Conditional Least Squares Estimator Research Articles

A note on the asymptotic behavior of a mildly unstable integer-valued AR(1) model

This note examines the conditional least squares (CLS) estimators for integer-valued autoregressive (INAR) models, with a focus on the INAR(1) model. Our investigation reveals that the joint limiting distribution of the CLS estimators for the autoregressive coefficient and the mean of the model errors is degenerate in the case of mildly unstable INAR(1) models with moderate deviations from unity. This degeneracy may pose challenges for statistical inference. We provide commenting notes based on the asymptotic results to address this issue. Additionally, we conduct simulation studies and analyze a real-world dataset to validate the theoretical findings.

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.

Change-Point Detection in the Volatility of Conditional Heteroscedastic Autoregressive Nonlinear Models

This paper studies single change-point detection in the volatility of a class of parametric conditional heteroscedastic autoregressive nonlinear (CHARN) models. The conditional least-squares (CLS) estimators of the parameters are defined and are proved to be consistent. A Kolmogorov–Smirnov type-test for change-point detection is constructed and its null distribution is provided. An estimator of the change-point location is defined. Its consistency and its limiting distribution are studied in detail. A simulation experiment is carried out to assess the performance of the results, which are compared to recent results and applied to two sets of real data.

Two-Threshold-Variable Integer-Valued Autoregressive Model

In the past, most threshold models considered a single threshold variable. However, for some practical applications, models with two threshold variables may be needed. In this paper, we propose a two-threshold-variable integer-valued autoregressive model based on the binomial thinning operator and discuss some of its basic properties, including the mean, variance, strict stationarity, and ergodicity. We consider the conditional least squares (CLS) estimation and discuss the asymptotic normality of the CLS estimator under the known and unknown threshold values. The performances of the CLS estimator are compared via simulation studies. In addition, two real data sets are considered to underline the superior performance of the proposed model.

Two-step conditional least squares estimation for the bivariate Z -valued INAR(1) model with bivariate Skellam innovations

This article studies the two-step conditional least squares (CLS) estimation for the bivariate Z -valued INAR(1) model with bivariate Skellam innovations. For readers’ convenience, we first give a brief review of the bivariate Skellam distribution, bivariate signed thinning operator and the definition of the bivariate Z -valued INAR(1) model with bivariate Skellam innovations (denoted as the BSK-BINARS(1) model). Then, we discuss the stationarity and ergodicity of the BSK-BINARS(1) model, give some stochastic properties. Second, we discuss the two-step CLS estimate of the parameters and establish their large-sample properties. Third, we conduct a simulation study to illustrate the finite sample performances of the two-step CLS estimators, which are compared with those obtained by the plug-in method. Last but not least, we apply the BSK-BINARS(1) model on the zonal annual means temperature (*100).

Grouped spatial autoregressive model

With the development of the internet, network data with replications can be collected at different time points. The spatial autoregressive panel (SARP) model is a useful tool for analyzing such network data. However, in the traditional SARP model, all individuals are assumed to be homogeneous in their network autocorrelation coefficients, while in practice, correlations could differ for the nodes in different groups. Here, a grouped spatial autoregressive (GSAR) model based on the SARP model is proposed to permit network autocorrelation heterogeneity among individuals, while analyzing network data with independent replications across different time points and strong spatial effects. Each individual in the network belongs to a latent specific group, which is characterized by a set of parameters. Two estimation methods are studied: two-step naive least-squares estimator, and two-step conditional least-squares estimator. Furthermore, their corresponding asymptotic properties and technical conditions are investigated. To demonstrate the performance of the proposed GSAR model and its corresponding estimation methods, numerical analysis was performed on simulated and real data.

Asymptotic properties of conditional least-squares estimators for array time series

The paper provides a kind of Klimko–Nelson’s theorems alternative in the case of conditional least-squares and M-estimators for array time series when the assumptions of almost sure convergence cannot be established. We do not assume stationarity nor even local stationarity. Besides, we provide sufficient conditions for two of the assumptions and a procedure for the evaluation of the information matrix in array time series. In addition to time-dependent models, illustrations to a threshold model and a count data model are given.

Conditional least squares estimation for the SINAR(1, 1) process

This paper presents the conditional least squares (CLS) estimation procedure for the class of first-order spatial integer-valued autoregressive processes, denote by SINAR(1, 1). We derive the asymptotic properties of the CLS estimators of model parameters. Simulation results are presented to assess the performance of estimators under finite sample sizes and under equidispersion and overdispersion. To illustrate the proposed methodology, two grid sampled count data sets are analyzed.

A New Extension of Thinning-Based Integer-Valued Autoregressive Models for Count Data.

The thinning operators play an important role in the analysis of integer-valued autoregressive models, and the most widely used is the binomial thinning. Inspired by the theory about extended Pascal triangles, a new thinning operator named extended binomial is introduced, which is a general case of the binomial thinning. Compared to the binomial thinning operator, the extended binomial thinning operator has two parameters and is more flexible in modeling. Based on the proposed operator, a new integer-valued autoregressive model is introduced, which can accurately and flexibly capture the dispersed features of counting time series. Two-step conditional least squares (CLS) estimation is investigated for the innovation-free case and the conditional maximum likelihood estimation is also discussed. We have also obtained the asymptotic property of the two-step CLS estimator. Finally, three overdispersed or underdispersed real data sets are considered to illustrate a superior performance of the proposed model.

On shifted integer-valued autoregressive model for count time series showing equidispersion, underdispersion or overdispersion

In this paper, we introduce a first order integer-valued autoregressive process with Borel innovations based on the binomial thinning. This model is suitable to modeling zero truncated count time series with equidispersion, underdispersion and overdispersion. The basic properties of the process are obtained. To estimate the unknown parameters, the Yule-Walker (YW), conditional least squares (CLS) and conditional maximum likelihood (CML) methods are considered. The asymptotic distribution of CLS estimators are obtained and hypothesis tests to test an equidispersed model against an underdispersed or overdispersed model are formulated. A Monte Carlo simulation is presented analyzing the estimators performance in finite samples. Two applications to real data are presented to show that the Borel INAR(1) model is suited to model underdispersed and overdispersed data counts.

Parameter change test for periodic integer-valued autoregressive process

This paper studies a parameter change testing problem in periodic integer-valued autoregressive process of order with period (). The process is transformed into a multivariate process for which basic probabilistic and statistical properties are researched. The cumulative sum test based on conditional least-squares estimators is employed to test the parameter change of process. That returns to tests for the parameter change of - models. The power of test is evaluated through simulation results.

An estimation procedure for the Hawkes process

In this paper, we present a nonparametric estimation procedure for the multivariate Hawkes point process. The timeline is cut into bins and—for each component process—the number of points in each bin is counted. As a consequence of earlier results in Kirchner [Stoch. Process. Appl., 2016, 162, 2494–2525], the distribution of the resulting ‘bin-count sequences’ can be approximated by an integer-valued autoregressive model known as the (multivariate) INAR(p) model. We represent the INAR(p) model as a standard vector-valued linear autoregressive time series with white-noise innovations (VAR(p)). We establish consistency and asymptotic normality for conditional least-squares estimation of the VAR(p), respectively, the INAR(p) model. After appropriate scaling, these time-series estimates yield estimates for the underlying multivariate Hawkes process as well as corresponding variance estimates. The estimates depend on a bin-size and a support s. We discuss the impact and the choice of these parameters. All results are presented in such a way that computer implementation, e.g. in R, is straightforward. Simulation studies confirm the effectiveness of our estimation procedure. In the second part of the paper, we present a data example where the method is applied to bivariate event-streams in financial limit-order-book data. We fit a bivariate Hawkes model on the joint process of limit and market order arrivals. The analysis exhibits a remarkably asymmetric relation between the two component processes: incoming market orders excite the limit-order flow heavily whereas the market-order flow is hardly affected by incoming limit orders. For the estimated excitement functions, we observe power-law shapes, inhibitory effects for lags under 0.003 s, second periodicities and local maxima at 0.01, 0.1 and 0.5 s.

Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model

We study the estimation of a stable Cox–Ingersoll–Ross model, which is a special subcritical continuous-state branching process with immigration. The exponential ergodicity and strong mixing property of the process are proved by a coupling method. The regular variation properties of distributions of the model are studied. The key is to establish the convergence of some point processes and partial sums associated with the model. From those results, we derive the consistency and central limit theorems of the conditional least squares estimators (CLSEs) and the weighted conditional least squares estimators (WCLSEs) of the drift parameters based on low frequency observations. The theorems show that the WCLSEs are more efficient than the CLSEs and their errors have distinct decay rates n−(α−1)/α and n−(α−1)/α2, respectively, as the sample sizes n goes to infinity. The arguments depend heavily on the recent results on the construction and characterization of the model in terms of stochastic equations.

Asymptotic behavior of CLS estimators for 2-type doubly symmetric critical Galton–Watson processes with immigration

In this paper, the asymptotic behavior of the conditional least squares (CLS) estimators of the offspring means $(\alpha,\beta)$ and of the criticality parameter $\varrho:=\alpha+\beta$ for a $2$-type critical doubly symmetric positively regular Galton-Watson branching process with immigration is described.

Test for parameter changes in generalized random coefficient autoregressive model

In this paper, we study the problem of testing for parameter changes in generalized random coefficient autoregressive model (GRCA). The testing method is based on the monitoring scheme proposed by Na et al. (Stat. Methods Appl. 20:171-199, 2011), and the test statistic relies on the conditional least-squares estimator of an unknown parameter. Furthermore, under mild conditions, we obtain the asymptotic property of the test statistic. Some simulation studies are also conducted to investigate the finite sample performances of the proposed test.

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.

Approximate conditional least squares estimation of a nonlinear state-space model via an unscented Kalman filter

The problem of estimating a nonlinear state-space model whose state process is driven by an ordinary differential equation (ODE) or a stochastic differential equation (SDE), with discrete-time data is studied. A new estimation method is proposed based on minimizing the conditional least squares (CLS) with the conditional mean function computed approximately via the unscented Kalman filter (UKF). Conditions are derived for the UKF–CLS estimator to preserve the limiting properties of the exact CLS estimator, namely, consistency and asymptotic normality, under the framework of infill asymptotics, i.e. sampling is increasingly dense over a fixed domain. The efficacy of the proposed method is demonstrated by simulation and a real application.

Binomial AR(1) processes: moments, cumulants, and estimation

The modelling and analysis of count-data time series are areas of emerging interest with various applications in practice. We consider the particular case of the binomial AR(1) model, which is well suited for describing binomial counts with a first-order autoregressive serial dependence structure. We derive explicit expressions for the joint (central) moments and cumulants up to order 4. Then, we apply these results for expressing moments and asymptotic distribution of the squared difference estimator as an alternative to the sample autocovariance. We also analyse the asymptotic distribution of the conditional least-squares estimators of the parameters of the binomial AR(1) model. The finite-sample performance of these estimators is investigated in a simulation study, and we apply them to real data about computerized workstations.

Variance Estimators in Branching Processes with Non-Homogeneous Immigration

Branching processes with immigration were proposed to study the temporal development of populations of differentiated cells in [3] and more recently in [1]. More specifically, terminally differentiated oligodendrocytes of the central nervous system and leukemia cells were analyzed. In both cases the cell population expanded through both division of existing (progenitor) cells and differentiation of stem cells. The population’s viability was preserved by allowing the immigration distribution to vary in time. We construct conditional least-squares estimators for the offspring variance assuming that the immigration mean increases to infinity over time. The asymptotic normality of the proposed estimators is established. Part of the results was published in ...

Conditional Least Squares Estimators for the Offspring Mean in a Subcritical Branching Process with Immigration

Consider a Bienayme–Galton–Watson process with generation-dependent immigration, whose mean and variance vary regularly with non negative exponents α and β, respectively. We study the estimation problem of the offspring mean based on an observation of population sizes. We show that if β <2α, the conditional least squares estimator (CLSE) is strongly consistent. Conditions which are sufficient for the CLSE to be asymptotically normal will also be derived. The rate of convergence is faster than n −1/2, which is not the case in the process with stationary immigration.