Linear Process Errors Research Articles

Partially Functional Linear Models with Linear Process Errors

In this paper, we focus on the partial functional linear model with linear process errors deduced by not necessarily independent random variables. Based on Mercer’s theorem and Karhunen–Loève expansion, we give the estimators of the slope parameter and coefficient function in the model, establish the asymptotic normality of the estimator for the parameter and discuss the weak convergence with rates of the proposed estimators. Meanwhile, the penalized estimator of the parameter is defined by the SCAD penalty and its oracle property is investigated. Finite sample behavior of the proposed estimators is also analysed via simulations.

Inference for the VEC(1) model with a heavy-tailed linear process errors*

This article studies the first-order vector error correction (VEC(1)) model when its noise is a linear process of independent and identically distributed (i.i.d.) heavy-tailed random vectors with a tail index . We show that the rate of convergence of the least squares estimator (LSE) related to the long-run parameters is n (sample size) and its limiting distribution is a stochastic integral in terms of two stable random processes, while the LSE related to the short-term parameters is not consistent. We further propose an automated approach via adaptive shrinkage techniques to determine the cointegrating rank in the VEC(1) model. It is demonstrated that the cointegration rank r 0 can be consistently selected despite the fact that the LSE related to the short-term parameters is not consistently estimable when the tail index . Simulation studies are carried out to evaluate the performance of the proposed procedure in finite samples. Last, we use our techniques to explore the long-run and short-run behavior of the monthly prices of wheat, corn, and wheat flour in the United States.

Statistical Inference for Estimators in a Semiparametric EV Model with Linear Process Errors and Missing Responses

This paper concentrates on the properties of estimators in a semiparametric EV model, particularly considering the effects of missing data and linear process errors according to the actual situation. The missing data are processed by three different methods: the direct deletion method, imputation (interpolation fill) method, and regression surrogate method. Also, the corresponding estimators of the slope parameter β and the nonparameter variable g(⋅) are obtained. All the estimators are asymptotically normal, and the consistency rates for which can achieve o(n−1/6 log n). Besides, the performance of the estimators is investigated by one sample experiment.

A Berry-Ess$\acute{e}$n bound of wavelet estimation for a nonparametric regression model under linear process errors based on LNQD sequence

By using some inequalities for linearly negative quadrant dependent random variables, Berry-Ess$\acute{e}$en bound of wavelet estimation for a nonparametric regression model is investigated under linear process errors based on linearly negative quadrant dependent sequence. The rate of uniform asymptotic normality is presented and the rate of convergence is near $O(n^{-\frac{1}{6}})$ under mild conditions, which generalizes or extends the corresponding results of Li et al.(2008) under associated random samples in some sense.

Common breaks in means for panel data under short-range dependence

In this paper, we revisit the model studied in Bai (2010), in which a common break in means for panel data with linear-process errors was studied. The errors admit the form of for series i. Bai (2010) conducted the change-point analysis under the situation where for all i. In this paper, we suppose and for all i, that is, the errors are short-range dependence. Then the least squares estimator of the common break is examined. In comparison to Bai (2010), the theoretical results in this paper reveal that: (1) Short-range dependence does not have the impact on the consistency of the estimator of the common break under the case of fixed magnitude of breaks. (2) It also does not have the impact on the convergence rate while it contaminates the limiting distribution of the estimator of the common break under the case of shrinking magnitude of breaks. Monte Carlo simulations are conducted to examine the finite-sample performance of the estimator. Our theoretical findings are supported by the Monte Carlo simulations.

The Berry–Esseen type bounds of the weighted estimator in a nonparametric model with linear process errors

In this paper, the Berry–Esseen type bounds of the weighted estimator in a nonparametric regression model are investigated under some mild conditions when random errors are from a linear process generated by $$\varphi $$ -mixing random variables. In particular, the rate of uniform normal approximation is near to $$O(n^{-\frac{3}{16}})$$ by the choice of some constants, which generalizes and improves the corresponding results of Li et al. (Stat Probab Lett 81:103–110, 2011) and Ding et al. (J Inequal Appl 2018:10, 2018). Finally, the simulation study is provided to verify the validity of the theoretical results.

Berry-Esseen type bounds of the estimators in a semiparametric model under linear process errors with α-mixing dependent innovations

ABSTRACTIn this paper, we mainly study the Berry-Esseen type bounds of estimators in a semiparametric model , where β is an unknown parameter, are nonrandom design points, are the response variables, is an unknown function designed on the closed interval , and the errors are formed by with and are α-mixing random variables. Under appropriate conditions, the Berry-Esseen type bounds for estimators of β and can attain nearly and , respectively, and under further restriction for the weights, the Berry-Esseen type bound for estimator of can also achieve nearly . A numerical simulation is also carried out to verify the validity of the theoretical results.

The Phillips unit root tests for polynomials of integrated processes revisited

We show that the Phillips (1987) unit root tests have nuisance parameter free limiting distributions when applied to polynomials of integrated processes driven by linear process errors. This substantially generalizes a similar result of Wagner (2012) allowing only for serially uncorrelated errors. The result is based on novel kernel weighted sum limit results involving powers of integrated processes. These results allow us to consider additional modifications of the Phillips (1987) tests applicable to polynomials of integrated processes.

Modified Unit Root Tests with Nuisance Parameter Free Asymptotic Distributions

It is well-known that the asymptotic distributions of the Dickey-Fuller (DF) tests for a unit root with linear process errors are not free of nuisance parameters. In this paper, we introduce a consistent estimator for the nuisance parameters and then use it to modify the DF tests, denoted as R-tests. Under fairly mild moment and summability conditions on the errors, we show that the asymptotic distributions of the R-tests are of the same as the Dickey-Fuller distributions. In Monte Carlo experiments, the R-tests are shown to have improved size properties.

Empirical Likelihood for Linear Models Under Linear Process Errors

In this article, we study the construction of confidence intervals for regression parameters in a linear model under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ2-type distributed. The result is used to obtain EL based confidence regions for regression parameters. The finite-sample performance of the method is evaluated through a simulation study.

Empirical Likelihood for Nonparametric Models Under Linear Process Errors

SummaryIn this paper, we study the construction of confidence intervals for a nonparametric regression function under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically distributed. The result is used to obtain EL based confidence intervals for the nonparametric regression function. The finite‐sample performance of the method is evaluated through a simulation study.

Empirical likelihood confidence regions for semi-varying coefficient models with linear process errors

In this paper, we apply the empirical likelihood method to semi-varying coefficient models with linear process errors, propose two empirical log-likelihood ratio statistics and show that their limiting distributions are two weighted sums of independent chi-square distributions with 1 degree of freedom. By estimating the unknown weights consistently, we not only construct the empirical likelihood confidence regions for the parametric component, but also construct the point-wise empirical likelihood confidence regions and the simultaneous empirical likelihood confidence bands for the varying-coefficient functions. Monte Carlo simulation results show that the proposed empirical likelihood confidence regions have better coverage probabilities and shorter median lengths than the normal approximation confidence regions ignoring the correlation information.

Berry-Esseen bounds for wavelet estimator in semiparametric regression model with linear process errors

Consider the semiparametric regression model Yi = xi b +g (ti) + ei, i= 1, ..., n, where the linear process errors ei = � ∞=−∞ ajei−j with � ∞=−∞ � a j � < ∞ , and {ei} are identically distributed and strong mixing innovations with zero mean. Under appropriate conditions, the Berry-Esseen type bounds of wavelet estimators for b and g(·) are established. Our results obtained generalize the results of nonparametric regression model by Li et al. to semiparametric regression model. Mathematical Subject Classification: 62G05; 62G08.

Berry–Esseen bounds for wavelet estimator in a regression model with linear process errors

In this paper, we derive the Berry–Esseen bounds of the wavelet estimator for a nonparametric regression model with linear process errors generated by φ -mixing sequences. As application, by the suitable choice of some constants, the convergence rate O ( n − 1 / 6 ) of uniformly asymptotic normality of the wavelet estimator is obtained. Our results generalize some known results in the literature.

Empirical likelihood inference for semiparametric model with linear process errors

The purpose of this article is to use the empirical likelihood method to study the confidence regions construction for the parameters of interest in semiparametric model with linear process errors under martingale difference. It is shown that the adjusted empirical log-likelihood ratio at the true parameters is asymptotically chi-squared. A simulation study indicates that the adjusted empirical likelihood works better than a normal approximation-based approach.

Berry–Esseen type bounds of estimators in a semiparametric model with linear process errors

Consider the semiparametric regression model y i = x i β + g ( t i ) + V i , 1 ≤ i ≤ n , where β is an unknown parameter of interest, ( x i , t i ) are nonrandom design points, y i are the response variables, g ( ⋅ ) is an unknown function defined on the closed interval [ 0 , 1 ] , and the correlated errors V i = ∑ j = − ∞ ∞ ψ j e i − j , with ∑ j = − ∞ ∞ | ψ j | < ∞ , and e i are negatively associated random variables. Under appropriate conditions, in this paper, we derive Berry–Esseen type bounds for estimators of β and g ( ⋅ ) . As a corollary, by making a certain choice of the weights, we give the Berry–Esseen type bounds for estimators of β and g ( ⋅ ) ; they are O ( n − 1 / 4 ( log n ) 3 / 4 ) and O ( n − 3 / 28 ( log n ) 9 / 28 ) , respectively, and under further restriction for the weights, the Berry–Esseen type bound for estimator of g ( ⋅ ) can also attain O ( n − 1 / 4 ( log n ) 3 / 4 ) .

On the Strong Mixing Property for Linear Sequences

On the Strong Mixing Property for Linear Sequences