Quantifying Cross-Correlations between Economic Policy Uncertainty and Bitcoin Market: Evidence from Multifractal Analysis

We investigate the dynamic correlation between the Bitcoin price (BTC) and the U.S. economic policy uncertainty index (USEPU) from the perspective of multifractality. Utilizing the multifractal detrended cross-correlation analysis (MF-DCCA), we conrm a long-range cross-correlation between BTC and USEPU. Moreover, the empirical results of MF-DCCA show that the power-law properties and multifractal characteristics between BTC and USEPU are signicant. We further examine the long-range dependency of cross-correlation between BTC and USEPU series via the Hurst exponent test and conrm the durable crosscorrelation. Finally, we introduce another multifractal indicator and examine the extent of multifractality among time series. e empirical results indicate that the BTC series, USEPU series, and the cross-correlation of BTC-USEPU present apparent multifractality, where BTC shows the strongest degree of multifractality.


Introduction
In the context of the subprime mortgage crisis and the public's growing concerns over government policy changes, Bitcoin was introduced by Nakamoto [1]. Bitcoin is an adequately decentralized cryptocurrency depending on sophisticated protocols. Since then, the cryptocurrency market represented by Bitcoin has developed rapidly and aroused the widespread interest of regulators, scholars, and investors. A series of studies have conducted in-depth research on price formation [2][3][4], market e ciency [5][6][7], stylized facts [8,9], and price dynamics of Bitcoin [10][11][12][13][14]. Additionally, because of Bitcoin's independence over authority and signi cant uctuations in its prices, the debates over whether it is a safe-haven asset and its relationship with government policy changes have caused controversy [14][15][16][17][18][19]. Especially during the trade war and COVID- 19, there has been a sudden increase in economic policy uncertainty (EPU).
Recent literature examines whether Bitcoin can hedge from uncertainty based on the wavelet-based quantile-onquantile regressions, ordinary least-squares regressions, and the Bayesian Graphical Structural VAR model [20,21]. ey argue that Bitcoin has safe haven properties and can serve as a suitable hedge against di erent kinds of uncertainty. Wang et al. [22] also con rm this argument and nd that EPU (this paper takes three indexes as proxies for EPU, including VIX, equity market uncertainty, and US EPU) exhibits generally weak risk spillover e ects on Bitcoin. In contrast, Conlon and McGee [23] document that the downside risk of a portfolio containing Bitcoin increased signi cantly relative to investing in the S&P 500 alone during the Covid-19 pandemic. ese studies usually focus on the linear properties between variables or simple nonlinear relationships. However, it has been widely accepted that the dynamic relationships between asset prices are inherently nonlinear and even multifractal [24][25][26][27][28][29][30][31][32]. Furthermore, various nonlinear cross-correlations between di erent time series have been confirmed, including EPU and stock market volatility [33], online searches and Bitcoin [34,35], different capital markets [36][37][38][39][40][41], and global stock markets [24,26]. With this view, we attempt to study the dynamic correlation between Bitcoin price (BTC) and EPU from the perspective of multifractality.
Since the financial crisis, the impact of EPU on the economy and finance has also become a hot topic for scholars [42][43][44][45][46]. Based on the frequency of newspaper coverage, Baker et al. [47] proposed a novel EPU index and found that the U.S. EPU index (USEPU) is strongly linked to macro and micro variables such as stock market volatility and employment. At present, North America and Western Europe are the regions with the highest recognition of Bitcoin. e daily EPU data of the United States are publicly available. erefore, we connect USEPU (the uncertainty of the U.S. economy and policy) with BTC to explore the joint dynamics between them. Some recent literature studies the impact of EPU on the Bitcoin market from the perspectives of risk spillovers, volatility effects, and dynamic correlations [48][49][50][51]. On the one hand, the empirical results of the impact of EPU on the Bitcoin market are relatively mixed. On the other hand, Wang et al. [50] demonstrated that there is a dynamic correlation between EPU and bit using the DCC-GARCH model. Inspired by these works of literature, we attempt to use the MF-DCCA method (traditional econometric methods mainly study the correlation between time series from a linear perspective; however, the characteristics of multifractality are common in socioeconomic systems. Also, it is not reliable and comprehensive to analyze financial markets with traditional linear analysis methods. e long memory of financial time series makes it unable to meet the application conditions of the traditional correlation coefficient method. On the contrary, by calculating the multifractal spectrum of the time series, the purpose of understanding the internal complex structure of the fractal can be achieved. erefore, this paper adopts the widely recognized MF-DCCA method to study multifractal features) to examine the dynamic cross-correlations between them. As we know, the advantage of the MF-DCCCA method is to measure the fractal characteristics of time series under different scales on the basis of eliminating local trends and further explore the cross-correlation and linear complex features between different time series [52][53][54].
Hence, we mainly adopt multifractal cross-correlation analysis (MF-DCCA) to investigate the multifractal characteristics between the USEPU series and the BTC series. e empirical evidence shows that there exists a long-range cross-correlation between the USEPU series and the BTC series. Moreover, using a log-log plot, we also find that no matter how the scaling order changes, the cross-correlations within the series demonstrate power-law character with the growing value of the fluctuation function. Furthermore, we calculate the Hurst exponent to test whether the crosscorrelation between the BTC and USEPU series is enduring and find that it is persistent. Additionally, the results show that none of the Hurst exponent lines is constant, providing the preliminary evidence for multifractality. To confirm the multifractality between BTC and USEPU, we calculate another proxy for multifractality and perform the spectrum singularity check. e empirical results indicate that neither the Rényi exponent nor the singularity spectral curve exhibits a typical linear shape, which confirms the existence of multifractality between BTC and USEPU. Finally, we calculate the Hölder exponent difference and find that the BTC series shows the strongest multifractality character with the largest value of the Hölder exponent difference.
Our main contributions to the growing literature are twofold: first, different from the existing literature on the impact of EPU on BTC [48][49][50][51], our study is the first to study the dynamic relationship between BTC and EPU from the perspective of multifractality. Based on the MF-DCCA, we provide strong evidence for power-law cross-correlation and apparent multifractal features between USEPU and BTC. Our findings complement the existing studies on the impact of uncertainty on the Bitcoin market and provide a comprehensive understanding of their joint dynamics. Second, our study enriches the literature on multifractality analysis. e previous studies suggest that nonlinear correlations between EPU and traditional assets (stock, oil, and gold) are persistent [55,56]. Unlike traditional assets, Bitcoin is a decentralized currency that is not dominated by any authority. However, contrary to intuitive expectations, EPU also exhibits robust multifractal characteristics with BTC. Moreover, BTC shows a continuous upward trend, especially after 2020, with a sharp rise and violent fluctuations. Our findings inform the dynamics, long-range autocorrelation, and multifractal characteristics of BTC. We expect that our study could shed light on the risk management of Bitcoin and help investors have a more comprehensive understanding of the innovative financial asset. e remainder of this study is organized as follows. Section 2 introduces the data and discusses summary statistics of the BTC and USEPU series. In Section 3, we describe our research methodology, including crosscorrelation analysis and MF-DCCA. Section 4 presents the detailed results and discusses the main empirical findings. Finally, we conclude in Section 5.

Data
Our sample is mainly composed of two-time series: the U.S. economic policy uncertainty index (USEPU) and Bitcoin price (BTC). Our selection of USEPU follows Baker et al. [47]. e index currently covers 27 major economies globally and has been adopted by many well-known institutions and academic research. As stated by the website, USEPU quantifies the coverage of various newspapers collected from Access World New's NewsBank source according to three sets of terms: (a) the terms related to "economic" or "economy" (e.g., "monetary policy"), (b) the terms related to "uncertain" or "uncertainty" (e.g., "COVID-19"), and (c) the terms related to "government actions" (e.g., "deficit"). We obtain data on the daily USEPU from the public website https://policyuncertainty.com, while the daily BTC is crawled from the website https://finance.yahoo.com, 2 Discrete Dynamics in Nature and Society which provides Bitcoin news, transaction data, and professional message. Since we study the relationship between U.S. economic policy uncertainty and Bitcoin price, the U.S. dollar is chosen as the exchange currency for Bitcoin (Bitcoin price data are usually the latest transaction price of Bitcoin against the US dollar). In this study, we conduct all of our analyses from September 2014 to August 2021. Table 1 presents descriptive statistics of our data sample, which consists of 2,527 observations. Both USEPU and BTC series show right-skew characters, exhibiting greater averages of the USEPU and BTC (124.90 and 8551.19) compared with their medians (91.44 and 5067.11, respectively), and implying distributions within the series is right-skewed. e skewness (2.63 and 2.48) also shows more extreme values at the right end of the series. For the kurtosis, it is accessible that the values of USEPU and BTC are well over 3 (11.52 and 8.81). In other words, the data distributions are steeper than the normal distribution. en, we conduct Jarque-Bera (JB) tests to test whether the series of USEPU and BTC demonstrates normality properties. e results reject the null hypothesis that the time series obeys a normal distribution since their JB statistics are much greater than 0 (10,540.42 and 6,148.18). e statistical results of the BTC and USEPU series provide strong evidence that neither series was normally distributed during our sample period. Figures 1 and 2 document the overall fluctuations of BTC and USEPU, respectively. As a representative of virtual currency, BTC has shown a general upward trend, especially after 2017. As shown in Figure 1, from January 2020 to July 2021, BTC fluctuates significantly, which may be affected by government regulation of digital currencies, transaction demand, investor sentiment, etc. For USEPU in Figure 2, the series exhibits relatively stable fluctuations on average from 2014 to 2019. However, in early 2020, USEPU rises rapidly and then gradually normalizes. e sudden outbreak of COVID-19 and the US election may be the drivers of the violent fluctuations. From the evolution patterns of the BTC and USEPU series, we find that both series present fat-tailed characteristics, especially for the USEPU series, which is consistent with the results in descriptive statistics.

Methodology
To examine whether there is multifractality between BTC and USEPU, we follow Zhou [52] and conduct the prevailing MF-DCCA (multifractal cross-correlation analysis), which combines the advantages of DFA and DCCA. First, we conduct the DCCA proposed by Podobnik and Stanley [57] to examine the long-range characteristics of cross-correlations between BTC and USEPU series. en, we apply the MF-DCCA to quantitatively analyze the power-law crosscorrelation and multifractal features within BTC and USEPU series.

Cross-Correlation Analysis.
Bitcoin is often seen as a safe haven due to its decentralized nature, and at the same time, Bitcoin has been an attractive risk asset in recent years. Hence, is there a linkage between Bitcoin's price and economic policy uncertainty? To acquire a universal understanding of the cross-correlation properties, we initially introduce the cross-correlation statistic Q cc (m). Consider two equal-length series x k and y k , where k � 1, 2, . . . , N. Specifically, Q cc (m) is defined as en, C i represents the cross-correlation function and can be constructed through the following equation: where m represents the degree of freedom; N is the length of the time series x k and y k . If the value of Q cc (m) exactly matches the critical value of χ 2 (m), a statistical cross-correlation cannot be found. Otherwise, we can infer that there is a statistically significant correlation.

MF-DCCA.
Next, we apply the MF-DCCA to quantitatively analyze the power-law cross-correlation and multifractal features within BTC and USEPU series. Wang et al.  Discrete Dynamics in Nature and Society [41] established the widespread MF-DCCA method to investigate the multifractal power-law cross-correlation features between two nonstationary sequences with autocorrelation. Since the MF-DCCA method has been widely recognized, we only briefly introduce the steps as follows.
First, to remove the drift feature of each time series, we transform authentic series, x k and y k , with a deduction of the constant shift (x and y), respectively. e full calculation operation has been demonstrated in the following equations: Second, we break the detrended time series X(i) and Y(i) into N s nonoverlapping equal parts with s observations to acquire a micro understanding of every single series. Taking N as the whole number of sample observations, the formation process of N s is as follows: int(·) speaks on behalf of the integer function (because the time series length N may not always be an integer multiple of scale s, a relatively short portion will be discarded at the end of each profile. Hence, to include the remaining small segments, we do the same for the reverse order of the time series). ird, with split N s parts, we examine the local trend of every single part, ]. Specifically, the detrended covariance of every single part is marked as F 2 (s, v) and can be obtained from the following equations.
For ] � 1, 2, 3, . . ., N s , that is, For ] � N s + 1, N s + 2, N s + 3, . . ., 2N s , we have where X and Y is the nth order polynomial fitness check of part ]. Fourth, we average the local covariances of all detrended segments to get the qth order fluctuation function, F q (s).
us, the fluctuation function can be constructed by the following equations: when q � 0 and It is evident that the movement of F q (s) is affected by the selection of q and s. To acquire a full perception of the multifractality within series, we carry out calculation processes from steps 2 to 4 on different scale s and then continue to the fifth and ending procedure.
Fifth, with the multiselection of scale s, the scaling character of the fluctuation function F q (s) could be found through the gradient analysis for every single q. If a longrange multifractality feature can be discovered within the detrended series, the power-law relationship can be depicted by the following equation: where H xy (q) is the generalized Hurst exponent, which is the slope of the log(Fq(s))∼log(s) function graph (it is evident from equation (9) that the gradient of function varies with the scaling order q. Specifically, considering the case where q takes a value of 2, if the value of gradient H xy (2) is above 0.5, the cross-correlation between time series X(i) and Y(i) is persistent. If H xy (2) is lower than 0.5, the crosscorrelation would be antipersistent).
To have a further study of multifractality within series, we begin to introduce another proxy, Rényi exponent. e proxy is designated as τ xy (q) presented in the following equation: Finally, we execute singularity spectrum analysis as an extra investigation to improve multifractality reliability. rough the Legendre transformation, the singularity spectrum, f xy (α), could be acquired and presented by the subsequent formulas, based on the involvement of the Rényi (τ xy (q)) and Hölder (α xy ) exponents, respectively.
where α xy is defined as the Hölder exponent, representing the strength of the singularity spectrum. erefore, we utilize the strength difference or the spectrum width of the Hölder exponent Δα xy (Δα xy � α xy,max − α xy,min ), as the measurement for multifractality degree. It is easy to learn from the equation that the greater value of Δα xy would represent higher levels of multifractality.

Cross-Correlation
Test. Following the steps described in the previous section, we first conduct the cross-correlation test introduced in Section 3.1 and obtain general knowledge of the relationship between BTC and USEPU series. e results are presented in Figure 3. Specifically, the significance level is 5%, and the degrees of freedom range from 1 to N − 1.
e green and red lines represent cross-correlation statistics Q cc (m) and the chi-square critical value χ 2 (m). e pattern of Figure 3 implies that Q cc (m) of BTC and USEPU is always higher than χ 2 (m) with a distinct gap. us, the pattern of Figure 3 preliminarily demonstrates a long-range crosscorrelation between BTC series and USEPU series.

Multifractal Detrended Cross-Correlation Analysis.
So far, our study has proven the long-range cross-correlation between BTC and USEPU through the cross-correlation test. Next, we utilize the MF-DCCA [52] to perform a comprehensive quantitative examination of the nonlinear characteristics. Following Zhang et al. [34], we assign values to q at intervals of 1, with a minimum value of −10 and a maximum of 10. According to the statement in Section 3.2, we can infer that if q < 0, the time series exhibits weak volatility. In contrast, strong fluctuations are highlighted. e evolution trend of F q (s) for BTC and USEPU is plotted in Figure 4. e results show that no matter how q changes, all lines show a similar evolution pattern, moving upward as time length s gradually increases, implying a long-range power-law cross-correlation between BTC and USEPU.
Furthermore, to examine whether the cross-relationship between the BTC and USEPU series has persistent and multifractal characteristics, we further compute the Hurst exponent H xy (q). As shown in Figure 5, the Hurst exponents of the BTC-USEPU, BTC, and USEPU are significantly greater than 0.5, which is the critical value. In other words, the correlation within the BTC series, the correlation within USEPU series, and the cross-correlation of BTC-USEPU are evidently persistent. From the trend of the curve, we can learn that Hurst exponents for the BTC-USEPU and the correlation within BTC decrease with the increase of q. For USEPU, the overall trend shows an S-shape curve that gradually increases as q varies from −10 to −2, peaks at q � 6, and then decreases. We can infer that the cross-correlations among the three pairs above are strictly multifractal. Specifically, when q is less than 0, the Hurst exponent of the correlation within BTC is greater than BTC-USEPU and the correlation within USEPU. at is, correlation within BTC exhibits the highest degree of durability. When q > 0, the blue line (the correlation within USEPU) climbs abruptly, exceeding the Hurst index of BTC-USEPU. In general, the results of Figure 5 demonstrate the persistence of the crosscorrelation between BTC and USEPU and provide preliminary evidence for multifractality.
To further demonstrate the existence of multifractality, we then investigate the pattern of Rényi exponents, including the cross-correlation of BTC-USEPU (red line), the correlation within the BTC series (green line), and the correlation within USEPU series (blue line). e results are presented in Figure 6. It is obvious that all three curves are distinct from typical linear shapes, confirming that BTC-USEPU cross-correlation conforms to multifractal characteristics.
Finally, we measure the degree of multifractality of the cross-correlation between BTC and USEPU using the singularity spectrum test, and the results are documented in Figure 7. e patterns indicate that all singularity spectrums are distinct from linear shapes, which is also consistent with the characteristics of multifractality. Furthermore, as shown in Table 2, we calculate the gap between minimum and maximum Hölder exponents (Δα) to represent the degree of multifractality. e delta values of BTC-USEPU, USEPU, and BTC are 0.417, 0.603, and 1.008, respectively. at is, BTC shows the strongest multifractality.

Conclusions
Our study examines the dynamic correlations between BTC and USEPU from September 2014 to August 2021. We mainly employ MF-DCCA to obtain a comprehensive insight into the long-range cross-correlation properties. First, we utilize cross-correlation analysis to preliminarily prove the long-range cross-correlation between BTC and USEPU. To further examine the nonlinear cross-correlation properties, we calculate the qth order fluctuation functions with an interval of one. All the curves demonstrate a similar evolution pattern, moving upward while time length s gradually increased, indicating the power-law properties and multifractal characteristics between BTC and USEPU. en, we calculate the Hurst exponents to test the persistence of the cross-correlation, and the results support the existence of persistence. Moreover, we calculated another indicator of multifractality and performed a spectral singularity check to revalidate the multifractality between BTC and USEPU. Due to the unique design of Bitcoin's decentralization, the existing research on whether Bitcoin can be used as a safehaven asset is still under discussion. Our findings inform the dynamics, long-range autocorrelation, and multifractal characteristics of BTC. Also, we confirm the dynamic crosscorrelation between BTC and USEPU. Based on the multifractal analysis of BTC and USEPU, our findings can provide valuable insights into the understanding of Bitcoin price dynamics and potential suggestions to market participants on the risk management of Bitcoin.

Data Availability
We affirm that the Bitcoin price and economic policy uncertainty data used to support the findings of this study are available at https://finance.yahoo.com and https:// policyuncertainty.com/ which is widely subscribed by the financial researchers, respectively.  Figure 7: Spectrum singularity of f(α) versus α for BTC and USEPU cross-correlation, the correlation of BTC, and the correlation of USEPU.