*2.1. Data Sources and Preprocessing*

In the present study, a data set of 24 financial time series representing contracts for difference (CFDs) from the Dukascopy trading platform [58] is considered. Unlike many other trading platforms, Dukascopy offers freely the high-frequency recordings of many financial instruments, which is the main reason it has been chosen as the data source. CFDs are characterised by the price movements that are close to the price movements of the original instruments, so we consider them as reliable proxies. Apart from the two highest capitalized cryptocurrencies, BTC and ETH, it includes the most important traditional financial instruments: 12 fiat currencies (Australian dollar—AUD, Canadian dollar—CAD, Swiss franc—CHF, Chinese yuan—CNH, euro—EUR, British pound—GBP, Japanese yen— JPY, Mexican peso—MXN, Norwegian krone—NOK, New Zealand dollar—NZD, Polish zloty—PLN, and South African rand—ZAR), 4 commodities (WTI crude oil—CL, high grade copper—HG, silver—XAG, and gold—XAU), 4 US stock market indices (Nasdaq 100—NQ100, S&P500, Dow Jones Industrial Average—DJI, and Russell 2000—RUSSEL), German stock index—DAX 40—DAX, and the Japanese stock index—Nikkei 225—NIKKEI. All these instruments except for the non-US stock indices are expressed in USD (thus there is no USD in the data set) and their quotes cover a period from 1 January 2020 to 28 October 2022. Each week the quotes were recorded over the whole trading hours, i.e., from Sunday 22:00 to Friday 20:15 with a break between 20:15 and 22:00 each trading day (UTC) [58].

Cumulative log-returns of all the instruments considered are plotted in Figure 1 against time. The original price changes, sampled every Δ*t* = 10 s, were transformed into logarithmic returns: *r*(*tm*) = ln *Pi*(*tm*+1) − ln *Pi*(*tm*), where *Pi*(*tm*) is a price quote recorded at time *tm* (*m* = 1, ... , *T*) and *i* stands for a particular financial instrument. We use this particular time interval Δ*t*, because such a data set was available from the source. However, it is satisfactory because it allows us to avoid excessive null returns, which lower reliability of the detrended analysis (see below). In order to obtain the indicative relationships among all the time series, the Pearson correlation coefficient *C* [59] was calculated for the logreturns *r*(*tm*) from January to October 2022, when the joint bear market mentioned above was observed. A correlation matrix obtained for 24 financial instruments is shown in Figure 2. While the Pearson coefficient is one of the most widely applied measures of time series dependencies (and this is why we also exploited it in our study), the results obtained with it have to be taken with some reserve in our context. This is because the statistical tests that we carried out, i.e., the Jacque-Bera test for normality and the ARCH test for no heteroskedasticity, both rejected the respective null hypotheses with high confidence (*p*-value < 0.00001), which means that the data under study was both heavy-tailed and heteroskedastic. Obviously, such a result is not surprising, because fat tails of the return distributions and volatility clustering are well-known effects observed in the financial time series [60–62]. Nevertheless, the very long time series that were analysed here and the statistical significance of the obtained results convinced us that the Pearson coefficient can still serve as an effective measure of the time series correlations even in such circumstances. Taking all this into account, a standard naming convention: small (0.1 ≤ *C* < 0.3), medium (0.3 ≤ *C* < 0.), and large (0.5 ≤ *C* ≤ 1.0) correlation was used to describe the coefficient values. The strongest cross-correlations (large, *C* > 0.6) can be seen for the stock indices, for BTC and ETH, for AUD, NZD and CAD, for XAU and XAG, and for EUR and GBP. If we consider the cross-correlations between BTC and the traditional instruments, the strongest ones can be seen for NQ100 and S&P500 (medium, *C* ≈ 0.32), DJI and RUSSEL (medium, *C* ≈ 0.29), DAX (small, *C* ≈ 0.24) and NIKKEI, (small *C* ≈ 0.23). The Pearson coefficient above 0.1 (small), is observed for BTC on the one side and HG, GBP and EUR, as well as the so-called commodity currencies: AUD, CAD, NZD, MXN, NOK, and ZAR, on the other side. The cross-correlations between ETH and the other instruments are even higher: *C* ≈ 0.38 (medium) for SP500 and NQ100, *C* ≈ 0.35 (medium) for DJI and RUSSEL, *C* ≈ 0.29 (medium) for DAX, and *C* ≈ 0.27 (small) for NIKKEI. The same is true for the cross-correlations between ETH and the commodity currencies: *C* ≈ 0.22 (small) for AUD, CAD, NZD, *C* ≈ 0.17 (small) for MXN, *C* ≈ 0.13 (small) for NOK, and *C* ≈ 0.12 (small) for ZAR. Among the commodities analyzed here, ETH is the most correlated with HG (*C* ≈ 0.15, small). The statistical significance of the coefficient values presented in Figure 2 has been checked by calculating the range: *<sup>C</sup>*¯ <sup>±</sup> *<sup>σ</sup>C*, where *<sup>C</sup>*¯ denotes mean and *<sup>σ</sup><sup>C</sup>* denotes standard deviation of *C*, from 100 independent realisations of the shuffled time series. The

statistically insignificant correlation region is very close to 0 (the third decimal place), all the presented values, except DAX vs. JPY, are thus significant.

**Figure 1.** Evolution of the cumulative log-returns of the cryptocurrencies *R*cum (**a**), the stock market indices (**b**), the fiat currencies (**c**), and the commodities (**d**) over a period from 1 January 2020 to 28 October 2022. Periods for which significant correlations between the cryptocurrencies and the US stock indices are distinguished by grey vertical strips. The most characteristic periods are denoted by Roman numerals: a COVID-19-related crash in March 2020 and a quick bounce in April–May 2020 (period I), new all-time highs of NQ100 and S&P500 and a September 2021 correction (period II), a bear phase in the cryptocurrency and stock markets since November 2021 (period III), and another downward wave of US stock indices after holiday upward correction along with the appreciating USD and inflation fears (period IV).

**Figure 2.** Correlation matrix of Pearson coefficients calculated for all possible pairs of the time series considered in this study (January to October 2022). All the values are statistically significant with *p*-value < 0.00001, except DAX vs. JPY, where *p* = 0.1.
