1. Introduction
Functional data analysis has become an important field of modern statistics, and now an abundant literature on this topic exists. The monograph of [
1] is an introductory source with very valuable examples and techniques for the analysis of functional data. Another standard book about functional data analysis is [
2]. As a good introduction to dependent functional data and functional time series, we recommend the monographs of [
3,
4]. Recently, the classical notion of causality in the sense of Granger (see [
5]) has been extended to the functional time series cases. This extension is important and useful to the statistical community. The notion of Granger causality test can be explained as a statistical hypothesis test that measures the usefulness of adding a variable to forecast another variable. Granger causality is a predictive notion of causality. The causality for functional time series has been studied from a Granger point of view by Saumard [
6]. The procedure was used in two application research articles; Almanjahie et al. [
7] studied the relations between oil prices and the gross domestic product and Sancetta [
8] with an application in financial econometrics. In this article, we provide a new testing procedure based on dynamic functional principal components.
Since the pioneering work of [
5,
9], an abundant literature has become available on causality of classical time series (for a broader review see [
10,
11]). In fact, there are various possible approaches for defining causality, notably causality in the frequency domain [
12], a nonparametric approach [
13], and causality on Bayesian networks [
14]. Furthermore, the wide range of applications (causality in econometrics, neuroscience, social science, biomedical, signals) indicates the central role of causality in science.
In this article, we recall the notion of causality for functional stationary time series and propose tests of non-causality. There are nowadays different tools to analyze dependency on functional data, such as a mixing, linear process [
3,
15], where a definition of dependency on functional data has been proposed that generalizes the
m-dependency, called
-approximable sequences. This notion is exploited to theoretically justify the “F-causality” test. “F-causality” stands for functional causality. This first test exploits the functional nature of the data and adapts a test of equality between operators for stationary functional time series (see [
16]) to the context of causality. This test is explained in detail in [
6]. The proposed test relies on functional dynamic principal components, which are able to capture the variability of functional time series better than functional principal components as demonstrated in [
17]. Once we obtain the dynamic FPCA scores, we make the decision by using a multivariate Granger test on the scores. Aue et al. [
18] use the same idea for predicting functional time series. Our dynamic-FPCA based procedure is publicly available at
https://github.com/PyMattAI/DynamicFPCA_Causality.git (accessed on 7 June 2021) In order to compare, we studied another procedure that does not use the functional nature of the data, which is called the classical test. This procedure is based on differentiating the time series and using a multivariate Granger test.
A functional time series is a sequence of functional objects that are dependent. For example, in
Figure 1, a classical time series of monthly sea surface temperatures (in °C) from January 1982 to December 2018 is plotted. These sea surface temperatures were measured by Moore buoys in the ‘Niño region’. The functional versions of the same data points are represented in
Figure 2. One functional observation represents the sea surface temperature for one year. From the
data of the classical time series, we generate 37 observations of a functional time series that are observed each month. More generally, a functional time series
, where
(
N is the sample size of the functional time series) and
(where there are
m points by functional observations we observe at the points
from
) can be generated from a classical time series
where
, with
m being the number of observations points. We have the relation
.
The functional time series must be stationary in order for our test to perform well. It is worth mentioning that stationarity for functional time series and stationarity for classical time series are quite different. For example, we could have a stationary functional time series from a classical time series that is not stationary. Mainly, it is the case when the classical time series has a seasonality over a period T, and we cut this series at these intervals of time.The classical time series appears non-stationary, but if we consider the data as functional observations, it can be stationary. Hence, our test provides a natural framework to test causality for classical non-stationary time series. The application of a traditional test and our procedure to the real dataset highlight this fact.
In
Section 2, the definition of causality for functional stationary time series and an example in the autoregressive functional processes are introduced. In
Section 3, we recall how we can estimate the dynamic FPCA and propose three algorithms for testing the non-causality including the dynamic FPCA-based testing procedure. We conclude this article with
Section 4 and
Section 5, which are developed to empirically analyze the testing procedure, and an application of the procedure to explain the relation between electricity demand and temperature in Australia.