Smoothed maximum score estimation with nonparametrically generated covariates

Abstract

This paper develops a two-stage semiparametric procedure to estimate the preference parameters of a binary choice model under uncertainty. In the model, the agent’s decision rule is affected by the conditional expectation. We nonparametrically estimate the conditional expectation in the first stage. Then, in the second stage, the preference parameters are estimated by the smoothed maximum score method. We establish the consistency and asymptotic distribution of the two-stage estimator. Furthermore, we also characterize the conditions under which the first-stage nonparametric estimation will not affect the asymptotic distribution of the smoothed maximum score estimator. Monte Carlo simulation results demonstrate that our proposed estimator performs well in finite samples.

Keywords:

• Binary choice model
• generated covariate
• smoothed maximum score estimator

JEL CLASSIFICATION:

• C13
• C14

Acknowledgment

We are grateful to the editor and the anonymous referee for excellent comments and suggestions that lead to substantial improvement. The usual disclaimer certainly applies.

Notes

1 For example, they can represent the strategic effects in a 2 × 2 incomplete information game (Aradillas-Lopez, 2010). Additionally, they can be the effects of the endogenous covariates (e.g., price) in a multinomial discrete-choice model with a subset of choices (Fox, 2007).

2 See Assumption 6 of Krief (2014).

3 In reality, the agent will use her observed information $(X,Z,ϵ)$ to form the subjective expectation. However, there is an underlying assumption here, which requires the expectations to be fulfilled and conditioned only on variables observed by the researcher. This is the key identifying condition in the literature on the binary choice model under uncertainty that facilitates the two-step estimation procedure (see, e.g., Ahn, 1997; Ahn and Manski, 1993; Chen, Lee, and Sung, 2014). In fact, the vector V here acts as a sufficient statistic in the conditional expectation (Manski, 1991) and thus $\mathbb{E}(V|Z,X,ϵ)=\mathbb{E}(V|Z).$ We thank a referee for highlighting this.

4 For example, according to Corollary 1 of Krief (2014), $\gamma (Z)$ needs to admit at least 18 derivatives for the SMS estimator to achieve the $n^{-4/9}$ rate.

Additional information

Funding

Financial support from the Humanities Development Fund of Nankai University (Grant No.ZB21BZ0311) and the Fundamental Research Funds for the Central Universities (Grant No.180-63192105) are gratefully acknowledged by Gao. Chen appreciates receiving finanical support for his research from the Fundamental Research Funds for the Central Universities (Grant No.CXTD11-01), the 111 Project from the State Administration of Foreign Experts Affairs (Grant No.B18014) and the Humanities and Social Sciences Fund of Ministry of Education of China (Grant No.19YJC790013).