A Nonparametric Estimator for the Covariance Function of Functional Data

Abstract

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 BoxPierce 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.

Greedy algorithms for prediction

“…This is not to say that functions in L cannot represent non-linear functions. For example, the set of regressors X could include functions that are dense in some set, or generally be a subset of some dictionary (e.g., Mallat and Zhang [50], Barron et al [6], Sancetta [63]).…”

“…The Constrained Greedy Algorithm (CGA) is a variation of the RGA. It is used in Sancetta [63] in a slightly different context. The Frank-Wolfe Algorithm (FWA) (Frank and Wolfe [35]; see Clarkson [26], Jaggi [42], Freund, Grigas and Mazumder [36], for recent results on its convergence) is a well-known algorithm for the optimization of functions under convex constraints.…”

“…This can be avoided by letting B → ∞ with the sample size. The following can then be used to bound the error when B < B, if µ 0 ∈ L(B) (Sancetta [63]).…”

“…The third algorithm is a version of the Hilbert space projection algorithm studied by Jones [43] and Barron [5] with the version studied in this paper taken from Barron et al [6], and called the Relaxed Greedy Algorithm (RGA). Adding a natural restriction to the RGA, the algorithm leads to the solution of the Lasso problem, which appears to be relatively new (see Sancetta [63]). This constrained version will be called Constrained Greedy Algorithm (CGA).…”

Estimation for the Prediction of Point Processes With Many Covariates

“…, with the standard modification when r = ∞. The following is a re-adaptation of a result in Sancetta (2015) and can be used to control the approximation error of the estimator. When g 0 / ∈ L (B), define the best uniform approximation g B = arg inf |g − g 0 | ∞ , where the infimum is over L (B).…”