed. A method to test the constancy of the autoregressive parameters is provided. As an illustration, a simulation study is conducted to illustrate the performances of these estimators and present an empirical analysis of monthly counts of break and enter non-dwelling in Bellingen.

Estimating item parameters in multistage designs with the tmt package in R

Various likelihood-based methods are available for the parameter estimation of item response theory models (IRT), leading to comparable parameter estimates. Considering multistage testing (MST) designs, Glas (1988; https://doi.org/10.2307/1164950) stated that the conditional maximum likelihood (CML) method in its original formulation leads to severely biased parameter estimates. A modified CML estimation method for MST designs proposed by Zwitser and Maris (2015; https://doi.org/10.1007/s11336-013-9369-6) finally provides asymptotically unbiased item parameter estimates. Steinfeld and Robitzsch (2021b; https://doi.org/10.31234/osf.io/ew27f) complemented this method to MST designs with probabilistic routing strategies. For both proposed modifications additional software solutions are required since design-specific information must be incorporated into the estimation process. An R package that has implemented both modifications is "tmt". In this article, first, the proposed solutions of the CML estimation in MST designs are illustrated, followed by the main part, the demonstration of the CML item parameter estimation with the R package "tmt". The demonstration includes the process of model specification, data simulation, and item parameter estimation, considering two different routing types of deterministic and probabilistic MST designs.

First-order random coefficient binomial AR process with dependent counting series

A first-order random coefficient binomial AR process with dependent counting series is introduced. We derive some basic properties of the process. Conditional least squares, weighted conditional least squares and conditional maximum likelihood methods are employed to estimate the parameters of the process. Some asymptotic properties are established. Numerical results of the estimators are studied to assess the performances. An example is provided for practical application.

Multivariate threshold integer-valued autoregressive processes with explanatory variables

To capture the multivariate count time series showing piecewise phenomena, we introduce a class of first-order multivariate threshold integer-valued autoregressive process. The component of autoregressive coefficient is driven by explanatory variables via a logistic link function. Basic probabilistic and statistical properties of the model are discussed. The model parameters are estimated by means of conditional least squares and conditional maximum likelihood methods. A new algorithm to estimate the threshold parameter of the model is also provided. Moreover, the nonlinearity test of the model and existence test of explanatory variables are well addressed using a Wald-type test. Forecasting problem and model extension problem are also considered. Finally, some numerical results of the estimates and a real data example are presented.

First-order spatial dependent count integer-valued autoregressive (Sp-DCINAR(1,1)) process

In this article, we propose a new model to model the spatial count data on a two-dimensional regular grid using binomial thinning operator with dependent Bernoulli counting series. The model is called “first-order spatial dependent count integer-valued autoregressive (Sp-DCINAR(1,1)) model.” Some of its properties have been derived and the estimation of the parameters of the model are obtained by the Yule-Walker method, conditional least squares method and conditional maximum likelihood method. Finally, numerical results are presented together with an application to a two-dimensional practical data set.

A New INAR(1) Model for ℤ-Valued Time Series Using the Relative Binomial Thinning Operator

Abstract A new first-order integer-valued autoregressive process (INAR(1)) with extended Poisson innovations is introduced based on a signed version of the thinning operator, called relative binomial thinning operator, which can be considered as an extension of standard binomial thinning operator introduced by Steutel, F.W. and van Harn, K. (1979. Discrete analogues of self-decomposability and stability. Ann. Probab. 7: 893–899). It is appropriate for modeling Z $\mathbb{Z}$ -valued time series and either positive or negative correlations. Some properties of the process are established. Conditional least squares, Yule–Walker and conditional maximum likelihood methods are considered for the parameter estimation of the model. Moreover, simulation experiments are carried out to attest to the performance of the estimation methods. The applicability of the proposed model is investigated through a practical data set of the Saudi stock market.

A statistical study for some classes of first-order mixed generalized binomial autoregressive models

In order to solve the modeling problems of integer-valued time series with complex and dependent structures, this article proposes some classes of first-order mixed generalized binomial autoregressive models. Basic probabilistic and statistical properties of the models are discussed. The parameters of models are estimated by the conditional maximum likelihood method. Also, the asymptotic properties and the numerical results of these estimators are obtained. Finally, the good performances of these models are illustrated, among other competitive models in the literature, by an application to the data of weekly rainy days.

On Periodic Generalized Poisson INAR(1) Model

In this paper, we introduce a first-order Periodic Generalized Poisson Integer-Valued Autoregressive model which has been shown to be useful to describe overdispersion, equidispersion and underdispersion feature encountered in periodically correlated Integer-Valued time series. Some probabilistic and statistical properties are established, such as the periodically correlated stationarity conditions, in the first and the second moments are provided and the closed-forms of these moments are, under these conditions, derived. Moreover, the structure of the periodic autocovariance is obtained. The estimation problem is addressed through the Yule-Walker the Two-Stage Conditional Least Squares and the Conditional Maximum Likelihood methods. The performance of these methods is done through an intensive simulation study and an application on real data set is accomplished. Keywords and phrases: Periodic Generalized Poisson, Integer-Valued Autoregressive, Periodically correlated process, periodically stationary condition.

Improving estimation efficiency for two-phase, outcome-dependent sampling studies

Two-phase outcome dependent sampling (ODS) is widely used in many fields, especially when certain covariates are expensive and/or difficult to measure. For two-phase ODS, the conditional maximum likelihood (CML) method is very attractive because it can handle zero Phase 2 selection probabilities and avoids modeling the covariate distribution. However, most existing CML-based methods use only the Phase 2 sample and thus may be less efficient than other methods. We propose a general empirical likelihood method that uses CML augmented with additional information in the whole Phase 1 sample to improve estimation efficiency. The proposed method maintains the ability to handle zero selection probabilities and avoids modeling the covariate distribution, but can lead to substantial efficiency gains over CML in the inexpensive covariates, or in the influential covariate when a surrogate is available, because of an effective use of the Phase 1 data. Simulations and a real data illustration using NHANES data are presented.

A new minification integer‐valued autoregressive process driven by explanatory variables

SummaryThe discrete minification model based on the modified negative binomial operator, as an extension to the continuous minification model, can be used to describe an extreme value after few increasing values. To make this model more practical and flexible, a new minification integer‐valued autoregressive process driven by explanatory variables is proposed. Ergodicity of the new process is discussed. The estimators of the unknown parameters are obtained via the conditional least squares and conditional maximum likelihood methods, and the asymptotic properties are also established. A testing procedure for checking existence of the explanatory variables is developed. Some Monte Carlo simulations are given to illustrate the finite‐sample performances of the estimators under specification and misspecification and the test, respectively. A real example is applied to illustrate the performance of our model.

First-order binomial autoregressive processes with Markov-switching coefficients

In this article, a new autoregressive process for finite-range time series of counts is proposed to analyse the finite-range integer-valued data based on an invisible Markov chain. We derive the probabilistic and statistical properties of the model. Conditional least squares (CLS) method and conditional maximum likelihood (CML) method are employed to estimate the parameters of interest. Furthermore, the forecasting problem is addressed. In addition, multiple simulation studies are performed to investigate the finite sample performance of parameter estimators and to compare the proposed estimation methods. The proposed model is applied to a finite-range data series of measles infections in Germany in 2004–2005.

Bivariate Poisson 2Sum-Lindley Distributions and the Associated BINAR(1) Processes

Discrete-valued time series modeling has witnessed numerous bivariate first-order integer-valued autoregressive process or BINAR(1) processes based on binomial thinning and different innovation distributions. These BINAR(1) processes are mainly focused on over-dispersion. This paper aims to propose new bivariate distributions and processes based on a recently proposed over-dispersed distribution: the Poisson 2S-Lindley distribution. The new bivariate distributions, denoted by the abbreviations BP2S-L(I) and BP2S-L(II), are then used as innovation distributions for the BINAR(1) process. Properties are investigated for both distributions as well as for the BINAR(1) processes. The distribution parameters are estimated using the maximum likelihood method, and the BINAR(1)BP2S-L(I) and BINAR(1)BP2S-L(II) process parameters are estimated using the conditional least squares and conditional maximum likelihood methods. Monte Carlo simulation experiments are conducted to study large and small sample performances and for the comparison of the estimation methods. The Pittsburgh crime series and candy sales datasets are then used to compare the new BINAR(1) processes to some other existing BINAR(1) processes in the literature.