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A Comparison of Tests for Heteroscedasticity in Linear Models

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A Comparison of Tests for Heteroscedasticity in Linear Models A comparison of tests for heteroscedasticity in linear models Antonella Plaia University of Palermo, Institute of Statistics, Faculty of Economics Viale delle Scienze 90128 Palermo, Italy E-mail: [email protected] Rosaria Russo University of Palermo, Institute of Statistics, Faculty of Economics Viale delle Scienze 90128 Palermo, Italy 1. Introduction A plenty of tests for heteroscedasticity, that can be applied in different situations, have been proposed since Neyman and Pearson ‘s one in 1931. It was Anscombe, in his paper published in 1961, the first to analyse the problem in the context of (multiple) linear regression models. In the present paper, starting from a comparison proposed by Lion and Tsai (1995), we consider some tests based on the likelihood function, to be applied to the residuals of ordinary least squares. The presence of nuisance parameters (the regression coefficients and the scale parameter), and the necessity to neutralise their effects on the inference on the parameters of the variance function, suggests the use of some modified likelihood functions: the profile likelihood, the modified profile l. (Cox and Reid, 1987), the adjusted profile l. (Cox and Reid, 1993), the conditional profile l. (Honda, 1989), the residual l. (Verbyla, 1993). All these functions, built basing on the principles of the conditional inference (Cox, 1988, Cox and Hinkley, 1974) with the aim to remove or to reduce the effects of the nuisance parameters, produced different likelihood ratio tests and score tests, all asymptotically χ2 distributed, with the proper degrees of freedom. Our aim is to carry out, by Monte-Carlo simulation, a comparative analysis that, beside the common evaluation of the power and the robustness of the different tests in the presence of non- normal errors, considers the peculiar aspects and problems when verifying heteroscedasticity in the regression models. Actually, the use of the likelihood function, in its original or modified version, needs for the specification of the variance function under the alternative hypothesis , and it is important to assess the sensitiveness of the different tests specification errors of the variance function. The above mentioned tests are generally applied to the sample ordinary least squares residuals that, due to the estimation process, are correlated and with non-constant variance, even for a model with independent and homoscedastic errors; therefore it is necessary to verify if all the tests are sufficiently and equally sensitive to these distributional aspects of residuals. Moreover, it is important: a) to verify, conditions being equal, the adequacy of the approximation to the asymptotic distribution for the different tests, especially on the distribution tails, even for moderate samples; b) then, estimating, for different sample sizes, the real significance level of each test, to find, by comparing it with the nominal one, which test, and in what measure, performs better in the presence of moderate or small samples, and, moreover, what sample size assures a good degree of approximation. In this context the so called small sample asymptotic techniques have been proposed in literature, among which the “Bartlett type corrections” (Bartlett 1937), and the “saddlepoint approximations” (Daniels, 1954). But their application to tests based on likelihood function is not so easy; our aim is to apply Bartlett type corrections to improve the test, or to consider saddlepoint approximations to get a better critical value, to those of the above mentioned tests that appear optimal from some theoretical point of view or to be preferred in real situations. REFERENCES Anscombe, S. J. (1961). Examination of residuals. Proc. 4th Berkeley Symp. On mathematical Statistics and Probability, 1-37. Barndoff-Nielsen O. E. (1994) Inference and Asymptotics. Chapman & Hall Cox. D.R. (1988). Some aspects of conditional and asymptotic inference: a review. Sunkya vol. 50. A. part III, 314-337. Cox, D.R., Hinkley, D.V. (1974). Theoretical Statistics. London: Chapman and Hall. Cox. D.R.; Reid. N. (1987) Parameter orthogonality and approximate conditional inference, J. R. Statist. Soc. B. 49, 1-39 Honda. Y. (1989) On the optimality of some tests of the error covariance matrix in the linear regression model. J. R. Statist. Soc. B, 51,. 71-79 Lyon. J. D.; Tsai. C. L. (1996) A comparison of tests for heteroscedasticity. The Statistician 45, 3, 337-349 Verbyla A. P. (1993) Modelling variance heterogeneity: residual maximum likelihood and diagnostics. J. R. Statist. Soc. B 55, 493-508. RESUME Dans cette communication on propose l’application des corrections genre Bartlett et des approximations “saddlepoint” à quelques tests pour le contrôle de l’homogénéité de la variance des erreurs dans modeles lineaires..
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