A semiparametric single index model with heterogeneous impacts on an unobserved variable

This paper proposes a single-index semiparametric model in which the unknown function has cross-sectional unit specific weights. The initial motivation comes from the search for a better measure of liquidity in stock trading which is captured by the unknown function here. The model is estimated by semiparametric least squares developed by Ichimura (1993) and Ichimura and Lee (1991). The proposed technique differs from theirs in at least two aspects. First, I show that the estimator has desirable asymptotic properties under less restrictive assumptions on data. Second, the form of the unknown function is fixed; however, the coefficients are allowed to differ across the cross-sectional units.