Improving Estimation in Functional Linear Regression With Points of Impact: Insights Into Google AdWords

Abstract

The functional linear regression model with points of impact (PoI) is a recent augmentation of the classical functional linear model with many practically important applications. In this article, 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 PoI 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 R-package FunRegPoI and additional R-codes are provided in the online supplementary materials.

Keywords:

• Functional data analysis
• Functional linear regression
• Online advertising
• Points of impact

Supplementary Materials

R-package and R-codes: The R-package FunRegPoI contains an implementation of our estimation algorithm. The package also contains the dataset used in our real data application. The provided R-codes facilitate the reproduction of the results in our simulation study and our application. (supplement.zip)

Acknowledgments

The authors wish to thank Prof. Alois Kneip (University of Bonn), Dominik Poss (University of Bonn), and Prof. Rolf Tschernig (University of Regensburg) for their valuable suggestions that helped to improve this research work. Special thanks go to Crealytics (www.crealytics.com) for providing the data, as well as stimulating and inspiring discussions, and for posing the statistical problem considered in our application. Furthermore, we are grateful to the referees and the editors for their constructive comments which helped to improve our article.

Notes

¹ Online ad campaigns use text corpora populations of relevant search keywords (e.g., outdoor jacket, mountain boots, etc., in the case of an outdoor equipment campaign) to identify potential customers by their Google searches (see Section 4 for more details).

² Remember that the FPCA-basis, based on the eigendecomposition of the covariance operator of X, is the optimal empirical basis to approximate X, but generally not the optimal basis to approximate $\beta (·)$.

³ Note that it is impossible to compute estimation errors for nonfound τ_s.

⁴ Sponsored impressions link to the advertised homepage—they are similar to, but distinguishable from ordinary Google search results.