IDEAS home Printed from https://ideas.repec.org/p/cir/cirwor/2001s-25.html
   My bibliography  Save this paper

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

Author

Listed:
  • 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.

Suggested Citation

  • Jean-Thomas Bernard & Jean-Marie Dufour & Ian Genest & Lynda Khalaf, 2001. "Simulation-Based Finite-Sample Tests for Heteroskedasticity and ARCH Effects," CIRANO Working Papers 2001s-25, CIRANO.
  • Handle: RePEc:cir:cirwor:2001s-25
    as

    Download full text from publisher

    File URL: http://www.cirano.qc.ca/files/publications/2001s-25.pdf
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    as
    1. Dufour, Jean-Marie & Kiviet, Jan F., 1996. "Exact tests for structural change in first-order dynamic models," Journal of Econometrics, Elsevier, vol. 70(1), pages 39-68, January.
    2. Kiviet, Jan F. & Dufour, Jean-Marie, 1997. "Exact tests in single equation autoregressive distributed lag models," Journal of Econometrics, Elsevier, vol. 80(2), pages 325-353, October.
    3. Godfrey, Leslie G., 1996. "Some results on the Glejser and Koenker tests for heteroskedasticity," Journal of Econometrics, Elsevier, vol. 72(1-2), pages 275-299.
    4. Dufour, Jean-Marie & Khalaf, Lynda, 2002. "Exact tests for contemporaneous correlation of disturbances in seemingly unrelated regressions," Journal of Econometrics, Elsevier, vol. 106(1), pages 143-170, January.
    5. Harvey, A C, 1976. "Estimating Regression Models with Multiplicative Heteroscedasticity," Econometrica, Econometric Society, vol. 44(3), pages 461-465, May.
    6. Dufour, Jean-Marie & Khalaf, Lynda & Bernard, Jean-Thomas & Genest, Ian, 2004. "Simulation-based finite-sample tests for heteroskedasticity and ARCH effects," Journal of Econometrics, Elsevier, vol. 122(2), pages 317-347, October.
    7. Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, vol. 64(2), pages 413-430, March.
    8. Dufour, Jean-Marie, 1989. "Nonlinear Hypotheses, Inequality Restrictions, and Non-nested Hypotheses: Exact Simultaneous Tests in Linear Regressions," Econometrica, Econometric Society, vol. 57(2), pages 335-355, March.
    9. F. Cribari-Neto & S. G. Zarkos, 1999. "Bootstrap methods for heteroskedastic regression models: evidence on estimation and testing," Econometric Reviews, Taylor & Francis Journals, vol. 18(2), pages 211-228.
    10. Harrison, M J, 1982. "Tables of Critical Values for a Beta Approximation to Szroeter's Statistic for Testing for Heteroscedasticity," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 44(2), pages 159-167, May.
    11. Ali, Mukhtar M. & Giaccotto, Carmelo, 1984. "A study of several new and existing tests for heteroscedasticity in the general linear model," Journal of Econometrics, Elsevier, vol. 26(3), pages 355-373, December.
    12. Evans, Merran, 1992. "Robustness of size of tests of autocorrelation and heteroscedasticity to nonnormality," Journal of Econometrics, Elsevier, vol. 51(1-2), pages 7-24.
    13. Lee, John H. H., 1991. "A Lagrange multiplier test for GARCH models," Economics Letters, Elsevier, vol. 37(3), pages 265-271, November.
    14. Dagenais, Marcel G & Dufour, Jean-Marie, 1991. "Invariance, Nonlinear Models, and Asymptotic Tests," Econometrica, Econometric Society, vol. 59(6), pages 1601-1615, November.
    15. Godfrey, Leslie G., 1978. "Testing for multiplicative heteroskedasticity," Journal of Econometrics, Elsevier, vol. 8(2), pages 227-236, October.
    16. Andrews, Donald W K, 2001. "Testing When a Parameter Is on the Boundary of the Maintained Hypothesis," Econometrica, Econometric Society, vol. 69(3), pages 683-734, May.
    17. Hansen, Bruce E., 2000. "Testing for structural change in conditional models," Journal of Econometrics, Elsevier, vol. 97(1), pages 93-115, July.
    18. Christiano, Lawrence J, 1992. "Searching for a Break in GNP," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(3), pages 237-250, July.
    19. Griffiths, W. E. & Surekha, K., 1986. "A Monte Carlo evaluation of the power of some tests for heteroscedasticity," Journal of Econometrics, Elsevier, vol. 31(2), pages 219-231, March.
    20. Honda, Yuzo, 1988. "A size correction to the Lagrange multiplier test for heteroskedasticity," Journal of Econometrics, Elsevier, vol. 38(3), pages 375-386, July.
    21. Andrews, Donald W K, 1993. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Econometrica, Econometric Society, vol. 61(4), pages 821-856, July.
    22. Dufour, Jean-Marie, 1990. "Exact Tests and Confidence Sets in Linear Regressions with Autocorrelated Errors," Econometrica, Econometric Society, vol. 58(2), pages 475-494, March.
    23. White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-838, May.
    24. Dufour, Jean-Marie & Torres, Olivier, 2000. "Markovian processes, two-sided autoregressions and finite-sample inference for stationary and nonstationary autoregressive processes," Journal of Econometrics, Elsevier, vol. 99(2), pages 255-289, December.
    25. Andrews, Donald W K & Ploberger, Werner, 1994. "Optimal Tests When a Nuisance Parameter Is Present Only under the Alternative," Econometrica, Econometric Society, vol. 62(6), pages 1383-1414, November.
    26. Jean-Marie Dufour & Jan F. Kiviet, 1998. "Exact Inference Methods for First-Order Autoregressive Distributed Lag Models," Econometrica, Econometric Society, vol. 66(1), pages 79-104, January.
    27. Dufour, Jean-Marie & Khalaf, Lynda, 2002. "Simulation based finite and large sample tests in multivariate regressions," Journal of Econometrics, Elsevier, vol. 111(2), pages 303-322, December.
    28. Cribari-Netoa, Francisco & Ferrari, Silvia L. P., 1995. "Bartlett-corrected tests for heteroskedastic linear models," Economics Letters, Elsevier, vol. 48(2), pages 113-118, May.
    29. Lee, John H H & King, Maxwell L, 1993. "A Locally Most Mean Powerful Based Score Test for ARCH and GARCH Regression Disturbances," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(1), pages 17-27, January.
    30. Sharma, Subhash C. & Giaccotto, Carmelo, 1991. "Power and robustness of jackknife and likelihood-ratio tests for grouped heteroscedasticity," Journal of Econometrics, Elsevier, vol. 49(3), pages 343-372, September.
    31. Koenker, Roger & Bassett, Gilbert, Jr, 1982. "Robust Tests for Heteroscedasticity Based on Regression Quantiles," Econometrica, Econometric Society, vol. 50(1), pages 43-61, January.
    32. Szroeter, Jerzy, 1978. "A Class of Parametric Tests for Heteroscedasticity in Linear Econometric Models X1-ab," Econometrica, Econometric Society, vol. 46(6), pages 1311-1327, November.
    33. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
    34. Farebrother, R. W., 1987. "The statistical foundations of a class of parametric tests for heteroscedasticity," Journal of Econometrics, Elsevier, vol. 36(3), pages 359-368, November.
    35. Harrison, M J, 1980. "The Small Sample Performance of the Szroeter Bounds Test for Heteroscedasticity and a Simple Test for Use When Szroeter's Test is Inconclusive," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 42(3), pages 235-250, August.
    36. Maekawa, Koichi, 1988. "Comparing the Wald, LR and LM tests for heteroscedasticity in a linear regression model," Economics Letters, Elsevier, vol. 26(1), pages 37-41.
    37. Evans, Merran A. & King, Maxwell L., 1985. "A point optimal test for heteroscedastic disturbances," Journal of Econometrics, Elsevier, vol. 27(2), pages 163-178, February.
    38. Koenker, Roger, 1981. "A note on studentizing a test for heteroscedasticity," Journal of Econometrics, Elsevier, vol. 17(1), pages 107-112, September.
    39. Robert F. Engle & David F. Hendry & David Trumble, 1985. "Small-Sample Properties of ARCH Estimators and Tests," Canadian Journal of Economics, Canadian Economics Association, vol. 18(1), pages 66-93, February.
    40. Binkley, James K, 1992. "Finite Sample Behavior of Tests for Grouped Heteroskedasticity," The Review of Economics and Statistics, MIT Press, vol. 74(3), pages 563-568, August.
    41. MacKinnon, James G. & White, Halbert, 1985. "Some heteroskedasticity-consistent covariance matrix estimators with improved finite sample properties," Journal of Econometrics, Elsevier, vol. 29(3), pages 305-325, September.
    42. Breusch, T S & Pagan, A R, 1979. "A Simple Test for Heteroscedasticity and Random Coefficient Variation," Econometrica, Econometric Society, vol. 47(5), pages 1287-1294, September.
    43. Demos, Antonis & Sentana, Enrique, 1998. "Testing for GARCH effects: a one-sided approach," Journal of Econometrics, Elsevier, vol. 86(1), pages 97-127, June.
    44. King, Maxwell L, 1981. "A Note on Szroeter's Bounds Test," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 43(3), pages 315-321, August.
    45. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    Full references (including those not matched with items on IDEAS)

    More about this item

    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;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cir:cirwor:2001s-25. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Webmaster). General contact details of provider: http://edirc.repec.org/data/ciranca.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.