Bitcoin and global financial stress: A copula-based approach to dependence and causality in the quantiles

TL;DR: In this paper, the authors apply different techniques and uncover the quantile conditional dependence between the global financial stress index and Bitcoin returns from July 18, 2010, to December 29, 2017.

Abstract: We apply different techniques and uncover the quantile conditional dependence between the global financial stress index and Bitcoin returns from July 18, 2010, to December 29, 2017. The results from the copula-based dependence show evidence of right-tail dependence between the global financial stress index and Bitcoin returns. We focus on the conditional quantile dependence and indicate that the global financial stress index strongly Granger-causes Bitcoin returns at the left and right tail of the distribution of the Bitcoin returns, conditional on the global financial stress index. Finally, we use a bivariate cross-quantilogram approach and show only limited directional predictability from the global financial stress index to Bitcoin returns in the medium term, for which Bitcoin can act as a safe-haven against global financial stress.

Summary

1. Introduction

• Bitcoin was first designed in 2009 to allow users to send and receive payments on a peer-to-peer basis.
• From July 2010 to December 2018, Bitcoin has quickly increased from less than one USD to more than fourteen thousand US dollars1.
• On the data level, this study considers the global financial stress index (GFSI) that was recently introduced by Bank of America Merrill Lynch.
• Further analysis indicates that Bitcoin can act as a safe-haven against global financial stress from a medium-term perspective.

2. Research background

• Bitcoin is an innovative peer-to-peer electronic payment network that uses a cryptography protocol to secure transactions.
• The building block of the network relies on an underlying blockchain technology that records and secures all Bitcoin transactions.
• Luther and Salter (2017) show that interest in Bitcoin substantially increased following the March 16, 2013, announcement that Cyprus would accept a bailout.
• Bouri et al. (2017b) indicate that Bitcoin can serve as an effective diversifier for major world stock indices, bonds, oil, gold, the general commodity index and the US dollar index.
• Based on the above, it appears that the GFSI is an essential tool for market participants to make better investment and risk management decisions.

3.1 Data

• Bitcoin prices are collected from CoinDesk (www.coindesk.com/price) and represent the average price of Bitcoin across leading exchanges (Bouri et al., 2017a).
• Introduced in November 2010 by Bank of America Merrill Lynch, the GFSI aggregates 23 measures of stress covering three types of financial market stress (risk, hedging demand, and AC CE PT ED M AN US CR IP T 6 investor appetite for risk) across five asset classes (credit, equity, interest rates, forex and commodity markets) and various geographies.

3.2 Methods

• The empirical analyses rely on three main approaches to uncover the quantile conditional dependence between GFSI and Bitcoin returns (RBC).
• The first one is the dependence via copulas, which can characterize the average movements and the joint extreme movements between the two examined variables.
• The second approach is the out-of-sample approach of Hong and Li (2005), called the Granger causality in distribution (GCD), which captures the Granger causality in distributions in each conditional quantile.
• The third one is the crossquantilogram approach of Han et al. (2016), which allows to measure of directional predictability in quantiles.

3.2.1 Modelling dependence using copulas

• It is well documented that copula functions provide both flexibility and effectiveness in characterizing such movement patterns, allowing obtaining valuable information on the average dependence and tail dependence.
• There are at least three advantages of using copulas in analyzing the dependence.
• Importantly, the tail dependence can measure the probability of simultaneous extreme losses for investors.
• This separation facilitates both the model specification and the model estimation.
• Copulas can jointly combine different univariate models through their copula functions.

3.2.2 Granger causality in distribution (GCD) test

• After identifying the appropriateness of adopting copula models and modelling the average dependence and tail dependence, the authors proceed to uncover the causality dynamic between the GFSI and RBC by computing the quantile forecasts that rely on the inversion of the parametric conditional copula distribution.
• From the modelling perspective, it is more informative to explore the causal relationship between the GFSI and RBC using the GCD test, which can model the causal relation at the extremes of the return distributions rather AC CE PT ED M AN US CR IP 8 than only at the centre.
• Indeed, this hypothesis is supported by the empirical evidence the authors provide later.
• The conditional quantile 𝑞𝛼(𝑌𝑡|ℱ𝑡) is derived from the inverse function of a conditional distribution function: 𝑞𝛼(𝑌𝑡|ℱ𝑡) = 𝐹𝑌 −1(𝛼|ℱ𝑡) (3) AC CE PT ED M AN US CR IP T 9 where 𝐹𝑌(𝑌𝑡|ℱ𝑡) is the predicted conditional distribution function of Yt .
• The quantile forecasting models 𝑞𝛼(𝑌𝑡|ℱ𝑡) are computed by solving the equation 𝐶𝑢(𝐹𝑋(𝑥𝑡+1), 𝐹𝑌(𝑞𝛼(𝑌𝑡|ℱ𝑡)) = 𝛼 (5) To evaluate the predictive ability of those quantile forecasting models 𝑞𝛼(𝑌𝑡|ℱ𝑡) obtained from the seven (I = 7) copula functions for C(u; v), the authors use the “check" loss function of Koenker and Bassett (1978)4.

3.2.3 Directional predictability test

• The authors employ the recent directional predictability test of Han et al. (2016) to complement the GCD test because investors may want to use the GFSI to predict the movement of RBC; this follows the need to access the forecasting performance of RBC using the GFSI as a predictor.
• Furthermore, the use of the bootstrap technique allows for the use of large lags in the directional predictability test.
• The cross-quantilogram proposed by Han et al. (2016) can provide a quantile-to-quantile relationship from the GSFI to RBC.
• To estimate the critical values from the limiting distribution, the authors could use the nonparametric estimation using the stationary bootstrap (SB) of Politis and Romano (1994).
• Second, the directional predictability test allows researchers to select arbitrary quantiles for the GFSI and RBC rather than pre-set quantiles, as in the case of GCD.

4.1 Results for marginal and copula models

• The authors rank the pair of series in ascending order and then divide each series evenly into 10 bins.
• Cell (10,1) has a high number, and there is strong negative left-tail dependence (lower-right corner, Table 2).
• It is interesting to observe that the Bitcoin market can perform well when the global financial markets are in depression.
• The Q-statistics suggest no serial correlation in the standardized residuals of GSFI and RBC.
• In accordance with Patton (2006), the authors calculate the copula likelihood for each candidate copula.

4.2 Results of the GCD test

• From subsection 4.1, the authors find evidence of a right tail dependence by the Gumbel copula, whilst all other copulas indicate no tail dependence.
• The analysis provides no information about the causality between the GFSI and RBC.the authors.
• Therefore, this section seeks to provide a more informative test to examine the GCD as a tool to explore a causal relationship between the GSFI and the return of Bitcoin.
• The authors can observe that a quantile forecasting model with no Granger causality in the quantile is rejected in many quantiles, except for the quantile at 40%, 50%, and 60% with evidence at 1 percent significance level.
• This result shows that the GSFI strongly Granger-causes the RBC at the left tail (poor performance) and right tail (superior performance) but not at the centre (usual performance) of the distribution of the RBC conditional on the GSFI.

4.3 Results of the directional predictability test

• This finding means that when the GSFI is lower than the median, it is less likely to have a very large negative loss for Bitcoin.
• The authors results exhibit a more complete quantile-to-quantile relationship between financial stress and Bitcoin return and show how the relationship changes for different lags.

5. Conclusion

• Initially introduced as an electronic payment system equivalent to cash that could be used nearly anonymously in e-commerce, Bitcoin has quickly gained ground as an investment asset.
• A great deal of attention has been devoted to the technological, cryptographic, and legal aspects of Bitcoin.
• Empirical evidence of its economic and financial aspects, particularly its role as a safe haven against global financial stress, is relatively scarce.
• This extension is important and useful to practitioners and policy-makers in an era of potentially high global financial stress.

Figures

Table 5. Testing for GCQ.

Frequently asked questions

Q1. Why do the authors use the model by Lee and Yang?

The authors use the model by Lee and Yang (2014) to examine the dependence between the GFSI and RBC using a parametric copula because the linear Granger causality test cannot model the asymmetric dependence between the GFSI and RBC, possibly because of the existence of nonlinearity and structural breaks. 

Q2. What is the definition of a distributed ledger?

Blockchain is a distributed ledger made of an unchangeable chain of data blocks spread across multiple sites but chained together cryptographically. 

Q3. What is the significance of the cross quantilograms?

The cross-quantilogram p ̂_α (k) for α1 = 0.7, 0.8, 0.9, and 0.95 is negative and significant for most lags, indicating that when GSFI is very low, it is less likely to have a very large positive gain for Bitcoin. 

Q4. What is the way to model GCQ?

The small p-values of the reality check signal the rejection of the null hypothesis, indicating that there is a copula function to model GCQ and produce a better quantile forecast of the RBC by conditioning on the GSFI. 

Q5. What is the directional predictability test used by Han et al.?

The directional predictability test of Han et al. (2016) was used by Jiang et al. (2016) to investigate the daily, overnight, intraday, and rolling return spillovers of four key agricultural commodities—soybeans, wheat, corn, and sugar— between the U.S. and Chinese futures markets. 

Q6. What is the model for the quantile forecasting of the RBC?

The authors can observe that a quantile forecasting model with no Granger causality in the quantile is rejected in many quantiles, except for the quantile at 40%, 50%, and 60% with evidence at 1 percent significance level. 

Q7. What is the probability of having negative losses to Bitcoin for a maximum of 50-60 days?

This finding implies that when financial stress is higher than the 0.9 quantile, there is an increased likelihood of having very large negative losses to Bitcoin for a maximum of 50-60 days. 

Q8. What is the importance of the copula structure?

the increased knowledge of which risks are essential and against which to hedge them in the different quantiles whilst explaining the copula dependence structure are two crucial aspects of successful investing. 

Q9. What is the main difference between copulas and the Sklar theorem?

copulas are invariant to increasing and continuous transformations (Ning, 2010), such as the scaling of logarithm returns, which is commonly used in economic and finance studies. 

Q10. What is the expected check loss for a quantile forecast?

For each copula distribution function Ck(u; v), the authors also denote the corresponding quantile forecast as 𝑞𝛼,𝑘(𝑌𝑡|ℱ𝑡) and its expected check loss as Qk(α).