This dissertation aims at investigating the theory and application of semiparametric econometrics. I first inspect the selection of optimal bandwidth using the cross-validation method for the kernel estimation of cumulative distribution/survivor functions. Then, I analyze the determination of the number of factors with the methods of principal component and information criteria. I also show the application of semiparametric methods to "purchasing power parity" puzzle. Firstly, I propose a data-driven least squares cross-validation method to optimally select smoothing parameters for the nonparametric estimation of cumulative distribution/ survivor functions. The general multivariate covariates can be continuous, discrete/ordered categorical or a mix of either. I establish the asymptotic optimality of least squares cross-validation method. Also, I show that the estimators of cumulative distribution/survivor functions using the smoothing parameters selected by the proposed method is asymptotically normally distributed. Monte Carlo simulation verifies the finite-sample properties of the least squares cross-validation method. Secondly, I provide some discussions on the econometric theory for factor models of large dimensions where the number of factors (r) is allowed to increase as the two dimensions, cross-sections (N) and time dimensions (T) increase. I mainly focus on the determination of the number of factors. I extend the existing panel criteria to high dimension case where r may be increasing with N or T. I show that the number of factors can be consistently estimated using the criteria. Also, Monte-Carlo simulation demonstrates the nite sample properties of the proposed estimating method. Lastly, I consider an empirical application of semiparametric econometrics to the problem of purchasing power parity (hereafter PPP) hypothesis test. Traditional linear cointegration tests of PPP hypothesis often lead to rejection of the PPP hypothesis. More recent studies allowing for some sort of nonlinearity in econometric modelings suggest mixed results and leave this problem as an unresolved issue. Therefore, I analyze PPP hypothesis within a semiparametric framework using the varying coe cient model with integrated variables, which can capture the nonlinearity of the economic structures. Applying the semiparametric functional cointegration test method, I conduct the cointegration test of PPP hypothesis between U.S. and Canada, U.S. and Japan, and U.S. and U.K., respectively to test the PPP hypothesis. In contrast to the usual ndings based on linear model PPP hypothesis testing, the semiparametric model based tests provide supporting evidence of the PPP hypothesis.
