A sequence of improved standard errors under heteroskedasticity of unknown form

The linear regression model is commonly used by practitioners to model the relationship between the variable of interest and a set of explanatory variables. The assumption that all error variances are the same (homoskedasticity) is oftentimes violated. Consistent regression standard errors can be computed using the heteroskedasticity-consistent covariance matrix estimator proposed by White (1980). Such standard errors, however, typically display nonnegligible systematic errors in finite samples, especially under leveraged data. Cribari-Neto et al. (2000) improved upon the White estimator by defining a sequence of bias-adjusted estimators with increasing accuracy. In this paper, we improve upon their main result by defining an alternative sequence of adjusted estimators whose biases vanish at a much faster rate. Hypothesis testing inference is also addressed. An empirical illustration is presented.     

GMM inference in spatial autoregressive models

In this study, we investigate the finite sample properties of the optimal generalized method of moments estimator (OGMME) for a spatial econometric model with a first-order spatial autoregressive process in the dependent variable and the disturbance term (for short SARAR(1, 1)). We show that the estimated asymptotic standard errors for spatial autoregressive parameters can be substantially smaller than their empirical counterparts. Hence, we extend the finite sample variance correction methodology of Windmeijer (2005 Windmeijer, F. (2005). A finite sample correction for the variance of linear efficient two-step GMM estimators. Journal of Econometrics 126(1):25–51.[Crossref], [Web of Science ®] , [Google Scholar]) to the OGMME for the SARAR(1, 1) model. Results from simulation studies indicate that the correction method improves the variance estimates in small samples and leads to more accurate inference for the spatial autoregressive parameters. For the same model, we compare the finite sample properties of various test statistics for linear restrictions on autoregressive parameters. These tests include the standard asymptotic Wald test based on various GMMEs, a bootstrapped version of the Wald test, two versions of the C(α) test, the standard Lagrange multiplier (LM) test, the minimum chi-square test (MC), and two versions of the generalized method of moments (GMM) criterion test. Finally, we study the finite sample properties of effects estimators that show how changes in explanatory variables impact the dependent variable.     

GMM estimation of spatial autoregressive models in a system of simultaneous equations with heteroskedasticity

This paper proposes a GMM estimation framework for the SAR model in a system of simultaneous equations with heteroskedastic disturbances. Besides linear moment conditions, the proposed GMM estimator also utilizes quadratic moment conditions based on the covariance structure of model disturbances within and across equations. Compared with the QML approach, the GMM estimator is easier to implement and robust under heteroskedasticity of unknown form. We derive the heteroskedasticity-robust standard error for the GMM estimator. Monte Carlo experiments show that the proposed GMM estimator performs well in finite samples.     

Consistent GMM Residuals-Based Tests of Functional Form

This paper presents a consistent Generalized Method of Moments (GMM) residuals-based test of functional form for time series models. By relating two moments we deliver a vector moment condition in which at least one element must be nonzero if the model is misspecified. The test will never fail to detect misspecification of any form for large samples, and is asymptotically chi-squared under the null, allowing for fast and simple inference. A simulation study reveals randomly selecting the nuisance parameter leads to more power than supremum-tests, and can obtain empirical power nearly equivalent to the most powerful test for even relatively small n.     

Approximate testing in two-stage nonlinear mixed models

We investigate here small sample properties of approximate F-tests about fixed effects parameters in nonlinear mixed models. For estimation of population fixed effects parameters as well as variance components, we apply the two-stage approach. This method is useful and popular when the number of observations per sampling unit is large enough. The approximate F-test is constructed based on large-sample approximation to the distribution of nonlinear least-squares estimates of subject-specific parameters. We recommend a modified test statistic that takes into consideration approximation to the large-sample Fisher information matrix (See [Volaufova J, Burton JH. Note on hypothesis testing in mixed models. Oral presentation at: LINSTAT 2012/21st IWMS; 2012; Bedlewo, Poland]). Our main focus is on comparing finite sample properties of broadly used approximate tests (Wald test and likelihood ratio test) and the modified F-test under the null hypothesis, especially accuracy of p-values (See [Volaufova J, LaMotte L. Comparison of approximate tests of fixed effects in linear repeated measures design models with covariates. Tatra Mountains. 2008;39:17–25]). For that purpose two extensive simulation studies are conducted based on pharmacokinetic models (See [Hartford A, Davidian M. Consequences of misspecifying assumptions in nonlinear mixed effects models. Comput Stat and Data Anal. 2000;34:139–164; Pinheiro J, Bates D. Approximations to the log-likelihood function in the non-linear mixed-effects model. J Comput Graph Stat. 1995;4(1):12–35]).     

GMM estimation of a realized stochastic volatility model: A Monte Carlo study

This article investigates alternative generalized method of moments (GMM) estimation procedures of a stochastic volatility model with realized volatility measures. The extended model can accommodate a more general correlation structure. General closed form moment conditions are derived to examine the model properties and to evaluate the performance of various GMM estimation procedures under Monte Carlo environment, including standard GMM, principal component GMM, robust GMM and regularized GMM. An application to five company stocks and one stock index is also provided for an empirical demonstration.     

GMM estimation in partial linear models with endogenous covariates causing an over-identified problem

ABSTRACT

We study partial linear models where the linear covariates are endogenous and cause an over-identified problem. We propose combining the profile principle with local linear approximation and the generalized moment methods (GMM) to estimate the parameters of interest. We show that the profiled GMM estimators are root? n consistent and asymptotically normally distributed. By appropriately choosing the weight matrix, the estimators can attain the efficiency bound. We further consider variable selection by using the moment restrictions imposed on endogenous variables when the dimension of the covariates may be diverging with the sample size, and propose a penalized GMM procedure, which is shown to have the sparsity property. We establish asymptotic normality of the resulting estimators of the nonzero parameters. Simulation studies have been presented to assess the finite-sample performance of the proposed procedure.     

Simultaneous confidence bands for nonlinear regression models

The maximization and minimization procedure for constructing confidence bands about general regression models is explained. Then, using an existing confidence region about the parameters of a nonlinear regression model and the maximization and minimization procedure, a generally conservative simultaneous confidence band is constructed about the model. Two examples are given, and some problems with the procedure are discussed     

Generalized mixed estimator for nonlinear models: a maximum likelihood approach

This paper considers the problem of estimating a nonlinear statistical model subject to stochastic linear constraints among unknown parameters. These constraints represent prior information which originates from a previous estimation of the same model using an alternative database. One feature of this specification allows for the disign matrix of stochastic linear restrictions to be estimated. The mixed regression technique and the maximum likelihood approach are used to derive the estimator for both the model coefficients and the unknown elements of this design matrix. The proposed estimator whose asymptotic properties are studied, contains as a special case the conventional mixed regression estimator based on a fixed design matrix. A new test of compatibility between prior and sample information is also introduced. Thesuggested estimator is tested empirically with both simulated and actual marketing data.     

Adjusted Pearson residuals in exponential family nonlinear models

In this paper, we give matrix formulae of order 𝒪(n ^?1), where n is the sample size, for the first two moments of Pearson residuals in exponential family nonlinear regression models [G.M. Cordeiro and G.A. Paula, Improved likelihood ratio statistic for exponential family nonlinear models, Biometrika 76 (1989), pp. 93–100.]. The formulae are applicable to many regression models in common use and generalize the results by Cordeiro [G.M. Cordeiro, On Pearson's residuals in generalized linear models, Statist. Prob. Lett. 66 (2004), pp. 213–219.] and Cook and Tsai [R.D. Cook and C.L. Tsai, Residuals in nonlinear regression, Biometrika 72(1985), pp. 23–29.]. We suggest adjusted Pearson residuals for these models having, to this order, the expected value zero and variance one. We show that the adjusted Pearson residuals can be easily computed by weighted linear regressions. Some numerical results from simulations indicate that the adjusted Pearson residuals are better approximated by the standard normal distribution than the Pearson residuals.     

Corrected maximum likelihood estimators in heteroscedastic symmetric nonlinear models

In this article, we derive general matrix formulae for second-order biases of maximum likelihood estimators (MLEs) in a class of heteroscedastic symmetric nonlinear regression models, thus generalizing some results in the literature. This class of regression models includes all symmetric continuous distributions, and has a wide range of practical applications in various fields such as engineering, biology, medicine and economics, among others. The variety of distributions with different kurtosis coefficients than the normal may give more flexibility in the choice of an appropriate distribution, particularly to accommodate outlying and influential observations. We derive a joint iterative process for estimating the mean and dispersion parameters. We also present simulation studies for the biases of the MLEs.     

On nonlinear random effects models for repeated measurements

Linear random effects models for longitudinal data discussed by Laird and Ware (1982), Jennrich and Schluchter (1986), Lange and Laird (1989), and others are extended in a straight forward manner to nonlinear random effects models. This results in a simple computational approach which accommodates patterned covariance matrices and data insufficient for fitting each subject separately. The technique is demonstrated with an interesting medical data set, and a short, simple SAS PROC IML program based on the EM algorithm is presented.     

A comparison of two exact confidence regions for partially nonlinear regression models

Exact confidence regions for all the parameters in nonlinear regression models can be obtained by comparing the lengths of projections of the error vector into orthogonal subspaces of the sample space. In certain partially nonlinear models an alternative exact region is obtained by replacing the linear parameters by their conditional estimates in the projection matrices. An ellipsoidal approximation to the alternative region is obtained in terms of the tangent-plane coordinates, similar to one previously obtained for the more usual region. This ellipsoid can be converted to an approximate region for the original parameters and can be used to compare the two types of exact confidence regions.     

Empirical likelihood for nonlinear models with missing responses

In this paper, a nonlinear model with response variables missing at random is studied. In order to improve the coverage accuracy for model parameters, the empirical likelihood (EL) ratio method is considered. On the complete data, the EL statistic for the parameters and its approximation have a χ² asymptotic distribution. When the responses are reconstituted using a semi-parametric method, the empirical log-likelihood on the response variables associated with the imputed data is also asymptotically χ². The Wilks theorem for EL on the parameters, based on reconstituted data, is also satisfied. These results can be used to construct the confidence region for the model parameters and the response variables. It is shown via Monte Carlo simulations that the EL methods outperform the normal approximation-based method in terms of coverage probability for the unknown parameter, including on the reconstituted data. The advantages of the proposed method are exemplified on real data.     

Estimation and inference for varying coefficient partially nonlinear models

In this paper, we extend the varying coefficient partially linear model to the varying coefficient partially nonlinear model in which the linear part of the varying coefficient partially linear model is replaced by a nonlinear function of the covariates. A profile nonlinear least squares estimation procedure for the parameter vector and the coefficient function vector of the varying coefficient partially nonlinear model is proposed and the asymptotic properties of the resulting estimators are established. We further propose a generalized likelihood ratio (GLR) test to check whether or not the varying coefficients in the model are constant. The asymptotic null distribution of the GLR statistic is derived and a residual-based bootstrap procedure is also suggested to derive the p-value of the GLR test. Some simulations are conducted to assess the performance of the proposed estimating and testing procedures and the results show that both the procedures perform well in finite samples. Furthermore, a real data example is given to demonstrate the application of the proposed model and its estimating and testing procedures.     

Semiparametric Bayesian hierarchical models for heterogeneous population in nonlinear mixed effect model: application to gastric emptying studies

Gastric emptying studies are frequently used in medical research, both human and animal, when evaluating the effectiveness and determining the unintended side-effects of new and existing medications, diets, and procedures or interventions. It is essential that gastric emptying data be appropriately summarized before making comparisons between study groups of interest and to allow study the comparisons. Since gastric emptying data have a nonlinear emptying curve and are longitudinal data, nonlinear mixed effect (NLME) models can accommodate both the variation among measurements within individuals and the individual-to-individual variation. However, the NLME model requires strong assumptions that are often not satisfied in real applications that involve a relatively small number of subjects, have heterogeneous measurement errors, or have large variation among subjects. Therefore, we propose three semiparametric Bayesian NLMEs constructed with Dirichlet process priors, which automatically cluster sub-populations and estimate heterogeneous measurement errors. To compare three semiparametric models with the parametric model we propose a penalized posterior Bayes factor. We compare the performance of our semiparametric hierarchical Bayesian approaches with that of the parametric Bayesian hierarchical approach. Simulation results suggest that our semiparametric approaches are more robust and flexible. Our gastric emptying studies from equine medicine are used to demonstrate the advantage of our approaches.     

Empirical likelihood-based inference for parameter and nonparametric function in partially nonlinear models

This paper is concerned with statistical inference for partially nonlinear models. Empirical likelihood method for parameter in nonlinear function and nonparametric function is investigated. The empirical log-likelihood ratios are shown to be asymptotically chi-square and then the corresponding confidence intervals are constructed. By the empirical likelihood ratio functions, we also obtain the maximum empirical likelihood estimators of the parameter in nonlinear function and nonparametric function, and prove the asymptotic normality. A simulation study indicates that, compared with normal approximation-based method and the bootstrap method, the empirical likelihood method performs better in terms of coverage probabilities and average length/widths of confidence intervals/bands. An application to a real dataset is illustrated.     

A robust class of homoscedastic nonlinear regression models

In this paper, we examine a nonlinear regression (NLR) model with homoscedastic errors which follows a flexible class of two-piece distributions based on the scale mixtures of normal (TP-SMN) family. The objective of using this family is to develop a robust NLR model. The TP-SMN is a rich class of distributions that covers symmetric/asymmetric and lightly/heavy-tailed distributions and is an alternative family to the well-known scale mixtures of skew-normal (SMSN) family studied by Branco and Dey [35]. A key feature of this study is using a new suitable hierarchical representation of the family to obtain maximum-likelihood estimates of model parameters via an EM-type algorithm. The performances of the proposed robust model are demonstrated using simulated and some natural real datasets and also compared to other well-known NLR models.     

Mem squared error estimation in finite populations under nonlinear models

Four estimators of the prediction mean squared error (MSB) of an estimated finite population total for a zero-one characteristic are examined. The characteristic associated with each population unit is modeled as the realization of a Bernoulli random variable whose expected value is a nonlinear function of a parameter vector and a set of known auxiliary variables. To compare the estimators, a simulation study is conducted using a population of hospitals. The MSB estimator Implied by the form of the assumed model underestimates the mean squared error in each of the cases studied and produces confidence lntervals with less than the nominal coverage probabilities. Of the three alternative MSE estimators presented, a linear approximation to the jackknife produces the best results and improves upon the model-specific estimator.