Search

Nonparametric smoothing, a method of estimating smooth functions, has gained increasing popularity in statistics and application literature during the last few decades. This dissertation has focused primarily on the nonparametric estimation in quantile regression (Chapter 1) and an application of nonparametric estimation to financial asset pricing (Chapter 2).In the first essay (Chapter 1), we consider the estimation problem of conditional quantile when multi-dimensional covariates are involved. To overcome the "curse of dimensionality" yet retain model flexibility, we propose two partially linear models for conditional quantiles: partially linear single-index models (QPLSIM) and partially linear additive models (QPLAM). The unknown univariate functions are estimated by penalized splines. An approximate iteratively reweighted penalized least square algorithm is developed. To facilitate model comparisons, we develop effective model degrees of freedom for penalized spline conditional quantiles. Two smoothing parameter selection criteria, Generalized Approximate Cross-validation (GACV) and Schwartz-type Information Criterion (SIC) are studied. Some asymptotic properties are established. Finite sample properties are investigated through simulation studies. Application to the Boston Housing data demonstrates the success of proposed approach. Both simulations and real applications show encouraging results of the proposed estimators.In the second essay (Chapter 2), we investigate whether the conditional CAPM helps explain the value premium using the single-index varying-coefficient model. Our empirical specification has two novel advantages relative to those commonly used in the previous studies. First, it not only allows for a flexible dependence of conditional beta on state variables but also modeling heteroskedasticity. Second, from a large set of candidate state variables, we identify the most influential ones through an exhaustive variable selection method. We have also developed statistics to test the functional form of conditional beta and alpha, which provides justifications for or against the practices of letting conditional beta depend linearly on state variables and assuming constant alpha. Consistent with the notion that the value premium tends to be riskier during business recessions than during business expansions, we find that its conditional beta co-moves with unemployment and inflation, the two most closely watched gauges of aggregate economy by the Federal Reserve, and the price-earnings ratio. Realized beta does not subsume all the other explanatory variables when we include the realized beta as a state variable. The alpha is smaller for the conditional CAPM than for the unconditional CAPM; nevertheless, neither model fully explains the value premium.