Empirical Likelihood-based Inference for Stationary-ergodicity of the Generalized Random Coefficient Autoregressive Model

In this article, we consider the application of the empirical likelihood method to the generalized random coefficient autoregressive (GRCA) model. When the order of the model is 1, we derive an empirical likelihood ratio test statistic to test the stationary-ergodicity. Some simulation studies are also conducted to investigate the finite sample performances of the proposed test.     

Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity

The main contribution of this paper is a proof of the asymptotic validity of the application of the bootstrap to AR(∞) processes with unmodelled conditional heteroskedasticity. We first derive the asymptotic properties of the least-squares estimator of the autoregressive sieve parameters when the data are generated by a stationary linear process with martingale difference errors that are possibly subject to conditional heteroskedasticity of unknown form. These results are then used in establishing that a suitably constructed bootstrap estimator will have the same limit distribution as the least-squares estimator. Our results provide theoretical justification for the use of either the conventional asymptotic approximation based on robust standard errors or the bootstrap approximation of the distribution of autoregressive parameters. A simulation study suggests that the bootstrap approach tends to be more accurate in small samples.     

Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity

The main contribution of this paper is a proof of the asymptotic validity of the application of the bootstrap to AR(∞) processes with unmodelled conditional heteroskedasticity. We first derive the asymptotic properties of the least-squares estimator of the autoregressive sieve parameters when the data are generated by a stationary linear process with martingale difference errors that are possibly subject to conditional heteroskedasticity of unknown form. These results are then used in establishing that a suitably constructed bootstrap estimator will have the same limit distribution as the least-squares estimator. Our results provide theoretical justification for the use of either the conventional asymptotic approximation based on robust standard errors or the bootstrap approximation of the distribution of autoregressive parameters. A simulation study suggests that the bootstrap approach tends to be more accurate in small samples.     

Asymptotic results for random coefficient bifurcating autoregressive processes

The purpose of this paper is to study the asymptotic behaviour of the weighted least-squares estimators of the unknown parameters of random coefficient bifurcating autoregressive processes. Under suitable assumptions on the immigration and the inheritance, we establish the almost sure convergence of our estimators, as well as a quadratic strong law and central limit theorems. Our study mostly relies on limit theorems for vector-valued martingales.     

First-order random coefficient integer-valued autoregressive processes

A first-order random coefficient integer-valued autoregressive (RCINAR(1)) model is introduced. Ergodicity of the process is established. Moments and autocovariance functions are obtained. Conditional least squares and quasi-likelihood estimators of the model parameters are derived and their asymptotic properties are established. The performance of these estimators is compared with the maximum likelihood estimator via simulation.     

On equalities between BLUES,WLSEs, and SLSEs

Necessary and sufficient conditions for equalities between the best linear unbiased estimator, the weighted least-squares estimator, and the simple least-squares estimator of the expectation vector in a general Gauss-Markoff model are given in some alternative formulations. The main result states, somewhat surprisingly, that the weighted least-squares estimator cannot be identical with the simple least-squares estimator unless they both coincide with the best linear unbiased estimator.     

The asymptotic behaviour of a class of L.-estimators under long-range dependence

This paper obtains asymptotic representations of a class of L-estimators in a linear regression model when the errors are a function of long-range-dependent Gaussian random variables. These representations are then used to address some of the efficiency robustness properties of L-estimators compared to the least-squares estimator. It is observed that under the Gaussian error distribution, each member of the class has the same asymptotic efficiency as that of the least-squares estimator. The results are obtained as a consequence of the asymptotic uniform linearity of some weighted empirical processes based on long-range-dependent random variables.     

The Empirical Likelihood for First-Order Random Coefficient Integer-Valued Autoregressive Processes

This article studies the empirical likelihood method for the first-order random coefficient integer-valued autoregressive process. The limiting distribution of the log empirical likelihood ratio statistic is established. Confidence region for the parameter of interest and its coverage probabilities are given, and hypothesis testing is considered. The maximum empirical likelihood estimator for the parameter is derived and its asymptotic properties are established. The performances of the estimator are compared with the conditional least squares estimator via simulation.     

Precise Asymptotics in Complete Moment Convergence of Parameter Estimator in the Gaussian Autoregressive Process

In this article, we study the precise asymptotic behaviors of the least-squares estimator in the Gaussian autoregressive process. Two kinds of complete moment convergence of this estimator can be obtained by the methods of deviation inequalities for this estimator and nonuniform Berry-Esseen bound for martingales.     

Testing for residual correlation of any order in the autoregressive process

We are interested in the implications of a linearly autocorrelated driven noise on the asymptotic behavior of the usual least-squares estimator in a stable autoregressive process. We show that the least-squares estimator is not consistent and we suggest a sharp analysis of its almost sure limiting value as well as its asymptotic normality. We also establish the almost sure convergence and the asymptotic normality of the estimated serial correlation parameter of the driven noise. Then, we derive a statistical procedure enabling to test for correlation of any order in the residuals of an autoregressive modelling, giving clearly better results than the commonly used portmanteau tests of Ljung–Box and Box–Pierce, and appearing to outperform the Breusch–Godfrey procedure on small-sized samples.     

Statistical inference for varying-coefficient partially linear errors-in-variables models with missing data

Abstract

The purpose of this paper is twofold. First, we investigate estimations in varying-coefficient partially linear errors-in-variables models with covariates missing at random. However, the estimators are often biased due to the existence of measurement errors, the bias-corrected profile least-squares estimator and local liner estimators for unknown parametric and coefficient functions are obtained based on inverse probability weighted method. The asymptotic properties of the proposed estimators both for the parameter and nonparametric parts are established. Second, we study asymptotic distributions of an empirical log-likelihood ratio statistic and maximum empirical likelihood estimator for the unknown parameter. Based on this, more accurate confidence regions of the unknown parameter can be constructed. The methods are examined through simulation studies and illustrated by a real data analysis.     

Efficient Estimation in Heteroscedastic Partially Linear Varying Coefficient Models

This article considers statistical inference for the heteroscedastic partially linear varying coefficient models. We construct an efficient estimator for the parametric component by applying the weighted profile least-squares approach, and show that it is semiparametrically efficient in the sense that the inverse of the asymptotic variance of the estimator reaches the semiparametric efficiency bound. Simulation studies are conducted to illustrate the performance of the proposed method.     

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.     

Bayesian inference for random coefficient dynamic panel data models

We develop a hierarchical Bayesian approach for inference in random coefficient dynamic panel data models. Our approach allows for the initial values of each unit's process to be correlated with the unit-specific coefficients. We impose a stationarity assumption for each unit's process by assuming that the unit-specific autoregressive coefficient is drawn from a logitnormal distribution. Our method is shown to have favorable properties compared to the mean group estimator in a Monte Carlo study. We apply our approach to analyze energy and protein intakes among individuals from the Philippines.     

Approaches to regression analysis with multiple measurements from individual sampling units

Application of ordinary least-squares regression to data sets which contain multiple measurements from individual sampling units produces an unbiased estimator of the parameters but a biased estimator of the covariance matrix of the parameter estimates. The present work considers a random coefficient, linear model to deal with such data sets: this model permits many senses in which multiple measurements are taken from a sampling unit, not just when it is measured at several times. Three procedures to estimate the covariance matrix of the error term of the model are considered. Given these, three procedures to estimate the parameters of the model and their covariance matrix are considered; these are ordinary least-squares, generalized least-squares, and an adjusted ordinary least-squares procedure which produces an unbiased estimator of the covariance matrix of the parameters with small samples. These various procedures are compared in simulation studies using three examples from the biological literature. The possibility of testing hypotheses about the vector of parameters is also considered. It is found that all three procedures for regression estimation produce estimators of the parameters with bias of no practical consequence, Both generalized least-squares and adjusted ordinary least-squares generally produce estimators of the covariance matrix of the parameter estimates with bias of no practical consequence, while ordinary least-squares produces a negatively biased estimator. Neither ordinary nor generalized least-squares provide satisfactory hypothesis tests of the vector of parameter estimates. It is concluded that adjusted ordinary least-squares, when applied with either of two of the procedures used to estimate the error coveriance matrix, shows promise for practical application with data sets of the nature considered here.     

Asymptotics of the weighted least squares estimation for AR(1) processes with applications to confidence intervals

For the first-order autoregressive model, we establish the asymptotic theory of the weighted least squares estimations whether the underlying autoregressive process is stationary, unit root, near integrated or even explosive under a weaker moment condition of innovations. The asymptotic limit of this estimator is always normal. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. An empirical likelihood confidence interval is proposed for interval estimations of the autoregressive coefficient. The results improve the corresponding ones of Chan et al. (Econ Theory 28:705–717, 2012). Some simulations are conducted to illustrate the proposed method.     

Preliminary estimators for robust non-linear regression estimation

In this paper, the robustness of weighted non-linear least-squares estimation based on some preliminary estimators is examined. The preliminary estimators are the Lnorm estimates proposed by Schlossmacher, by El-Attar et al., by Koenker and Park, and by Lawrence and Arthur. A numerical example is presented to compare the robustness of the weighted non-linear least-squares approach when based on the preliminary estimators of Schlossmacher (HS), El-Attar et al. (HEA), Koenker and Park (HKP), and Lawrence and Arthur (HLA). The study shows that the HEA estimator is as robust as the HKP estimator. However, the HEA estimator posed certain computational problems and required more storage and computing time.     

A likelihood based estimator for vector autoregressive processes

A onestep estimator, which is an approximation to the unconditional maximum likelihood estimator (MLE) of the coefficient matrices of a Gaussian vector autoregressive process is presented. The onestep estimator is easy to compute and can be computed using standard software. Unlike the computation of the unconditional MLE, the computation of the onestep estimator does not require any iterative optimization and the computation is numerically stable. In finite samples the onestep estimator generally has smaller mean square error than the ordinary least squares estimator. In a simple model, where the unconditional MLE can be computed, numerical investigation shows that the onestep estimator is slightly worse than the unconditional MLE in terms of mean square error but superior to the ordinary least squares estimator. The limiting distribution of the onestep estimator for processes with some unit roots is derived.     

Improved autoregressive forecasts in the presence of non-normal errors

This paper is concerned with obtaining more accurate point forecasts in the presence of non-normal errors. Specifically, we apply the residual augmented least-squares (RALS) estimator to autoregressive models to utilize the additional moment restrictions embodied in non-normal errors. Monte Carlo experiments are performed to compare our RALS forecasts to forecasts based on the ordinary least-squares estimator and the least absolute deviations (LAD) estimator. We find that the RALS approach provides superior forecasts when the data are skewed. Compared to the LAD forecast, the RALS forecast has smaller mean squared prediction errors in the baseline case with normal errors.     

A note on the efficiency of composite quantile regression

Composite quantile regression (CQR) is motivated by the desire to have an estimator for linear regression models that avoids the breakdown of the least-squares estimator when the error variance is infinite, while having high relative efficiency even when the least-squares estimator is fully efficient. Here, we study two weighting schemes to further improve the efficiency of CQR, motivated by Jiang et al. [Oracle model selection for nonlinear models based on weighted composite quantile regression. Statist Sin. 2012;22:1479–1506]. In theory the two weighting schemes are asymptotically equivalent to each other and always result in more efficient estimators compared with CQR. Although the first weighting scheme is hard to implement, it sheds light on in what situations the improvement is expected to be large. A main contribution is to theoretically and empirically identify that standard CQR has good performance compared with weighted CQR only when the error density is logistic or close to logistic in shape, which was not noted in the literature.