Robust Misspecification Tests for the Heckman's Two-Step Estimator

Abstract

This article constructs and evaluates Lagrange multiplier (LM) and Neyman's C(α) tests based on bivariate Edgeworth series expansions for the consistency of the Heckman's two-step estimator in sample selection models, that is, for marginal normality and linearity of the conditional expectation of the error terms. The proposed tests are robust to local misspecification in nuisance distributional parameters. Monte Carlo results show that testing marginal normality and linearity of the conditional expectations separately have a better size performance than testing bivariate normality. Moreover, the robust variants of the tests have better empirical size than nonrobust tests, which determines that these tests can be successfully applied to detect specific departures from the null model of bivariate normality. Finally, the tests are applied to women's labor supply data.

Keywords:

• Heckman's two-step
• LM tests
• Neyman's C(α) tests

JEL Classification:

• C12
• C24

ACKNOWLEDGMENTS

I am in debt to Anil Bera, Roger Koenker, Todd Elder, Zhonjgun Qu, Thomas Mroz, Esfandiar Maasoumi, an Associate Editor, and two anonymous referees for helpful comments and suggestions. A previous version of this article appeared as a chapter in my Ph.D. in Economics Dissertation at the University of Illinois Urbana-Champaign. All remaining errors are mine.

Notes

¹As an example, assume that e ∼ N(0, 1) and u = ρ · e + ψ, where ψ and e are independent, and E(ψ) = 0. Then, both conditions are satisfied, but bivariate normality is not, if ψ follows a non-Gaussian distribution.

²For skewness: κ ₃₀ = µ₃₀, κ ₂₁ = µ₂₁, κ ₁₂ = µ₁₂, and κ ₀₃ = µ₀₃. For kurtosis: κ ₄₀ = µ₄₀ − 3, κ ₄₀ = µ₃₁ −3ρ, κ ₂₂ = µ₂₂ − 2ρ² − 1, κ ₁₃ = µ₁₃ − 3ρ, and κ ₀₄ = µ₀₄ − 3. Here we use the standard notation μ _ij  = E(u ⁱ e ^j ).

³We consider the regularity conditions stated in Lee (1984, footnote 5, p. 854) to get the asymptotic distribution of the LM tests. Bera and Yoon (1993) statistics require the same conditions because they work under local departures from the joint null hypothesis.

⁴This condition may be exploited in certain occasions to show that standard LM tests can be used without having to control for misspecification in some parameters. In our case, some covariance terms among the Hermite polynomials are zero under the assumption of bivariate normality. This determines that some off-diagonal terms of the IM may be asymptotically negligible. We do not exploit this line of research.

⁵See Bera and Bilias (2001) for an excellent discussion on Neyman's C(α) tests.

⁶As stated in Jaggia and Trivedi (1994), Neyman's C(α) tests are special cases of their conditional score tests.

⁷Only κ₃₀ and κ₄₀ can be consistently estimated.

Notes: Rejection rates based on 1000 replications. See text for details.

Notes: Rejection rates based on 1000 replications. See text for details.

Notes: Rejection rates based on 1000 replications. See text for details.

⁸Note that we can only evaluate ρ = 0.2, 0.4, 0.6.

Notes: Rejection rates based on 1000 replications. See text for details.

Notes: Rejection rates based on 1000 replications. See text for details.

⁹Lee (1984) studies the effect of being in a labor union on the workers wage. In this case, there is a strong selectivity for individuals who are in a union vs. those that are not affiliated. Using the OPG method to estimate the IM, this author found strong evidence to reject and , that is, the hypothesis that all cumulants are zero, although he cannot reject . Given our Monte Carlo results we may conclude that rejection rates in standard LM tests occur too often and therefore, they may not be used as evidence to reject the validity of two-step estimation methods.

Notes: Standard errors in parenthesis. Critical values for : 4.61 (10%), 5.99 (5%), 9.21 (1%); : 7.78 (10%), 9.49 (5%), 13.28 (1%); : 14.68 (10%), 16.92 (5%), 21.67 (1%).

Notes: Standard errors in parenthesis. Critical values for : 4.61 (10%), 5.99 (5%), 9.21 (1%); : 7.78 (10%), 9.49 (5%), 13.28 (1%); : 14.68 (10%), 16.92 (5%), 21.67 (1%).