A Joint Score Test for Heteroscedasticity in the Two Way Error Components Model

This article extends the work by Holly and Gardiol (2000) (A score test for individual heteroscedasticity in a one-way error component model. In: Krishnakumar, J., Ronchetti, E., Eds. Panel Data Econometrics: Future Directions. Elsevier, North-Holland, Amsterdam, pp. 199–211, Ch. 10) to the two-way error components model. It deals exclusively with a joint heteroscedasticity test by first deriving Rao's efficient score statistics. Then, based on appropriate set of assumptions, we deduce the asymptotic distribution of the score under contiguous alternatives. Finally, we provide the expression for the score test statistic in the presence of heteroscedasticity and discuss its asymptotic local power.     

Tests of heteroscedasticity and correlation in multivariate t regression models with AR and ARMA errors

Heteroscedasticity checking in regression analysis plays an important role in modelling. It is of great interest when random errors are correlated, including autocorrelated and partial autocorrelated errors. In this paper, we consider multivariate t linear regression models, and construct the score test for the case of AR(1) errors, and ARMA(s,d) errors. The asymptotic properties, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied. Based on modified profile likelihood, the adjusted score test is also developed. The finite sample performance of the tests is investigated through Monte Carlo simulations, and also the tests are illustrated with two real data sets.     

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.     

Improved Score Tests in Symmetric Linear Regression Models

The class of symmetric linear regression models has the normal linear regression model as a special case and includes several models that assume that the errors follow a symmetric distribution with longer-than-normal tails. An important member of this class is the t linear regression model, which is commonly used as an alternative to the usual normal regression model when the data contain extreme or outlying observations. In this article, we develop second-order asymptotic theory for score tests in this class of models. We obtain Bartlett-corrected score statistics for testing hypotheses on the regression and the dispersion parameters. The corrected statistics have chi-squared distributions with errors of order O(n ^?3/2), n being the sample size. The corrections represent an improvement over the corresponding original Rao's score statistics, which are chi-squared distributed up to errors of order O(n ^?1). Simulation results show that the corrected score tests perform much better than their uncorrected counterparts in samples of small or moderate size.     

Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most sotuations the robust test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and IM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the QMLE appears to be relatively small. An empirical application to stock return volatility illustrates the potential imprtance of computing robust statistics in practice.     

On the asymptotic normality of rank tests for independence

We show that the commonly used linear rank tests for independence have certain continuity properties with respect to the score functions which are applied to the ranks. Using these properties, we derive the asymptotic normality of the test statistics under general conditions. These conditions are closely related to those under which simple linear rank statistics are known to be asymptotically normal.     

Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most sotuations the robust test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and IM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the QMLE appears to be relatively small. An empirical application to stock return volatility illustrates the potential imprtance of computing robust statistics in practice.     

Testing for multivariate heteroscedasticity

In this article, we propose a testing technique for multivariate heteroscedasticity, which is expressed as a test of linear restrictions in a multivariate regression model. Four test statistics with known asymptotical null distributions are suggested, namely the Wald, Lagrange multiplier (LM), likelihood ratio (LR) and the multivariate Rao F-test. The critical values for the statistics are determined by their asymptotic null distributions, but bootstrapped critical values are also used. The size, power and robustness of the tests are examined in a Monte Carlo experiment. Our main finding is that all the tests limit their nominal sizes asymptotically, but some of them have superior small sample properties. These are the F, LM and bootstrapped versions of Wald and LR tests.     

Nonparametric two-sample tests for dispersion differences based on placements

Two different two-sample tests for dispersion differences based on placement statistics are proposed. The means and variances of the test statistics are derived, and asymptotic normality is established for both. Variants of the proposed tests based on reversing the X and Y labels in the test statistic calculations are shown to have different small-sample properties; for both pairs of tests, one member of the pair will be resolving, the other nonresolving. The proposed tests are similar in spirit to the dispersion tests of both Mood and Hollander; comparative simulation results for these four tests are given. For small sample sizes, the powers of the proposed tests are approximately equal to the powers of the tests of both Mood and Hollander for samples from the normal, Cauchy and exponential distributions. The one-sample limiting distributions are also provided, yielding useful approximations to the exact tests when one sample is much larger than the other. A bootstrap test may alternatively be performed. The proposed test statistics may be used with lightly censored data by substituting Kaplan-Meier estimates for the empirical distribution functions.     

JACKKNIFE TESTS FOR HETEROSCEDASTICITY IN THE GENERAL LINEAR MODEL

In this paper, tests based on the Jackknife technique are proposed to test for heteroscedasticity in the linear regression model when the errors are non-normal. These are the Jackknifed Goldfeld-Quandt (GQ), and jack-knife related variations of White (H), Lagrange multiplier (LM), Glejser (GL) and Bickel (B) tests. The power of the proposed tests is compared with that of GQ, H, LM, GL and B tests; and the robustness to the error distribution is analyzed under several heteroscedastic assumptions. The GQ test is by far the best test if the error distribution is close to normal, however, GQ test is not robust against non-normal errors. By applying the jackknife technique to the regression a more robust statistic (GQJRG) is produced but the cost is a loss in power. The GQJRG statistic generally is not M powerful as the Bickel (BlOLS) and Glejser (GLlOLS) statistics.     

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.     

Testing for Common Features

This article introduces a class of statistical tests for the hypothesis that some feature that is present in each of several variables is common to them. Features are data properties such as serial correlation, trends, seasonality, heteroscedasticity, autoregressive conditional hetero-scedasticity, and excess kurtosis. A feature is detected by a hypothesis test taking no feature as the null, and a common feature is detected by a test that finds linear combinations of variables with no feature. Often, an exact asymptotic critical value can be obtained that is simply a test of overidentifying restrictions in an instrumental variable regression. This article tests for a common international business cycle.     

A non-parametric statistic for testing conditional heteroscedasticity for unobserved component models

When prediction intervals are constructed using unobserved component models (UCM), problems can arise due to the possible existence of components that may or may not be conditionally heteroscedastic. Accurate coverage depends on correctly identifying the source of the heteroscedasticity. Different proposals for testing heteroscedasticity have been applied to UCM; however, in most cases, these procedures are unable to identify the heteroscedastic component correctly. The main issue is that test statistics are affected by the presence of serial correlation, causing the distribution of the statistic under conditional homoscedasticity to remain unknown. We propose a nonparametric statistic for testing heteroscedasticity based on the well-known Wilcoxon''s rank statistic. We study the asymptotic validation of the statistic and examine bootstrap procedures for approximating its finite sample distribution. Simulation results show an improvement in the size of the homoscedasticity tests and a power that is clearly comparable with the best alternative in the literature. We also apply the test on real inflation data. Looking for the presence of a conditionally heteroscedastic effect on the error terms, we arrive at conclusions that almost all cases are different than those given by the alternative test statistics presented in the literature.     

Smoothed alternatives of the two-sample median and Wilcoxon's rank sum tests

We discuss smoothed rank statistics for testing the location shift parameter of the two-sample problem. They are based on discrete test statistics – the median and Wilcoxon's rank sum tests. For the one-sample problem, Maesono et al. [Smoothed nonparametric tests and their properties. arXiv preprint. 2016; ArXiv:1610.02145] reported that some nonparametric discrete tests have a problem with their p-values because of their discreteness. The p-values of Wilcoxon's test are frequently smaller than those of the median test in the tail area. This leads to an arbitrary choice of the median and Wilcoxon's rank sum tests. To overcome this problem, we propose smoothed versions of those tests. The smoothed tests inherit the good properties of the original tests and are asymptotically equivalent to them. We study the significance probabilities and local asymptotic powers of the proposed tests.     

An Adaptive Test for the Two-Sample Location Problem Based on U-Statistics

For the two-sample location problem with continuous data we consider a general class of tests, all members of it are based on U-statistics. The asymptotic efficacies are investigated in detail. We construct an adaptive test where all statistics involved are suitably chosen U-statistics. It is shown that the proposed adaptive test has good asymptotic and finite sample power properties.     

Asymptotic relative efficiency of score tests in Weibull models with measurement errors

The main object of this paper is to discuss properties of the score statistics for testing the null hypothesis of no association in Weibull model with measurement errors. Three different score statistics are considered. The efficient score statistics, a naive score statistics obtained by replacing the unobserved true covariate with the observed one and a score statistics based on the corrected score statistics. It is shown that corrected and naive score statistics are equivalent and the asymptotic relative efficiency between naive and efficient score statistics is derived.     

Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear Models

We propose two test statistics for testing serial correlation in semiparametric varying-coefficient partially linear models. The proposed test statistics are not only for testing zero first-order serial correlation, but also for testing higher-order serial correlations. Under the null hypothesis of no serial correlation, the test statistics are shown to have asymptotic normal or chi-square distributions. By using R, some Monte Carlo experiments are conducted to examine the finite sample performances of the proposed tests. Simulation results show that the estimated size and power of the proposed tests behave well.     

A Joint Test for Conditional Heteroscedasticity in Dynamic Panel Data Models

This article proposes a joint test for conditional heteroscedasticity in dynamic panel data models. The test is constructed by checking the joint significance of estimates of second to pth-order serial correlation in the squares sequence of the first differenced errors. To avoid any distribution assumptions of the errors and the effects, we adopt the GMM estimation for the parameter coefficient and higher order moment estimation for the errors. Based on the estimations, a joint test is constructed for conditional heteroscedasticity in the error. The resulted test is asymptotically chi-squared under the null hypothesis and easy to implement. The small sample properties of the test are investigated by means of Monte Carlo experiments. The evidence shows that the test performs well in dynamic panel data with large number n of individuals and short periods T of time. A real data is analyzed for illustration.     

Parametric bootstrap and approximate tests for two Poisson variates

The parametric bootstrap tests and the asymptotic or approximate tests for detecting difference of two Poisson means are compared. The test statistics used are the Wald statistics with and without log-transformation, the Cox F statistic and the likelihood ratio statistic. It is found that the type I error rate of an asymptotic/approximate test may deviate too much from the nominal significance level α under some situations. It is recommended that we should use the parametric bootstrap tests, under which the four test statistics are similarly powerful and their type I error rates are all close to α. We apply the tests to breast cancer data and injurious motor vehicle crash data.     

Testing heteroscedasticity in nonlinear and nonparametric regressions

This article presents new nonparametric tests for heteroscedasticity in nonlinear and nonparametric regression models. The tests have an asymptotic standard normal distribution under the null hypothesis of homoscedasticity and are robust against any form of heteroscedasticity. A Monte Carlo simulation with critical values obtained from the wild bootstrap procedure is provided to asses the finite sample performances of the tests. A real application of testing interest rate volatility functions illustrates the usefulness of the tests proposed. The Canadian Journal of Statistics © 2009 Statistical Society of Canada