# Dynamic Functional Principal Components for Testing Causality
## 3 Testing Linear Causality
### Three Procedures for Testing the Causality
### Comparison of the Three Tests

Since the nominal level is well respected for the two first tests, it seems that the second test Dynamic-FPCA is more powerful than the two other tests in all the scenarios except for model 1. We can explain this by observing that the Dynamic-FPCA captures the properties of functional time series more, as shown in [17]. To understand the fact that model 1 is better for the F-causality test, we can say the departure from the null hypothesis is weak, as \(\rho_{12}\) is nearly zero. As the classical test does not reach the level of significance, and the Dynamic-FPCA seems more powerful than the F-causality test, we recommend that the Dynamic FPCA test is performed. Figure 5: The trajectory of two functional time series with a causal relation. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Fcausality Test**} & \multicolumn{4}{c}{**DynamicFPCA Test**} & \multicolumn{4}{c}{**Classical Test**} \\ \hline
**Model** & **N** & **\(K\) Optimal** & **nd** & \multicolumn{4}{c}{**\(p\)**} & \multicolumn{4}{c}{**\(\mathbf{Classical}\) Test**} \\ \hline
1 & 100 & 0.047 & 0.057 & 0.06 & 0.074 & 0.537 & 0.237 & 0.043 & 0.038 & 0.014 & 0.009 \\  & 200 & 0.076 & 0.062 & 0.059 & 0.068 & 0.523 & 0.229 & 0.05 & 0.034 & 0.011 & 0.008 \\
2 & 100 & 0.054 & 0.054 & 0.06 & 0.068 & 0.537 & 0.243 & 0.058 & 0.038 & 0.016 & 0.011 \\  & 200 & 0.078 & 0.064 & 0.057 & 0.061 & 0.514 & 0.216 & 0.057 & 0.036 & 0.02 & 0.012 \\
3 & 100 & 0.052 & 0.059 & 0.054 & 0.063 & 0.54 & 0.257 & 0.052 & 0.039 & 0.022 & 0.012 \\  & 200 & 0.099 & 0.061 & 0.06 & 0.063 & 0.532 & 0.244 & 0.061 & 0.04 & 0.019 & 0.011 \\
4 & 100 & 0.041 & 0.048 & 0.072 & 0.002 & 0.538 & 0.217 & 0.062 & 0.102 & 0.09 & 0.086 \\  & 200 & 0.055 & 0.055 & 0.071 & 0.001 & 0.539 & 0.236 & 0.07 & 0.101 & 0.088 & 0.083 \\ \hline \hline \end{tabular}
\end{table} 
Table 2. Nominal Level 0.05. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Fcausality Test**} & \multicolumn{4}{c}{**DynamicFPCA Test**} & \multicolumn{4}{c}{**Classical Test**} \\ \hline
**Model** & **N** & **\(K\)** & \multicolumn{4}{c}{**\(\mathbf{Classical}\) Test**} \\ \hline
1 & 100 & 0.761 & 0.918 & 0.865 & 0.735 & 0.489 & 0.214 & 0.069 & 0.153 & 0.004 & 0.552 & 0.231 & 0.073 & 0.1 & 0.09 & 0.086 \\  & 200 & 0.891 & 0.998 & 0.989 & 0.973 & 0.914 & 0.82 & 0.096 & 0.271 & 0.003 & 0.558 & 0.266 & 0.081 & 0.107 & 0.083 & 0.091 \\
2 & 100 & 0.779 & 0.924 & 0.888 & 0.756 & 0.509 & 0.242 & 0.18 & 0.925 & 0.741 & 0.541 & 0.26 & 0.074 & 0.109 & 0.091 & 0.093 \\  & 200 & 0.895 & 0.996 & 0.992 & 0.972 & 0.915 & 0.826 & 0.287 & 0.999 & 0.954 & 0.314 & 0.082 & 0.126 & 0.093 & 0.09 \\
3 & 100 & 0.733 & 0.907 & 0.857 & 0.75 & 0.487 & 0.241 & 0.988 & 0.987 & 0.98 & 0.665 & 0.223 & 0.371 & 0.434 & 0.472 & 0.487 \\  & 200 & 0.858 & 0.99 & 0.989 & 0.957 & 0.9 & 0.788 & 0.997 & 0.999 & 0.668 & 0.383 & 0.419 & 0.453 & 0.495 & 0.495 \\
4 & 100 & 0.696 & 0.647 & 0.518 & 0.344 & 0.133 & 0.062 & 1 & 0.999 & 1 & 0.95 & 0.812 & 0.629 & 0.792 & 0.793 & 0.779 \\  & 200 & 0.949 & 0.939 & 0.917 & 0.779 & 0.507 & 0.303 & 1 & 1 & 1 & 0.993 & 0.989 & 0.963 & 0.982 & 0.975 & 0.967 \\
5 & 100 & 0.662 & 0.727 & 0.566 & 0.368 & 0.15 & 0.071 & 0.971 & 0.96 & 0.93 & 0.677 & 0.124 & 0.293 & 0.385 & 0.424 & 0.458 \\  & 200 & 0.886 & 0.954 & 0.916 & 0.793 & 0.539 & 0.321 & 0.996 & 0.997 & 0.994 & 0.653 & 0.232 & 0.33 & 0.422 & 0.455 & 0.478 \\ \hline \hline \end{tabular}
\end{table} 
Table 3. Power on five different models. ## 5 Real Data Illustration

The dataset we used is available in the R package \"fds\" [22] and is already studied in [23]. The article [23] deals with the electricity demands and the temperature in Adelaide. This dataset consists of half-hourly electricity demands in Adelaide and temperatures measured at the Adelaide airport from Sunday to Saturday between 6 July 1997 and 31 March 2007 (see Figure 6 for the electricity demand on Thursday and Figure 7 for the temperature on Thursday).