Financial stress is defined as the force on financial markets and possible loss expectations. Financial stress and risk directly affect companies, financial markets, intermediary institutions and organizations, and the real economy. It arises because of the overvaluation of financial crises and causes disruptions in the functioning of financial markets (Hakkio and Keeton, 2009; cit. in Alsu, 2020). Although financial stress is often used as an indicator for predicting financial crises, it also adversely affects financial markets and economic growth and development (Barut et al., 2016). Financial stress has recently involved developing and developed financial markets and economies (Zhang and Wang, 2021).

There exists a vast literature on financial stress's effect on macroeconomics, financial markets, and the spillover among countries. [e.g., Sum (2012); Nazlıoğlu et al. (2015); Apostolakis (2016); MacDonald et al. (2018); Gil-Alana and Abakah (2019); Liu et al. (2021)]. However, the literature on the possible relationship between cryptocurrencies and financial stress is limited (Zhang and Wang, 2021). Attention to cryptocurrencies, considered risky investment instruments, is increasing daily. Therefore, it is essential to investigate whether the variance resulting from sudden changes in cryptocurrencies has a spillover relationship with the conflict resulting from sudden changes in the financial stress index, which shows the possible loss in financial markets. This paper examines the volatility modelling of Bitcoin, the financial stress index and the volatility spillover among them. The results are expected to have implications on investors' risk predictions.

This paper examines the volatility model of Bitcoin and the financial stress index for the period between the 7th of January 2011 and the 24th of December 2021 and consists of four sections. The introduction in the first section provides general information on the subject. The second section covers the Literature Review. The third section, Methodology, presents the research model and related empirical findings. Finally, the study is concluded by evaluating these empirical findings.

The literature review on the subject clearly shows a limited number of studies examining the volatility spillover among the financial stress index and cryptocurrencies. In contrast, the studies on the volatility spillover among macroeconomic variables, stock markets, financial stress index and cryptocurrencies are outnumbered, as given below.

Studies on volatility modelling and volatility spillover in the financial stress index:

Nazlıoğlu et al.'s (2015) paper examines the volatility spillover between WTI crude oil prices and Cleveland financial stress index for the pre-crisis, in-crisis, and post-crisis periods between 1991 and 2014, pointing to a causality relationship that was found in oil prices to financial stress index in the pre-crisis period and from financial stress to oil prices in the post-crisis period. Besides, the impulse response analysis concludes that the volatility spillover pattern has similar dynamics in the pre-and post-crisis periods. Apostolakis (2016) examines the spillover of financial stress in five Asian countries (China, South Korea, Malaysia, Thailand, and the Philippines), empirically finding that China impacts financial stress in other countries. In their study, MacDonald et al. (2018) examine the volatility spillover among Eurozone economies and financial markets using financial stress indices employing the GARCH model. Their findings reveal that financial stress spillover significantly affects the Eurozone banking and money markets. In their study, Gil-Alana and Abakah (2021) examine the stochastic characteristics of financial stress indices of 10 Asian countries and how they are transmitted among countries. They found that shocks will have temporary but long-term effects for all nations, and the spillover of financial stress among Japan and smaller economies is faster than in China. Zhang and Wang's (2021) study examines the volatility spillover of the financial stress index in China and the U.S. between gold and Bitcoin. They report that the financial stress index has a medium-term effect on gold prices in the U.S. during periods of uncertainty. It has a short-term impact on Bitcoin, while the financial stress index in China has a medium-term impact on Bitcoin. Apostolakis et al. (2021) examined the relationship between non-financial Brent oil market prices and financial stress and economic policy uncertainties in Group of Seven (G7) countries. Findings show that oil price volatility has a more significant spillover to financial stress and monetary policy uncertainty during COVID-19. Oil price uncertainty is associated with more financial stress for some G7 countries. In their study, Liu et al. (2021) examined the effects of different oil price shocks on China's financial stress index. They found that oil supply shocks have a significant positive impact on financial stress in case of low volatility, while demand shocks have a detrimental effect on financial stress.

Studies on volatility modelling and volatility spillover in cryptocurrencies:

Bouri et al. (2018) studied the volatility spillover among Bitcoin and stocks, commodities, foreign currency, and bonds from 2010 to 2017 under bear and bull market conditions. They reveal that the volatility spillover from stocks, commodities and exchange rates to Bitcoin is more significant than the volatility spillover from Bitcoin to stocks, commodities and exchange rates. Kumar and Anandarao (2019) examine the volatility spillover among Bitcoin, Ethereum, Ripple, and Litecoin for 2015-2018 using the dynamic conditional correlation-GARCH method. They identify a volatility spillover between Ethereum and Litecoin. In their study, Vardar and Aydoğan (2019) examined the volatility spillover between Bitcoin and conventional assets such as equities, bonds, and currencies between 2010 and 2018 using VAR-GARCH and BEKK methods. Their findings point out a one-way spillover effect from the bond market to Bitcoin. Katsiampa et al.'s (2019) study examined the volatility structure and the volatility spillover among Bitcoin, Litecoin, and Ethereum. They identify a two-way volatility spillover between Bitcoin and Ethereum and Bitcoin and Litecoin. Finally, Zhang and He (2021) examined the spillover effect among Bitcoin, gold, crude oil, and stock markets. They found that Bitcoin does not have a significant spillover effect compared to other assets, and there is a one-way volatility spillover between gold and stock markets.

This paper aims to examine the volatility model of Bitcoin and the financial stress index for the period between the 7th of January 2011 and the 24th of December 2021 using weekly data and the volatility spillover among them.

Firstly, continuous returns are calculated using the formula in equation 1 using the Bitcoin weekly price series.

St. Louis Fed Financial Stress Index is a proxy for the financial stress variable. St. Louis Fed Financial Stress Index is designed with an average zero value. Zero indicates normal market conditions, negative values indicate below-average financial market stress, and positive values indicate above-average financial market stress (fred.stlouisfed.org). Secondary data on the financial stress index and the Bitcoin price variables are obtained from fred.stlouisfed.org and investing.com websites.

The stationarity of the series is checked by ADF and P.P. tests to examine the volatility spillover between the financial stress index and Bitcoin.

The equations for the ADF test are as follows:

The equation for the PP test is as follows:

After checking the stationarity of the series, the best fitting ARMA model is determined according to the appropriate lag duration based on the Schwarz information criterion. Finally, the ARMA process is based on the assumption that the series is stationary, and A.R. (p), M.A. (q) and their combination ARMA (p,q) are applied to stationary processes.

The equation for the AR process is as follows:

The equation for the MA process is as follows:

The equation is as follows when the Y time series fits both AR and MA processes:

After determining the best fitting initial ARMA model, we examined whether heteroscedasticity and autocorrelation problems related to the model's error term and whether the series contained nonlinearity. Since the series have heteroscedasticity, autocorrelation problems, and nonlinearity, it is determined that the ARCH/GARCH models are required for volatility forecasting instead of the ARMA model. Various symmetric and asymmetric GARCH models select the most fitted model for volatility forecasting. The equations of the models that exceed only the significance and parameter constraints used in the study are provided below:

The equation for the GARCH (p, q) model is as follows:

The equation for the IGARCH (p, q) model is as follows:

Following the forecasting of volatility for the series, residual volatility series (GARCH conditional variance series) is created for each variable through IGARCH (1,2), GARCH (1,1) models to identify the volatility spillover among series [Nazlıoğlu et al. (2015); Jan and Jebran (2015)]. Then, spillover, impulse-response, and variance decomposition analyses are performed on these series. First, the volatility spillover among series is examined using the Diagonal VECH method.

The equation for the diagonal VECH model is as follows:

A standard VAR model is created to perform impulse-response and variance decomposition analyses among volatility series. The equations related to the VAR model are as follows:

An impulse-effect analysis is performed using the VAR model. The equation regarding to the impulse-effect analysis is as follows:

Following the impulse-response analysis, variance decomposition analysis determines which variables most affect a particular variable. The equation is represented as follows:

The empirical findings provided are given below in the following sections.

The descriptive statistics related to the Financial Stress Index (FSI) and Bitcoin (BTC) series are provided in Table 1 due to the analysis performed for volatility forecasting.

Descriptive statistics show that the average value for the BTC series is 1.878 and -2.297 for the FSI series, and the standard deviation in the BTC series is higher than in the FSI series. The J-B probability of the series is less than 0.05, which is the critical value. Thus, the series is not normally distributed. Then, the stationarity of the series is checked. The graphs of BTC series return and FSI series index values are demonstrated in Figure 1.

Series graphs show an upward and downward trend and fluctuate around a fixed average value, and there is a deviation from the average in the FSI index only in 2020. Therefore, even though it can be deducted from the graphs that the series is stationary, this result needs to be supported by unit root tests. The unit root test results are shown in Table 2.

When ADF and P.P. test results for the BTC and FSI series are examined, it is observed that the probability values calculated for both tests, both in constant and inconstant and trend, are smaller than 0.05, which is the critical value. Therefore, the null hypothesis of "there is a unit root" in both series is rejected, and it is concluded that both sequences are stationary. As a result of the stationarity analyses, the most fitting initial ARMA model for the series was determined based on the Schwarz Information Criteria. Combinations calculated up to the fifth lag are shown in Table 3.

According to the Schwarz Information Criterion, the ARMA (4,4) model with the lowest coefficient for the BTC series and the ARMA (3,2) model with the lowest coefficient for the FSI series are identified as the best fitting initial models. After determining the best appropriate ARMA model for the series, heteroscedasticity, autocorrelation problems, and whether the series include nonlinearity are examined using ARCH LM heteroscedasticity, error terms correlograms, and BDS linearity tests. The test results are shown in Table 4.

When the ARCH LM heteroscedasticity test results for the BTC and FSI series are examined, it is observed that the probability value calculated for the 30^th lag and following values is less than the critical value of 0.05. When the error terms correlograms of the series are examined, it is observed that the Q statistics probability values calculated for the 30^th lag and the following ones are smaller than 0.05. These results indicate that both series have heteroscedasticity and autocorrelation problems. When the BDS test results of the series are examined, it is observed that the test probability value is less than the critical value of 0.05 in both series, and there is nonlinearity in both series. Since the series contained heteroscedasticity, autocorrelation problems, and nonlinearity, ARCH/GARCH models are needed for volatility forecasting instead of the ARMA model. To this end, various degrees of symmetric and asymmetrical GARCH models are tried for volatility forecasting, and models that fulfil significance and parameter constraints are selected. Only the GARC (1,1) model fulfilled the significance and parameter constraints for the BTC series. For the FSI series, the GARCH (1,1), GARCH (1,2), IGARCH (1,1), and IGARCH (1,2) models fulfil the significance and parameter constraints. For the FSI series, the IGARCH (1,2) model, which has the smallest Theil Inequality Coefficient, is the most fitting model. The models are presented in Table 5.

The models shown in Table 5 fulfil the significance and parameter constraints. In this context, the GARCH (1,1) model is the most fitting model for the BTC series for volatility forecasting. For the model to be valid, the 𝛼₀ coefficient should have to be significant and positive, the 𝛼₁ve β₁ coefficients should have to be substantially positive, and their sum must be smaller than one. The model fulfils these constraints. When the coefficients are examined, the 𝛼₁ coefficient is observed as 0.256, and it can be said that past shocks cause about 25% of the shocks affecting the BTC series. The β₁ coefficient is observed as 0.718, and it can be said that past shocks cause about 71% of the shocks affecting the BTC series. The most suitable model for the FSI series is IGARCH (1,2), and for the model to be valid, the 𝛼₁, β₁ and β₂ coefficients must be significant and positive, and their sum should have to be smaller than one. The model fulfils these restrictions, and the 𝛼₁ coefficient is 0.315, the β₁ coefficient is 0.257, and the β₂ coefficient is 0.425. Therefore, it can be said that about 31% of the shocks affecting the FSI index are caused by past shocks, and 67% are caused by previous shocks. In addition, it can be said that shocks that affect volatility in both indices have a short-term, lasting effect. Whether the models solve the problems of heteroscedasticity and autocorrelation is examined using ARCH-LM and error terms correlograms. The results of the tests are shown in Table 6.

The F statistical and Q statistical probability values for the GARCH (1,1) and the IGARCH (1,2) models of the BTC and FSI series, respectively, are more significant than 0.05, the critical value. Therefore, it is determined that the models solve the problem of heteroscedasticity and autocorrelation. Following the volatility modelling, it is examined whether the asymmetric model is sufficient for volatility modelling of the series using the Engle-Ng (1993) Sign Bias test. The test results are shown in Table 7.

According to the Sign Bias test results, the test probability values of both indices are greater than 0.05, which is the critical value. Therefore, the null hypothesis that there is no leverage effect in the series cannot be rejected. In this context, it can be said that there is no leverage effect on the series and that the models selected for volatility forecasting are valid. The conditional heteroscedasticity graphs created for the GARCH (1,1) and IGARCH (1,2) models for the BTC series and the FSI series are shown in Figure 2.

According to the dependent heteroscedasticity graphs of the series, it can be said that volatility took place in the BTC series in 2011, 2013 and 2014 and in the FSI series in 2020. The news impact curve (NIC) for the GARCH (1,1) and IGARCH (1,2) models for the BTC series and the FSI series is shown in Figure 3.

Figure 3 shows that the news impact curves for the GARCH (1,1) model for the BTC series and the IGARCH (1,2) model for the FSI series are symmetrical, so the time series are asymmetric.

Following the forecasting of volatility for the series, residual volatility series (GARCH conditional variance series) are created for each variable through IGARCH (1,2), GARCH (1,1) models to identify the volatility spillover among series [Nazlıoğlu et al. (2015); Jan and Jebran (2015)]. Finally, the volatility spillover between the series is examined using the Diagonal VECH GARCH method through the generated sequence.

The standard ARCH and GARCH parameters offer information about the shared variance among the variables. The fact that the sum of the coefficients for the parameters is less than 1, positive, and significant indicates volatility spillover among the variables (Yaman and Korkmaz, 2020: 697). According to the analysis results, the sum of the coefficients of the familiar ARCH and GARCH parameters is less than 1, positive and significant. Therefore, it can be said that there is a positive volatility spillover from the FSI variable to the BTC variable.

Impact-response and variance decomposition analyses are performed on residual volatility series created for each variable. In this context, the VAR model was created first. The VAR model lag duration test results are shown in Table 9.

The lag duration of the VAR model is determined to be eight according to the L.R., FPE, and AIC information criteria. VAR model results are shown in Table 10.

Before proceeding to the VAR model's impact-response and variance decomposition analysis, the autocorrelation assumption regarding the stationarity and error terms of the model was tested. The inverse roots of the A.R. characteristic polynomial for the stationarity of the model are shown in Figure 4.

The graph of the stationarity of the VAR model shows that the inverse A.R. roots are within the unit circle, and the model is stationary. The autocorrelation test results are shown in Table 11.

According to the L.M. test results, the test probability values are more significant than the critical value of 0.05. Therefore, it was determined that there is no autocorrelation problem in the model. After examining the assumptions related to the model, the impulse-response analysis is carried out. The results are shown in Figure 5.

According to the results of the impulse-response analysis, a shock in the FSI volatility series caused a negative shock at week 5 in the BTC volatility series. This effect disappears as of week 35, coming close to zero. Conversely, a surprise in the BTC volatility series causes a positive shock at week 5 in the FSI volatility series. This effect disappears as of week 30, coming close to zero. Following the impulse-response analysis, the variance decomposition analysis is performed to determine what percentage of the changes in the BTC volatility series are caused by itself and the FSI volatility series, and what percentage of the changes in the FSI volatility series are driven by itself and the BTC volatility series. Analysis results are shown in Tables 12 and 13.

According to the variance decomposition analysis results, in the first period, all of the changes in the BTC volatility series are caused by themselves. In the second period, the variation outside the FSI volatility series, about 0.16%, is caused by the FSI volatility series. As of the 15^th period, approximately 1% is caused by the FSI volatility series, and this rate increased. Therefore, it can be said that the changes in the BTC volatility series are caused mainly by itself, and the series is most affected by its shocks. The results of the variance decomposition test for the FSI volatility series are shown in Table 13.

According to the FSI volatility series analysis results, almost all of the changes in the FSI volatility series are caused by itself in the first period. During the second period, the variables outside the BTC volatility series, about 2%, are caused by the BTC volatility series. As of the 15^th period, approximately 3% is caused by the BTC volatility series, which was found to increase. Therefore, it is safe to say that the FSI volatility series is also most affected by its shocks. Following the impulse-response and variance decomposition analyses of the residual series of the volatility models, examining the effect between Bitcoin return series and financial stress index values is thought to contribute to the findings. In this context, first, the VAR model is created for the Bitcoin return series and the financial stress index, and the results of the lag duration of the VAR model are shown in Table 14.

The lag duration of the VAR model is determined to be two according to the L.R., FPE, AIC, and H.Q. information criteria. VAR model results are shown in Table 15.

The inverse roots of the A.R. characteristic polynomial for the stationarity of the VAR model are shown in Figure 6.

The graph of the stationarity of the VAR model shows that the inverse A.R. roots are within the unit circle, and the model is stationary. The autocorrelation test results are shown in Table 16.

L.M. test results show that the test probability values are more significant than 0.05, the critical value for the 10^th lag and following lags. Therefore, it is determined that there is no autocorrelation problem in the model. After examining the assumptions related to the model, the impulse-response analysis is carried out. The results are shown in Figure 7.

The impact-response analysis results of the BTC return series and Financial Stress Index values show that a shock in the FSI series causes a negative surprise in week 5 in the BTC return series. However, this effect disappears as of week 10, coming close to zero. In contrast, a shock in the BTC return series caused a negative surprise in the FSI series in week 5. However, this effect disappeared as of week 20, coming close to zero. Therefore, it can be said that there is a longer-term effect among volatility series. Following the impulse-response analysis, the variance decomposition analysis is performed to determine what percentage of the changes in the BTC return series are caused by itself and the FSI series, and what percentage of the changes in the FSI series are caused by itself and the BTC return series. The analysis results on the BTC return series are shown in Table 17.

According to the variance decomposition analysis results, in the first period, all of the changes in the BTC return series are caused by itself. However, the variables outside the FSI series, in the second period, about 0.04%, were caused by the FSI series. Finally, as of the 15^th period, approximately 0.24% is caused by the FSI series, and this rate was found to increase. The results of the variance decomposition test for the FSI series are shown in Table 18.

According to the results of the variance decomposition analysis of the FSI series, in the first period, 99% of the changes in the FSI series are caused by itself and the variables outside the BTC return series. In the second period, about 0.68% is caused by the BTC return series. As of the 15^th period, the BTC return series caused approximately 1%. This rate is found to be increasing. Therefore, it can be said that the changes in the BTC return series and FSI series are caused mainly by themselves, and the series are most affected by their shocks. By comparing the variance decomposition of the volatility series with the analysis results, it can be said that the changes in the volatility series are caused mainly by each other.

Financial stress refers to the changes that cause disruptions in financial markets that negatively impact financial markets and the entire economy. Recent studies on financial stress show that financial stress can reflect uncertainties and unexpected shocks in markets.

This paper aims to examine the volatility model of Bitcoin and the financial stress index for the period between the 7th of January 2011 and the 24th of December 2021. Firstly, a volatility model is employed for the series, and the most fitting volatility models are found to be GARCH (1,1) and IGARC (1,2) for the BTC and FSI series, respectively. Volatility is observed in 2011, 2013, and 2014 in the BTC series and 2020 in the FSI series. Following the forecasting of volatility for the series, residual volatility series (GARCH conditional variance series) are created for each variable through IGARCH (1,2) and GARCH (1,1) models to identify the volatility spillover among series (Nazlıoğlu et al., (2015); Jan and Jebran (2015)). Using the Diagonal VECH GARCH method, the volatility spillover between the series is examined through the series, and a positive volatility spillover effect from the FSI variable to the BTC variable was found. Then, impulse-response and variance decomposition analyses are performed on residual volatility series created for each variable. In this context, the VAR model is created first. According to the results of the impulse-response analysis performed on the VAR model, a shock in the FSI volatility series causes a negative shock at week 5 in the BTC volatility series. This effect disappears as of week 35, coming close to zero. A surprise in the BTC volatility series caused a positive shock at week 5 in the FSI volatility series. This effect disappeared as of week 30, coming close to zero. According to the results of the variance decomposition analysis performed after the impulse-response examination, it is found that the series are most affected by their shocks. Following the analyses of the volatility series, impulse-response and variance analysis analyses are performed on the BTC return series and Financial Stress Index values to render the findings comparable and supportable. Similar results are obtained with the analyses made on the volatility series.

The financial stress index and Bitcoin volatility's mutual influence show that an economic crisis may affect the cryptocurrency market, and the risks experienced in the cryptocurrency market may cause financial problems. Overall, it can be said that there is a relationship between the financial stress index and Bitcoin, and the findings are essential for investors using Bitcoin in their investment strategies, fund managers, and policymakers.