Testing Additive Separability of Error Term in Nonparametric Structural Models

Abstract

This article considers testing additive error structure in nonparametric structural models, against the alternative hypothesis that the random error term enters the nonparametric model nonadditively. We propose a test statistic under a set of identification conditions considered by Hoderlein et al. (2012), which require the existence of a control variable such that the regressor is independent of the error term given the control variable. The test statistic is motivated from the observation that, under the additive error structure, the partial derivative of the nonparametric structural function with respect to the error term is one under identification. The asymptotic distribution of the test is established, and a bootstrap version is proposed to enhance its finite sample performance. Monte Carlo simulations show that the test has proper size and reasonable power in finite samples.

Keywords:

• Additive separability
• Hypotheses testing
• Nonparametric structural equation
• Nonseparable models

JEL Classification:

• C12
• C13
• C14

ACKNOWLEDGMENTS

The authors gratefully thank the Co-editors and two anonymous referees for their many constructive comments on the previous version of the article. They are also thankful to Rosa Matzkin and Aditi Bhattacharya for some discussions on the subject matter of this article.

Notes

¹Interestingly, LW show that by imposing monotonicity in unobservables for the nonparametric structural function, they can establish the equivalence between the conditional independence and additive separability hypotheses. In this case, their test is also consistent.

²The sample median is random but converges to the population median at the parametric rate. Noting that our test is of nonparametric nature and has power against local alternatives converging to the null at the nonparametric rate, which implies that one can treat the sample median as the population median without affecting the asymptotic theory studied below.

³Here and below we restrict (x, z, e) to 𝒳₀ × 𝒵₀ × ℰ₀ because we need to estimate G(e | x, z) and its inverse G ⁻¹(· | x, z) which cannot be estimated sufficiently well if G(e|x, z) is close to either 0 or 1, say, when (x, z, e) lies at the boundary of its support 𝒳 × 𝒵 × ℰ.

⁴Alternatively one can follow HSW and apply the local polynomial method to obtain all necessary estimates. But we find that the local constant method is less computational expensive than the latter.

⁵We abuse the notation a little bit. The multivariate kernel function L can be defined either on ℝ ^d for U _i or ℝ ^d+1 for W _i , which is self evident from its argument.

⁶When G and G ⁻¹ are estimated by the local polynomial regressions, the asymptotic distributions of , , and are quite complicated and studied in HSW.

⁷Even though X _i , Z _i , Y _i , and ϵ _i all enter the definition of ζ₀, we can still use to summarize these variables because ϵ _i  = m ⁻¹(X _i , Y _i ) is measurable under Assumption I.1 and the continuity of m(·, ·).

⁸We abuse the notation Φ a little bit here: Φ _α(z) = α^{−d _Z} Φ(z/α) and Φ _α(x) = α^{−d _X} Φ(z/α). So the argument of Φ can be of dimension d _X or d _Z . The bandwidths here are all set according to the Silverman's rule of thumb in our simulations below.

⁹Hoderlein and Mammen (2007) show that an average over the marginal effects can be identified without the monotonicity assumption.

¹⁰Write , say. A careful calculation suggests that both ζ_1ij and ζ_2ij contribute to the asymptotic bias of J _1na but only ζ_1ij contributes to the asymptotic variance of J _1na .