Mathematical Problems in Engineering
Volume 2021 (2021), Article ID 8842877, 9 pages
https://doi.org/10.1155/2021/8842877

Research Article

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

Chien-Yun Chang,¹ Chien-Chien Lo,² Jui-Chang Cheng,³ Tzer-Long Chen,⁴ Liang-Yun Chi,⁵ and Chih-Cheng Chen6,7

¹Department of Fashion Business and Merchandising, Ling Tung University, Taichung, Taiwan
²Department of International Business, Providence University, Taichung 43301, Taiwan
³Department of Leisure and Recreation Management, National Taichung University of Science and Technology, Taichung, Taiwan
⁴Department of Finance, Providence University, Taichung, Taiwan
⁵Department of Finance, National Taichung University of Science and Technology, Taichung, Taiwan
⁶Department of Automatic Control Engineering, Feng Chia University, Taichung 40724, Taiwan
⁷Department of Aeronautical Engineering, Chaoyang University of Technology, Taichung 413310, Taiwan

Correspondence should be addressed to Tzer-Long Chen; tlchen1976@pu.edu.com and Chih-Cheng Chen; ccc@gm.cyut.edu.tw

Received 29 October 2020; Revised 21 February 2021; Accepted 5 April 2021; Published 29 April 2021

Academic Editor: Teen-Hang Meen

Copyright © 2021 Chien-Yun Chang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The Bitcoin exchange rate (BER) is influenced by many variables such as human speculation and policies and, thus, is dependent on the financial system. The fluctuation of BER submitted has been extensively investigated. However, the correlation analysis of the short- and long-term effects by indicators of online sentiment is unexplored. Therefore, this study establishes a VAR model for BER which provides a framework to the Google search volume index (SVI), the investor fear gauge (VIX), and the S&P500 Index. The findings of the analysis suggest that BER and Google SVI have a Granger causality feedback relationship in both the short- and long-term co-integration equilibrium, and the VIX is significantly related to BER in the long-term co-integration.

1. Introduction

The 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].

Figure 1: Bitcoin exchange rate chart (10/01/2014–06/22/2018).

The 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. Thus, behavioral financial theories emerged to explain irrational decisions. For example, Barber and Odean proposed the “Attention Theory” 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]. The 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].

This research employs the vector autoregression (VAR) model to explore the long- and short-term relationship between the Google SVI and BER. The 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. The vector autoregression (VAR) analysis of the short- and long-term impacts and Granger causal relationship ([5–7]) are employed to develop a prediction model for BER change for providing a reference for Bitcoin hedging operation to investors. The 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. The forecasting model of BER can offer mostly relation financial products and be applied for hedge, arbitrage for investors. The result expected the SVI should have higher relationship in the short and long periods.

This 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.

2. Literature Review

2.1. Google SVI

Urquhart [8] and Bleher and Dimpfl [9] adopted Google SVI as a proxy explanatory variable for “investor attention” for studying 12 cryptocurrencies including Bitcoin. The other researchers also employed Google SVI to measure investor attention [10, 11].

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. The 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. The search items are searched for their trends in a single country or all over the world. The 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:

The 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.

Figure 2: Google search engine market share worldwide (2013/1–2019/4, Source: http://gs.statcounter.com).

2.2. VIX

The 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. The 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 “The 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.

This 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.

2.3. Bitcoin

Kurka studied the interaction between Bitcoin and traditional finance assets [1]. The results showed that BER was independent of any financial system assets. Though 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]. The 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. The conclusion supported the existence of bidirectional causality between Bitcoin and investment returns.

This study analyzed the short- and long-term relationship between Google SVI and BER by the VAR model. The long-term relationship among BER, VIX, and the S&P500 index was also investigated for a co-integration phenomenon.

3. Methodology and Model

The 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]. Therefore, the co-integration analysis and vector error correction model (VECM) validated the long-term relationship between the variables.

Figure 3: Flow chart of this research [23].

3.1. VAR Model and VECM

The VAR model in this study is defined as follows:

where is a (n×1) vector of endogenous variables, c = () is the (n×1) intercept vector of the VAR, is the i-th (n×n) matrix of autoregressive coefficients for i = 1, 2, ..., n, and  =  is the (n×1) generalization of a white noise process [24].

The following equation is the co-integration transformation of equation (2).

.

where and .

If has a co-integration relationship, then and equation (3) is expressed as follows:

where  = is the error correction term for a long-term equilibrium relationship between variables. Equation (4), then, becomes

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. The 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]. The 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. The number of periods corresponding to the minimum value of the measurement of AIC or SC is the optimal number of lagged terms. The 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.

3.2. Multivariate Granger Causality Analysis

Multivariate Granger causality analysis is performed by fitting a VAR model to the time series. Let () be a d-dimensional multivariate time series. Multivariate Granger causality is measured by fitting a VAR model to L time lags as follows:

where is a white Gaussian random vector and is a matrix for every λ. A time series Xi is called a Granger cause of another time series Xj, if at least one of the elements is significantly larger than zero.

3.3. Data Source and Sample Collection Period

This research mainly discussed the key factors that affect the changes in BER by referring to Cai et al. [4], Kurka [1], and Hsieh [27]. The main variables included Google SVI, the VIX of Chicago Board of Exchange, S&P500 index, gold prices, US dollar index, and Japanese exchange rate. The period for sampling was from April 29, 2013, to June 22, 2018. The data sources were Yahoo Finance and Bloomberg. The 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.

4. Analysis Results and Discussion

The 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. Then we used the co-integration analysis and vector error correction model (VECM) to validate the long-term relationship between the nonstationary variables. The descriptive statistics, VAR model analysis, and the co-integration relationship were performed and illustrated below.

Figure 4: Granger causality between short-term variables.

4.1. Descriptive Statistics

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.

Table 1: Variable raw data statistics.

Variables
Average value
Standard deviation
Minimum value
Maximum value

BTC
1916.304
3281.969
69.66000
18972.32

SVI
7.727833
13.16746
1.000000
100.0000

VIX
14.62402
3.864276
9.140000
40.74000

SP500
2130.599
303.6529
1573.090
2872.870

GOLD
1252.709
76.27864
1049.400
1469.250

USD
86.37288
6.655887
74.52690
96.86650

JPY
110.0013
7.632938
94.21000
125.2200

4.2. Short-Term VAR Analysis

4.2.1. Unit Root Verification

The 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.

Table 2: The natural logarithm of each variable and the first-order difference unit root verification results.

Variables
ADF
PP

P
T
P
T

BTC
0.0000
−18.13386
0.0000
−38.34844

GOLD
0.0000
−37.81181
0.0000
−37.76925

JPY
0.0000
−35.90829
0.0000
−35.95482

SP500
0.0000
−36.56911
0.0000
−36.69663

SVI
0.0000
−35.99486
0.0000
−36.22507

USD
0.0000
−34.30820
0.0000
−34.47007

VIX
0.0000
−37.13990
0.0001
−42.94997

Significant level of 1%.

4.2.2. Optimal Lagging Period Selection

The research employed the AIC to determine the optimal number of two lagged periods. Table 3 shows the details of the optimal lagged periods.

Table 3: The optimal lagged periods selected by stationary sequence.

Lagged periods
Information criterion

AIC
SC

0
−35.68981
−35.66180

1
−35.87884
−35.65473

2
−35.88120
−35.46100

3
−35.86150
−35.24522

4
−35.83668
−35.02430

Selected lagged periods.

4.2.3. Results of Analysis of the VAR Model

The 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. The 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%. The 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%. The S&P500 index had a negative impact (−0.108234) on the leading two-period S&P500 index at a significant level of 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.

Table 4: Results of analysis by the VAR model.

BTC
GOLD
JPY
SP500
SVI
USD
VIX

BTC (−2)
Coefficient
0.108278
−0.005290
−0.000852
−0.001416
−0.100983
−8.92E−05
−0.001668

Standard deviation
0.02758
0.00859
0.00217
0.00294
0.04277
0.00089
0.02952

t value
3.92650
−0.61583
−0.39197
−0.48229
−2.36124
−0.09973
−0.05650

GOLD (−2)
Coefficient
−0.089612
−0.064190
−0.005214
−0.013245
0.048425
0.005545
0.073877

Standard deviation
0.09084
0.02830
0.00716
0.00967
0.14088
0.00295
0.09724

t value
−0.98646
−2.26838
−0.72842
−1.36920
0.34373
1.88266
0.75975

JPY (−2)
Coefficient
−0.533897
−0.106320
0.018735
−0.068497
−0.098955
0.014686
0.520942

Standard deviation
0.37632
0.11723
0.02965
0.04007
0.58362
0.01220
0.40282

t value
−1.41873
−0.90697
0.63185
−1.70925
−0.16955
1.20365
1.29325

SP500 (−2)
Coefficient
0.133665
0.068168
0.007063
−0.108234
0.384718
−0.001683
0.375067

Standard deviation
0.45540
0.14186
0.03588
0.04850
0.70627
0.01477
0.48747

t value
0.29351
0.48053
0.19685
−2.23182
0.54472
−0.11399
0.76942

SVI (−2)
Coefficient
0.064947
0.002600
−0.000551
−0.002003
0.006269
0.000247
0.026340

Standard deviation
0.01803
0.00562
0.00142
0.00192
0.02797
0.00058
0.01930

t value
3.60159
0.46282
−0.38802
−1.04325
0.22417
0.42245
1.36461

USD (−2)
Coefficient
0.599314
−0.362583
0.019072
0.057129
0.452971
0.003090
−1.319591

Standard deviation
0.90763
0.28273
0.07151
0.09665
1.40761
0.02943
0.97154

t value
0.66031
−1.28242
0.26669
0.59107
0.32180
0.10501
−1.35825

VIX (−2)
Coefficient
−0.010045
−0.004217
−0.000970
−0.007474
0.007971
−0.000770
−0.026743

Standard deviation
0.04539
0.01414
0.00358
0.00483
0.07039
0.00147
0.04858

t value
−0.22130
−0.29825
−0.27119
−1.54626
0.11323
−0.52347
−0.55045

C
Coefficient
0.002676
1.58E−05
9.02E−05
0.000486
0.001907
1.74E−05
−0.000270

Standard deviation
0.00205
0.00064
0.00016
0.00022
0.00318
6.7E−05)
0.00220)

t value
1.30416
0.02475
0.55767
2.22614
0.59935
0.26144]
−0.12296]

R squared
0.025671
0.007505
0.001804
0.009159
0.004947
0.004295
0.008862

Adj. R squared
0.020376
0.002111
−0.003621
0.003774
−0.000461
−0.001116
0.003476

Significant level of 1%, 5%, 10% and the coefficients means the estimated coefficients.

The Granger causality diagram of the short-term variables affected the BER. The results demonstrated that the feedback relationship appeared only in the BER and the Google SVI among the Granger causality as shown in Figure 4.

4.3. Long-Term Co-Integration Analysis

4.3.1. Johansen Maximum Likelihood Co-Integration Test

Nelson and Plosser supposed that a different process caused the time series to lose long-term information [22]. That 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. Therefore, 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.

Table 5: Johansen maximum likelihood co-integration test result.

Hypothesized no. of CE(s)
Eigenvalue
Trace statistics
Critical value at α = 0.05
Probability

None@
0.065359
226.9521
134.6780
0.0000

At most 1@
0.04826
139.6904
103.8473
0.0000

At most 2
0.0282
75.8209
76.9727
0.0610

At most 3
0.0144
38.8515
54.0790
0.5287

At most 4
0.0077
20.0129
35.1927
0.7259

At most 5
0.0055
9.9263
20.2618
0.6471

At most 6
0.0020
2.6882
9.16454
0.6404

Significant level of 1%; @the number of co-integrated equations.

Table 6: Co-integration vector equation and the VECM.

Co-integrating equation
Equation (2)
Equation (3)

BTC
1.0000
0.0000

SVI
0.0000
1.0000

VIX
Coefficient
16.0725
11.3168

Standard deviation
2.2836
1.5958

t value
7.0381
7.0912

C
− 49.2158
− 31.4336

Error correction:
D(BTC)
D(SVI)
D(VIX)

Co-integrating equation (2)
Coefficient
− 0.0139
0.0208
− 0.0051

Standard deviation
0.0037
0.0057
0.0039

t value
− 3.7744
3.6354
− 1.2972

Co-integrating equation (3)
Coefficient
0.0186
− 0.0333
0.0021

Standard deviation
0.0052
0.0081
0.00560

t value
3.5346
− 4.0969
0.3765

BTC
Coefficient
0.1037
− 0.1032
− 0.0081

Standard deviation
0.0274
0.0423
0.0291

t value
0.7826
− 2.4376
− 0.2786

SVI
Coefficient
0.0527
0.0208
0.0192

Standard deviation
0.0181
0.0280
0.0193

t value
2.8992
0.7410
0.9942

VIX
Coefficient
− 0.0098
0.0028
− 0.0388

Standard deviation
0.0262
0.04047
0.02785

t value
− 0.3752
0.0698
− 1.3941

C
Coefficient
0.0027
0.0020
− 5.13E−05

Standard deviation
0.0020
0.0031
0.0021

t value
1.3436
0.6529
− 0.0237

R squared
0.0353
0.0225
0.0325

Adj. R squared
0.0316
0.0187
0.0288

Significant level of 1%, 5%, and 10%.

4.3.2. Vector Error Correction Estimates Model

Through the Johansen Co-integration test, two sets of co-integration vector equations were obtained by using the trace test method. The co-integration vector equation and the VECM are shown in Table 6. The co-integration vector equations show that equations (2) and (3) are based on the BER and Google SVI, respectively. The coefficients of the equation variables are highly significant to each other.

4.3.3. Long-Term Granger Causality

Granger causality is a statistical concept of causality that is based on VAR. The 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. The results of the granger causality test are shown in Table 7. The BER influenced the S&P500 index and Google SVI. The Google SVI Granger caused the BER index. Therefore, the BER and SVI presents intergranger causality. The S&P500 SVI Granger influenced Google SVI. The VIX Granger contributed to the S&P500 index, the BER index, and the Google SVI.

Table 7: Granger causality test results for the variables in natural logarithm.

Null hypothesis
Prob.

GOLD does not influence BTC
0.9397

BTC does not influence GOLD.
0.3118

JPY does not influence BTC
0.5978

BTC does not influence JPY
0.9319

SP500 does not influence BTC
0.2092

BTC does not influence SP500
0.0325

SVI does not influence BTC
4.E−05

BTC does not influence SVI
0.0018

USD does not influence BTC
0.3888

BTC does not influence USD
0.8713

VIX does not influence BTC
0.0373

BTC does not influence VIX
0.1173

SVI does not influence SP500
0.4206

SP500 does not influence SVI
0.0469

VIX does not influence SP500
0.0098

SP500 does not influence VIX
0.2837

VIX does not influence SVI
0.0074

SVI does not influence VIX
0.3751

Significant level of 1%, 5%, 10%.

Figure 5 shows all variables that affect BER in the long-term. The “feedback relationship” of the Granger causality appears in BER and the Google SVI. The VIX has a moderately significant Granger causality on the Bitcoin price in the long term.

Figure 5: Granger Causality diagram among long-term variables.

5. Conclusions

The 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. The VIX influences mostly the BER in the long-term co-integration.

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. The 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. The 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. The 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. The price of Bitcoin fell back to the equilibrium value, which verifies the investor attention theory. The results help investors hedging or arbitrating engaged Bitcoin-related risks.

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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