Statistical inference in a panel data semiparametric regression model with serially correlated errors

We consider a panel data semiparametric partially linear regression model with an unknown vector β of regression coefficients, an unknown nonparametric function g(·) for nonlinear component, and unobservable serially correlated errors. The correlated errors are modeled by a vector autoregressive process which involves a constant intraclass correlation. Applying the pilot estimators of β and g(·), we construct estimators of the autoregressive coefficients, the intraclass correlation and the error variance, and investigate their asymptotic properties. Fitting the error structure results in a new semiparametric two-step estimator of β, which is shown to be asymptotically more efficient than the usual semiparametric least squares estimator in terms of asymptotic covariance matrix. Asymptotic normality of this new estimator is established, and a consistent estimator of its asymptotic covariance matrix is presented. Furthermore, a corresponding estimator of g(·) is also provided. These results can be used to make asymptotically efficient statistical inference. Some simulation studies are conducted to illustrate the finite sample performances of these proposed estimators.     

Asymptotic distribution of the weighted least squares estimator

This paper derives the asymptotic distribution of the weighted least squares estimator (WLSE) in a heteroscedastic linear regression model. A consistent estimator of the asymptotic covariance matrix of the WLSE is also obtained. The results are obtained under weak conditions on the design matrix and some moment conditions on the error distributions. It is shown that most of the error distributions encountered in practice satisfy these moment conditions. Some examples of the asymptotic covariance matrices are also given.     

Delete-group Jackknife Estimate in Partially Linear Regression Models with Heteroscedasticity

Abstract Consider a partially linear regression model with an unknown vector parameter β,an unknownfunction g(.),and unknown heteroscedastic error variances.Chen,You proposed a semiparametric generalizedleast squares estimator(SGLSE)for β,which takes the heteroscedasticity into account to increase efficiency.Forinference based on this SGLSE,it is necessary to construct a consistent estimator for its asymptotic covariancematrix.However,when there exists within-group correlation, the traditional delta method and the delete-1jackknife estimation fail to offer such a consistent estimator.In this paper, by deleting grouped partial residualsa delete-group jackknife method is examined.It is shown that the delete-group jackknife method indeed canprovide a consistent estimator for the asymptotic covariance matrix in the presence of within-group correlations.This result is an extension of that in[21].     

Jackknifing in partially linear regression models with serially correlated errors

In this paper jackknifing technique is examined for functions of the parametric component in a partially linear regression model with serially correlated errors. By deleting partial residuals a jackknife-type estimator is proposed. It is shown that the jackknife-type estimator and the usual semiparametric least-squares estimator (SLSE) are asymptotically equivalent. However, simulation shows that the former has smaller biases than the latter when the sample size is small or moderate. Moreover, since the errors are correlated, both the Tukey type and the delta type jackknife asymptotic variance estimators are not consistent. By introducing cross-product terms, a consistent estimator of the jackknife asymptotic variance is constructed and shown to be robust against heterogeneity of the error variances. In addition, simulation results show that confidence interval estimation based on the proposed jackknife estimator has better coverage probability than that based on the SLSE, even though the latter uses the information of the error structure, while the former does not.     

一般半相依回归系统的协方差改进估计

本文讨论了由两个等阶的回归方程组成的半相依系统,运用协方差改进法获得了参数的一个迭代估计序列,并证明了它的协方差阵已知时,处处收敛到最佳线性无偏估计,同时其协方差阵在矩阵偏序意义下单调性,并且给出了当迭代次数亦趋于无穷时,保证其具有相合性的一个条件。     

纵向数据边际模型非参数光滑方法的比较

对于纵向数据边际模型的均值函数, 有很多非参数估计方法, 其中回归样条, 光滑样条, 似乎不相关(SUR)核估计等方法在工作协方差阵正确指定时具有最小的渐近方差. 回归样条的渐近偏差与工作协方差阵无关, 而SUR核估计和光滑样条估计的渐近偏差却依赖于工作协方差阵. 本文主要研究了回归样条, 光滑样条和SUR核估计的效率问题. 通过模拟比较发现回归样条估计的表现比较稳定, 在大多数情况下比光滑样条估计和SUR核估计的效率高.     

Theoretical and empirical estimates of mean–variance portfolio sensitivity

This paper studies properties of an estimator of mean–variance portfolio weights in a market model with multiple risky assets and a riskless asset. Theoretical formulas for the mean square error are derived in the case when asset excess returns are multivariate normally distributed and serially independent. The sensitivity of the portfolio estimator to errors arising from the estimation of the covariance matrix and the mean vector is quantified. It turns out that the relative contribution of the covariance matrix error depends mainly on the Sharpe ratio of the market portfolio and the sampling frequency of historical data. Theoretical studies are complemented by an investigation of the distribution of portfolio estimator for empirical datasets. An appropriately crafted bootstrapping method is employed to compute the empirical mean square error. Empirical and theoretical estimates are in good agreement, with the empirical values being, in general, higher.     

HAC estimation and strong linearity testing in weak ARMA models

In the framework of ARMA models, we consider testing the reliability of the standard asymptotic covariance matrix (ACM) of the least-squares estimator. The standard formula for this ACM is derived under the assumption that the errors are independent and identically distributed, and is in general invalid when the errors are only uncorrelated. The test statistic is based on the difference between a conventional estimator of the ACM of the least-squares estimator of the ARMA coefficients and its robust HAC-type version. The asymptotic distribution of the HAC estimator is established under the null hypothesis of independence, and under a large class of alternatives. The asymptotic distribution of the proposed statistic is shown to be a standard χ² under the null, and a noncentral χ² under the alternatives. The choice of the HAC estimator is discussed through asymptotic power comparisons. The finite sample properties of the test are analyzed via Monte Carlo simulation.     

Iterative Weighted Semiparametric Least Squares Estimation in Repeated Measurement Partially Linear Regression Models

Consider a repeated measurement partially linear regression model with an unknown vector parameter β, an unknown function g(.), and unknown heteroscedastic error variances. In order to improve the semiparametric generalized least squares estimator (SGLSE) of β, we propose an iterative weighted semiparametric least squares estimator (IWSLSE) and show that it improves upon the SGLSE in terms of asymptotic covariance matrix. An adaptive procedure is given to determine the number of iterations. We also show that when the number of replicates is less than or equal to two, the IWSLSE can not improve upon the SGLSE. These results are generalizations of those in [2] to the case of semiparametric regressions.     

A note on improving quadratic inference functions using a linear shrinkage approach

In some commonly used longitudinal clinical trials designs, the quadratic inference functions (QIF) method fails to work due to non-invertible estimation of the optimal weighting matrix. We propose a modified QIF method, in which the optimal weighting matrix is estimated by a linear shrinkage estimator, replacing the sample covariance matrix. We prove that the linear shrinkage estimator is consistent and asymptotically optimal under the expected quadratic loss, and will have more stable numerical performance than the sample covariance matrix. Simulations show that numerical improvements are acquired in light of a higher percentage of convergence, and smaller standard errors and mean square errors of parameter estimates.     

Testing inference in heteroskedastic fixed effects models

This paper considers the issue of performing testing inference in fixed effects panel data models under heteroskedasticity of unknown form. We use numerical integration to compute the exact null distributions of different quasi-t test statistics and compare them to their limiting counterpart. The test statistics use different heteroskedasticity-consistent standard errors. Our results reveal that the asymptotic approximation is usually poor in small samples when the test statistic is based on the covariance matrix estimator proposed by Arellano (1987). The quality of the approximation is greatly increased when the standard error is obtained using other heteroskedasticity-consistent estimators, most notably the CHC4 estimator. Our results also reveal that the performance of Arellano’s test improves considerably when standard errors are computed using restricted residuals.     

Covariance estimation under spatial dependence

Correlated multivariate processes have a dependence structure which must be taken into account when estimating the covariance matrix. The natural estimator of the covariance matrix is introduced and is shown that to be biased under the dependence structure. This bias is studied under two different asymptotic models, namely increasing the domain by increasing the number of observations, and increasing the number of observations in the fixed domain. Using the first asymptotic model, we quantify the convergence rate of the bias and of the covariance between the components of the estimated covariance matrix. The second asymptotic model serves to derive a fast and accurate bias correction. As shown, under mild hypotheses, the asymptotic normality of the estimated covariance matrix holds and can be used to test whether the bias is significant, for example, in the sense that the eigenvectors of the estimated and true covariance matrices are significantly different.     

The Distribution of Robust Distances

Mahalanobis-type distances in which the shape matrix is derived from a consistent, high-breakdown robust multivariate location and scale estimator have an asymptotic chi-squared distribution as is the case with those derived from the ordinary covariance matrix. For example, Rousseeuw's minimum covariance determinant (MCD) is a robust estimator with a high breakdown. However, even in quite large samples, the chi-squared approximation to the distances of the sample data from the MCD center with respect to the MCD shape is poor. We provide an improved F approximation that gives accurate outlier rejection points for various sample sizes.     

Differentially Private Precision Matrix Estimation

In this paper, we study the problem of precision matrix estimation when the dataset contains sensitive information. In the differential privacy framework, we develop a differentially private ridge estimator by perturbing the sample covariance matrix. Then we develop a differentially private graphical lasso estimator by using the alternating direction method of multipliers (ADMM) algorithm. Furthermore, we prove theoretical results showing that the differentially private ridge estimator for the precision matrix is consistent under fixed-dimension asymptotic, and establish a convergence rate of differentially private graphical lasso estimator in the Frobenius norm as both data dimension p and sample size n are allowed to grow. The empirical results that show the utility of the proposed methods are also provided.     

L_1 regression estimate and its bootstrap

We study the asymptotic distribution of the L1 regression estimator under general condi-tions with matrix norming and possibly non i.i.d.errors.We then introduce an appropriate bootstrap procedure to estimate the distribution of this estimator and study its asymptotic properties.It is shown that this bootstrap is consistent under suitable conditions and in other situations the bootstrap limit is a random distribution.     

Hypothesis tests for high-dimensional covariance structures

We consider hypothesis testing for high-dimensional covariance structures in which the covariance matrix is a (i) scaled identity matrix, (ii) diagonal matrix, or (iii) intraclass covariance matrix. Our purpose is to systematically establish a nonparametric approach for testing the high-dimensional covariance structures (i)–(iii). We produce a new common test statistic for each covariance structure and show that the test statistic is an unbiased estimator of its corresponding test parameter. We prove that the test statistic establishes the asymptotic normality. We propose a new test procedure for (i)–(iii) and evaluate its asymptotic size and power theoretically when both the dimension and sample size increase. We investigate the performance of the proposed test procedure in simulations. As an application of testing the covariance structures, we give a test procedure to identify an eigenvector. Finally, we demonstrate the proposed test procedure by using a microarray data set.

     

Use of prior information in the consistent estimation of regression coefficients in measurement error models

A multivariate ultrastructural measurement error model is considered and it is assumed that some prior information is available in the form of exact linear restrictions on regression coefficients. Using the prior information along with the additional knowledge of covariance matrix of measurement errors associated with explanatory vector and reliability matrix, we have proposed three methodologies to construct the consistent estimators which also satisfy the given linear restrictions. Asymptotic distribution of these estimators is derived when measurement errors and random error component are not necessarily normally distributed. Dominance conditions for the superiority of one estimator over the other under the criterion of Löwner ordering are obtained for each case of the additional information. Some conditions are also proposed under which the use of a particular type of information will give a more efficient estimator.     

Second-order generalized estimating equations for correlated count data

Generalized estimating equations have been widely used in the analysis of correlated count data. Solving these equations yields consistent parameter estimates while the variance of the estimates is obtained from a sandwich estimator, thereby ensuring that, even with misspecification of the so-called working correlation matrix, one can draw valid inferences on the marginal mean parameters. That they allow misspecification of the working correlation structure, though, implies a limitation of these equations should scientific interest also be in the covariance or correlation structure. We propose herein an extension of these estimating equations such that, by incorporating the bivariate Poisson distribution, the variance-covariance matrix of the response vector can be properly modelled, which would permit inference thereon. A sandwich estimator is used for the standard errors, ensuring sound inference on the parameters estimated. Two applications are presented.     

Nonparametric regression estimation with general parametric error covariance

The asymptotic distribution for the local linear estimator in nonparametric regression models is established under a general parametric error covariance with dependent and heterogeneously distributed regressors. A two-step estimation procedure that incorporates the parametric information in the error covariance matrix is proposed. Sufficient conditions for its asymptotic normality are given and its efficiency relative to the local linear estimator is established. We give examples of how our results are useful in some recently studied regression models. A Monte Carlo study confirms the asymptotic theory predictions and compares our estimator with some recently proposed alternative estimation procedures.