COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1612

Tilted Nonparametric Estimation of Volatility Functions
with Empirical Applications

Ke-Li Xu and Peter C.B. Phillips

June 2007
Revised July 2010

This paper proposes a novel positive nonparametric estimator of the conditional variance function without reliance on logarithmic or other transformations. The estimator is based on an empirical likelihood modification of conventional local level nonparametric regression applied to squared mean regression residuals. The estimator is shown to be asymptotically equivalent to the local linear estimator in the case of unbounded support but, unlike that estimator, is restricted to be non-negative in finite samples. It is fully adaptive to the unknown conditional mean function. Simulations are conducted to evaluate the finite sample performance of the estimator. Two empirical applications are reported. One uses cross section data and studies the relationship between occupational prestige and income. The other uses time series data on Treasury bill rates to fit the total volatility function in a continuous-time jump diffusion model.

Keywords: Conditional variance function, Empirical likelihood, Conditional heteroskedasticity, Jump diffusion, Local linear estimator, Heteroskedastic nonparametric regression, Volatility

JEL Classification: C13, C14, C22