# Partial linear modelling with multi-functional covariates

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## Abstract

This paper takes part on the current literature on semi-parametric regression modelling for statistical samples composed of multi-functional data. A new kind of partially linear model (so-called MFPLR model) is proposed. It allows for more than one functional covariate, for incorporating as well continuous and discrete effects of functional variables and for modelling these effects as well in a nonparametric as in a linear way. Based on the continuous specificity of functional data, a new method is proposed for variable selection (so-called PVS method). In addition, from this procedure, new estimates of the various parameters involved in the partial linear model are constructed. A simulation study illustrates the finite sample size behavior of the PVS procedure for selecting the influential variables. Through some real data analysis, it is shown how the method is reaching the three main objectives of any semi-parametric procedure. Firstly, the flexibility of the nonparametric component of the model allows to get nice predictive behavior; secondly, the linear component of the model allows to get interpretable outputs; thirdly, the low computational cost insures an easy applicability. Even if the intent is to be used in multi-functional problems, it will briefly discuss how it can also be used in uni-functional problems as a boosting tool for improving prediction power. Finally, note that the main feature of this paper is of applied nature but some basic asymptotics are also stated in a final “Appendix”.

## Keywords

Semi-parametrics Functional data analysis Multi-functional covariates Partial linear model Variable selection## JEL Classification

C14## Notes

### Acknowledgments

The authors wish to express their great gratitude to the Editors and the Reviewers who have provided very interesting comments. Their suggestions were of great help when revising this work and will certainly increase its impact. In particular the reviewing procedure has been the opportunity to have highly interesting cross exchanges with one Referee about the interest and the meaning of our model (see Remark 2), which have greatly contributed to improve this work, at least by pointing our some challenging issues for the future. The research of G. Aneiros was partially supported by Grants MTM2011-22392 and CN2012/130 from Spanish Ministerio de Economía y Competitividad and Xunta Galicia, respectively.

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