A Nonparametric Estimator of the Covariance Function for Functional Data

Many quantities of interest in economics and finance can be represented as partially observed functional data. Examples include structural business cycle estimation, implied volatility smile, the yield curve. Having embedded these quantities into continuous random curves, estimation of the covariance function is needed to extract factors, perform dimensionality reduction, and conduct inference on the factor scores. A series expansion for the covariance function is considered. Under summability restrictions on the absolute values of the coefficients in the series expansion, an estimation procedure that is resilient to overfitting is proposed. Under certain conditions, the rate of consistency for the resulting estimator achieves the minimax rate, allowing the observations to be weakly dependent. When the domain of the functional data is K(>1) dimensional, the absolute summability restriction of the coefficients avoids the so called curse of dimensionality. As an application, a Box–Pierce statistic to test independence of partially observed functional data is derived. Simulation results and an empirical investigation of the efficiency of the Eurodollar futures contracts on the Chicago Mercantile Exchange are included.