## 4 Applications to Real Data
### Metal  
Our last data field of interest is the field of metal prices, in which we study the time series gold (GOL), silver (SIL), platinum (PLA), ruthenium (RUT), and palladium (PAL). Note that for all types of metal, we use average asking prices, which is in line with the literature (see [77]). All series cover the time period from November 17, 1994 to March 29, 2019 at daily frequency, which means each time series comprises  
Figure 11: Standardized finance time series of S&P 500 index (S&P), federal funds rate (FFR), Deutscher Aktienindex (DAX), Dollar/Euro exchange rate (DEE), and bitcoin (BIT) (**left**) and pairwise distances applying the generalized causality algorithm (**right**). The number in bold symbol represents the lowest cost.  
Figure 12: Local costs and optimal causal path of Deutscher Aktienindex (DAX) and S&P 500 index (S&P). The large bar on the x-axis represents an structural break with corresponding 95 percent confidence interval (small bars).  
6060 entries. RUT depicts an enormous peak in 2006 and 2007 as a result of a price bubble. It was caused by a misinterpretation of the demand for ruthenium in the solar industry and the assumption that it could be used in medicine as an active ingredient for cancer therapy. The left-hand side of Figure 13 shows the development of the normalized data over the examined time period. The right-hand part of Figure 13 gives an overview of the distances between all time series after having applied the generalized causality algorithm. Gold and silver possesses minimal total distance.  
Figure 14 shows the causal path and local costs for the gold price (GOL) and the silver price (SIL). Again, the path wanders around the diagonal for the entire period of time we consider. In the last quarter of 2010, silver starts to lead gold until the end of the time period we study. One possible reason for this structural break is that the dollar was rather weak in 2010, which usually increases silver prices [78; 79]. In contrast to [79; 80], which see gold as a driving factor for silver prices, our generalized causality algorithm detects a contrary relationship. This finding is in line with other empirical evidence for silver driving gold prices (see [77]).  
To summarize Section 4.4, the generalized causality algorithm also performs well in the data field of metal prices. The pair yielding minimal total distance exhibits only one structural break in the causal path, which goes along in time with an economic event that might have had an influence on the structural break.  
Figure 14: Local costs and optimal causal path of gold (GOL) and silver (SIL). The large bar on the x-axis represents an structural break with corresponding 95 percent confidence interval (small bars).  
Figure 13: Standardized metal time series of gold (GOL), silver (SIL), platinum (PLA), ruthenium (RUT), and palladium (PAL) (**left**) and pairwise distances applying the generalized causality algorithm (**right**). The number in bold symbol represents the lowest cost.  
In general it can be said that the generalized causality algorithm outperforms in different areas of application, yielding sound results in line with economic and political events explaining occurring structural breaks.