Correcting double outward box distributed residuals by WCEV

Double outward box distributed residuals are another type of non monotonic heteroscedasticity that severely violates homoscedasticity assumption. In this study Çelik's (2015 Çelik, R. (2015). Stabilizing heteroscedasticity for butterfly-distributed residuals by the weighting absolute centered external variable. J. Appl. Stat. 42(4):705–721.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) WCEV is applied to double outward box distributed residuals to provide homoscedasticity for simple and multiple regression models.     

A new test to detect monotonic and non-monotonic types of heteroscedasticity

A direct parametric test is proposed to detect monotonic and non-monotonic types of heteroscedasticity. After giving brief information about non-monotonic types of heteroscedasticity, the test algorithm is introduced. Proposed test and usual heteroscedasticity tests are compared on monotonic and non-monotonic types of heteroscedasticity in real and artificial data.     

RCEV heteroscedasticity test based on the studentized residuals

All the usual heteroscedasticity tests in the statistics and econometrics literature are based on raw residuals. Although the raw residuals are heteroscedastic, studentized residuals can still be homoscedastic. In this study, the version of Çelik’s RCEV heteroscedasticity test which is based on studentized residuals is introduced.     

Empirical distribution function under heteroscedasticity

Neglecting heteroscedasticity of error terms may imply the wrong identification of a regression model (see appendix). Employment of (heteroscedasticity resistent) White's estimator of covariance matrix of estimates of regression coefficients may lead to the correct decision about the significance of individual explanatory variables under heteroscedasticity. However, White's estimator of covariance matrix was established for least squares (LS)-regression analysis (in the case when error terms are normally distributed, LS- and maximum likelihood (ML)-analysis coincide and hence then White's estimate of covariance matrix is available for ML-regression analysis, tool). To establish White's-type estimate for another estimator of regression coefficients requires Bahadur representation of the estimator in question, under heteroscedasticity of error terms. The derivation of Bahadur representation for other (robust) estimators requires some tools. As the key too proved to be a tight approximation of the empirical distribution function (d.f.) of residuals by the theoretical d.f. of the error terms of the regression model. We need the approximation to be uniform in the argument of d.f. as well as in regression coefficients. The present paper offers this approximation for the situation when the error terms are heteroscedastic.     

Heteroscedasticity diagnostics in varying-coefficient partially linear regression models and applications in analyzing Boston housing data

It is important to detect the variance heterogeneity in regression model because efficient inference requires that heteroscedasticity is taken into consideration if it really exists. For the varying-coefficient partially linear regression models, however, the problem of detecting heteroscedasticity has received very little attention. In this paper, we present two classes of tests of heteroscedasticity for varying-coefficient partially linear regression models. The first test statistic is constructed based on the residuals, in which the error term is from a normal distribution. The second one is motivated by the idea that testing heteroscedasticity is equivalent to testing pseudo-residuals for a constant mean. Asymptotic normality is established with different rates corresponding to the null hypothesis of homoscedasticity and the alternative. Some Monte Carlo simulations are conducted to investigate the finite sample performance of the proposed tests. The test methodologies are illustrated with a real data set example.     

Alternative tests for heteroscedasticity of disturbances:a comparative study

This paper presents three small sample tests for testing the heteroscedasticity among regression disturbances. The power of these tests are compared with two of the leading tests for this hypothesis, one by Goldfeld and Quandt [5] and the other by Theil [17]. We also provide a heuristic method of selecting the number of middle observations to be deleted for the Goldfeld-Quandt type of tests.     

A note on glejser's test for heteroscedasticity

Glejser published a test on the residuals of a regression model where the parameters are estimated by OLS that purports to detect “mixed” heteroscedasticity. This note addresses the problem of detecting this type heteroscedasticity from,both a theoretical and pragmatic point of view. We conclude that “mixed” heteroscedasticity cannot be separated from non zero expected errors and thus cannot be detected using Glejser s technique.     

Comparing the mean vectors of two independent multivariate log-normal distributions

The multivariate log-normal distribution is a good candidate to describe data that are not only positive and skewed, but also contain many characteristic values. In this study, we apply the generalized variable method to compare the mean vectors of two independent multivariate log-normal populations that display heteroscedasticity. Two generalized pivotal quantities are derived for constructing the generalized confidence region and for testing the difference between two mean vectors. Simulation results indicate that the proposed procedures exhibit satisfactory performance regardless of the sample sizes and heteroscedasticity. The type I error rates obtained are consistent with expectations and the coverage probabilities are close to the nominal level when compared with the other method which is currently available. These features make the proposed method a worthy alternative for inferential analysis of problems involving multivariate log-normal means. The results are illustrated using three examples.     

An outlier-resistant test for heteroscedasticity in linear models

The presence of contamination often called outlier is a very common attribute in data. Among other causes, outliers in a homoscedastic model make the model heteroscedastic. Moreover, outliers distort diagnostic tools for heteroscedasticity such that it may not be correctly identified. In this article, we show how outliers affect heteroscedasticity diagnostics. We then proposed a robust procedure for detecting heteroscedasticity in the presence of outliers by robustifying the non-robust component of the Goldfeld–Quandt (GQ) test. The performance of the proposed procedure is examined using simulation experiment and real data sets. The proposed procedure offers great improvement where the conventional GQ and other procedures fail.     

Modeling healthcare costs in simultaneous presence of asymmetry,heteroscedasticity and correlation

Highly skewed outcome distributions observed across clusters are common in medical research. The aim of this paper is to understand how regression models widely used for accommodating asymmetry fit clustered data under heteroscedasticity. In a simulation study, we provide evidence on the performance of the Gamma Generalized Linear Mixed Model (GLMM) and log-Linear Mixed-Effect (LME) model under a variety of data-generating mechanisms. Two case studies from health expenditures literature, the cost of strategies after myocardial infarction randomized clinical trial on the cost of strategies after myocardial infarction and the European Pressure Ulcer Advisory Panel hospital prevalence survey of pressure ulcers, are analyzed and discussed. According to simulation results, the log-LME model for a Gamma response can lead to estimations that are biased by as much as 10% of the true value, depending on the error variance. In the Gamma GLMM, the bias never exceeds 1%, regardless of the extent of heteroscedasticity, and the confidence intervals perform as nominally stated under most conditions. The Gamma GLMM with a log link seems to be more robust to both Gamma and log-normal generating mechanisms than the log-LME model.     

A consistent test for heteroscedasticity in semi-parametric regression with nonparametric variance function based on the kernel method

It is important to detect the variance heterogeneity in regression models. Heteroscedasticity tests have been well studied in parametric and nonparametric regression models. This paper presents a consistent test for heteroscedasticity for nonlinear semi-parametric regression models with nonparametric variance function based on the kernel method. The properties of the test are investigated through Monte Carlo simulations. The test methods are illustrated with a real example.     

Optimal Sequential Design in a Controlled Non-parametric Regression

Abstract. In a non‐parametric regression, the heteroscedasticity (dependence of the variance of the regression error on the predictor) can be a serious complication in estimation or visualization of an underlying regression function. If a controlled sampling is permitted, then the statistician can choose the design of predictors which attenuates the effect of heteroscedasticity. It is proposed to use a design which minimizes the mean integrated squared error of the regression function estimation. Then the corresponding optimal design density is proportional to the standard deviation of the regression error (the so‐called scale function). Because in general the statistician does not know an underlying scale function, the natural question is as follows: is it possible to suggest a sequential design which performs as well as an oracle that knows the underlying scale function? The answer is ‘yes’, and a corresponding sequential procedure is developed. It is proved, for the first time in the literature, that a data‐driven sequential design, together with an adaptive regression estimator, can mimic the oracle and be sharp minimax. Further, it is shown that the suggested method is feasible for small samples.     

AMEMIYA'S FORM OF THE WEIGHTED LEAST SQUARES ESTIMATOR

Amemiya's estimator is a weighted least squares estimator of the regression coefficients in a linear model with heteroscedastic errors. It is attractive because the heteroscedasticity is not parametrized and the weights (which depend on the error covariance matrix) are estimated nonparametrically. This paper derives an asymptotic expansion for Amemiya's form of the weighted least squares estimator, and uses it to discuss the effects of estimating the weights, of the number of iterations, and of the choice of the initial estimate. The paper also discusses the special case of normally distributed errors and clarifies the particular consequences of assuming normality.     

Re-weighting estimation of the coefficients in the varying coefficient model with heteroscedastic errors

ABSTRACT

This article explores the estimation problem of the coefficients in the varying coefficient model with heteroscedastic errors. Specifically, we first present a method for estimating the variance function of the error term and the resulting estimator is proved to be consistent. Then, motivated by the generalized least-squares procedure for dealing with heteroscedasticity in the linear regression literature, we re-weight each squared residual term in the local linear smoother with the inverse of the corresponding estimated error variance to construct estimates of the coefficients. Simulation experiments and practical data analysis conducted demonstrate that the re-weighting approach can improve the accuracy of the coefficient estimates under a finite sample size, especially when the error heteroscedasticity is strong.     

A nonparametric hypothesis test for heteroscedasticity

In this paper, a hypothesis test for heteroscedasticity is proposed in a nonparametric regression model. The test statistic, which uses the residuals from a nonparametric fit of the mean function, is based on an adaptation of the well-known Levene's test. Using the recent theory for analysis of variance when the number of factor levels goes to infinity, the asymptotic distribution of the test statistic is established under the null hypothesis of homocedasticity and under local alternatives. Simulations suggest that the proposed test performs well in several situations, especially when the variance is a nonlinear function of the predictor.     

Directional dependence via Gaussian copula beta regression model with asymmetric GARCH marginals

This article proposes a new directional dependence by using the Gaussian copula beta regression model. In particular, we consider an asymmetric Generalized AutoRegressive Conditional Heteroscedasticity (GARCH) model for the marginal distribution of standardized residuals to make data exhibiting conditionally heteroscedasticity to white noise process. With the simulated data generated by an asymmetric bivariate copula, we verify our proposed directional dependence method. For the multivariate direction dependence by using the Gaussian copula beta regression model, we employ a three-dimensional archemedian copula to generate trivariate data and then show the directional dependence for one random variable given two other random variables. With West Texas Intermediate Daily Price (WTI) and the Standard & Poor’s 500 (S&P 500), our proposed directional dependence by the Gaussian copula beta regression model reveals that the directional dependence from WTI to S&P 500 is greater than that from S&P 500 to WTI. To validate our empirical result, the Granger causality test is conducted, confirming the same result produced by our method.     

A Locally Most Mean Powerful Based Score Test for ARCH and GARCH Regression Disturbances

This article considers the twin problems of testing for autoregressive conditional heteroscedasticity (ARCH) and generalized ARCH disturbances in the linear regression model. A feature of these testing problems, ignored by the standard Lagrange multiplier test, is that they are onesided in nature. A test that exploits this one-sided aspect is constructed based on the sum of the scores. The small-sample-size and power properties of two versions of this test under both normal and leptokurtic disturbances are investigated via a Monte Carlo experiment. The results indicate that both versions of the new test typically have superior power to two versions of the Lagrange multiplier test and possibly also more accurate asymptotic critical values.     

Validation of A Heteroscedastic Hazards Regression Model

A Cox-type regression model accommodating heteroscedasticity, with a power factor of the baseline cumulative hazard, is investigated for analyzing data with crossing hazards behavior. Since the approach of partial likelihood cannot eliminate the baseline hazard, an overidentified estimating equation (OEE) approach is introduced in the estimation procedure. Its by-product, a model checking statistic, is presented to test for the overall adequacy of the heteroscedastic model. Further, under the heteroscedastic model setting, we propose two statistics to test the proportional hazards assumption. Implementation of this model is illustrated in a data analysis of a cancer clinical trial.     

Heteroscedasticity diagnostics in two-phase linear regression models

In two-phase linear regression models, it is a standard assumption that the random errors of two phases have constant variances. However, this assumption is not necessarily appropriate. This paper is devoted to the tests for variance heterogeneity in these models. We initially discuss the simultaneous test for variance heterogeneity of two phases. When the simultaneous test shows that significant heteroscedasticity occurs in the whole model, we construct two individual tests to investigate whether or not both phases or one of them have/has significant heteroscedasticity. Several score statistics and their adjustments based on Cox and Reid [D. R. Cox and N. Reid, Parameter orthogonality and approximate conditional inference. J. Roy. Statist. Soc. Ser. B 49 (1987), pp. 1–39] are obtained and illustrated with Australian onion data. The simulated powers of test statistics are investigated through Monte Carlo methods.     

Bounds on the effect of heteroscedasticity on the chow test for structural change

This paper considers the effect of heteroscedastic regression errors on the size of the Chow test for structural stability. We show that bounds can be placed on the true size of this test in the light of such misspecification, and on the true critical value needed to achieve any desired significance level when using the test under various degrees of heteroscedasticity. These bounds are data-independent, and some cases are tabulated. Examples are given to illustrate the practical application of the critical value bounds.