Improving Estimation in Functional Linear Regression with Points of Impact: Insights into Google AdWords

The functional linear regression model with points of impact is a recent augmentation of the classical functional linear model with many practically important applications. In this work, however, we demonstrate that the existing data-driven procedure for estimating the parameters of this regression model can be very instable and inaccurate. The tendency to omit relevant points of impact is a particularly problematic aspect resulting in omitted-variable biases. We explain the theoretical reason for this problem and propose a new sequential estimation algorithm that leads to significantly improved estimation results. Our estimation algorithm is compared with the existing estimation procedure using an in-depth simulation study. The applicability is demonstrated using data from Google AdWords, today's most important platform for online advertisements. The \textsf{R}-package \texttt{FunRegPoI} and additional \textsf{R}-codes are provided in the online supplementary material

Partially observed functional data: The case of systematically missing parts

New estimators for the mean and the covariance function for partially observed functional data are proposed using a detour via the fundamental theorem of calculus. The new estimators allow for a consistent estimation of the mean and covariance function under specific violations of the missing-completely-at-random assumption. The requirements of the estimation procedure can be tested using a sequential multiple hypothesis test procedure. An extensive simulation study compares the new estimators with the classical estimators from the literature in different missing data scenarios. The proposed methodology is motivated by the practical problem of estimating the mean price curve in the German Control Reserve Market. In this auction market, price curves are only partially observable, and the underlying missing data mechanism depends on systematic trading strategies which clearly violate the missing-completely-at-random assumption. In contrast to the classical estimators, the new estimators lead to useful estimates of the mean and covariance functions. Supplementary materials are provided online.(1) (C) 2018 Elsevier B.V. All rights reserved