Abstract: To make the conventional synthetic control method more flexible to estimate the average treatment effect (ATE), this article proposes a quasi synthetic control method for nonlinear models under the index model framework with possible high-dimensional covariates, together with a suggestion of using the minimum average variance estimation (MAVE) method to estimate parameters and the LASSO-type procedure to choose high-dimensional covariates. We derive the asymptotic distribution of the proposed ATE estimators for both finite and diverging dimensions of covariates. A properly designed Bootstrap method is proposed to obtain confidence intervals and its theoretical justification is provided. When the dimension of covariates is greater than the sample size, we suggest using the robust version of sure independence screening procedure based on the distance correlation to first reduce the dimensionality and then apply the MAVE approach to estimate parameters. Finally, Monte Carlo simulation studies are conducted to examine the finite sample performance of our proposed estimators and Bootstrap procedure. In addition, an empirical application to reanalyzing data from the National Supported Work Demonstration demonstrates the practical usefulness of our proposed method.

Key words and phrases: Average treatment effect, bootstrap inference, index model, semiparametric estimation, synthetic control method.