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Simulation-Based Finite-Sample Tests for Heteroskedasticity and ARCH Effects

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  • Jean-Thomas Bernard
  • Jean-Marie Dufour

    ()

  • Ian Genest
  • Lynda Khalaf

Abstract

A wide range of tests for heteroskedasticity have been proposed in the econometric and statistics literatures. Although a few exact homoskedasticity tests are available, the commonly employed procedures are quite generally based on asymptotic approximations which may not provide good size control in finite samples. There has been a number of recent studies that seek to improve the reliability of common heteroskedasticity tests using Edgeworth, Bartlett, jackknife and bootstrap methods. Yet the latter remain approximate. In this paper, we describe a solution to the problem of controlling the size of homoskedasticity tests in linear regression contexts. We study procedures based on the standard test statistics [e.g., the Goldfeld-Quandt, Glejser, Bartlett, Cochran, Hartley, Breusch-Pagan-Godfrey, White and Szroeter criteria] as well as tests for autoregressive conditional heteroskedasticity (ARCH-type models). We also suggest several extensions of the existing procedures (sup-type or combined test statistics) to allow for unknown breakpoints in the error variance. We exploit the technique of Monte Carlo tests to obtain provably exact p-values, for both the standard and the new tests suggested. We show that the MC test procedure conveniently solves the intractable null distribution problem, in particular those raised by the sup-type and combined test statistics as well as (when relevant) unidentified nuisance parameter problems under the null hypothesis. The method proposed works in exactly the same way with both Gaussian and non-Gaussian disturbance distributions [such as heavy-tailed or stable distributions]. The performance of the procedures is examined by simulation. The Monte Carlo experiments conducted focus on: (1) ARCH, GARCH and ARCH-in-mean alternatives; (2) the case where the variance increases monotonically with: (i) one exogenous variable, and (ii) the mean of the dependent variable; (3) grouped heteroskedasticity; (4) breaks in variance at unknown points. We find that the proposed tests achieve perfect size control and have good power. Un grand éventail de tests d'hétéroskédasticité a été proposé en économétrie et en statistique. Bien qu'il existe quelques tests d'homoskédasticité exacts, les procédures couramment utilisées sont généralement fondées sur des approximations asymptotiques qui ne procurent pas un bon contrôle du niveau dans les échantillons finis. Plusieurs études récentes ont tenté d'améliorer la fiabilité des tests d'hétéroskédasticité usuels, sur base de méthodes de type Edgeworth, Bartlett, jackknife et bootstrap. Cependant, ces méthodes demeurent approximatives. Dans cet article, nous décrivons une solution au problème de contrôle du niveau des tests d'homoskédasticité dans les modèles de régression linéaire. Nous étudions des procédures basées sur les critères de test standards [e.g., les critères de Goldfeld-Quandt, Glejser, Bartlett, Cochran, Hartley, Breusch-Pagan-Godfrey, White et Szroeter], de même que des tests pour l'hétéroskédasticité autorégressive conditionnelle (les modèles de type ARCH). Nous suggérons plusieurs extensions des procédures usuelles (les statistiques de type-sup ou combinées) pour tenir compte de points de ruptures inconnus dans la variance des erreurs. Nous appliquons la technique des tests de Monte Carlo (MC) de façon à obtenir des seuils de signification marginaux (les valeurs-p) exacts, pour les test usuels et les nouveaux tests que nous proposons. Nous démontrons que la procédure de MC permet de résoudre les problèmes des distributions compliquées sous l'hypothèse nulle, en particulier ceux associés aux statistiques de type-sup, aux statistiques combinées et aux paramètres de nuisance non-identifiés sous l'hypothèse nulle. La méthode proposée fonctionne exactement de la même manière en présence de lois Gaussiennes et non-Gaussiennes [comme par exemple les lois aux queues épaisses ou les lois stables]. Nous évaluons la performance des procédures proposées par simulation. Les expériences de Monte Carlo que nous effectuons portent sur: (1) les alternatives de type ARCH, GARCH and ARCH-en-moyenne; (2) le cas où la variance augmente de manière monotone en fonction: (i) d'une variable exogène, et (ii) de la moyenne de la variable dépendante; (3) l'hétéroskédasticité groupée; (4) les ruptures en variance à des points inconnus. Nos résultats montrent que les tests proposés permettent de contrôler parfaitement le niveau et ont une bonne puissance.

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Paper provided by CIRANO in its series CIRANO Working Papers with number 2001s-25.

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Date of creation: 01 Apr 2001
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Handle: RePEc:cir:cirwor:2001s-25

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Keywords: Heteroskedasticity; homoskedasticity; linear regression; Monte Carlo test; exact test; finite-sample test; specification test; ARCH; GARCH; ARCH in mean; stable distribution; structural stability; hétéroskédasticité; homoskédasticité; régression linéaire; test de Monte Carlo; test exact; test valide en échantillon fini; test de spécification; ARCH; GARCH; ARCH-en-moyenne; distribution stable; stabilité structurelle;

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