Consistency Of Least Squares Estimator Research Articles

On the Consistency of Least Squares Estimator in Models Sampled at Random Times Driven by Long Memory Noise: The Jittered Case

On the Consistency of Least Squares Estimator in Models Sampled at Random Times Driven by Long Memory Noise: The Jittered Case

Sufficient and necessary conditions for the strong consistency of LS estimators in simple linear EV regression models

ABSTRACT In this paper, the sufficient and necessary conditions for the strong consistency of least squares estimators in simple linear EV regression models are proved under widely orthant dependent random errors. The results obtained in the paper improve and extend some existing ones in the literature. Simulation studies are also carried out to support the consistency of the estimators under dependence assumption and finite samples.

Strong Consistency of Estimators in a Partially Linear Model with Asymptotically Almost Negatively Associated Errors

This paper studies a heteroscedastic partially linear regression model in which the errors are asymptotically almost negatively associated (AANA, in short) random variables with not necessarily identical distribution and zero mean. Under some mild conditions, we establish the strong consistency of least squares estimators, weighted least squares estimators, and the ultimate weighted least squares estimators for the unknown parameter, respectively. In addition, the strong consistency of the estimator for nonparametric component is also investigated. The results derived in the paper include the corresponding ones of independent random errors and some dependent random errors as special cases. At last, two simulations are carried out to study the numerical performance of the strong consistency for least squares estimators and weighted least squares estimators of the unknown parametric and nonparametric components in the model.

Strong and weak consistency of least squares estimators in simple linear EV regression models

For a simple linear errors-in-variables regression model, we obtain a necessary and sufficient condition for the convergence rate in the strong consistency of the least squares estimator for each of the unknown parameters. We also obtain a necessary and sufficient condition for the convergence rate in the weak consistency.

Strong consistency of the least squares estimator in regression models with adaptive learning

This paper looks at the strong consistency of the ordinary least squares (OLS) estimator in linear regression models with adaptive learning. It is a companion to Christopeit & Massmann (2018) which considers the estimator’s convergence in distribution and its weak consistency in the same setting. Under constant gain learning, the model is closely related to stationary, (alternating) unit root or explosive autoregressive processes. Under decreasing gain learning, the regressors in the model are asymptotically collinear. The paper examines, first, the issue of strong convergence of the learning recursion: It is argued that, under constant gain learning, the recursion does not converge in any probabilistic sense, while for decreasing gain learning rates are derived at which the recursion converges almost surely to the rational expectations equilibrium. Secondly, the paper establishes the strong consistency of the OLS estimators, under both constant and decreasing gain learning, as well as rates at which the estimators converge almost surely. In the constant gain model, separate estimators for the intercept and slope parameters are juxtaposed to the joint estimator, drawing on the recent literature on explosive autoregressive models. Thirdly, it is emphasised that strong consistency is obtained in all models although the near-optimal condition for the strong consistency of OLS in linear regression models with stochastic regressors, established by Lai & Wei (1982a), is not always met.

Strong Consistency of the Least Squares Estimator in Regression Models with Adaptive Learning

This paper looks at the strong consistency of the ordinary least squares (OLS) estimator in a stereotypical macroeconomic model with adaptive learning. It is a companion to Christopeit & Massmann (2017, Econometric Theory) which considers the estimator’s convergence in distribution and its weak consistency in the same setting. Under constant gain learning, the model is closely related to stationary, (alternating) unit root or explosive autoregressive processes. Under decreasing gain learning, the regressors in the model are asymptotically collinear. The paper examines, first, the issue of strong convergence of the learning recursion: It is argued that, under constant gain learning, the recursion does not converge in any probabilistic sense, while for decreasing gain learning rates are derived at which the recursion converges almost surely to the rational expectations equilibrium. Secondly, the paper establishes the strong consistency of the OLS estimators, under both constant and decreasing gain learning, as well as rates at which the estimators converge almost surely. In the constant gain model, separate estimators for the intercept and slope parameters are juxtaposed to the joint estimator, drawing on the recent literature on explosive autoregressive models. Thirdly, it is emphasised that strong consistency is obtained in all models although the near-optimal condition for the strong consistency of OLS in linear regression models with stochastic regressors, established by Lai & Wei (1982), is not always met.

Moment inequalities for m-negatively associated random variables and their applications

The moment inequalities for m-NA random variables, especially the Marcinkiewicz–Zygmund type inequality and Rosenthal type inequality are established and the Khintchine–Kolmogorov convergence theorem and the three series theorem for m-NA random variables are also obtained. As one application of the moment inequalities, we study the large deviation for least squares estimator in nonlinear regression models under some general conditions. As another application, we investigate the strong consistency for least squares estimator in multiple linear regression models based on m-NA random variables.

Strong consistency of least squares estimates with i.i.d. errors with mean values not necessarily defined

We establish strong consistency of the least squares estimates in multiple regression models discarding the usual assumption of the errors having null mean value. Thus, we required them to be i.i.d. with absolute moment of order r, 0<r<2, and null mean value when r>1. Only moderately restrictive conditions are imposed on the model matrix. In our treatment, we use an extension of the Marcinkiewicz–Zygmund strong law to overcome the errors mean value not being defined. In this way, we get a unified treatment for the case of i.i.d. errors extending the results of some previous papers.

Strong consistency of least squares estimates in multiple regression models with random regressors

The strong consistency of the least squares estimator in multiple regression models is established assuming the randomness of the regressors and errors with infinite variance. Only moderately restrictive conditions are imposed on the stochastic model matrix and the errors will be random variables having moment of order \(r,\,1 \leqslant r \leqslant 2\). In our treatment, we use Etemadi’s strong law of large numbers and a sharp almost sure convergence for randomly weighted sums of random elements. Both techniques permit us to extend the results of some previous papers.

On sufficient conditions for the strong consistency of least-squares estimates

The strong consistency of the least-squares estimates in regression models is obtained when the errors are i.i.d. with absolute moment of order r, 0<r⩽ 2. The assumptions presented for the random error sequence will permit us to obtain improvements of the conditions on the regressors in order to obtain the strong consistency of the least-squares estimates in linear and nonlinear regression models.

Strong consistency of estimators in partially linear models for longitudinal data with mixing-dependent structure

For exhibiting dependence among the observations within the same subject, the paper considers the estimation problems of partially linear models for longitudinal data with the φ-mixing and ρ-mixing error structures, respectively. The strong consistency for least squares estimator of parametric component is studied. In addition, the strong consistency and uniform consistency for the estimator of nonparametric function are investigated under some mild conditions.

Nonlinear regression models with increasing numbers of unknown parameters

Regression models with an increasing number of unknown parameters and with unknown and different observation error variances are of interest in important applications. The reason is that, with an increased number of unknown parameters, the unknown function can be approximated more accurately in experiments. Moreover, in some applications, repeated tests at a single point are costly (financially and technically), which hampers the estimation of the unknown error variance, which is different at different observation points. Regression models have been addressed in numerous publications [1‐3], but models with an increasing number of unknown parameters have received little attention, which motivates our interest in this subject. In [4, 5] various approaches were suggested for estimating the covariance matrix of the deviation vector of the unknown parameter for linear models of finite dimension. Linear regression models of an increasing dimension were studied in [6], where necessary and sufficient conditions were found for the consistency of least squares estimators and the covariance matrix of the deviation vector. In this paper, an approach is suggested for determining least squares estimators and covariance matrix elements. The results can be used to construct a confidence interval for the unknown function in nonlinear regression models with an increasing number of unknown parameters.

A note on the strong consistency of least squares estimates

The strong consistency of least squares estimates in multiples regression models with i.i.d. errors is obtained under assumptions on the design matrix and moment restrictions on the errors.

Asymptotic Consistency of Least Squares Estimators in Fuzzy Regression Model

This paper deals with the properties of the fuzzy least squares estimators for fuzzy linear regression model. Especially fuzzy triangular input-output model including error term is proposed. The error term is considered as a fuzzy random variable. The asymptotic unbiasedness and the consistency of the estimators are proved using a suitable metric.

Sufficient conditions for the strong consistency of least squares estimator with α-stable errors

Sufficient conditions for the strong consistency of least squares estimator with α-stable errors

Uniqueness, consistency and optimality in spherical regression experiments

For designed experiments based on the spherical regression model of Chang (Ann. Statist. 14 (1986) 907) we provide results on the minimum number of covariate directions that are necessary and sufficient for uniqueness and consistency of least squares estimates and on minimizing confidence regions.

Consistency of least squares estimates in a system of linear correlation models

We consider a system of linear response models with random explanatory variables in which the global matrix parameter is subject to arbitrary constraints. A generalized least squares estimate (GLSE) of the global parameter is defined by its property of mi

Studies on consistency of LSE in China

In this paper, we survey-some results obtained by Chinese statisticians in recent years, on the consistency of Least Squares Estimate (LSE) in the linear regression model.

Strong consistency in nonlinear stochastic regression models

The class of nonlinear stochastic regression models includes most of the linear and nonlinear models used in time series, stochastic control and stochastic approximation schemes.The consistency of least squares estimators was established first by Lai.We present another set of sufficient conditions for consistency, which avoid the use of partial derivatives and are closer in spirit to the conditions presented by Wu for non-stochastic regression models with independent errors.

Strong Law of Large Numbers with Operator Normalizations for Martingales and Sums of Orthogonal Random Vectors

We establish the strong law of large numbers with operator normalizations for vector martingales and sums of orthogonal random vectors. We describe its applications to the investigation of the strong consistency of least-squares estimators in a linear regression and the asymptotic behavior of multidimensional autoregression processes.