Consistent estimation of scaled coefficient

RECEIV 7 This paper studies the estimation of coefficients ^ in single index models such that E(y | X)=F(a+X'^) , where the function F is misspecified or unknown. A general connection between behavioral derivatives and covariance estimators is established, which shows how 8 can be estimated up to scale using information on the marginal distribution of X. A sample covariance estimator and an instrumental variables slope coefficient vector are proposed, which are constructed using appropriately defined score vectors of the X distribution. The framework is illustrated using several common limited dependent variable odels, and extended to multiple index models, including models of selection bias and multinomial discrete choice. The asymptotic bias in the OLS coefficients of y regressed on X are analyzed. The asymptotic distribution of the instrumental variables estimator is established, when the X distribution is modeled up to a finite parameterization