A Test for Slope Heterogeneity in Fixed Effects Models

Typical panel data models make use of the assumption that the regression parameters are the same for each individual cross-sectional unit. We propose tests for slope heterogeneity in panel data models. Our tests are based on the conditional Gaussian likelihood function in order to avoid the incidental parameters problem induced by the inclusion of individual fixed effects for each cross-sectional unit. We derive the Conditional Lagrange Multiplier test that is valid in cases where N → ∞ and T is fixed. The test applies to both balanced and unbalanced panels. We expand the test to account for general heteroskedasticity where each cross-sectional unit has its own form of heteroskedasticity. The modification is possible if T is large enough to estimate regression coefficients for each cross-sectional unit by using the MINQUE unbiased estimator for regression variances under heteroskedasticity. All versions of the test have a standard Normal distribution under general assumptions on the error distribution as N → ∞. A Monte Carlo experiment shows that the test has very good size properties under all specifications considered, including heteroskedastic errors. In addition, power of our test is very good relative to existing tests, particularly when T is not large.     

Heteroskedastic linear regression model with compositional response and covariates

Compositional data are known as a sort of complex multidimensional data with the feature that reflect the relative information rather than absolute information. There are a variety of models for regression analysis with compositional variables. Similar to the traditional regression analysis, the heteroskedasticity still exists in these models. However, the existing heteroskedastic regression analysis methods cannot apply in these models with compositional error term. In this paper, we mainly study the heteroskedastic linear regression model with compositional response and covariates. The parameter estimator is obtained through weighted least squares method. For the hypothesis test of parameter, the test statistic is based on the original least squares estimator and corresponding heteroskedasticity-consistent covariance matrix estimator. When the proposed method is applied to both simulation and real example, we use the original least squares method as a comparison during the whole process. The results implicate the model's practicality and effectiveness in regression analysis with heteroskedasticity.     

A new heteroskedasticity-consistent covariance matrix estimator for the linear regression model

The assumption that all random errors in the linear regression model share the same variance (homoskedasticity) is often violated in practice. The ordinary least squares estimator of the vector of regression parameters remains unbiased, consistent and asymptotically normal under unequal error variances. Many practitioners then choose to base their inferences on such an estimator. The usual practice is to couple it with an asymptotically valid estimation of its covariance matrix, and then carry out hypothesis tests that are valid under heteroskedasticity of unknown form. We use numerical integration methods to compute the exact null distributions of some quasi-t test statistics, and propose a new covariance matrix estimator. The numerical results favor testing inference based on the estimator we propose.     

Heteroskedasticity-consistent interval estimators

The linear regression model is commonly used in applications. One of the assumptions made is that the error variances are constant across all observations. This assumption, known as homoskedasticity, is frequently violated in practice. A commonly used strategy is to estimate the regression parameters by ordinary least squares and to compute standard errors that deliver asymptotically valid inference under both homoskedasticity and heteroskedasticity of an unknown form. Several consistent standard errors have been proposed in the literature, and evaluated in numerical experiments based on their point estimation performance and on the finite sample behaviour of associated hypothesis tests. We build upon the existing literature by constructing heteroskedasticity-consistent interval estimators and numerically evaluating their finite sample performance. Different bootstrap interval estimators are also considered. The numerical results favour the HC4 interval estimator.     

Applying neural network Poisson regression to predict cognitive score changes

In this study, we combined a Poisson regression model with neural networks (neural network Poisson regression) to relax the traditional Poisson regression assumption of linearity of the Poisson mean as a function of covariates, while including it as a special case. In four simulated examples, we found that the neural network Poisson regression improved the performance of simple Poisson regression if the Poisson mean was nonlinearly related to covariates. We also illustrated the performance of the model in predicting five-year changes in cognitive scores, in association with age and education level; we found that the proposed approach had superior accuracy to conventional linear Poisson regression. As the interpretability of the neural networks is often difficult, its combination with conventional and more readily interpretable approaches under the generalized linear model can benefit applications in biomedicine.     

Finite-sample refinement of GMM approach to nonlinear models under heteroskedasticity of unknown form

It is quite common to observe heteroskedasticity in real data, in particular, cross-sectional or micro data. Previous studies concentrate on improving the finite-sample properties of tests under heteroskedasticity of unknown forms in linear models. The advantage of a heteroskedasticity consistent covariance matrix estimator (HCCME)-type small-sample improvement for linear models does not carry over to the nonlinear model specifications since there is no obvious counterpart for the diagonal element of the projection matrix in linear models, which is crucial for implementing the finite-sample refinement. Within the framework of nonlinear models, we develop a straightforward approach by extending the applicability of HCCME-type corrections to the two-step GMM method. The Monte Carlo experiments show that the proposed method not only refines the testing procedure in terms of the error of rejection probability, but also improves the coefficient estimation based on the mean squared error (MSE) and the mean absolute error (MAE). The estimation of a constant elasticity of substitution (CES)-type production function is also provided to illustrate how to implement the proposed method empirically.     

Accurately sized test statistics with misspecified conditional homoskedasticity

We study the finite-sample performance of test statistics in linear regression models where the error dependence is of unknown form. With an unknown dependence structure, there is traditionally a trade-off between the maximum lag over which the correlation is estimated (the bandwidth) and the amount of heterogeneity in the process. When allowing for heterogeneity, through conditional heteroskedasticity, the correlation at far lags is generally omitted and the resultant inflation of the empirical size of test statistics has long been recognized. To allow for correlation at far lags, we study the test statistics constructed under the possibly misspecified assumption of conditional homoskedasticity. To improve the accuracy of the test statistics, we employ the second-order asymptotic refinement in Rothenberg [Approximate power functions for some robust tests of regression coefficients, Econometrica 56 (1988), pp. 997–1019] to determine the critical values. The simulation results of this paper suggest that when sample sizes are small, modelling the heterogeneity of a process is secondary to accounting for dependence. We find that a conditionally homoskedastic covariance matrix estimator (when used in conjunction with Rothenberg's second-order critical value adjustment) improves test size with only a minimal loss in test power, even when the data manifest significant amounts of heteroskedasticity. In some specifications, the size inflation was cut by nearly 40% over the traditional heteroskedasticity and autocorrelation consistent (HAC) test. Finally, we note that the proposed test statistics do not require that the researcher specify the bandwidth or the kernel.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.     

Empirical likelihood estimation for linear regression models with AR(p) error terms with numerical examples

Linear regression models are useful statistical tools to analyze data sets in different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors. If error terms are correlated the Conditional Maximum Likelihood (CML) estimation method under normality assumption is often used to estimate the parameters of interest. The CML estimation method is required a distributional assumption on error terms. However, in practice, such distributional assumptions on error terms may not be plausible. In this paper, we propose to estimate the parameters of a linear regression model with autoregressive error term using Empirical Likelihood (EL) method, which is a distribution free estimation method. A small simulation study is provided to evaluate the performance of the proposed estimation method over the CML method. The results of the simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.     

A robust test for autocorrelation in the presence of a structural break in variance

It has been known that when there is a break in the variance (unconditional heteroskedasticity) of the error term in linear regression models, a routine application of the Lagrange multiplier (LM) test for autocorrelation can cause potentially significant size distortions. We propose a new test for autocorrelation that is robust in the presence of a break in variance. The proposed test is a modified LM test based on a generalized least squares regression. Monte Carlo simulations show that the new test performs well in finite samples and it is especially comparable to other existing heteroskedasticity-robust tests in terms of size, and much better in terms of power.     

Small sample behavior of a robust heteroskedasticity consistent covariance matrix estimator

In heteroskedastic regression models, the least squares (OLS) covariance matrix estimator is inconsistent and inference is not reliable. To deal with inconsistency one can estimate the regression coefficients by OLS, and then implement a heteroskedasticity consistent covariance matrix (HCCM) estimator. Unfortunately the HCCM estimator is biased. The bias is reduced by implementing a robust regression, and by using the robust residuals to compute the HCCM estimator (RHCCM). A Monte-Carlo study analyzes the behavior of RHCCM and of other HCCM estimators, in the presence of systematic and random heteroskedasticity, and of outliers in the explanatory variables.     

Process Capability Analysis in Non Normal Linear Regression Profiles

When the distribution of a process characterized by a profile is non normal, process capability analysis using normal assumption often leads to erroneous interpretations of the process performance. Profile monitoring is a relatively new set of techniques in quality control that is used in situations where the state of product or process is represented by a function of two or more quality characteristics. Such profiles can be modeled using linear or nonlinear regression models. In some applications, it is assumed that the quality characteristics follow a normal distribution; however, in certain applications this assumption may fail to hold and may yield misleading results. In this article, we consider process capability analysis of non normal linear profiles. We investigate and compare five methods to estimate non normal process capability index (PCI) in profiles. In three of the methods, an estimation of the cumulative distribution function (cdf) of the process is required to analyze process capability in profiles. In order to estimate cdf of the process, we use a Burr XII distribution as well as empirical distributions. However, the resulted PCI with estimating cdf of the process is sometimes far from its true value. So, here we apply artificial neural network with supervised learning which allows the estimation of PCIs in profiles without the need to estimate cdf of the process. Box-Cox transformation technique is also developed to deal with non normal situations. Finally, a comparison study is performed through the simulation of Gamma, Weibull, Lognormal, Beta and student-t data.     

On the restricted almost unbiased two-parameter estimator in linear regression model

Özkale and Kaçiranlar introduced the restricted two-parameter estimator (RTPE) to deal with the well-known multicollinearity problem in linear regression model. In this paper, the restricted almost unbiased two-parameter estimator (RAUTPE) based on the RTPE is presented. The quadratic bias and mean-squared error of the proposed estimator is discussed and compared with the corresponding competitors in literatures. Furthermore, a numerical example and a Monte Carlo simulation study are given to explain some of the theoretical results.     

Partially adaptive quantile estimators

This paper contrasts two approaches to estimating quantile regression models: traditional semi-parametric methods and partially adaptive estimators using flexible probability density functions (pdfs). While more general pdfs could have been used, the skewed Laplace was selected for pedagogical purposes. Monte Carlo simulations are used to compare the behavior of the semi-parametric and partially adaptive quantile estimators in the presence of possibly skewed and heteroskedastic data. Both approaches accommodate skewness and heteroskedasticity which are consistent with linear quantiles; however, the partially adaptive estimator considered allows for non linear quantiles and also provides simple tests for symmetry and heteroskedasticity. The methods are applied to the problem of estimating conditional quantile functions for wages corresponding to different levels of education.     

Theoretical evaluation of prediction error in linear regression with a bivariate response variable containing missing data

Methods for linear regression with multivariate response variables are well described in statistical literature. In this study we conduct a theoretical evaluation of the expected squared prediction error in bivariate linear regression where one of the response variables contains missing data. We make the assumption of known covariance structure for the error terms. On this basis, we evaluate three well-known estimators: standard ordinary least squares, generalized least squares, and a James–Stein inspired estimator. Theoretical risk functions are worked out for all three estimators to evaluate under which circumstances it is advantageous to take the error covariance structure into account.     

Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

Nonlinear unbiased estimation in the linear regression model with nonnormal disturbances

In the application of the linear regression model there continues to be wide-spread use of the Least Squares Estimator (LSE) due to its theoretical optimality. For example, it is well known that the LSE is the best unbiased estimator under normality while it remains best linear unbiased estimator (BLUE) when the normality assumption is dropped. In this paper we extend an approach given in Knautz (1993) that allows improvement of the LSE in the context of nonnormal and nonsymmetric error distributions. It will be shown that there exist linear plus quadratic (LPQ) estimators, consisting of linear and quadratic terms in the dependent variable, which dominate the LS estimator, depending on second, third and fourth moments of the error distribution. A simulation study illustrates that this remains true if the moments have to be estimated from the data. Computation of confidence intervals using bootstrap methods reveal significant improvement compared with inference based on the LS especially for nonsymmetric distributions of the error term.     

A modified ridge m-estimator for linear regression model with multicollinearity and outliers

The ordinary least-square estimators for linear regression analysis with multicollinearity and outliers lead to unfavorable results. In this article, we propose a new robust modified ridge M-estimator (MRME) based on M-estimator (ME) to deal with the combined problem resulting from multicollinearity and outliers in the y-direction. MRME outperforms modified ridge estimator, robust ridge estimator and ME, according to mean squares error criterion. Furthermore, a numerical example and a Monte Carlo simulation experiment are given to illustrate some of the theoretical results.     

Robust parameter estimation of regression model with AR(p) error terms

In this article, we consider a linear regression model with AR(p) error terms with the assumption that the error terms have a t distribution as a heavy-tailed alternative to the normal distribution. We obtain the estimators for the model parameters by using the conditional maximum likelihood (CML) method. We conduct an iteratively reweighting algorithm (IRA) to find the estimates for the parameters of interest. We provide a simulation study and three real data examples to illustrate the performance of the proposed robust estimators based on t distribution.     

Empirical likelihood based diagnostics for heteroscedasticity in partially linear errors-in-variables models

A standard assumption in regression analysis is homogeneity of the error variance. Violation of this assumption can have adverse consequences for the efficiency of estimators. In this paper, we propose an empirical likelihood based diagnostic technique for heteroscedasticity in the partially linear errors-in-variables models. Under mild conditions, a nonparametric version of Wilk's theorem is derived. Simulation results reveal that our test performs well in both size and power.