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Testing Normality: A GMM Approach

Author

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  • Christian Bontemps
  • Nour Meddahi
Abstract
In this paper, we consider testing marginal normal distributional assumptions. More precisely, we propose tests based on moment conditions implied by normality. These moment conditions are known as the Stein (1972) equations. They coincide with the first class of moment conditions derived by Hansen and Scheinkman (1995) when the random variable of interest is a scalar diffusion. Among other examples, Stein equation implies that the mean of Hermite polynomials is zero. The GMM approach we adopted is well suited for two reasons. It allows us to study in detail the parameter uncertainty problem, i.e., when the tests depend on unknown parameters that have to be estimated. In particular, we characterize the moment conditions that are robust against parameter uncertainty and show that Hermite polynomials are special examples. This is the main contribution of the paper. The second reason for using GMM is that our tests are also valid for time series. In this case, we adopt a Heteroskedastic-Autocorrelation-Consistent approach to estimate the weighting matrix when the dependence of the data is unspecified. We also make a theoretical comparison of our tests with Jarque and Bera (1980) and OPG regression tests of Davidson and MacKinnon (1993). Finite sample properties of our tests are derived through a comprehensive Monte Carlo study. Finally, three applications to GARCH and realized volatility models are presented. Dans cet article, nous testons des hypothèses de normalité marginale. Plus précisément, nous proposons des tests fondés sur des conditions de moments connues sous le nom d?équations de Stein. Ces conditions coïncident avec la première classe de conditions de moments obtenues par Hansen et Scheinkman (1995) quand la variable d?intérêt est une diffusion. L?équation de Stein implique, par exemple, que l?espérance de chaque polynôme de Hermite est nulle. L?approche GMM est utile pour deux raisons. Elle nous permet de tenir compte du problème d?incertitude des paramètres préalablement estimés. En particulier, nous caractérisons les conditions de moments qui sont robustes à ce problème et montrons que c?est le cas des polynômes de Hermite. C?est la principale contribution de l?article. Le second avantage de l?approche GMM est que nos tests sont aussi valides pour des séries temporelles. Dans ce cas, nous adoptons une approche HAC (Heteroskedastic-Autocorrelation-Consistent) pour estimer la matrice de poids qui intervient dans la statistique de test quand la forme sérielle des données n?est pas spécifiée. Nous comparons nos tests de manière théorique avec les tests de Jarque et Bera (1981) et les tests dits OPG de Davidson et MacKinnon (1993). Les propriétés de petits échantillons de nos tests sont étudiées par simulation. Finalement, nous appliquons nos tests à trois exemples de modèles de volatilité GARCH et volatilité réalisée.

Suggested Citation

  • Christian Bontemps & Nour Meddahi, 2002. "Testing Normality: A GMM Approach," CIRANO Working Papers 2002s-63, CIRANO.
    • Handle: RePEc:cir:cirwor:2002s-63
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    References listed on IDEAS

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    More about this item

    Keywords

    Normality; Stein-Hansen-Scheinkman equation; GMM; Hermite polynomials; parameter uncertainty; HAC; OPG regression; Normalité; équation de Stein-Hansen-Scheinkman; GMM; polynômes de Hermite; incertitude des paramètres; HAC; régression OPG;
    All these keywords.

    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
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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