## 4 Applications to Real Data  
### Data set  
In this section, we apply the generalized causality algorithm developed in the previous sections to real-world data to check for causality relations. Table 1 provides an overview of the self-assembled data set we use. For all time series, it shows the frequency, start and end points, as well as their source. We study three data subsets, namely macroeconomics, finance, and metal data. Regarding the first data subset, we examine macroeconomic data of the United States, using the following time series: consumer price index (CPI), gross domestic product (GDP), federal government tax receipts (FGT), and civilian unemployment rate (CUR). In addition, we use the economic policy uncertainty (EPU) index developed by [66]. According to the authors, EPU can be used as an indicator for economic uncertainty related to political events. As it is common for macroeconomic time series, most of the ones in this subset are of monthly or quarterly frequency. Our finance data subset comprises the S&P 500 index (S&P), the US federal funds rate (FFR), the Deutschen Aktienindex (DAX), \begin{table}
\begin{tabular}{l|l|l|l|l|l} \hline
**Data** & **Time Series** & **Frequency** & **Period Source** & **Period Article** & **Source** \\ \hline \multirow{8}{*}{Macro} & Consumer price index (CPI) & Monthly & 01/1947 \(-\) 06/2019 & 01/1985 \(-\) 01/2019 & FRED1 \\  & Gross domestic product (GDP) & Quarterly & 07/1947 \(-\) 04/2019 & 01/1985 \(-\) 01/2019 & FRED1 \\  & Federal government tax receipts (FGT) & Quarterly & 07/1947 \(-\) 01/2019 & 01/1985 \(-\) 01/2019 & FRED1 \\  & Civilian unemployment rate (CUR) & Monthly & 07/1948 \(-\) 07/2019 & 01/1985 \(-\) 01/2019 & FRED1 \\  & Economic policy uncertainty (EPU) & Monthly & 07/1985 \(-\) 07/2019 & 01/1985 \(-\) 01/2019 & FRED1 \\ \hline \multirow{8}{*}{Finance} & S\&P 500 index (S\&P) & Daily & 01/1950 \(-\) 07/2019 & 07/2010 \(-\) 07/2019 & Yahoo2 \\  & Federal funds rate (FFR) & Daily & 07/1954 \(-\) 07/2019 & 07/2010 \(-\) 07/2019 & FRED1 \\  & Deutscher Aktienindex (DAX) & Daily & 12/1987 \(-\) 07/2019 & 07/2010 \(-\) 07/2019 & Yahoo2 \\  & Dollar/Euro exchange rate (DEE) & Daily & 01/1999 \(-\) 07/2019 & 07/2010 \(-\) 07/2019 & FRED1 \\  & Bitcoin (BIT) & Daily & 07/2010 \(-\) 07/2019 & 07/2010 \(-\) 07/2019 & Yahoo2 \\ \hline \multirow{8}{*}{Metal} & Gold (GOL) & Daily & 01/1975 \(-\) 03/2019 & 11/1994 \(-\) 03/2019 & Perth Mit\({}^{3}\) \\  & Silver (SII) & Daily & 01/1975 \(-\) 03/2019 & 11/1994 \(-\) 03/2019 & Perth Mit\({}^{3}\) \\ \cline{1-1}  & Platinum (PLA) & Daily & 06/1991 \(-\) 03/2019 & 11/1994 \(-\) 03/2019 & Perth Mit\({}^{3}\) \\ \cline{1-1}  &uthenium (RUT) & Daily & 07/1992 \(-\) 07/2019 & 11/1994 \(-\) 03/2019 & Quandl4 \\ \cline{1-1}  & Palladium (PAL) & Daily & 11/1994 \(-\) 03/2019 & 11/1994 \(-\) 03/2019 & Perth Mit\({}^{3}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Overview of the data subsets examined. The data are from the sources FRED1 ([[https://fred.stlouisfed.org/](https://fred.stlouisfed.org/)]([https://fred.stlouisfed.org/](https://fred.stlouisfed.org/))), Yahoo2 ([[https://de.finance.yahoo.com](https://de.finance.yahoo.com)]([https://de.finance.yahoo.com](https://de.finance.yahoo.com))), Perth Mit3 ([[https://www.perthmint.com/](https://www.perthmint.com/)]([https://www.perthmint.com/](https://www.perthmint.com/))), and Quandl4 ([[https://www.quandl.com/](https://www.quandl.com/)]([https://www.quandl.com/](https://www.quandl.com/))). Figure 7: Local costs and optimal causal path of time series \(x\) and \(y\). The left side represents a step function in case the algorithm does not identify plausible relationships. The right side displays two optimal warping paths as a result of identical costs. the exchange rate between US Dollar and Euro (DEE), and the bitcoin price (BIT). Note that we could use the interest time series as part of the macroeconomic data subset as well, but the daily frequency fits better to the finance data.