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.     

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.     

Diagnosing explainable heterogeneity of variance in random‐effects models

Data‐analytic tools for models other than the normal linear regression model are relatively rare. Here we develop plots and diagnostic statistics for nonconstant variance for the random‐effects model (REM). REMs for longitudinal data include both within‐ and between‐subject variances. A basic assumption is that the two variance terms are constant across subjects. However, we often find that these variances are functions of covariates, and the data set has what we call explainable heterogeneity, which needs to be allowed for in the model. We characterize several types of heterogeneity of variance in REMs and develop three diagnostic tests using the score statistic: one for each of the two variance terms, and the third for a form of multivariate nonconstant variance. For each test we present an adjusted residual plot which can identify cases that are unusually influential on the outcome of the test.     

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.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

Testing for heteroscedasticity of exponential correlation mixed-effects linear models based on M-estimation

Homogeneity of variance is a basic assumption in longitudinal data analysis. However, the assumption is not necessarily appropriate. In this paper, Fisher scoring method is applied to get M-estimator in the exponential correlation mixed-effects linear model. The score tests for heteroscedasticity and correlation coefficient based on M-estimator are then studied. Monte Carlo method is applied to investigate the properties of test statistics. At last, the methods and properties are illustrated by an actual data example.     

Improved likelihood inference for the shape parameter in Weibull regression

We obtain adjustments to the profile likelihood function in Weibull regression models with and without censoring. Specifically, we consider two different modified profile likelihoods: (i) the one proposed by Cox and Reid [Cox, D.R. and Reid, N., 1987, Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B, 49, 1–39.], and (ii) an approximation to the one proposed by Barndorff–Nielsen [Barndorff–Nielsen, O.E., 1983, On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.], the approximation having been obtained using the results by Fraser and Reid [Fraser, D.A.S. and Reid, N., 1995, Ancillaries and third-order significance. Utilitas Mathematica, 47, 33–53.] and by Fraser et al. [Fraser, D.A.S., Reid, N. and Wu, J., 1999, A simple formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 655–661.]. We focus on point estimation and likelihood ratio tests on the shape parameter in the class of Weibull regression models. We derive some distributional properties of the different maximum likelihood estimators and likelihood ratio tests. The numerical evidence presented in the paper favors the approximation to Barndorff–Nielsen's adjustment.     

Varying Dispersion Diagnostics for Inverse Gaussian Regression Models

Homogeneity of dispersion parameters is a standard assumption in inverse Gaussian regression analysis. However, this assumption is not necessarily appropriate. This paper is devoted to the test for varying dispersion in general inverse Gaussian linear regression models. Based on the modified profile likelihood (Cox & Reid, 1987), the adjusted score test for varying dispersion is developed and illustrated with Consumer- Product Sales data (Whitmore, 1986) and Gas vapour data (Weisberg, 1985). The effectiveness of orthogonality transformation and the properties of a score statistic and its adjustment are investigated through Monte Carlo simulations.     

The effects of missing serial effects and/or heteroscedastic errors on mixed models using repeated growth data

When a two-level multilevel model (MLM) is used for repeated growth data, the individuals constitute level 2 and the successive measurements constitute level 1, which is nested within the individuals that make up level 2. The heterogeneity among individuals is represented by either the random-intercept or random-coefficient (slope) model. The variance components at level 1 involve serial effects and measurement errors under constant variance or heteroscedasticity. This study hypothesizes that missing serial effects or/and heteroscedasticity may bias the results obtained from two-level models. To illustrate this effect, we conducted two simulation studies, where the simulated data were based on the characteristics of an empirical mouse tumour data set. The results suggest that for repeated growth data with constant variance (measurement error) and misspecified serial effects (ρ > 0.3), the proportion of level-2 variation (intra-class correlation coefficient) increases with ρ and the two-level random-coefficient model is the minimum AIC (or AIC_c) model when compared with the fixed model, heteroscedasticity model, and random-intercept model. In addition, the serial effect (ρ > 0.1) and heteroscedasticity are both misspecified, implying that the two-level random-coefficient model is the minimum AIC (or AIC_c) model when compared with the fixed model and random-intercept model. This study demonstrates that missing serial effects and/or heteroscedasticity may indicate heterogeneity among individuals in repeated growth data (mixed or two-level MLM). This issue is critical in biomedical research.     

Impact study of volatility modelling of Bangladesh stock index using non-normal density

This article examines a wide variety of popular volatility models for stock index return, including the random walk (RW), autoregressive, generalized autoregressive conditional heteroscedasticity (GARCH), and asymmetric GARCH models with normal and non-normal (Student's t and generalized error) distributional assumption. Fitting these models to the Chittagong stock index return data from the period 2 January 1999 to 29 December 2005, we found that the asymmetric GARCH/GARCH model fits better under the assumption of non-normal distribution than under normal distribution. Non-parametric specification tests show that the RW-GARCH, RW-TGARCH, RW-EGARCH, and RW-APARCH models under the Student's t-distributional assumption are significant at the 5% level. Finally, the study suggests that these four models are suitable for the Chittagong Stock Exchange of Bangladesh. We believe that this study would be of great benefit to investors and policy makers at home and abroad.     

Heteroscedasticity and/or autocorrelation diagnostics in nonlinear models with AR(1) and symmetrical errors

In this paper, we discuss tests of heteroscedasticity and/or autocorrelation in nonlinear models with AR(1) and symmetrical errors. The symmetrical errors distribution class includes all symmetrical continuous distributions, such as normal, Student-t, power exponential, logistic I and II, contaminated normal, so on. First, score test statistics and their adjustment forms of heteroscedasticity are derived. Then, the asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied. The properties of test statistics are investigated through Monte Carlo simulations. Finally, a real data set is used to illustrate our test methods.     

Tests for homogeneity of variance

Six procedures which convert tests of homogeneity of variance into tests for mean equality for independent groups are compared. The tests are the analysis of variance (ANOVA) and Welch F statistics. The Welch statistics are included since it was anticipated that ANOVA would not provide a robust test when samples of unequal sizes are obtained from non-normal populations. However, the Welch tests are not found to be uniformly preferrable. In addition, a prior recommendation for Miller's jackknife procedure is not supported for the unequal sample size case. The data indicates that the current tests for variance heterogeneity are either sensitive to non-normality or, if robust, lacking in power. Therefore, these tests cannot be recommended for the purpose of testing the validity of the ANOVA homogeneity assumption.     

A non-parametric maximum test for the Behrens–Fisher problem

Non-normality and heteroscedasticity are common in applications. For the comparison of two samples in the non-parametric Behrens–Fisher problem, different tests have been proposed, but no single test can be recommended for all situations. Here, we propose combining two tests, the Welch t test based on ranks and the Brunner–Munzel test, within a maximum test. Simulation studies indicate that this maximum test, performed as a permutation test, controls the type I error rate and stabilizes the power. That is, it has good power characteristics for a variety of distributions, and also for unbalanced sample sizes. Compared to the single tests, the maximum test shows acceptable type I error control.     

Comparison of ANOVA-F and ANOM tests with regard to type I error rate and test power

A Monte Carlo simulation was conducted to compare the type I error rate and test power of the analysis of means (ANOM) test to the one-way analysis of variance F-test (ANOVA-F). Simulation results showed that as long as the homogeneity of the variance assumption was satisfied, regardless of the shape of the distribution, number of group and the combination of observations, both ANOVA-F and ANOM test have displayed similar type I error rates. However, both tests have been negatively affected from the heterogeneity of the variances. This case became more obvious when the variance ratios increased. The test power values of both tests changed with respect to the effect size (Δ), variance ratio and sample size combinations. As long as the variances are homogeneous, ANOVA-F and ANOM test have similar powers except unbalanced cases. Under unbalanced conditions, the ANOVA-F was observed to be powerful than the ANOM-test. On the other hand, an increase in total number of observations caused the power values of ANOVA-F and ANOM test approach to each other. The relations between effect size (Δ) and the variance ratios affected the test power, especially when the sample sizes are not equal. As ANOVA-F has become to be superior in some of the experimental conditions being considered, ANOM is superior in the others. However, generally, when the populations with large mean have larger variances as well, ANOM test has been seen to be superior. On the other hand, when the populations with large mean have small variances, generally, ANOVA-F has observed to be superior. The situation became clearer when the number of the groups is 4 or 5.     

Testing Heterogeneity for Frailty Distribution in Shared Frailty Model

Abstract

The frailties, representing extra variations due to unobserved measurements, are often assumed to be iid in shared frailty models. In medical applications, however, a speculation can arise that a data set might violate the iid assumption. In this paper we investigate this conjecture through an analysis of the kidney infection data in McGilchrist and Aisbett (McGilchrist, C. A., Aisbett, C. W. (1991). Regression with frailty in survival analysis. Biometrics 47:461–466). As a test procedure, we consider the cusum of squares test which is frequently used for monitoring a variance change in statistical models. Our result strongly sustains the heterogeneity of the frailty distribution.     

Heteroscedasticity in non-stationary time series,some Monte Carlo evidence

A frequent question raised by practitioners doing unit root tests is whether these tests are sensitive to the presence of heteroscedasticity. Theoretically this is not the case for a wide range of heteroscedastic models. However, for some limiting cases such as degenerate and integrated heteroscedastic processes it is not obvious whether this will have an effect. In this paper we report a Monte Carlo study analyzing the implications of various types of heteroscedasticity on three types of unit root tests: The usual Dickey-Fuller test, Phillips' (1987) semi-parametric test and finally a Dickey-Fuller type test using White's (1980) heteroscedasticity consistent standard errors. The sorts of heteroscedasticity we examine are the GARCH model of Bollerslev (1986) and the Exponential ARCH model of Nelson (1991). In particular, we call attention to situations where the conditional variances exhibit a high degree of persistence as is frequently observed for returns of financial time series, and the case where, in fact, the variance process for the first class of models becomes degenerate.     

An approximate degrees of freedom test for heteroscedastic two-way ANOVA

Heteroscedastic two-way ANOVA are frequently encountered in real data analysis. In the literature, classical F-tests are often blindly employed although they are often biased even for moderate heteroscedasticity. To overcome this problem, several approximate tests have been proposed in the literature. These tests, however, are either too complicated to implement or do not work well in terms of size controlling. In this paper, we propose a simple and accurate approximate degrees of freedom (ADF) test. The ADF test is shown to be invariant under affine-transformations, different choices of contrast matrix for the same null hypothesis, or different labeling schemes of cell means. Moreover, it can be conducted easily using the usual F-distribution with one unknown degree of freedom estimated from the data. Simulations demonstrate that the ADF test works well in various cell sizes and parameter configurations but the classical F-tests work badly when the cell variance homogeneity assumption is violated. A real data example illustrates the methodologies.     

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.     

The performance of tests when observations have different variances

It is common to test if there is an effect due to a treatment. The commonly used tests have the assumption that the observations differ in location, and that their variances are the same over the groups. Different variances can arise if the observations being analyzed are means of different numbers of observations on individuals or slopes of growth curves with missing data. This study is concerned with cases in which the unequal variances are known, or known to a constant of proportionality. It examines the performance of the ttest, the Mann–Whitney–Wilcoxon Rank Sum test, the Median test, and the Van der Waerden test under these conditions. The t-test based on the weighted means is the likelihood ratio test under normality and has the usual optimality properties. The other tests are compared to it. One may align and scale the observations by subtracting the mean and dividing by the standard deviation of each point. This leads to other, analogous test statistics based on these adjusted observations. These statistics are also compared. Finally, the regression scores tests are compared to the other procedures.     

Testing the Goodness of Fit of Parametric Regression Models with Random Toeplitz Forms

ABSTRACT: We introduce a class of Toeplitz‐band matrices for simple goodness of fit tests for parametric regression models. For a given length r of the band matrix the asymptotic optimal solution is derived. Asymptotic normality of the corresponding test statistic is established under a fixed and random design assumption as well as for linear and non‐linear models, respectively. This allows testing at any parametric assumption as well as the computation of confidence intervals for a quadratic measure of discrepancy between the parametric model and the true signal g;. Furthermore, the connection between testing the parametric goodness of fit and estimating the error variance is highlighted. As a by‐product we obtain a much simpler proof of a result of 34 ) concerning the optimality of an estimator for the variance. Our results unify and generalize recent results by 9 ) and 15 , 16 ) in several directions. Extensions to multivariate predictors and unbounded signals are discussed. A simulation study shows that a simple jacknife correction of the proposed test statistics leads to reasonable finite sample approximations.