Many quantities of interest in economics and finance can be
represented as partially observed functional data. Examples include
structural business cycles 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 over-fitting is proposed. Under certain conditions, the rate of consistency for the resulting estimator is nearly the parametric rate when the observations are 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.