The Theory And Development Of Nonparametric Series Methods

This dissertation focuses on nonparametric series estimation methods. After summing up the existing results in the literature, we propose two developments on the theory of series methods.First, the asymptotic properties of the series estimator on the functional of the conditional mean,θ|^ =a(g|^), for independently and identically distributed data have been studied a lot in the literature. The corresponding properties ofθ|^ for weakly dependent data have been derived only for the case where a(·) is a smooth functional (Chen and Shen (1998)) under some high level conditions. The asymptotic properties of the estimatorθ|^ for weakly dependent data for the case where a(·) is not a smooth functional, however, have not been reported in published papers. This dissertation will compute the convergence rates and derive the asymptotic normality ofθ|^ for stationary andρ-mixing data and for any functional a(·) . Our derivations are similar to those in Newey (1997) but ours are more complicated because of data dependence.Second, through estimating the nonparametric model by series regression, Hong and White (1995) proposed two consistent one-sided test statistics for i.i.d data which are showed to have good power and perform quite well in some situations by a Monte Carlo study. However, the desired asymptotic properties of these test statistics depend on a strong assumption that the underling regression function can be approached by approximating functions quite fast under the null hypothesis. This dissertation presents an alternative test which has the same asymptotic properties without imposing the assumption mentioned above. The new test statistic behaves like Hong and White's in finite sample for flat regression functions but much better for fluctuant ones.