Testing Normality: A GMM Approach

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.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

Set identified linear models

We analyze the identification and estimation of parameters β satisfying the incomplete linear moment restrictions E(z T (x β−y)) = E(z Tu(z)) where z is a set of instruments and u(z) an unknown bounded scalar function. We first provide empirically relevant examples of such a set-up. Second, we show that these conditions set identify β where the identified set B is bounded and convex. We provide a sharp characterization of the identified set not only when the number of moment conditions is equal to the number of parameters of interest but also in the case in which the number of conditions is strictly larger than the number of parameters. We derive a necessary and sufficient condition of the validity of supernumerary restrictions which generalizes the familiar Sargan condition. Third, we provide new results on the asymptotics of analog estimates constructed from the identification results. When B is a strictly convex set, we also construct a test of the null hypothesis, β 0 ε B, whose size is asymptotically correct and which relies on the minimization of the support function of the set B − { β 0 }. Results of some Monte Carlo experiments are presented.

Connectedness of the space of smooth actions of Zn\Z^n on the interval

We prove that the spaces of \Cinf orientation-preserving actions of $\Z^n$ on $[0,1]$ and nonfree actions of $\Z^2$ on the circle are connected

Testing Normality : A GMM Approach

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

Testing Distributional Assumptions: A GMM Approach

We consider testing distributional assumptions by using moment conditions. A general class of moment conditions satisfied under the null hypothesis is derived and connected to existing moment-based tests. The approach is simple and easy to implement, yet reasonably powerful. In addition, we provide moment tests that are robust against parameter estimation error uncertainty in the general case which includes the case of serial correlation. In particular, we consider the location-scale model for which we derive robust moment tests, regardless of the forms of the conditional mean and variance. We study in detail the Student and inverse Gaussian distributions. Simulation experiments are conducted to assess the finite sample properties of the tests. We provide two empirical examples on foreign exchange rates by testing the Student distributional assumption of T-GARCH daily returns and on daily realized variance by testing the inverse Gaussian distributional assumption

A Geometric Approach to Inference in Set-Identified Entry Games

In this paper, we consider inference procedures for entry games with complete information. Due to the presence of multiple equilibria, we know that such a model may be set identified without imposing further restrictions. We complete the model with the unknown selection mechanism and characterize geometrically the set of predicted choice probabilities, in our case, a convex polytope with many facets. Testing whether a parameter belongs to the identified set is equivalent to testing whether the true choice probability vector belongs to this convex set. Using tools from the convex analysis, we calculate the support function and the extreme points. The calculation yields a finite number of inequalities, when the explanatory variables are discrete, and we characterized them once for all. We also propose a procedure that selects the moment inequalities without having to evaluate all of them. This procedure is computationally feasible for any number of players and is based on the geometry of the set. Furthermore, we exploit the specific structure of the test statistic used to test whether a point belongs to a convex set to propose the calculation of critical values that are computed once and independent of the value of the parameter tested, which drastically improves the calculation time. Simulations in a separate section suggest that our procedure performs well compared with existing methods