Relationship between Bitcoin Exchange Rate and Other Financial Indexes in Time Series

Department of Fashion Business and Merchandising, Ling Tung University, Taichung, Taiwan Department of International Business, Providence University, Taichung 43301, Taiwan Department of Leisure and RecreationManagement, National Taichung University of Science and Technology, Taichung, Taiwan Department of Finance, Providence University, Taichung, Taiwan Department of Finance, National Taichung University of Science and Technology, Taichung, Taiwan Department of Automatic Control Engineering, Feng Chia University, Taichung 40724, Taiwan Department of Aeronautical Engineering, Chaoyang University of Technology, Taichung 413310, Taiwan


Introduction
e Bitcoin exchange rate (BER) is extremely volatile. As shown in Figure 1, BER to the US dollar from October 01, 2013, to June 22, 2018, increased by nearly 2 million times from less than USD 0.01 to 19,345.49 on December 16, 2017. Kurka concluded that BER is independent of any financial asset classes in the system, but its spillover effect affected the traditional financial markets [1]. e pricing models of stock price or exchange rate have assumptions of transactions by rational investors. However, researchers pointed out that investment decisions made by investors are not necessarily rational. us, behavioral financial theories emerged to explain irrational decisions. For example, Barber and Odean proposed the "Attention eory" that explained investors' intention of buying stocks without having time to interpret the disclosure of massive information on the stocks [2]. Merton suggested the "Investor Recognition Hypothesis" that stated the information spillover effect of investors' stock buying on the firm's visibility [3]. e higher the visibility of a firm, the more the interest investors have in the stock. It yields higher price returns and greater trading volume. Cai et al. proved that Google SVI of Bitcoin had a significant impact on its price and transaction volume as the proxy explanatory variable of "investor focus" [4].
is research employs the vector autoregression (VAR) model to explore the long-and short-term relationship between the Google SVI and BER.
e key variables are studied by applying the theoretical basis of "investor attention" that influences BER. Based on the findings of the current research, the important variables of traditional financial assets are selected in consideration of the volatility index (VIX), the S & P500 index, and the Google SVI. e vector autoregression (VAR) analysis of the short-and longterm impacts and Granger causal relationship ( [5][6][7]) are employed to develop a prediction model for BER change for providing a reference for Bitcoin hedging operation to investors.
e BER is a new investment tool with higher volatility to the traditional financial product. Investors have little hedge and others reference target to avoid the risk of BER. e forecasting model of BER can offer mostly relation financial products and be applied for hedge, arbitrage for investors. e result expected the SVI should have higher relationship in the short and long periods.
is article is organized as follows: Section 2 reviews previous references and discusses the relation of the Google SVI and VIS, in variable explanation to Bitcoin. Section 3 presents the VAR for the data analysis as the research method. Section 4 describes the analysis results. Finally, Section 5 concludes this article.
Since "Google Trends" was first introduced, the number of searches was queried by entering the targeted word. Later on, Google launched "Google Insights for Search" which divided the data into detailed categories such as time and geographical regions. "Google Insights for Search" became the current Google Trend in September 2012. e time range for searching is customized or selected in the past 1, 4 hours, 1, 7, 30, 90, and 5 years. Up to 5 keywords are analyzed simultaneously. e search items are searched for their trends in a single country or all over the world. e categories are selected by the industry of interest or all categories. After determining the keywords, region, periods, and categories, Google Trends generate a trend chart with a standard quantitative range from 0 to 100 and calculate the SVI data based on the average time series, but the data frequency is a week. SVI calculation formula is as follows: SVI � total search volume maximum total search volume in a given time period × 100.
(1) e SVI may yield different results for the same keywords according to different query, time, date, and geographic location. When Google compiles SVI data, it does not search all data but randomly selects samples. However, different results did not change the research outcomes [12]. Takeda and Wakao present the SVI has a positive impact on the stock price and volume [13]. Aouadi et al. using Google search volume to evidence the relationship among the attention of French stock market investors and trading volume, stock market illiquidity, and volatility [10].
Zhang et al. pointed out that the long-term prices converged to the mean reversion when stock prices rose solely with increased investor attention [14]. To understand whether the mean reversion of BER is related to Google SVI, we selected Google SVI as the proxy interpretation variable for "investor focus". Details are shown in Figure 2.

VIX.
e Volatility Index (VIX) is compiled by the COBE Exchange that implies volatility of future options on the S&P500 and reflects the degree of risk in the stock market over the next 30 days. e VIX greater than 40 indicates that market investors expect a strong fluctuation of stock index in the future, that is, irrational panic. When the index is less than 15, it means the investors believe the volatility in the future stock market to be mild which shows irrational exuberance. As it reflects the traders' expectations on future stock price change, it is also called the "Fear Index" or " e Investor Fear Gauge." Qadan et al. believed that VIX reflected the investors' sentiment. In a period of extreme uncertainty, investors tend to be more risk-averse and therefore need higher idiosyncratic volatility (IVOL) premium at a high level of VIX [15]. Under the circumstance, investors avoid stocks of high IVOL, which results in lower investment returns with high IVOL than with low IVOL. Tsai et al. also found the reversed prices in the future led to significant negative investment loss when investors overestimated stock prices positively [16]. Simon and Wiggins showed that the VIX index was used to predict the future stock market [17]. Lee pointed the new sentiment of the current period has a positive relationship with investment reports, and the new sentiment of the lagged period has a negative relationship with investment aspirations [18]. Copeland and Copeland noted that COBE Volatility Index (VIX) can be treated as leading index of stock price index [19]. Dennis et al. treated VIX is a proxy variable of stock return volatility [20]. When the VIX soars and the market shows extreme fear, it would be the best opportunity to enter the stock market.
is research includes the VIX by Chicago Board Options Exchange (COBE) as a proxy explanatory variable for investor sentiment in an empirical study to understand the impact of investor sentiment on the BER.

Bitcoin.
Kurka studied the interaction between Bitcoin and traditional finance assets [1]. e results showed that BER was independent of any financial system assets. ough traditional financial assets had little impact on the BER trend, the spillover effect of BER was significant in the traditional financial market. Cai et al. adopted the Google SVI as the proxy explanatory variable for investors' decisions [4]. e results supported that the SVI had a vital influence on the price and volume of Bitcoin. Dastgir et al. surveyed Google Trends search flow on Bitcoin [21]. It employed the Granger causality to test the causal relationship between the interest in Bitcoin and investment returns. e conclusion supported the existence of bidirectional causality between Bitcoin and investment returns.
is study analyzed the short-and long-term relationship between Google SVI and BER by the VAR model. e longterm relationship among BER, VIX, and the S&P500 index was also investigated for a co-integration phenomenon.

Methodology and Model
e research process was divided into two aspects as shown in Figure 3. For the short-term impact analysis, the VAR model and Granger causality were employed to test the explanatory variables of BER and verify their relationship, respectively. For the long-term impact, Nelson and Plosser argued that the variable difference was lost in the implicit information on the long-term equilibrium through the stationary sequence [22]. erefore, the co-integration analysis and vector error correction model (VECM) validated the long-term relationship between the variables.

VAR Model and VECM.
e VAR model in this study is defined as follows: where y t is a (n×1) vector of endogenous variables, c � (c 1 , .. .., c n ) is the (n×1) intercept vector of the VAR, ∅ t is the i-th (n×n) matrix of autoregressive coefficients for i � 1, 2, ..., n, and ε t � (ε 1 . . . ε t ) is the (n×1) generalization of a white noise process [24]. e following equation is the co-integration transformation of equation (2). .
If y t has a co-integration relationship, then Πy t−1 ∼ I(0) and equation (3) is expressed as follows: where β ′ y t−1 � ECM t−1 is the error correction term for a long-term equilibrium relationship between variables. Equation (4)

Mathematical Problems in Engineering
Equation (5) is used for the vector error correction model (VECM) in which each equation belongs to an error correction model [25].
Before establishing the VAR model, it is necessary to choose the optimal lagged periods. e criteria include Akaike's Information Criterion (AIC), Hannan-Quinn Criterion (HQ), Schwarz Criterion (SC) likelihood ratio (LR), and so on. However, the Akaike proposed the AIC method and SC method which are most used [26]. e equations of the AIC and SC criteria are defined as follows: where k is the number of all parameters to be estimated by the VAR model. e number of periods corresponding to the minimum value of the measurement of AIC or SC is the optimal number of lagged terms. e AIC is more consistent than SC. However, as the estimated parameters of the AIC are less and the number of samples is larger than the SC, the SC is better than the AIC for this research.

Multivariate Granger Causality
Analysis. Multivariate Granger causality analysis is performed by fitting a VAR model to the time series. Let Χ(t) ∈ R d×1 (t � 1, . . . , T) be a d-dimensional multivariate time series. Multivariate Granger causality is measured by fitting a VAR model to L time lags as follows: where ] t is a white Gaussian random vector and Φ λ is a matrix for every λ. A time series X i is called a Granger cause of another time series X j , if at least one of the elements Φ λ (i, j) is significantly larger than zero.

Data Source and Sample Collection Period.
is research mainly discussed the key factors that affect the changes in BER by referring to Cai et al. [4], Kurka [1], and Hsieh [27]. e main variables included Google SVI, the VIX of Chicago Board of Exchange, S&P500 index, gold prices, US dollar index, and Japanese exchange rate. e period for sampling was from April 29, 2013, to June 22, 2018. e data sources were Yahoo Finance and Bloomberg. e data frequency was a day, and the data processing method did not affect the final results. If there was no observation value for any variable on the same day, the data of that day were deleted [28]. A total of 1,299 daily observation values for each variable were used.

Analysis Results and Discussion
e analysis was carried out in two processes as shown in Figure 4. We used the VAR model and Granger causality to test the explanatory stationary time series variables for the short-term impact. en we used the co-integration analysis and vector error correction model (VECM) to validate the long-term relationship between the nonstationary variables. e descriptive statistics, VAR model analysis, and the cointegration relationship were performed and illustrated below. Table 1 shows standard deviation, and maximum and minimum BER, gold price (GOLD), USD/JPY exchange rate (JPY), S&P500 index (SP500), Google SVI, US dollar index (USD), and VIX during the study period.

Unit Root Verification.
e Augmented Dickey-Fuller (ADF) and the Phillips-Perron (PP) tests ( [29,30]) were applied to the unit root sequence in this study. By taking the natural logarithm and first-order difference, all the variables were in a significant stationary sequence as shown in Table 2.

Optimal Lagging Period Selection.
e research employed the AIC to determine the optimal number of two lagged periods. Table 3 shows the details of the optimal lagged periods.

Results of Analysis of the VAR Model.
e first-order difference in the variables presents a stationary sequence and is used to analyze the short-term influence of the variables with the VAR model. e results are shown in Table 4. BER of the two lagged periods had a positive impact (0.108278) on the overall BER at a significant level of 1%. e Google SVI had a negative effect (−0.100983) on the BER at a significant level of 1%, while Google SVI has a positive impact (0.064947) on the BER at a significant level of 1%. e S&P500 index had a negative impact (−0.108234) on the leading two-period S&P500 index at a significant level of 1%. p < 1, 5, or 10% of the significant level indicates that the coefficient is not equal to zero, that is, the S&P500 index Granger causality causes the lag two-period S&P500 index. e Granger causality diagram of the short-term variables affected the BER. e results demonstrated that the feedback relationship appeared only in the BER and the Google SVI among the Granger causality as shown in

Johansen Maximum Likelihood Co-Integration Test.
Nelson and Plosser supposed that a different process caused the time series to lose long-term information [22]. at is, all nonstationary time series become stationary after the first-order difference operation, so it is used to study the short-term influence between variables.
Johansen co-integration test was used to test long-term relationship among the variables, and the Trace test to obtain the number of co-integration vector (CE) equation sets for variables of no difference ( [31,32]). As shown in Table 6, the hypothesis assumes that there is none CE, and the trace statistics shows 226.9521 and larger than the critical value 134.6780 which means reject that there is none CE. erefore, Table 5 shows that there are two sets of co-integration equations under 5% critical value that are most relevant to the goodness-of-fit test because the trace statistics shows 75.8209 and smaller than the critical value 76.9727 at most 2, which means that we cannot reject that there are two co-integration vectors.

Vector Error Correction Estimates Model.
rough the Johansen Co-integration test, two sets of co-integration vector equations were obtained by using the trace test method. e co-integration vector equation and the VECM are shown in Table 6. e co-integration vector equations     Mathematical Problems in Engineering 5 show that equations (2) and (3) are based on the BER and Google SVI, respectively. e coefficients of the equation variables are highly significant to each other.

Long-Term Granger Causality.
Granger causality is a statistical concept of causality that is based on VAR. e coefficient of granger causality test is assumed that does not influence between two variables under null hypothesis as shown in Table 7. If the probability is larger than 0.1, then we do not reject the null hypothesis, and these two variables do not have granger causality. e results of the granger causality test are shown in Table 7. e BER influenced the S&P500 index and Google SVI. e Google SVI Granger caused the BER index. erefore, the BER and SVI presents intergranger causality. e S&P500 SVI Granger influenced Google SVI. e VIX Granger contributed to the S&P500 index, the BER index, and the Google SVI. Significant level of * * * 1%, * * 5%, * 10% and the coefficients means the estimated coefficients.  Figure 5 shows all variables that affect BER in the long-term. e "feedback relationship" of the Granger causality appears in BER and the Google SVI. e VIX has a moderately significant Granger causality on the Bitcoin price in the long term.

Conclusions
e analysis results of this research suggested the co-integrated equilibrium and feedback relationship of the Granger causality between BER and Google SVI in the long term. e VIX influences mostly the BER in the long-term cointegration.
In addition to confirming the results of Cai et al. [4] and Dastgir et al. [21], the result confirmed the Granger causality between Google SVI and BER and the existence of the short-and long-term feedback between Google SVI and the BER.
e result also showed that there is a long-term co-integration relationship between the BER, VIX, and Google SVI. By observing Google SVI and the VIX, investors predict the future trend of the BER as BER has a long-term leading relationship with the S&P500 index. e  Significant level of * * * 1%, * * 5%, and * 10%. crypto-economy affects the real economy as the spillover effect of BER influences the S&P500 [1]. When BER rises, the S&P index should be viewed with optimism. Also, the VIX has a long-term leading relationship with BER as it drops with the rise of BER. e co-integration equilibrium between BER and Google SVI in the long-term leads to the feedback relationship of the Granger causality, which impacts Bitcoin price and causes the following: the prohibition of the Bitcoin trading exchange, the issuer's implementation of hard fork policy that damages investors' rights and confidence with no compensation, and information security issues. e price of Bitcoin fell back to the equilibrium value, which verifies the investor attention theory. e results help investors hedging or arbitrating engaged Bitcoin-related risks.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.