The leverage effect refers to the generally negative correlation between an asset return and its changes of volatility. A natural estimate consists in using the empirical correlation between the daily returns and the changes of daily volatility estimated from high frequency data. The puzzle lies in the fact that such an intuitively natural estimate yields nearly zero correlation for most assets tested, despite the many economic reasons for expecting the estimated correlation to be negative. To better understand the sources of the puzzle, we analyze the different asymptotic biases that are involved in high frequency estimation of the leverage effect, including biases due to discretization errors, to smoothing errors in estimating spot volatilities, to estimation error, and to market microstructure noise. This decomposition enables us to propose novel bias correction methods for estimating the leverage effect.

With modern technology, massive data can easily be collected in a form of multiple sets of curves. New statistical challenge includes testing whether there is any statistically significant difference among these sets of curves. In this article we propose some new tests for comparing two groups of curves based on the adaptive Neyman test and the wavelet thresholding techniques introduced earlier by Fan. We demonstrate that these tests inherit the properties outlined by Fan and that they are simple and powerful for detecting differences between two sets of curves. We then further generalize the idea to compare multiple sets of curves, resulting in an adaptive high-dimensional analysis of variance, called HANOVA. These newly developed techniques are illustrated by using a dataset on pizza commercials where observations are curves and an analysis of cornea topography in ophthalmology where images of

Technological innovations have revolutionized the process of scientific research and knowledge discovery. The availability of massive data and challenges from frontiers of research and development have reshaped statistical thinking, data analysis and theoretical studies. The challenges of high-dimensionality arise in diverse fields of sciences and the humanities, ranging from computational biology and health studies to financial engineering and risk management. In all of these fields, variable selection and feature extraction are crucial for knowledge discovery. We first give a comprehensive overview of statistical challenges with high dimensionality in these diverse disciplines. We then approach the problem of variable selection and feature extraction using a unified framework: penalized likelihood methods. Issues relevant to the choice of penalty functions are addressed. We demonstrate that for a host of statistical problems, as long as the dimensionality is not excessively large, we can estimate the model parameters as well as if the best model is known in advance. The persistence property in risk minimization is also addressed. The applicability of such a theory and method to diverse statistical problems is demonstrated. Other related problems with high-dimensionality are also discussed.

This article deals with statistical inferences based on the varying-coefficient models proposed by Hastie and Tibshirani. Local polynomial regression techniques are used to estimate coefficient functions, and the asymptotic normality of the resulting estimators is established. The standard error formulas for estimated coefficients are derived and are empirically tested. A goodness-of-fit test technique, based on a nonparametric maximum likelihood ratio type of test, is also proposed to detect whether certain coefficient functions in a varying-coefficient model are constant or whether any covariates are statistically significant in the model. The null distribution of the test is estimated by a conditional bootstrap method. Our estimation techniques involve solving hundreds of local likelihood equations. To reduce the computational burden, a one-step Newton-Raphson estimator is proposed and implemented. The resulting one

Several new tests are proposed for examining the adequacy of a family of parametric models against large nonparametric alternatives. These tests formally check if the bias vector of residuals from parametric fits is negligible by using the adaptive Neyman test and other methods. The testing procedures formalize the traditional model diagnostic tools based on residual plots. We examine the rates of contiguous alternatives that can be detected consistently by the adaptive Neyman test. Applications of the procedures to the partially linear models are thoroughly discussed. Our simulation studies show that the new testing procedures are indeed powerful and omnibus. The power of the proposed tests is comparable to the <i>F</i>-test statistic even in the situations where the <i>F</i> test is known to be suitable and can be far more powerful than the <i>F</i>-test statistic in other situations. An application to testing linear models versus