**3. Results and Discussion**

The aforementioned ability of *ρq*(*s*) to quantify cross-correlation for various time scales (*s*-dependence) and fluctuation size (*q*-dependence) is documented in Figures 3 and 4, where the values of *ρq*(*s*) calculated for BTC and ETH versus the traditional instruments (the same as Figure 1) calculated for the log-returns *r*(*tm*) from January to October 2022 is shown. One can immediately notice two properties: (i) the correlation strength increases with scale *s* for most financial instruments, and (2) the correlation strength is lower for *q* = 4 (i.e., for large fluctuations Figure 4). These properties, observed here for BTC and ETH versus the other instruments, are typical for the financial markets in general [68,69].

**Figure 3.** The *q*-dependent detrended cross-correlation coefficient *ρq*(*s*) between BTC/USD (**right**) and ETH/USD (**left**) versus selected traditional financial instruments for *q* = 1, which does not favor any specific amplitude range. *ρq*(*s*) for a range of time scales from *s* = 12 (2 min) to *s* = 32, 000 (∼4 trading days) is presented based on data from January to October 2022. The statistically insignificant correlation region (dotted green line) is given as ± standard deviation of *ρq*(*s*) calculated from 100 independent realizations of the shuffled time series.

**Figure 4.** The *q*-dependent detrended cross-correlation coefficient *ρq*(*s*) between BTC/USD (**right**) and ETH/USD (**left**) versus selected traditional financial instruments for *q* = 4, which amplifies the large return contributions. *ρq*(*s*) for a range of time scales from *s* = 12 (2 min) to *s* = 32, 000 (∼4 trading days) is presented based on data from January to October 2022. The statistically insignificant correlation region (dotted green line) is given as ± standard deviation of *ρq*(*s*) calculated from 100 independent realizations of the shuffled time series.

As in the case of the Pearson coefficient, the strongest cross-correlations measured by *ρq*(*s*) for *q* = 1 (Figure 3) are BTC and ETH versus the stock indices NQ100 and S&P500. It is different for DJI, RUSSEL, DAX, and NIKKEI, which are less cross-correlated with the cryptocurrencies. What is interesting is that these correlations were stronger for ETH than for BTC, particularly on short time scales. For the shortest scale considered (*s* = 12 = 2 min), they started from *ρq*(*s*) ≈ 0.5 in the case of ETH vs. NQ100 and S&P500 and from *ρq*(*s*) ≈ 0.4 in the case of BTC vs. NQ100 and S&P500. For the longest scale considered (*s* = 32, 000 ≈ 4 trading days), the coefficient *ρq*(*s*) ≈ 0.75 for BTC and ETH vs. NQ100 and S&P500. The lowest correlations and the weakest scale dependence are observed for JPY, where *ρq*(*s*) ≈ 0. XAU and CL are slightly more correlated: *ρq*(*s*) ≈ 0.1 and 0.2 for the longest scale *s*. Above them are XAG and CHF for which the correlations increase with *s* from 0.1 to 0.3. The cross-correlations for remaining fiat currencies and HG start from *ρq*(*s*) ≈ 0.15 ÷ 0.25 for *s* = 12 and end at *ρq*(*s*) ≈ 0.35 ÷ 0.55 for *s* = 32, 000. If we focus on the large fluctuations and apply *q* = 4 (Figure 4), the cross-correlation levels are lower and approximately the same for BTC and ETH. Again, the most significant correlations are observed for the BTC and ETH vs. the US stock indices, but the correlations between BTC and NIKKEI are higher by ∼ 0.05 than for ETH and NIKKEI. In the range of scales 4000 ≤ *s* ≤ 10, 000 the correlations between BTC and NIKKEI are the strongest. Unlike for *q* = 1, the cross-correlations for BTC and ETH vs. DAX are on the same level as vs. AUD, CAD, MXN, NZD, and NOK. Only for *s* ≈ 20, 000 the negative values of *ρq*(*s*) can be found for BTC and ETH versus XAU. The statistical significance of *ρq*(*s*) in each case was determined by calculating the standard deviation of *ρq*(*s*) for 100 independent realizations of shuffled time series. This quantity is plotted in Figures 3 and 4 by green dotted lines. It

shows that the detrended cross-correlations are significant for all the instruments in the case of *q* = 1, except for the longest considered scales for JPY, while in the case of *q* = 4, the results for CL and XAU lack statistical significance for the longest considered scales.

Now, a time-dependent analysis of the cross-correlations measured by *ρq*(*s*) for BTC and ETH versus the traditional financial instruments: AUD, CAD, CHF, CL, DAX, EUR, HG, JPY, MXN, NIKKEI, NQ100, S&P500, XAG and XAU will be presented. A 5-day rolling window with a 1-day step was applied on two time scales: *s* = 12 (2 min) and *s* = 360 (60 min) in order to calculate *ρq*(*s*). A window of this length corresponds to a trading week. Figures 5 and 6 shows the results obtained for *q* = 1 and Figures 7 and 8 shows the results obtained for *q* = 4. The results for some assets presented in Figures 3 and 4 are omitted here because they are similar to the ones already shown. Our previous study [47] reported that before 2020 the cross-correlations for BTC and ETH versus the traditional instruments were close to 0. In this study, the period starting in 2020 is considered, thus. During these 2.5 unstable years, several important events that affected price changes in the financial markets could be observed.

**Figure 5.** Evolution of the *q*-dependent detrended cross-correlation coefficient *ρq*(*s*) with *q* = 1 and *s* = 2 min calculated in a 5-day rolling window with a 1-day step between 1 January 2020 and 28 October 2022 for the price returns of BTC/USD (**left**) and ETH/USD (**right**) versus the selected traditional assets: AUD, CAD, CHF, CL, DAX, EUR, HG, JPY, MXN, NIKKEI, NQ100, S&P500, XAG, and XAU. The statistically insignificant correlations are in the region *ρq*(*s*) = 0 ± 0.001.

**Figure 6.** Evolution of the *q*-dependent detrended cross-correlation coefficient *ρq*(*s*) with *q* = 1 and *s* = 60 min calculated in a 5-day rolling window with a 1-day step between 1 January 2020 and 28 October 2022 for the price returns of BTC/USD (**left**) and ETH/USD (**right**) versus the selected traditional assets: AUD, CAD, CHF, CL, DAX, EUR, HG, JPY, MXN, NIKKEI, NQ100, S&P500, XAG, and XAU. The statistically insignificant correlations are in the region *ρq*(*s*) = 0 ± 0.01.

The first event was the outburst of the COVID-19 pandemic that caused a crash in March 2020 on almost all the financial instruments expressed in USD. Only JPY and CHF gained in early March 2020, but later on they also lost value against the US dollar. This price behavior during period I (see Figure 1) resulted in the appearance of a significant positive cross-correlation for BTC and ETH versus the risky assets such as the stock indices, CL, HG, and the commodity currencies (AUD, NZD, CAD, MXN, NOK), which can be seen in Figures 5 and 6. The largest values of *ρq*(*s*) for BTC and ETH versus the stock indices are observed. In the case of *q* = 1 and *s* = 2 min, *ρq*(*s*) ≈ 0.2 and in the case of *q* = 1 and *s* = 60 min, *ρq*(*s*) ≈ 0.4. Such strong cross-correlations observed during the general meltdown are not that surprising, but still the joint behavior of the cryptocurrencies and, particularly, the stock indices is noteworthy because it has changed the view that the cryptocurrency market is independent. What is more interesting is the appearance of the even stronger positive cross-correlations for BTC and ETH versus almost all the other instruments except for JPY in the second half of 2020. The strongest cross-correlations are observed again for the stock indices, but very close were also those for CL, HG, XAG, XAU, and the commodity currencies. The highest values, *ρq*(*s*) > 0.5 for *q* = 1 and *s* = 60 min, were observed at the turn of September and October 2020 after the stock and cryptocurrency markets peaked and turned down at the beginning of September 2020. The third period of the significant cross-correlations for BTC and ETH versus the other instruments starts at the beginning of December 2021 after the November 2021's all-time highs on both the cryptocurrency and the US stock markets occurred. *ρq*(*s*) grew above 0.5 for *q* = 1 and *s* = 2 min and above 0.6 for *q* = 1 and *s* = 60 min in January 2022, when both markets experienced strong declines. BTC and ETH dropped 50% from their peak price down to 33,000 USD and 2300 USD, respectively, S&P500 dropped 8% down to 4200 USD and

NQ100 dropped 18% to 13,700 USD that were their local lows on 22 January 2022. At that time, there were also significant negative cross-correlations for BTC and ETH versus JPY, which is typically considered as a safe currency during the market meltdowns. After local peak of cross-correlations at the beginning of May 2022, when S&P500, NQ100, BTC, and ETH broke into new lows below 4150, 13,000, 35,000, and 2200 levels, respectively, the cross-correlations for BTC and ETH vs. the remaining instruments were significant at approximately the same levels until mid-August 2022, when the holiday upward correction in the US stock indices ended. From that moment on, one can distinguish period IV, when another downward wave of US exchange indices took place, which lasted until mid-October. This was accompanied by a strengthening of the USD, and the EUR/USD exchange rate fell below 1. During this period, the cross-correlation of BTC and ETH with all instruments denominated in USD has started to increase. They were even significantly positive in the case of the least correlated JPY at a level above 0.2 for *s* = 2 min and 0.4 for *s* = 60 min. The cross-correlations peaked in the last week of September, when for NQ100 and S&P500 they first exceeded the level of 0.6 and in the case of *s* = 60 min, they were close to 0.8. They were again slightly higher for ETH.

**Figure 7.** Evolution of the *q*-dependent detrended cross-correlation coefficient *ρq*(*s*) with *q* = 4 and *s* = 2 min calculated in a 5-day rolling window with a 1-day step between 1 January 2020 and 28 October 2022 for the price returns of BTC/USD (**left**) and ETH/USD (**right**) versus the selected traditional assets: AUD, CAD, CHF, CL, DAX, EUR, HG, JPY, MXN, NIKKEI, NQ100, S&P500, XAG, and XAU. Higher levels of cross-correlations, associated with the Consumer Price Index (CPI) readings, are marked (see the event description in Figure 9). The statistically insignificant correlations are in the region *ρq*(*s*) = 0 ± 0.001.

**Figure 8.** Evolution of the *q*-dependent detrended cross-correlation coefficient *ρq*(*s*) with *q* = 4 and *s* = 60 min calculated in a 5-day rolling window with a 1-day step between 1 January 2020 and 28 October 2022 for the price returns of BTC/USD (**left**) and ETH/USD (**right**) versus the selected traditional assets: AUD, CAD, CHF, CL, DAX, EUR, HG, JPY, MXN, NIKKEI, NQ100, S&P500, XAG, and XAU. Higher levels of cross-correlations, associated with the Consumer Price Index (CPI) readings, are marked (see the event description in Figure 9. The statistically insignificant correlations are in the region *ρq*(*s*) = 0 ± 0.01.

If large returns are considered (*q* = 4, Figures 7 and 8) the detrended cross-correlations for *s* = 2 min remain close to 0 and are statistically insignificant for most of the considered instruments until November 2021, when *ρq*(*s*) for BTC and ETH versus most currencies, especially MXN, CHF, and, to a lesser extent, for AUD, NZD, EUR and CNH turn negative for short periods of time. As in the case of *q* = 1, the cross-correlations versus the US indices became significantly positive starting from December 2021. What is most interesting is that from July 2022, the cross-correlation levels in some weekly windows exceed those obtained for *q* = 1. For *s* = 60 min they are even higher than 0.8 in the case of NQ100. There are also high correlations of BTC and ETH vs. precious metals: gold and silver. Unlike average fluctuations (*q* = 1), here BTC is slightly more strongly correlated with traditional financial instruments.

**Figure 9.** Evolution of the cumulative logarithmic returns *R*cum of selected financial instruments: BTC, ETH, AUD, CAD, CHF, CL, DAX, EUR, HG, JPY, MXN, NIKKEI, NQ100, S&P500, XAG, and XAU on specific dates, around the publication time of the Consumer Price Index (CPI) report in the US.

After careful checking of the exact start and end dates of the sliding window with increased correlations for *q* = 4 and the time of large fluctuations, it turned out that the correlation of BTC and ETH with traditional financial instruments is (directly or indirectly via other markets) influenced by the CPI inflation data published every month at 12:30 UTC. Cumulative price changes in days during the CPI publication date 13 July 2022, 10 August 2022, 13 September 2022, 10 October 2022, from 12:29 to 12:35 are presented in the Figure 9. It can be clearly seen that in all four cases US tech stocks and cryptocurrencies price changes behave in the same way just after 12:30 UTC. It happened regardless of whether the surprise with the CPI data was positive or negative. In three cases, inflation data was higher than expected and surprised markets negatively, leading to declines. This is especially well visible in the case of 10 October 2022, when apart from US indices, XAG also follows the same trajectory. In the roling windows containing this day, the correlations were the strongest: 0.6 for S&P500, 0.79 for XAG and 0.86 for NQ100 vs. BTC and 0.6 for S&P500, 0.72 for XAG and 0.82 for NQ100 vs. ETH for *s* = 60 min. In one case, 10 August 2022, the inflation was lower than expectations, which resulted in an increase in all instruments. This price behavior means that cryptocurrencies have started to respond to readings from the economy, just like traditional financial instruments. Despite the fact that our analysis of the cross-correlations was carried out by means of the measures, which were unable to detect the direction of influence, it seems natural to infer that these were the economical data releases that had direct or indirect impact on the cryptocurrency market rather than the opposite. That is why we concluded about the direction above.
