To achieve the aim of this study, we begin by identifying bubble periods for each cryptocurrency. We then investigate herding behavior during bubble periods and analyze whether explosivity in one digital currency leads to explosivity in others. Finally, we estimate factors behind the cryptocurrency bubble.

### Bubble estimation

Bubbles attract economists because explosive behavior dampens capital allocation in the economy by distorting market efficiency, as in the Dutch Tulip mania, the Mississippi bubble, the Internet bubble, and, more recently, the global housing bubble. Considering the negative impact of bubbles on the real economy, several attempts to detect explosive behavior in asset prices have been made. One strand of the literature attempted to measure bubbles based on a comparison between market and intrinsic values, determined based on the underlying asset’s net present value (Siegel 2003). However, calculating the capitalized value of future cash flows is quite difficult as expected cash flows may differ among investors and will continue for many years. Another approach for detecting bubbles is based on the explosive behavior characteristics of bubbles. Unit root tests, such as autoregressive unit root tests, can be employed to measure explosive behavior (Taipalus 2012; Phillips et al. 2011, 2014). The Markov-switching unit root test can also be used to detect bubbles (Hall et al. 1999). Econophysics models, such as the log-periodic power law model, are other tools that can be used to identify explosive behavior (Filimonov and Sornette 2013; Sornette and Cauwels 2014). Generally, these models concentrate on price increases rather than directly addressing asset prices and attempting to detect bubbles based on price increase rates.

In this study, we use a unit root test to detect cryptocurrency market bubbles. We then adopted Phillips et al. (2015a, b; hereafter PSY) methodology to detect possible bubbles. The PSY procedure adopts a recursive test methodology^{Footnote 4} and is frequently employed in finance literature to identify explosive behavior in various asset prices, including commodity, energy, real estate, and virtual and digital currencies (Dirk and Kristoffer, 2012; Bettendorf and Chen 2013; Cheung et al. 2015; Corbet et al. 2018; Geuder et al. 2019; Enoksen et al. 2020; Li et al. 2020; among others).

PSY postulates that bubbles exhibit slightly explosive behavior, reflecting an autoregressive nature. Therefore, they can be captured using the right-tailed ADF test, where the null hypothesis states that series have a unit root ((H_{0} :delta_{{r_{1} ,r_{2} }} = 1)) and are tested against an alternative hypothesis wherein time series have an explosive unit root ((H_{1} :delta_{{r_{1} ,r_{2} }} > 1)). As financial bubbles periodically emerge and conventional ADF unit root tests have limited capability of discovering recurring bubbles, PSY adopts a recursive approach containing a rolling window ADF regression sample that begins with the fraction (r_{1}) of the total sample (*T*) and ends at fraction (r_{2}), where (r_{2} = r_{1} + r_{w}), (r_{w} > 0) is the rolling window size. Regression model as follows:

$$Delta y_{t} = alpha_{{r_{1} ,r_{2} }} + delta_{{r_{1} ,r_{2} }} . y_{t – 1} + mathop sum limits_{i = 1}^{k} psi_{{r_{1} ,r_{2} }}^{i} Delta y_{t – i} + varepsilon_{t}$$

(4)

Here, *α*, *δ*, and *ψ* are parameters determined by the regression, and k is the lag order. (T_{w} = left[ {T_{w} } right]) is the total number of observations, where [.] is the floor function.

To consistently capture multiple bubble episodes, the PSY methodology employs a supremum ADF (SADF) test. In this estimation, window size (r_{w}) increases from (r_{0}) to 1, where (r_{0}) is the smallest sample window range. Conversely, 1 is the total sample size representing the largest window size in the recursion. In the SADF test, the initial point (r_{1}) is set to 0 and the endpoint of the subsamples equals (r_{2}), ranging from (r_{0 }) to 1. The SADF is robust against multiple breaks and is formulated as follows:

$$SADFleft( {{text{r}}_{0} } right) = sup_{{r_{2} in left[ {{text{r}}_{0} ,1} right] }} ADF_{ 0}^{{r_{2} }}$$

(5)

The SADF procedure is then recursively performed to construct a generalized supremum ADF (GSADF). The GSADF test allows window width to change to a predefined range by extracting more fractions of the entire sample. Therefore, this test is more flexible for determining multiple bubbles. Initial point (r_{1}) in GSADF ranges from 0 to (r_{2} – r_{0}) where (r_{2} in left[ {{text{r}}_{0} , 1} right]), and the endpoint of the subsamples equals (r_{2}) and ranges from (r_{0 }) to 1, and GSADF is defined as follows:

$${text{GSADF}}left( {r_{0} } right) = sup_{{mathop {{text{r}}_{1} in left[ {0,{text{r}}_{2} – {text{r}}_{0} } right]}limits^{{r_{2} in left[ {{text{r}}_{0} , 1} right]}} }} ADF_{{r_{1} }}^{{r_{2} }}$$

(6)

To date-stamp the origin and endpoints of financial bubbles, we also applied the backward supremum ADF (BSADF) test. The BSADF procedure employs either a fixed initial origin, as in SADF, or adjustable starting and endpoints. In the BSADF test, the initial point of the bubble is displayed as (T_{{r_{e} }}) where series crosses over the critical value, and the termination of the bubble is represented by (T_{{r_{f} }}) where series crosses the critical value downward. Estimates of the bubble period based on the GSADF are as follows:

$$check{R}_{e} = mathop {r_{2} in left[ {r_{0} ,1} right]}limits^{inf} left{ {r_{2} :BSADF_{{r_{2} }} left( {r_{0} } right) > cv_{{r_{2} }}^{{beta_{T} }} } right}$$

(7)

$$check{r}_{f} = mathop {r_{2} in left[ {check{r}_{e} ,1} right]}limits^{inf} left{ {r_{2} :BSADF_{{r_{2} }} left( {r_{0} } right) > cv_{{r_{2} }}^{{beta_{T} }} } right}$$

(8)

where (cv_{{r_{2} }}^{{ _{ } }}) is the critical value of the subsample (r_{2}), and (beta_{T}) is the significance level that depends on the size of the total sample *T*. BSADF(r_{0}) for (r_{2} in left[ {{text{r}}_{0} , 1} right]) is the BSADF statistics that relate to the GSADF statistic by the following relation:

$$GSADF(r_{0} ) = mathop {r_{2} in left[ {r_{0} ,1} right]}limits^{sup} left{ {BSADF_{{r_{2} }} left( {r_{0} } right)} right}$$

(9)

To identify explosive behavior in the 11 cryptocurrency prices with the highest market value, we perform a date-stamp GSADF test. Figure 1 depicts the PSY test results for each cryptocurrency. A small (large) dashed line indicates the 95 (90) percent level of the critical value of the bootstrapped Dickey-Fuller test statistics. The bubble was defined as a PSY test results (straight line) exceeding critical values. Most bubbles occurred in short windows. Table 3 also presents the number of bubble days for each cryptocurrency and its percentage in the sample. THETA, BTC, and ADA are the top three cryptocurrencies that experienced more bubble days. Conversely, XLM, XRP, and TRX are the bottom three cryptocurrencies with fewer bubble periods. When focusing on the percentage of bubble periods, cryptocurrencies tend to experience explosive behavior in 2021 compared with 2020. For instance, DOGE has more than 3 bubble days in 2021 compared with 2020. Both Fig. 1 and Table 3 emphasize that bubbles existed in all cryptocurrencies during the COVID-19 pandemic. These results are in line with previous studies presenting evidence of the presence of bubbles in cryptocurrencies (Cheah and Fry 2015; Geuder et al. 2019; Chaim and Laurini 2019; Kyriazis et al. 2020; Enoksen et al. 2020). Hence, we can proceed with our analysis to understand investors’ behavior during bubble periods by questioning herding behavior.

### Herding estimation

During explosive price movement periods, investment decisions tend to be affected by collective market behavior. Considering the extreme price movements in the cryptocurrency market during the COVID-19 pandemic, herding behavior may be associated with this explosive behavior. To detect herding behavior, prior studies referred to two widely used proxies. Christie and Huang (1995) proposed the first model, cross-sectional standard deviation (CSSD), and Chang et al. (2000) proposed the second model, cross-sectional absolute deviation (CSAD). Outliers easily affect the CSSD measure (Economou et al. 2011), and the CSSD model is suitable for a linear relationship between market returns and CSSD of returns (Dhall and Singh 2020). Chang et al. (2000), Mobarek et al. (2014), and Ballis and Drakos (2020) suggest that Newey and West’s (1987) standard error correction should be used to adjust estimation for autocorrelation and heteroskedasticity. Therefore, we decided to use the Newey-West standard error-corrected CSAD as our primary herd measure as follows:

$$CSAD_{t} = alpha + alpha_{1} left| {R_{m,t} } right| + alpha_{2} R_{m,t}^{2} + varepsilon_{t}$$

(10)

where *R*_{m,t} is the return of the CCI-30 index return, and *CSAD*_{t} is the return dispersion proxy, which is also determined as follows:

$$CSAD_{t} = frac{1}{N}sumnolimits_{i = 1}^{N} {left| {R_{i,t} – R_{m,t} } right|}$$

(11)

Here, *R*_{i,t} is the return of cryptocurrency *i* at time *t*, *R*_{m,t} is the return on the CCI-30 index, and *N* is the number of cryptocurrencies in the portfolio. To detect herding in the cryptocurrency market during the bubble, we modified our basic model as follows:

$$CSAD_{t} = alpha + alpha_{1} left| {R_{m,t} } right| + alpha_{2} R_{m,t}^{2} + alpha_{3} R_{m,t}^{2} * Bubble_{t} + varepsilon_{t}$$

(12)

where *Bubble* is a dummy that gets one if the bubble period and 0 otherwise. The coefficient of interest is (alpha_{3}) which should be significant and negative if herding behavior exists during the bubble period.

Table 4 presents the results of herding behavior analysis. Each column reports the estimation results of Eq. (12). For each cryptocurrency, we use a dummy variable to identify bubble periods. The dummy variable corresponds to bubbles in the underlying cryptocurrency, and the results determine whether herding behavior exists in the overall market. For instance, in the first column, we generated a dummy variable when there is a Bitcoin bubble and analyzed whether herding behavior exists in the entire sample. The same applies to other cryptocurrencies.

The negative and significant coefficients of (R_{m,t}^{2}) indicates that herding behavior exists in each cryptocurrency. This result is consistent with the literature on herding behavior in the cryptocurrency market (Bouri et al. 2019; Kallinterakis and Wang 2019; Kaiser and Stöckl 2020). However, we found striking results during the bubble period. Coefficients of (R_{m,t}^{2} {*}Bubble_{t} { }) are positive and statistically significant, suggesting that herding behavior diminishes in the overall market when a particular cryptocurrency has a bubble. This may contradict the common expectation that, during the bubble period, investors follow the crowd and invest accordingly instead of their strategies. However, this result does not indicate the opposite; rather, it indicates adverse herding owing to higher risk aversion during extreme periods (da Gama Silva et al. 2019). This result is also consistent with previous studies suggesting that herding behavior is obvious during normal periods, whereas it disappears during up and down periods (Susana et al. 2020). Supporting these findings are Vidal-Tomas et al. (2019) and Papadamou et al. (2021), who state that herding in the cryptocurrency market is stronger during down periods as most cryptocurrencies have experienced extreme price increases during the pandemic.

As a robustness analysis, we follow the literature and estimate herding behavior using CSSD and CSAD with generalized autoregressive conditional heteroscedasticity (GARCH) models as follows:

$$CSSD_{t} = alpha + alpha_{1} left| {R_{m,t} } right| + alpha_{2} R_{m,t}^{2} + varepsilon_{t}$$

(13)

where *R*_{m,t} is the return of the CCI-30 index return, and *CSSD*_{t} is the return dispersion proxy, which is also determined as follows:

$$CSSD_{t} = sqrt {frac{{sumnolimits_{i = 1}^{N} {left( {R_{i,t} – R_{m,t} } right)^{2} } }}{N – 1}}$$

(14)

where *R*_{i,t} is the return on cryptocurrency *i* at time *t*, *R*_{m,t} is the return on the CCI-30 index, and *N* is the number of cryptocurrencies in the portfolio. To detect herding in the cryptocurrency market during the bubble, we use the following basic model:

$$CSSD_{t} = alpha + alpha_{1} left| {R_{m,t} } right| + alpha_{2} R_{m,t}^{2} + alpha_{3} R_{m,t}^{2} * Bubble_{t} + varepsilon_{t}$$

(15)

Here, *Bubble* is a dummy that is equal to 1 during the bubble period and 0 if otherwise. (alpha_{3}), the coefficient of interest, should be significant and negative if herding behavior exists during the bubble period.

Table 5 presents the results of the CSSD measure of herding behavior using Eq. (15). Although issues are using CSSD, as mentioned earlier, coefficients of interest in Table 5 are similar to those in Table 4. Herding behavior declines during bubble periods in most cryptocurrencies.

We also offer another widely used estimation for herding behavior to eliminate sample heteroscedasticity. Goyal and Santa-Clara (2003) demonstrate that the herding coefficient of the CSAD regression captures the link between idiosyncratic volatility and market returns. Therefore, controlling for the effect of volatility on cryptocurrency returns using the GARCH model is vital. Specifically, we added a conditional variance variable to the CSAD mean equation model and estimated the following GARCH (1,1) mean model:

$$CSAD_{t} = alpha + alpha_{1} left| {R_{m,t} } right| + alpha_{2} R_{m,t}^{2} + alpha_{3} R_{m,t}^{2} * Bubble_{t} + theta_{1} sigma_{t}^{2} + varepsilon_{t}$$

(16)

$$sigma_{t}^{2} = omega_{0} + omega_{1} varepsilon_{t – 1}^{2} + omega_{2} sigma_{t – 1}^{2}$$

(17)

where *Bubble* is a dummy equal to 1 during the bubble period and 0 otherwise. (sigma_{t}^{2}) is the conditional variance of the residual *CSAD*_{t}.

Table 6 presents the estimation results for herding behavior using the GARCH (1,1) model in the mean equation. Our results are consistent with the CSAD and CSSD using the Newey-West standard error models. Therefore, we conclude that the herding behavior of the overall cryptocurrency market significantly diminishes during the cryptocurrency market bubbles. The results of the herding analysis led us to explore the contemporaneous relationship between cryptocurrencies during bubble periods.

As the empirical analysis reveals that herding behavior diminishes during the bubble period, determining whether a reverse relationship exists between herding and cryptocurrency market bubbles is important. Following Bouri et al. (2019), we estimated herding behavior using a 30-day rolling window and defined herding at a 10% significance level. Hence, we create a dummy variable equal to 1 if the rolling t-statistic on (alpha_{2} le – 1.645) and 0 otherwise. Once we have a herding behavior proxy, we employ a logistic regression and analyze the impact of speculative bubbles on herding for capturing the reverse effect. We adopt a time-series logistic regression, where the dependent variable is a binary variable that is equal to 1 if there is herding behavior and 0 otherwise. The dependent variable is the 1-day lag of the bubble in each cryptocurrency. Table 7 presents the results of the reverse relationship between herding and speculative bubbles. The findings show that only the speculative bubble in DOGE, VET, and THETA impacts herding in the cryptocurrency market, whereas the bubble in major cryptocurrencies does not impact herding behavior.

### Co-explosivity analysis

As herd behavior in the cryptocurrency market diminishes during bubble periods, the reason behind the bubbles in cryptocurrencies needs further investigation. Given that most cryptocurrencies facilitate similar technology and mining processes, the absence of fundamental techniques for calculating cryptocurrency value and low financial knowledge among cryptocurrency traders suggests that a bubble in one cryptocurrency can be transmitted to another (Bouri et al. 2019). Therefore, we explore co-explosivity in the cryptocurrency market to understand how bubbles in one cryptocurrency lead to explosive behavior in others.

We follow the procedure by Bouri et al. (2019) to investigate co-explosive price movements across cryptocurrencies by employing logistic regression after identifying bubble days in the prior section:

$$log left( {frac{{Pleft( {Y = 1|X} right)}}{{1 – Pleft( {Y = 1|X} right)}}} right) = beta_{0} + beta_{i} X_{i,t – 1} + varepsilon_{t}$$

(18)

where the dependent variable is a dummy variable *Y* equal to 1 if the day is a bubble day and 0 otherwise. *X*_{i,t-1} is a set of ten dummy variables that takes 1 if each remaining cryptocurrency has a bubble on the previous day.

Table 8 presents the results of the co-explosivity analysis.^{Footnote 5} Bubbles in XRP, LINK, LTC, and VET are least dependent on the existence of a bubble in other cryptocurrencies on the previous day. Presence of a bubble in ETH, LTC, and THETA increases the probability of a bubble in BTC, whereas the VET bubble has a negative impact. BTC, XRP, ADA, and DOGE increased the existence of a bubble in the ETH, whereas XLM and THETA had a negative effect. LTC is the only cryptocurrency that impacts the presence of bubbles in the XRP. Bubbles in DOGE are most affected by bubbles in other cryptocurrencies. ETH, XRP, LTC, THETA, and TRX increased the probability of bubble occurrence in DOGE, while XLM and VET had a negative impact. The probability of the presence of a bubble in the LTC increases when a bubble in the BTC, XRP, DOGE, and VET exists. LINK is only affected by TRX. The probability of a bubble in the VET increases with a bubble in the LTC. Regarding THETA, the probability of occurrence of the bubble increases when there is a bubble in BTC, DOGE, LINK, and TRX and decreases with ETH. Finally, the probability of a bubble in TRX increases with the presence of a bubble in ADA, DOGE, and LINK.

Overall, co-explosivity estimation results are consistent with the finding that the presence of bubble in one cryptocurrency significantly increases with the existence of a bubble in others. This suggests that one of the factors behind the cryptocurrency market bubble is co-explosivity in the cryptocurrency market (Ang et al. 2005; Bouri et al. 2019; Cagli 2019). Based on the existing results, an investor can follow co-explosive price movements by switching from one cryptocurrency to another to gain profit.

### Bubble predictors

As cryptocurrency bubbles are largely characterized by cryptocurrency co-movements rather than herding, analyzing cryptocurrency-specific factors that can predict the occurrence of bubbles in each cryptocurrency is vital. Thus, we apply the probit model to identify factors behind cryptocurrency bubbles. We employ panel model estimation with all cryptocurrencies and estimations of time-series models for each cryptocurrency separately.

$$Bubble_{i,t} = left{ {begin{array}{*{20}c} {1, ;i{text{f}};PSY_{i,t} left( {r_{0} } right) > cvartheta_{i,t} left( {beta_{T} } right)} \ {0, ;{text{if}};PSY_{i,t} left( {r_{0} } right) < cvartheta_{i,t} left( {beta_{T} } right)} \ end{array} } right.$$

(19)

We formulate time series and panel probit models as follows:

$$pleft( {Bubble_{t} = 1} right) = theta left( {beta x_{t – 1} } right)$$

(20)

$$pleft( {Bubble_{i,t} = 1} right) = theta left( {beta x_{i,t – 1} + vartheta_{i} } right)$$

(21)

where (theta left( . right)) indicates a normal cumulative distribution function. (x_{t – 1}) is a 1-day lagged variable consisting of factors that can be used to predict a bubble. (vartheta_{i}) is the random effect in the panel estimation. We follow Enoksen et al. (2020) and use random effects and robust standard errors clustered by cryptocurrency to eliminate autocorrelation and heteroscedasticity issues in panel estimation. In the time-series estimation, we use Newey and West’s (1987) robust standard errors. Variables are standardized by subtracting the sample mean and scaling it by standard deviation to obtain the coefficient so we can properly interpret variables’ economic impact. As mentioned, two sets of variables are available: the first set includes cryptocurrency-specific factors (i.e., lagged return, volume, and volatility) and the second comprises market-related factors (i.e., market return and Google Trends).

The bubble is given as a function of lagged return, volume, Google Trends, market return, and volatility:

$$begin{aligned} Bubble_{i,t} & = beta_{0} + beta_{1} Lagged;{text{Re}} turn_{i,t – 1} + beta_{2} Volume_{i,t – 1} + beta_{3} Google;Trend_{i,t – 1} \ & quad + beta_{4} Market; {text{Re}} turn_{i,t – 1} + beta_{5} Volatility_{i,t – 1} + varepsilon_{i,t} \ end{aligned}$$

(22)

The last column of Table 9 reports the panel probit estimation, and the other columns present the results of the time-series probit estimations. Because the variables are standardized, higher coefficients represent stronger economic effect on the bubble. Positive coefficients suggest a higher probability of predicting bubbles. Conversely, a negative coefficient indicates a lower probability of predicting bubbles. Consistent with Enoksen et al. (2020), we find that both crypto-specific and market-related factors can predict cryptocurrency market bubbles. Turning to individual factors, volume, Google Trends, and volatility were positively associated with bubbles in the panel probit estimation. Conversely, the 1-day lagged return of cryptocurrency and market returns cannot predict bubbles in panel regressions. Time-series estimations indicate that Google Trends can predict bubbles as it is positive and statistically significant in 7 out of 11 cryptocurrencies. The lack of fundamental information regarding cryptocurrencies leads investors to follow public interest using Google Trends, as stated in Choi and Varian (2009), Choi and Varian (2012), Bijl et al. (2016), and Molnár and Bašta (2017). Only DOGE has a negative Google Trends coefficient. This may be attributable to DOGE being mostly dominated by Twitter owing to the activities of well-known individuals (Ante 2021). Volume is also positively associated with bubbles, and this is significant for five cryptocurrencies, which is consistent with Enoksen et al. (2020). Lagged return and market return do not have considerable effect in the time series, and volatility has ambiguous results.

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