# Dynamic Functional Principal Components for Testing Causality
## 3 Testing Linear Causality
### Three Procedures for Testing the Causality
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(p=3\) & **Monday** & **Tuesday** & **Wednesday** & **Thursday** & **Friday** & **Saturday** & **Sunday** \\ \hline Monday & \(3.88\times 10^{-15}\) & \(<\)2.2 \(\times 10^{-16}\) & \(7.757\times 10^{-10}\) & \(1.964\times 10^{-7}\) & 0.03575 & 0.5818 & 0.516 \\ Tuesday & 0.0028 & \(<\)2.2 \(\times 10^{-16}\) & \(<\)2.2 \(\times 10^{-16}\) & \(1.1527\times 10^{-12}\) & 0.003014 & 0.3427 & 0.188 \\ Wednesday & 0.0383 & 0.001607 & \(<\)2.2 \(\times 10^{-16}\) & \(<\)2.2 \(\times 10^{-16}\) & 0.0002952 & 0.9729 & 0.691 \\ Thursday & 0.62 & 0.7831 & 0.5706 & \(<\)2.2 \(\times 10^{-16}\) & \(<\)2.2 \(\times 10^{-16}\) & \(4.6\times 10^{-12}\) & 0.172 \\ Friday & 0.107 & 0.3103 & 0.4036 & 0.03976 & \(<\)2.2 \(\times 10^{-16}\) & \(<\)2.2 \(\times 10^{-16}\) & 0.0221 \\ Saturday & 0.032 & 0.2931 & 0.1128 & 0.02072 & \(3.066\times 10^{-12}\) & \(<\)2.2 \(\times 10^{-16}\) & \(5.83\times 10^{-6}\) \\ Sunday & \(1.5\times 10^{-10}\) & \(6.873\times 10^{-5}\) & 0.2707 & 0.1951 & 0.01001 & \(1.027\times 10^{-5}\) & \(7.88\times 10^{-07}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Classical tests between temperature and electricity demands. We highlight the diagonal in red. * [PERSON] et al. (2012) [PERSON]; [PERSON] _Inference for Functional Data with Applications_; Springer: New York, NY, USA, 2012; Volume 200. * [PERSON] (1969) [PERSON]. Investigating Causal Relations by Econometric Models and Cross-spectral Methods. _Econometrica_**1969**, _37_, 424-438. [CrossRef]
* [PERSON] (2017) [PERSON] Linear causality in the sense of Granger with stationary functional time series. In _Functional Statistics and Related Fields_; [PERSON], [PERSON], [PERSON], [PERSON], [PERSON]; Springer International Publishing: Cham, Switzerland, 2017; pp. 225-231. * [PERSON] (2020) [PERSON], [PERSON] Functional Causality between Oil Prices and GDP Based on Big Data. _Comput. Mater. Contin._**2020**, _63_, 593-604. [CrossRef]
* [PERSON] (2019) [PERSON] Intraday End-of-Day Volume Prediction. _J. Financ. Econom._**2019**, nbz019. [CrossRef]
* [PERSON] (1956) [PERSON] The theory of prediction. _Mod. Math. Eng._**1956**, \(1\), 125-139. * [PERSON] (1997) [PERSON]; [PERSON] _Time Series and Dynamic Models_; Cambridge University Press: Cambridge, MA, USA, 1997; Volume 3. * [PERSON] et al. (2010) [PERSON]; [PERSON] Short and long run causality measures: Theory and inference. _J. Econom._**2010**, _154_, 42-58. [CrossRef]
* [PERSON] (1982) [PERSON] Measurement of linear dependence and feedback between multiple time series. _J. Am. Stat. Assoc._**1982**, _77_, 304-313. [CrossRef]
* [PERSON] et al. (2014) [PERSON]; [PERSON]; [PERSON] Nonparametric estimation and inference for conditional density based Granger causality measures. _J. Econom._**2014**, _180_, 251-264. [CrossRef]
* [PERSON] and [PERSON] (1995) [PERSON]; [PERSON] A theory of inferred causation. _Stud. Log. Found. Math._**1995**, _134_, 789-811. * [PERSON] and [PERSON] (2010) [PERSON]; [PERSON] Weakly dependent functional data. _Ann. Stat._**2010**, _38_, 1845-1884. [CrossRef]
* [PERSON] and [PERSON] (2015) [PERSON]; [PERSON] Two sample inference for the second-order property of temporally dependent functional data. _Bernoulli_**2015**, _21_, 909-929. [CrossRef]
* [PERSON] et al. (2015) [PERSON]; [PERSON]; [PERSON] Dynamic functional principal components. _J. R. Stat. Soc. Ser. B Stat. Methodol._**2015**, _77_, 319-348. [CrossRef]
* [PERSON] et al. (2015) [PERSON]; [PERSON]; [PERSON] On the prediction of stationary functional time series. _J. Am. Stat. Assoc._**2015**, _110_, 378-392. [CrossRef]
* [PERSON] and [PERSON] (2015) [PERSON]; [PERSON] _Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators_; John Wiley & Sons: West Sussex, UK, 2015. * [PERSON] et al. (2017) [PERSON]; [PERSON]; [PERSON]; [PERSON] fda: Functional Data Analysis. R Package Version 2.4.7. 2017.