The problem of estimating the density function and the regression function involving errors-in-variables in time series is considered. Under appropriate conditions, it is shown that the rates obtained in Fan (1991), Fan and Truong (1990) are also achievable in the context of dependent observations. Consequently, the results presented here extend our previous results for cross-sectional data to the longitudinal ones. oAbbreviated title. Measurement Errors in Time Series AMS 1980 subject classification. Primary 62G20. Secondary 62G05, 62J99.

Varying coefficient models are a useful extension of classical linear models. They arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age. The appeal of these models is that the coef. cient functions can easily be estimated via a simple local regression. This yields a simple one-step estimation procedure. We show that such a one-step method cannot be optimal when different coefficient functions admit different degrees of smoothness. This drawback can be repaired by using our proposed two-step estimation procedure. The asymptotic mean-squared error for the two-step procedure is obtained and is shown to achieve the optimal rate of convergence. A few simulation studies show that the gain by the two-step procedure can be quite substantial. The methodology is illustrated by an application to an environmental data set.

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.

I would like to congratulate Professor Antoniadis for successfully outlining the current state-of-art of wavelet applications in statistics. Since wavelet techniques were introduced to statistics in the early 90s, the applications of wavelet techniques have mushroomed. There is a vast forest of wavelet theory and techniques in statistics and one can find himself easily lost in the jungle. The review by Antoniadis, ranging from linear wavelets to nonlinear wavelets, addressing both theoretical issues and practical relevance, gives in-depth coverage of the wavelet applications in statistics and guides one entering easily into the realm of wavelets.

Variable selection is vital to statistical data analyses. Many of procedures in use are ad hoc stepwise selection procedures, which are computationally expensiveandignore stochastic errors in the variable selection process of previous steps. An automatic and simultaneous variable selection procedure can be obtained by using a penalized likelihood method. In traditional linear models, the best subset selection and stepwise deletion methods coincide with a penalized leastsquares method when design matrices are orthonormal. In this paper, we propose a few new approaches to selecting variables for linear models, robust regression models and generalized linear models based on a penalized likelihood approach. A family of thresholding functions are proposed. The LASSO proposed by Tibshirani (1996) is a member of the penalized leastsquares with the # #-penalty. A smoothly clipped absolute deviation (SCAD) penalty function is introduced to ameliorate the properties of # #-penalty. A unied algorithm is introduced, which is backed up by statistical theory. The new approaches are compared with the ordinary leastsquares methods, the garrote method by Breiman (1995) and the LASSO method by Tibshirani (1996). Our simulation results show that the newly proposed methods compare favorably with other approaches as an automatic variable selection technique. Because of simultaneous selection of variables and estimation of parameters, we are able to give a simple estimated standard error formula, which is tested to be accurate enough for practical applications. Two real data examples illustrate the versatility and eectiveness of the