estimator based on numerical derivatives is ˜ V ND n , the matrix whose element ( k, l ) is given by where e$_{k}$ is the k th unit vector in R d and where ϵ$_{n}$ = o (1) is a tuning parameter. Conditions under which this estimator is consistent are given in the following lemma. Lemma 1 Suppose Condition CRA holds and that ˆ θ$_{n}$ satisfies (8). If ϵ$_{n}$ → 0 and nϵ 3 n → ∞ , then ˜ V ND n → $_{P}$V 0 . The proof of the lemma goes beyond consistency and develops a Nagar-type mean squared error (MSE) expansion for ˜ V ND n under the additional assumption that M is four times continuously differentiable near θ 0 . The approximate MSE (AMSE) can be minimized by choosing ϵ$_{n}$ proportional to n − 1 / $^{7}$, the optimal factor of proportionality being a functional of the covariance kernel H and the fourth order derivatives of M evaluated at θ 0 . For details, see the supplemental appendix (Theorem A.2 and Section A.5), which also contains a brief discussion of alternative generic estimators of V 0 . Arguably the most prominent econometric example of an estimator exhibiting cube root asymptotics is the maximum score estimator of Manski (1975). To describe a version of this estimator, suppose { z$_{i}$ : 1 ≤ i ≤ n } is a random sample of z = ( y, x $^{′}$)′ generated by the binary response model where β 0 ∈ R d +1 is an unknown parameter of interest, x ∈ R d +1 is a vector of covariates, and F$_{ε}$$_{|}$ x ( ·| x ) is the conditional cumulative distribution function of the unobserved error term ε given x . Following Abrevaya and Huang (2005), we normalize the (unidentified) scale of β 0 by setting the first element of the parameter vector equal to unity. In other words, we employ the parameterization β 0 = (1 , θ ′ $_{0}$) $^{′}$, where θ 0 ∈ R d is unknown. Partitioning x conformably with β 0 as x = ( x 1 , x ′ $_{2}$) $^{′}$, a 4 The Maximum Score Estimator y = 1( x $^{′}$β 0 + ε ≥ 0) , F$_{ε}$$_{|}$ x (0 | x ) = 1 / 2 , (11) ˜ V ND n,kl = − 1 4 ϵ 2 n [ M$_{n}$ ( ˆ θ$_{n}$ + e$_{k}$ ϵ$_{n}$ + e$_{l}$ ϵ$_{n}$ ) − M$_{n}$ ( ˆ θ$_{n}$ + e$_{k}$ ϵ$_{n}$ − e$_{l}$ ϵ$_{n}$ ) − M$_{n}$ ( ˆ θ$_{n}$ − e$_{k}$ ϵ$_{n}$ + e$_{l}$ ϵ$_{n}$ ) + M$_{n}$ ( ˆ θ$_{n}$ − e$_{k}$ ϵ$_{n}$ − e$_{l}$ ϵ$_{n}$ ) ] , 8