Simple closed-form estimation of a binary latent variable model.

This article develops a closed-form non-parametric estimator, termed the eigen-decomposition estimator (EDE), for the conditional distribution function of a binary outcome given an unobserved scalar latent variable. Building on identification results from Hu and Schennach (HS), the EDE offers a computationally simpler alternative to the previously proposed sieve maximum likelihood estimator (MLE) by expressing all sieve coefficients in closed form via eigenvalue and eigenvector calculations, avoiding iterative optimization. The estimator is consistent under standard smoothness and invertibility assumptions and is particularly suited for models involving measurement error and unobserved heterogeneity. An empirical application to China’s Targeted Poverty Alleviation program illustrates the estimator’s use in a probit model with latent income subject to measurement error, revealing differences from conventional IV probit estimates and highlighting challenges in program targeting. The paper also discusses the estimator’s limitations, including its current restriction to scalar latent variables and boundary bias issues common in non-parametric methods.