Nonparametric deconvolution problems require one to recover an unknown density when the data are contaminated with errors. Optimal global rates of convergence are found under the weighted <i>L</i>_<i>p</i>loss (1 <i>p</i> ). It appears that the optimal rates of convergence are extremely low for supersmooth error distributions. To resolve this difficulty, we examine how high the noise level can be for deconvolution to be feasible, and for the deconvolution estimate to be as good as the ordinary density estimate. It is shown that if the noise level is not too high, nonparametric Gaussian deconvolution can still be practical. Several simulation studies are also presented.

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.

This article reviews the literature on sparse high-dimensional models and discusses some applications in economics and finance. Recent developments in theory, methods, and implementations in penalized least-squares and penalized likelihood methods are highlighted. These variable selection methods are effective in sparse high-dimensional modeling. The limits of dimensionality that regularization methods can handle, the role of penalty functions, and their statistical properties are detailed. Some recent advances in sparse ultra-high-dimensional modeling are also briefly discussed.

The estimation of large covariance and precision matrices is fundamental in modern multivariate analysis. However, problems arise from the statistical analysis of large panel economic and financial data. The covariance matrix reveals marginal correlations between variables, while the precision matrix encodes conditional correlations between pairs of variables given the remaining variables. In this paper, we provide a selective review of several recent developments on the estimation of large covariance and precision matrices. We focus on two general approaches: a rankbased method and a factormodelbased method. Theories and applications of both approaches are presented. These methods are expected to be widely applicable to the analysis of economic and financial data.

Suppose that we have n observations from the convolution model Y = X+, where X and are the independent unobservable random variables, and is measurement error with a known distribution. We will discuss the asymptotic normality for deconvolving kernel density estimators of the unknown density f_{X}(.) of X by assuming either the tail of the characteristic function of behaves as f_{X}(.) (which is called supersmooth error), or the tail of the characteristic function is of order f_{X}(.) (called ordinary smooth error). Asymptotic normality of estimating the functional f_{X}(.) is also addressed.