Hidehiko Ichimura's paper introduces a semiparametric least squares (SLS) estimator for single-index models, which are a class of econometric models that include censored and truncated Tobit models, binary choice models, and duration models with unobserved individual heterogeneity and random censoring. The SLS estimator is designed to be consistent and asymptotically normal, achieving a $1/\sqrt{n}$-consistency rate, where $n$ is the sample size. The paper also discusses a weighted SLS (WSLS) estimator that achieves the semiparametric efficiency bound, as proposed by Newey (1990). The WSLS estimator uses a weighting scheme that accounts for the semiparametric nature of the model, which involves both finite-dimensional and infinite-dimensional parameters. The paper provides a detailed definition of the single-index model, the geometric motivation for the SLS estimation method, and the identification conditions for the linear single-index model. It also includes proofs of the consistency and asymptotic normality of the SLS and WSLS estimators, along with a consistent estimator for the covariance matrix. The paper concludes with a Monte Carlo experiment to evaluate the small-sample properties of the SLS estimator and discusses future research directions.Hidehiko Ichimura's paper introduces a semiparametric least squares (SLS) estimator for single-index models, which are a class of econometric models that include censored and truncated Tobit models, binary choice models, and duration models with unobserved individual heterogeneity and random censoring. The SLS estimator is designed to be consistent and asymptotically normal, achieving a $1/\sqrt{n}$-consistency rate, where $n$ is the sample size. The paper also discusses a weighted SLS (WSLS) estimator that achieves the semiparametric efficiency bound, as proposed by Newey (1990). The WSLS estimator uses a weighting scheme that accounts for the semiparametric nature of the model, which involves both finite-dimensional and infinite-dimensional parameters. The paper provides a detailed definition of the single-index model, the geometric motivation for the SLS estimation method, and the identification conditions for the linear single-index model. It also includes proofs of the consistency and asymptotic normality of the SLS and WSLS estimators, along with a consistent estimator for the covariance matrix. The paper concludes with a Monte Carlo experiment to evaluate the small-sample properties of the SLS estimator and discusses future research directions.