## 4 Applications to Real Data
### Macroeconomics  
The first data field we want to draw attention to is macroeconomics. The time series consumer price index (CPI), gross domestic product (GDP), federal government tax receptions (FGT), civilian unemployment rate (CUR), and economic policy uncertainty (EPU) cover the time period from January 1, 1985 to January 1, 2019 at monthly frequency, which means 137 entries per time series. The reason why our examined period of time starts in 1985 is that EPU was set up starting at that time only. The left-hand side of Figure 8 shows the development of the normalized data over the examined time period. We see that GDP and CPI follow an almost linear trend, whereas the other time series, such as FGT underlie more fluctuation. Furthermore there is a co-movement of CUR and EPU, with some peaks of EPU being followed by a similar development of CUR. The right-hand part of Figure 8 gives an overview of the distances between all time series after having applied the generalized causality algorithm. The two pairs with minimal total distance are (i) GDP and CPI as well as (ii) CUR and EPU. It is not surprising that GDP and CPI have small total distance. Their development over time is similar, as can be seen in Figure 8, and the variables mutually depend on one another. The small distance between CUR and EPU is an interesting finding, as unemployment usually is accompanied with uncertainty. Figures 9 and 10 help to learn more about the causal relationship of these pairs. Please note that in this subsection as well as in Sections 4.3 and 4.4, we will only consider figures of type Figure 4, because they contain more information than graphics such as Figure 5.  
Figure 9 shows that gross domestic product (GDP) and consumer price index (CPI) most of the time move almost perfectly together, with the lead switching regularly. However, the lag is never more than a few months. There are two structural breaks in the lead-lag structure approximately at the end of 1997 (from CPI to GDP as the leader) and around the second third of 2010 (back to CPI as the leader). Incidences that might have had an impact on this relationship are the Asian financial crisis in 1997-1998, which influenced other economies of the world as well [71; 72] and the European financial crisis that was still prevalent in 2010 [73]. Taking a closer look at Figure 9, we see that there is an area of very low cost, i.e., a region of almost white color, approximately along the diagonal. This is not surprising if we look at the development of GDP and CPI over time as shown in the left part of Figure 8. The two time series move almost perfectly together, which means low cost along the diagonal. The similarity of the series is due to their (partial) mutual dependency. Since CPI is a measure of inflation and GDP naturally increases with higher prices (caused at least partly by inflation), rising prices imply larger values of both. We also take a closer look at the pair yielding the second smallest total distance according to the generalized causality algorithm, namely civilian unemployment rate (CUR) and economic policy uncertainty (EPU). Figure 10 shows the causal path for the two variables. It is salient that the series do not move as smoothly together as GDP and CPI do in the aforementioned case, which is shown by the heterogeneous cost matrix with no clear area of small cost along the diagonal. CUR leads EPU throughout the entire period of time observed, which means that whenever unemployment is high, EPU will rise after some time and vice versa. The size of the lag varies, however, in the way that it becomes smaller over time and is almost zero between approximately the end of 2008 and 2014, before it increases again. There are significant moments to the lead-lag structure in mid-1992 and at the end of 2014. Reasons for this might be the reflation of the economy initiated by former U.S. president [PERSON] in 1992, which caused unemployment to drop [74] and hence lose its prompt impact on economic policy uncertainty.