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Robust Estimation of Probit Models with Endogeneity. (2021). Zhelonkin, Mikhail ; Varadi, Mate ; Naghi, Andrea A.
In: Tinbergen Institute Discussion Papers.
RePEc:tin:wpaper:20210004.

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  1. — (1967), “The Behavior of Maximum Likelihood Estimates under Nonstandard Conditions,” in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, L.M. LeCam, J. Neyman (Eds.), Berkeley: University of California Press, 221–233.
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  2. — (2016), “Robust Inference in Sample Selection Models,” Journal of the Royal Statistical Society B, 78, 805–827. A Derivation of Proposition 1 We derive the IF of the Rivers-Vuong two-step estimator. The first step is OLS for which the IF is known. Using our notation we have IF{(x1, x2, y1); S, F} = − Z x1x> 1 x1x> x2x> 1 x2x>
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  3. Avella-Medina, M. (2020), “Privacy-Preserving Parametric Inference: A Case for Robust Statistics,” Journal of the American Statistical Association, to Appear.
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  4. Blundell, R. W. and Powell, J. L. (2004), “Endogeneity in Semiparametric Binary Response Models,” Review of Economic Studies, 71, 655–679.

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  9. Duncan, G. M. (1987), “A Simplified Approach to M-Estimation with Application to TwoStage Estimators,” Journal of Econometrics, 34, 373–389.

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  14. Hampel, F., Rousseeuw, P. J., and Ronchetti, E. (1981), “The Change-of-Variance Curve and Optimal Redescending M-Estimators,” Journal of the American Statistical Association, 76, 643–648.
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  16. Ψr 1(z1; γ)> , Ψr 2(z; γ, δ)> . Assume that the following regularity conditions (adapted from Duncan 1987) hold: 1. zi, for i = 1, . . . , N, is a sequence of independent identically distributed random vectors with distribution F defined on a space Z. 2. The parameter space Θ is a compact subset of R × R2p1 × Rp2 × [0, 1] × R+ . 3. R Ψr (z; θ)dF = 0 has a unique solution θ0 in the interior of Θ. 4. Ψr (z; θ) and ∂ ∂θ Ψr (z; θ) are measurable for each θ in Θ, continuous for each z in Z, and there exist F-integrable functions ξ1 and ξ2 such that for all θ ∈ Θ and z ∈ Z |Ψr (z; θ)Ψr (z; θ)T | ≤ ξ1 and | ∂ ∂θ Ψr (z; θ)| ≤ ξ2. 5. R Ψr (z; θ)Ψr (z; θ)> dF is non-singular for each θ ∈ Θ. 6. R ∂Ψr (z; θ0)/∂θdF is finite and non-singular.
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  17. Maronna, R. A., Martin, G. R., and Yohai, V. J. (2006), Robust Statistics: Theory and Methods, Chichester: John Wiley and Sons.
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  18. Orrenius, P. M. and Zavodny, M. (2015), “Does immigration affect whether US natives major in science and engineering?” Journal of Labor Economics, 33, S79–S108.

  19. Proof of Proposition 2. Consistency and asymptotic normality follow from Theorems 1-4 in Duncan (1987). The asymptotic variance reduces to two terms. This is because the error terms 1 in the first step and y2 − x̃> δ in the second step are independent by construction and thus Υ1(z)Υ2(z)> and Υ2(z)Υ1(z)> vanish after integration.
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  20. Rivers, D. and Vuong, Q. H. (1988), “Limited Information Estimators and Exogeneity Tests for Simultaneous Probit Models,” Journal of Econometrics, 39, 347–366.

  21. Rousseeuw, P. J. and Van Driessen, K. (1999), “A Fast Algorithm for the Minimum Covariance Determinant Estimator,” Technometrics, 41, 212–223.
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  22. Since MLE is the particular case of an M-estimator, its IF is known to be proportional to the score function (Hampel et al., 1986). Hence the IF is (un)bounded if the score function is (un)bounded. The score functions for different parameters are as follows: ∂l ∂γj = −y2 φ(u) Φ(u) ρxj/σ1 (1 − ρ2)1/2 + (1 − y2) φ(u) {1 − Φ(u)} ρxj/σ1 (1 − ρ2)1/2 + σ2 (y1 − x> 1iγ1 − x> 2iγ2)xj, (21) for j = 1, 2. ∂l ∂β = φ(u) y2 − Φ(u) Φ(u){1 − Φ(u)} x1 (1 − ρ2)1/2 , (22) ∂l ∂α = φ(u) y2 − Φ(u) Φ(u){1 − Φ(u)} y1 (1 − ρ2)1/2 . (23) The score functions in (22) and (23) are structurally the same as (6). It is clear that all the score functions in (21)-(23) are unbounded. C Assumptions and Proof of Proposition 2 Denote Ψr (z; θ) =
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  23. von Mises, R. (1947), “On the Asymptotic Distribution of Differentiable Statistical Functions,” The Annals of Mathematical Statistics, 18, 309–348.
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  24. White, H. (1980), “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity,” Econometrica, 48, 817–838.

  25. Wooldridge, J. M. (2010), Econometric Analysis of Cross Section and Panel Data, Cambridge, MA: The MIT Press, 2nd ed.

  26. Yohai, V. J. (1987), “High Breakdown-Point and High Efficiency Robust Estimates for Regression, ” The Annals of Statistics, 15, 642–656.
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  27. Zhelonkin, M. (2013), “Robustness in Sample Selection Models,” PhD Thesis, University of Geneva, Switzerland.
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  28. Zhelonkin, M., Genton, M. G., and Ronchetti, E. (2012), “On the Robustness of Two-Stage Estimators,” Statistics & Probability Letters, 82, 726–732.

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