Functional Linear Regression: Dependence and Error Contamination

Abstract

Functional linear regression is an important topic in functional data analysis. It is commonly assumed that samples of the functional predictor are independent realizations of an underlying stochastic process, and are observed over a grid of points contaminated by i.i.d. measurement errors. In practice, however, the dynamical dependence across different curves may exist and the parametric assumption on the error covariance structure could be unrealistic. In this paper, we consider functional linear regression with serially dependent observations of the functional predictor, when the contamination of the predictor by the white noise is genuinely functional with fully nonparametric covariance structure. Inspired by the fact that the autocovariance function of observed functional predictors automatically filters out the impact from the unobservable noise term, we propose a novel autocovariance-based generalized method-of-moments estimate of the slope function. We also develop a nonparametric smoothing approach to handle the scenario of partially observed functional predictors. The asymptotic properties of the resulting estimators under different scenarios are established. Finally, we demonstrate that our proposed method significantly outperforms possible competing methods through an extensive set of simulations and an analysis of a public financial dataset.

Functional instrumental variable regression with an application to estimating the impact of immigration on native wages

“…In this sense, the studies by Benatia et al (2017) and Chen et al (2020) are the most closely related to the present paper among the foregoing articles. With slight modifications, our results to be given can be adjusted for the case where the response variable is scalar-or vector-valued, and hence our paper is also related, to some degree, with Florens and Van Bellegem (2015) and the literature on many instruments (Morimune, 1983;Bekker, 1994;Chao and Swanson, 2005;Newey and Windmeijer, 2009;Carrasco, 2012).…”

“…One of those is an estimator proposed by Benatia et al (2017) when functional observations are independently and identically distributed (iid), and this may be viewed as an extension of their own least square-type estimator for the case where endogeneity is present. More recently, Chen et al (2020) considered endogeneity induced by measurement errors and proposed a GMM-type estimator using the autocovariance structure of serially dependent observations. In the case where the response variable is scalar-valued, the estimators proposed by Florens and Van Bellegem (2015) can also be used.…”

Factor modelling for high-dimensional functional time series

“…. , p. Examples include strong mixing conditions (Bosq;2000;Chen, Guo and Qiao;, cumulant mixing conditions (Panaretos and Tavakoli; and L q -m-approximability (Hörmann and Kokoszka;2010;. With the additional tail assumption and Lipschitz-continuity in Condition 5(ii), it is not difficult to apply the partition technique that reduces the problem from supremum over U 2 to the maximum over a grid of pairs, and hence the uniform convergence rate of n ´1{2 plog nq 1{2 in Condition 5(iii) can be achieved.…”

“…Functional time series, which refers to a sequential collection of curves observed over time exhibiting serial dependence, has recently received a great deal of attention. Despite progress being made in this field, existing literature has focused on the estimation based on a single or fixed number of functional time series, see, e.g., Bosq (2000); Hörmann and Kokoszka (2010); Bathia et al (2010); Panaretos and Tavakoli (2013); Hörmann et al (2015); Aue et al (2015); Li et al (2020); Chen, Guo and Qiao (2020) and among many others.…”

Binary response model with many weak instruments

“…This is because we accommodate the case where the instrumental variable is a functional variable such as the age specific fertility rate (Florens and Van Bellegem, 2015), occupation specific share of immigrants (Seong and Seo, 2021), and lagged cumulative intraday return trajectory of the Standard & Poor's 500 index (Chen et al, 2020). Such a functional instrumental variable has received considerable attention in econometrics, but it has mostly been used in the linear model, see, e.g., Carrasco (2012), Florens and Van Bellegem (2015), Carrasco andTchuente (2015a, 2015b), Benatia et al (2017), Chen et al (2020), and Seong and Seo (2021). However, in contrast with the case where the model is linear, our moment condition is nonlinear and not additively separable from the error term.…”