Some notes on statistic robustness of nonparametric bivariate probit model in a finite sample

Abstract

This article describes qualitatively some interesting statistic aspects of the nonparametric bivariate Probit model, which was examined in Aoki (2005) as a nonparametrically modified version of the estimator to test asymmetric information, originally proposed in Chiappori and Salanie (2000). My computation results and analysis show that even in a finite sample case the nonparametric version is very robust to the variable bandwidth, which is relatively smaller than the optimal bandwidth policy. This statistic characteristics enables the proposed nonparametric estimator to be put widely and conveniently into practical use, without applied researcher's necessity to pay too much attention to the precise value of optimal bandwidth.

Notes

¹ . Also,, which is clearly maximized at g(x_i ) = 0. Here the operator E_y or Var _y implies the expectation or the variance over the entire space of y, where Prob(y = 1|x) = Φ(g(x)).

²The leave-one-out estimator of is calculated by excluding the observation i.

³Asymptotically (N → ∞, h → 0, Nh → ∞), W ^h follows, under this hypothesis, a χ² distribution with degree of freedom one, χ²(1), in which the 5% significance level is 3.84.

⁴ H ₀: Cov(∊ _i , η _i ) = 0 implies . Also, . Then the construction of this test statistics is justified.

⁵ For example, the operator Cov _{x , y} or E_{x , y} imply the covariance or the expectation over the entire space of x and y (integral with PDF of x and y, (x, y)).

⁶ Since in reality the true decision function g(x_i ) is not known, the optimal bandwidth can be instead computed as a minimizer of the sample mean integrated squared residual:

(1.6′)

⁷ Obviously, the values of and , or of and are expected to be almost close.

⁸ Also, obviously , which was actually shown in computation results.

⁹ The operator E_x implies the expectation over the entire space of x (integral with PDF of x, p(x)).

¹⁰ Since here a leave-one-out estimator is calculated, does not contain any information about ∊ _i , while, of course, y_i does enough. Therefore and y_i are stochastically independent, and it is expected that, in calculating (1.4), the information about ∊ _i is well preserved, even if an extremely narrow bandwidth h is taken, without and y_i ’s offsetting the information with each other. As a matter of fact, however as far as computation results in the settings of this article are concerned, a nonleave-one-out estimator (that is, when the observation i is included for estimating g(x_i )) did not make statistically a big difference from cross-validation case.

¹¹ A non-leave-one-out estimator is assumed here.

¹² A leave-one-out estimator is assumed.

¹³ .