This paper considers a linear triangular simultaneous equations model with conditional quantile restrictions. The paper adjusts for endogeneity by adopting a control function approach and presents a simple two-step estimator that exploits the partially linear structure of the model. The first step consists of estimation of the residuals of the reduced-form equation for the endogenous explanatory variable. The second step is series estimation of the primary equation with the reduced-form residual included nonparametrically as an additional explanatory variable. This paper imposes no functional form restrictions on the stochastic relationship between the reduced-form residual and the disturbance term in the primary equation conditional on observable explanatory variables. The paper presents regularity conditions for consistency and asymptotic normality of the two-step estimator. In addition, the paper provides some discussions on related estimation methods in the literature and on possible extensions and limitations of the estimation approach. Finally, the numerical performance and usefulness of the estimator are illustrated by the results of Monte Carlo experiments and two empirical examples, demand for fish and returns to schooling. , and seminar participants at CREST for helpful comments. Special thanks go to Andrew Chesher for encouraging me to work on this and other related quantile regression projects. I am also grateful to Kathryn Graddy for kindly providing me with her data on demand for fish. To establish the n −1/2 -consistency and asymptotic normality of (β(τ ),γ(τ ) ) , we make the following assumptions: These familiar exclusion and inclusion conditions are necessary for identification. Assumption 3.3. The rectangle W has a nonempty interior and is contained in the interior of the support of W . ). That is, V is a trimming function on a compact interval corresponding to V . Assumption 3.4. The distribution of V is absolutely continuous with respect to Lebesgue measure. The density of V is bounded away from zero on V. 6 This restriction requires that the endogenous variable X be continuously distributed. Assumption 3.5. λ τ (v) is r-times continuously differentiable on V. Assumptions 3.4 and 3.5 imply that for both power series and splines, there exists Newey (1997) ). Assumption 3.6. For almost every x and z, F ε τ (0|x, z) = τ . There is a positive constant for all ε 1 and ε 2 in a neighborhood of zero and for all x and z. Also, there are constants c 2 > 0 and c 3 < ∞ such that c 2 ≤ f ετ (ε|x, z) ≤ c 3 for all ε in a neighborhood of zero and for all x and z. Among other things, Assumption 3.6 requires that f ετ (·|x, z) be bounded away from zero in a neighborhood of zero uniformly over x and z. The smallest eigenvalue of Φ κ is bounded away from zero for all κ, and the largest eigenvalue of Φ κ is bounded for all κ. Assumptions 3.7 and 3.8 insure the non-singularity of the second moment matrix of the estimator. Assumption 3.9. As functions of v, Assumption 3.9 implies that for both power series and splines, there exists a sequence as κ → ∞. Assumption 3.10. There is a ([d z + 1] × 1)-vector-valued function ∆ µ,π (x, z) such that (a) E[∆ µ,π (X, Z)] = 0,
REF considers a control function approach but deviates from the model (1).
Endogeneity in quantile regression models: a control function approach
Journal of Econometrics
Mathematics