Moment-based estimation of nonlinear regression models with boundary outcomes and endogeneity, with applications to nonnegative and fractional responses

ABSTRACT

In this article, we suggest simple moment-based estimators to deal with unobserved heterogeneity in a special class of nonlinear regression models that includes as main particular cases exponential models for nonnegative responses and logit and complementary loglog models for fractional responses. The proposed estimators: (i) treat observed and omitted covariates in a similar manner; (ii) can deal with boundary outcomes; (iii) accommodate endogenous explanatory variables without requiring knowledge on the reduced form model, although such information may be easily incorporated in the estimation process; (iv) do not require distributional assumptions on the unobservables, a conditional mean assumption being enough for consistent estimation of the structural parameters; and (v) under the additional assumption that the dependence between observables and unobservables is restricted to the conditional mean, produce consistent estimators of partial effects conditional only on observables.

KEYWORDS:

• Boundary outcomes
• endogeneity
• exponential regression
• fractional regression
• transformation regression models
• unobserved heterogeneity

JEL CLASSIFICATION:

• C25
• C51
• C26

Acknowledgment

The authors thank the Editor, the Associate Editor, the referees, and Joao Santos Silva for their valuable suggestions and remarks.

Notes

¹We use the term “neglected heterogeneity” to designate the case where the unobserved and the included covariates are independent.

²See Heckman (2000) for a rigorous definition of Marshallian causal functions.

³For example, if G₂(⋅) is an exponential function, then we may redefine the constant 𝜃₀ and the error term as and 𝜀 = exp(u)∕E[exp(u)], respectively.

⁴Note that as π₁ is fixed independently of the number of instruments, more instruments imply a higher overall fit of the instruments to the endogenous regressor x₁.

⁵Note that the highest value that we could add to y is any positive value below 0.002, given that the maximum value for Leverage in the sample is 0.998, see Table 2.

⁶R code to compute all estimators and tests discussed throughout the article is available at http://evunix.uevora.pt/∼jsr/ FRM.htm