Three Essays on Nonparametric Econometrics with Applications to Financial Economics and Insurance

Abstract

This dissertation includes three essays. The first essay concerns nonparametric kernel density estimation on the unit interval. The Kernel Density Estimator (KDE) suffers boundary biases when applied to densities on bounded supports, which are assumed to be the unit interval. Transformations mapping the unit interval to the real line can be used to remove boundary biases. However, this approach may induce erratic tail behaviors when the estimated density of transformed data is transformed back to its original scale. I propose a modified transformation based KDE that employs a tapered and tilted back-transformation. I derive the theoretical properties of the new estimator and show that it asymptotically dominates the naive transformation based estimator while maintains its simplicity. I then propose three automatic methods of smoothing parameter selection. Monte Carlo simulations demonstrate the good finite sample performance of the proposed estimator, especially for densities with poles near the boundaries. An example with real data is provided.

The second essay proposes a new kernel estimator of copula densities. The standard kernel estimator suffers boundary biases since copula densities are defined on a bounded support and often tend to infinity on the boundaries. A transformation based estimator aptly remedies both boundary biases and inconsistencies due to unbounded densities. This method, however, might entail undesirable boundary behaviors due to an unbounded multiplicative factor associated with the transformation. I propose a modified transformation-based estimator that employs an infinitesimal tapering device to mitigate the influence of the unbounded multiplier. I establish the asymptotic properties of our estimator and show that it dominates the original transformation estimator in terms of mean squared error due to bias correction. I present two practically simple methods of smoothing parameter selection. I further show that the proposed estimator admits higher order bias reduction for Gaussian copulas and provides outstanding performance for Gaussian and near Gaussian copulas. This appealing feature makes our estimator particularly suitable for financial data analyses. Extensive simulations corroborate our theoretical analysis and demonstrate outstanding performance of the proposed method relative to competing estimators.

Three empirical applications are provided. The third essay studies nonparametric estimation of crop yield distributions and crop insurance premium rates. Since U.S. crop yield data are typically available at county level for only a few decades, nonparametric estimation of yield distribution for individual counties suffers from small sample sizes. The fact that nearby counties share similarities in their yield distributions suggests possible efficiency gains through information pooling. I propose a weighted kernel density estimator subject to selected spatial moment restrictions. The weights are calculated using the method of empirical likelihood and the spatial moments are specified based on the consideration of flexibility and robustness. I further extend the proposed method to the adaptive kernel density estimation. My simulations demonstrate the outstanding performance of the proposed methods in the estimation of crop yield distributions and that of crop insurance premium rates. I apply these methods to estimate corn yield distributions and crop insurance premium rates for the ninety-nine counties in Iowa.

The second essay proposes a new kernel estimator of copula densities. The standard kernel estimator suffers boundary biases since copula densities are defined on a bounded support and often tend to infinity on the boundaries. A transformation-based estimator aptly remedies both boundary biases and inconsistencies due to unbounded densities. This method, however, might entail undesirable boundary behaviors due to an unbounded multiplicative factor associated with the transformation. I propose a modified transformation-based estimator that employs an infinitesimal tapering device to mitigate the influence of the unbounded multiplier. I establish the asymptotic properties of our estimator and show that it dominates the original transformation estimator in terms of mean squared error due to bias correction. I present two practically simple methods of smoothing parameter selection. I further show that the proposed estimator admits higher order bias reduction for Gaussian copulas and provides outstanding performance for Gaussian and near Gaussian copulas. This appealing feature makes our estimator particularly suitable for financial data analyses. Extensive simulations corroborate our theoretical analysis and demonstrate outstanding performance of the proposed method relative to competing estimators. Three empirical applications are provided.

The third essay studies nonparametric estimation of crop yield distributions and crop insurance premium rates. Since U.S. crop yield data are typically available at county level for only a few decades, nonparametric estimation of yield distribution for individual counties suffers from small sample sizes. The fact that nearby counties share similarities in their yield distributions suggests possible efficiency gains through information pooling. I propose a weighted kernel density estimator subject to selected spatial moment restrictions. The weights are calculated using the method of empirical likelihood and the spatial moments are specified based on the consideration of flexibility and robustness. I further extend the proposed method to the adaptive kernel density estimation. My simulations demonstrate the outstanding performance of the proposed methods in the estimation of crop yield distributions and that of crop insurance premium rates. I apply these methods to estimate corn yield distributions and crop insurance premium rates for the ninety-nine counties in Iowa.