Detecting Structural Changes in Bitcoin, Altcoins, and the S&P 500 Using the GSADF Test: A Comparative Analysis of 2024 Trends

Abstract

Understanding structural regime shifts in crypto asset markets is vital for early detection of systemic risk. This study applies the Generalized Sup Augmented Dickey–Fuller (GSADF) test to daily high-frequency price data of five major crypto assets—BTC, ETH, SOL, AAVE, and BCH—from 2023 to 2025. The results reveal asset-specific structural breaks: BTC and BCH aligned with macroeconomic shocks, while DeFi tokens (e.g., AAVE, SOL) exhibited fragmented, project-driven shifts. The S&P 500 index, in contrast, showed no persistent regime shifts, indicating greater structural stability. To examine inter-asset linkages, we construct co-occurrence matrices based on GSADF breakpoints. These reveal strong co-explosivity between BTC and other assets, and unexpectedly weak synchronization between ETH and AAVE, underscoring the sectoral idiosyncrasies of DeFi tokens. While the GSADF test remains central to our analysis, we also employ a Markov Switching Model (MSM) as a secondary tool to capture short-term volatility clustering. Together, these methods provide a layered view of long- and short-term market dynamics. This study highlights crypto markets’ structural heterogeneity and proposes scalable computational frameworks for real-time monitoring of explosive behavior.

1. Introduction

Asset prices in financial markets often appear to fluctuate randomly. However, beneath this apparent randomness lie identifiable patterns and structural changes—commonly referred to as regime shifts—that reflect deeper market dynamics not captured by simple random walk models. Particularly during the formation and bursting of asset bubbles, markets may transition from a stationary, mean-reverting state to a momentum-driven, non-stationary regime. These regime shifts can signal structural changes in the pricing mechanism, often triggered by macroeconomic shocks, speculative sentiment, or technological developments.

A broad array of econometric techniques has been proposed to detect speculative bubbles, yet they differ in assumptions and statistical robustness. Gürkaynak (2008) surveys these approaches and highlights the ongoing debate over the empirical detectability of bubbles. The inherent challenge lies in distinguishing rational exuberance from explosive price behavior. More recent techniques, such as the Generalized Supremum Augmented Dickey–Fuller (GSADF) test proposed by Phillips et al. (2015), address this issue by allowing for flexible, rolling-window tests that can detect multiple episodes of explosiveness within a single price series.

From a statistical perspective, the core problem is to determine whether observed price trends are stationary—reflecting temporary deviations—or non-stationary, indicating structural change. Non-stationary price processes often exhibit self-reinforcing dynamics, such as accelerating price movements without corresponding shifts in fundamentals. These dynamics are closely associated with speculative bubbles, particularly when investor expectations become unanchored and positive feedback loops dominate. Under the Efficient Market Hypothesis (EMH), prices should follow a martingale process:

$$\mathbb{E}[P_{t+1}\mid {\mathcal{F}}_t]=P_t$$

(1)

However, during bubble regimes, this expectation shifts:

$$\mathbb{E}[P_{t+1}\mid {\mathcal{F}}_t]>P_t$$

(2)

This departure from martingale behavior signals a breakdown in informational efficiency and a transition to a structurally altered market regime.

Identifying such transitions in real time has become increasingly important for risk management, regulatory oversight, and quantitative investment strategies. Econometric tests like the SADF by Phillips et al. (2011), and GSADF offer a theoretically grounded and empirically validated framework for bubble detection. Unlike traditional unit root tests, the GSADF allows for varying window sizes in both the starting and ending points of the estimation sample, making it particularly well suited for detecting transient but impactful episodes of explosiveness.

Despite its effectiveness, the GSADF method is computationally intensive, scaling on the order of $O\left(T^4\right)$ in naive implementations. In this study, to mitigate computational challenges, we utilize GPU acceleration techniques and MPI parallelization, which together enable practical analysis of high-frequency, multi-asset data. While this paper focuses on GSADF-based detection, other modeling approaches, such as Log-Periodic Power Law Singularity (LPPLS) analysis (Johansen & Sornette, 1999; Sornette, 2003), which targets bubble termination prediction, represent promising directions for future research. Incorporating such methods may complement GSADF testing by improving real-time detection and forecasting capabilities.

On the other hand, the MSM (Markov Switching Model), which does not estimate the position of the unit root but instead predicts the next state based on past states, is effective for capturing short-term regime changes, but it has limitations in identifying long-term structural changes. Additionally, the ZA test can only identify a single structural break, making it suitable for information on long-term structural changes that rely on detecting a single change point. This study focuses on identifying the beginning of structural changes in time series data using the GSADF test, specifically targeting the onset of bubbles. While GSADF aims to detect temporary excess reactions and explosive behavior in market price movements, LPPLS is used to predict the end of bubbles, addressing different stages of market dynamics. Thus, GSADF focuses on detecting the onset of bubbles, while LPPLS captures their end.

This study applies the GSADF test to high-frequency price data from five major cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), Bitcoin Cash (BCH), and Aave (AAVE)—over the 2023–2025 period. These are compared to the S&P 500 index to highlight differences between crypto and traditional financial markets. Our analysis reveals that BTC and BCH exhibit regime shifts that align closely with macroeconomic announcements and monetary policy changes, while DeFi tokens show more fragmented structural breaks, often associated with protocol-specific developments. Notably, ETH experienced a major structural break in April 2024, triggered by Layer-2 migration pressures and delays in scaling upgrades.

In April 2025, both the crypto asset market and traditional financial markets faced significant turbulence driven by escalating trade tensions originating in the United States and disruptions in international trade agreements, which heightened global economic uncertainty. However, the S&P 500 index’s fluctuations during this period did not result in persistent structural breaks but were interpreted as short-term volatility caused by transient macroeconomic shocks. These findings suggest that the time series structure of traditional equity indices like the S&P 500 is more stable compared to crypto asset markets, maintaining structural integrity even amid events such as presidential elections or temporary macroeconomic disturbances.

By combining structural break testing with high-frequency data, this work contributes to the ongoing debate about the maturity and systemic relevance of cryptocurrency markets. It also opens up avenues for real-time structural risk assessment using improved computational tools. The results have implications for investors, policymakers, and developers seeking to understand the evolving dynamics of crypto-financial systems in a post-pandemic, high-volatility era.

2. Materials and Methods

2.1. Detection of Explosive Episodes via the GSADF Test

The SADF test and its extension, the GSADF test, are powerful statistical tools used to detect bubbles and explosive behaviors in financial markets. The SADF test fixes the starting point of the window and varies the endpoint, allowing it to capture temporary and sharp fluctuations in price time series. This approach makes it possible to identify short-term explosive behaviors. The GSADF test further develops the SADF test by dynamically changing both the starting point and the endpoint of the window. This flexible approach retains the characteristics of the SADF test while adjusting both the starting and ending points, thereby enabling a clear identification of how short-term explosive fluctuations influence the long-term market trends. The GSADF test can capture both short-term episodes and long-term structural changes, providing a more comprehensive and detailed market analysis.

Traditional methods based on unit root processes (e.g., the ADF test) operate under the assumption that the data follows a unit root process (i.e., it is non-stationary in the long term). While this method can detect long-term non-stationarity in time series data, it may miss short-term sharp fluctuations or initial signs of bubbles due to temporary excessive reactions. This is because tests based on unit root processes often treat these short-term sharp fluctuations and temporary excessive reactions as part of the non-stationary state and fail to identify structural changes effectively.

The GSADF test is used to finely identify temporary excessive reactions that are not captured by traditional methods based on unit root processes. The data-generating process is defined as follows:

$$y_t=d{T}^{-\eta }+y_{t-1}+{ϵ}_t,{ϵ}_t\sim \mathrm{i}.\mathrm{i}.\mathrm{d}.(0,{\sigma }^2)$$

(3)

Here, $T^{-\eta }$ represents the drift term, where T is the sample size, $d\in \mathbb{R}$ is a constant, and $\eta >0$ is the localization parameter. The drift term introduces a deterministic trend component into the price process, while the parameter $\eta$ governs the speed and persistence of the asymptotic divergence, effectively controlling the localization of explosive behavior over time.

As T increases, the drift term gradually decreases:

$$\underset{T\to \infty }{lim}d{T}^{-\eta }=0$$

(4)

Expanding the above Equation (3), we obtain the following:

$$y_t=d{T}^{-\eta }+\sum _{j=0}^t{ϵ}_j+y_0$$

(5)

Here, ${\sum }_{j=0}^t{ϵ}_j$ represents the random walk behavior, that is, the martingale component. The variance of the martingale component is

$$Var\left(\sum _{j=0}^t{ϵ}_j\right)=t{\sigma }^2$$

(6)

Thus, the standard deviation is given as $\sqrt{t}\sigma$. Therefore, the martingale component increases over time, with its standard deviation increasing in proportion to $\sqrt{t}$.

The condition for the drift term, which decreases at the order of $T^{-\eta }$, and the martingale component, whose standard deviation increases at the order of $\sqrt{t}$, to remain of the same order is that both must increase or decrease at the same rate. From this condition, we obtain $T^{-\eta }\sim \sqrt{t}$, and taking the logarithm of both sides, we conclude that the data length T and the current time point t increase at roughly the same order, giving the critical value, $\eta ={\displaystyle \frac{1}{2}}$. Therefore, when $\eta <{\displaystyle \frac{1}{2}}$, the drift term decreases more slowly than the increase in the martingale component, and temporary excessive reactions in the market (e.g., rapid price increases) tend to persist for a longer period, leading to a situation where the market becomes more unstable. As a result, temporary excessive reactions in the market are more likely to be observed.

In actual time series tests, a difference-based regression model (ADF regression) is used to test for the presence of a unit root. The regression equation is as follows:

$$\mathrm{\Delta }{y}_t={\widehat{\alpha }}_{r_1,r_2}+{\widehat{\beta }}_{r_1,r_2}{y}_{t-1}+\sum _{i=1}^k{\widehat{\psi }}_{r_1,r_2}^i\mathrm{\Delta }{y}_{t-i}+{\widehat{ϵ}}_t$$

(7)

Here, $r_1,r_2$ represent the relative positions of the starting and ending points of the subsample, and k denotes the lag order. The null hypothesis of a unit root, $H_0:\beta =0$, is tested to determine whether the data is stationary or non-stationary.

The GSADF test extends the SADF framework by allowing both the starting point, $r_1$, and the ending point, $r_2$, of the estimation window to vary dynamically. This rolling window strategy enables the detection of multiple episodes of explosive behavior across different periods. The test statistic is defined as follows:

$$\mathrm{GSADF}\left(r_0\right)=\underset{\begin{array}{c}{r}_2\in [r_0,1]\\ r_1\in [0,r_2-r_0]\end{array}}{sup}AD{F}_{r_1,r_2}$$

(8)

This recursive formulation enhances the sensitivity of the test to complex market dynamics and structural instabilities. Unlike traditional ADF-based tests, the GSADF method is capable of capturing transient episodes of explosiveness, making it particularly well-suited for identifying early signals of bubble formation and abrupt regime shifts in financial time series.

However, since the GSADF test employs a rolling window with dynamically shifting subsamples, the sample size varies over time. As a result, the GSADF statistic does not converge to a standard asymptotic distribution but instead follows a non-standard limit distribution. This sensitivity to local nonstationarity and explosive behavior distinguishes it from traditional unit root tests. Consequently, Monte Carlo simulations are required to estimate critical values and ensure accurate statistical inference.

The GSADF test is therefore a valuable tool for the early detection of explosive dynamics and speculative bubbles in financial markets. Its recursive framework allows for continuous monitoring of structural instabilities in evolving time series.

Moreover, as an extension of the SADF test, GSADF is particularly effective at identifying multiple episodes of explosiveness that are temporally localized. This enables a more nuanced characterization of bubble dynamics, capturing early signs of regime shifts that conventional unit root tests often miss.

2.2. Markov Switching Model (MSM)

To provide a comparative benchmark against the GSADF test, we additionally applied a Markov Switching Model (MSM) framework to assess short-term regime shifts in both traditional and crypto markets. The MSM was applied to the BTC-USD and S&P 500 time series data to model the regime shifts. To improve the robustness of the MSM analysis, Kalman Filtering was applied to the log-returns of BTC-USD to remove noise from the data. Exogenous variables were carefully selected based on their causal relationship with the returns. The Granger Causality Test (via statmodels v0.15.0) was used to identify significant lags between log-returns and volatility, with logarithmic returns being the main dependent variable.

In traditional markets such as the S&P 500, strong causal relationships were found between volatility and logarithmic returns, with significant causality at $\mathrm{lag}=2(p=0.0013)$. In contrast, in the BTC-USD market, no significant causal relationship was observed between volatility and returns, as no p-value below $0.05$ was found. This lack of a clear causal relationship in the BTC-USD market can be attributed to factors such as information asymmetry, low liquidity in the trading, and market vulnerability to financial stress. The cryptocurrency market is highly affected by speculative trading, regulatory uncertainty, and investor sentiment, which can lead to market dynamics that are not easily captured by traditional volatility-returns relationships (Katsiampa, 2017; Corbet et al., 2019).

The ACF analysis for BTC-USD revealed that the log-returns at lag 1 and lag 2 were significantly correlated, with their ACF values falling outside the 95% confidence interval, suggesting their importance as exogenous variables. Consequently, the logarithmic returns at $t-1$ and $t-2$ were selected as exogenous variables for the MSM analysis.

The MSM aimed to capture short-term regime changes that influence market dynamics. The number of regimes was determined based on the Bayesian Information Criterion (BIC), where the optimal number of regimes was found to be 2. The regimes were categorized into stable (Regime 0) and unstable (Regime 1) periods, and the probability of being in Regime 1 was plotted over time.

2.3. Co-Explosivity Analysis Based on GSADF Statistics

To quantify the frequency of simultaneous explosive price behavior across multiple crypto assets, we constructed a co-explosivity matrix. Specifically, time points at which the GSADF statistic exceeded the 95th percentile critical value were treated as indicators of statistically significant structural change (i.e., bubble-like episodes). For each time point, we identified which asset pairs simultaneously exhibited such behavior and counted the number of overlapping days to construct a symmetric co-occurrence matrix.

The resulting matrix was row-normalized by dividing each element by the corresponding diagonal entry, representing the conditional co-explosivity rate—i.e., the relative frequency with which asset B was also in an explosive state, given that asset A was. This enables a normalized comparison of how often individual assets entered structural change regimes concurrently with others.

To capture temporal dynamics in co-explosivity patterns, we applied a 30-day rolling window, within which co-explosivity matrices were recomputed using the same procedure. This yielded a time-varying sequence of dynamic co-explosivity matrices, allowing us to analyze how bubble co-movements evolved over time.

The aim of this approach is to assess whether episodes of explosive behavior are idiosyncratic to each asset or reflect broader systemic linkages across markets. In particular, by examining not only intra-crypto co-explosivity but also linkages with the S&P 500 index, we seek to evaluate whether crypto markets are increasingly integrated into macro-financial systems shared with traditional assets—or whether they continue to follow distinct, self-contained dynamics.

2.4. Data Sources and Preprocessing

We applied the GSADF test to the following datasets:

• S&P 500 futures tick-level data from March 2000 to December 2019, obtained from Kaggle (2020).
• Cryptocurrency hourly data (BTC-USD, ETH-USD, SOL-USD, BCH-USD, AAVE-USD) and S&P 500 (USD) spot data for the most recent 730 days up to 30 April 2024, collected using the yfinance API.

For the S&P 500 futures, we focused on the period from March 2005 to December 2019, which includes major market events such as the 2008 financial crisis and the 2018 VIX shock. The tick data were aggregated into daily median-based dollar bars, and only the closing prices were used for constructing the time series. The ADF lag order was fixed to 1.

For cryptocurrencies and the S&P 500 spot index, the hourly data were also aggregated into daily dollar bars and converted into daily return series. The final sequence lengths were approximately 600 observations per asset, which allowed efficient analysis without HPC acceleration.

The BTC-USD and S&P 500 spot return series were additionally used in a comparative MSM analysis to evaluate short-term regime transitions. Co-explosivity matrices were constructed for the 2023–2025 crypto and equity datasets to investigate synchronized explosive episodes across assets.

2.5. Lag Order Selection

For the shorter cryptocurrency and spot index series, the optimal lag order k for the ADF regression was selected using the Bayesian Information Criterion (BIC). Specifically, for each series, autoregressive models with lag orders up to 5 were estimated, and the lag that minimized the BIC was chosen.

2.6. Monte Carlo Simulation for Critical Values

To obtain valid critical values for the GSADF statistic, we generated 3000 synthetic time series under the unit root null hypothesis for each real series length. From these Monte Carlo replications, the 90% and 95% quantiles of the GSADF distribution were computed and used as critical thresholds in subsequent analyses.

To assess the robustness of these thresholds, we conducted a sensitivity analysis with respect to random seed variation. Using an initial sample size of 1000, we found that the structural break episodes identified by the GSADF test were broadly consistent across different seeds. However, slight variations were observed near the critical thresholds—typically one- to two-bar shifts in the timing or duration of borderline episodes exceeding the 90% or 95% levels.

To reduce this sensitivity and improve threshold stability, we increased the number of Monte Carlo simulations to 3000. This resulted in smoother and more consistent critical value curves, mitigating the influence of simulation randomness. The enhanced stability confirms that increasing the simulation size yields more reliable detection of explosive episodes, particularly near marginal significance levels.

2.7. Computational Environment

To handle the long S&P 500 futures dataset (length > 5000), we developed a PyTorch-based GSADF pipeline optimized for execution on the Dawn high-performance computing system at the Cambridge Computing Center (Cambridge Open Zettascale Lab, 2023). Dawn consists of 256 Dell PowerEdge nodes, each equipped with multi-core Intel Xeon CPUs and four Intel Data Center GPUs, interconnected via Intel’s Xe-Link and MPI.

Our implementation utilizes two layers of parallelization: (1) MPI-based distributed computing, which partitions the GSADF sub-sample windows across multiple nodes, and (2) GPU acceleration, which executes batched ADF regressions simultaneously across time points using PyTorch’s GPU kernels. Performance is further enhanced by the Intel^® Extension for PyTorch* (version 2.0.120+xpu), which optimizes computations on Intel CPUs and GPUs, leveraging Intel AVX-512, VNNI, AMX, and XeMatrix Extensions (XMX). The Python environment used for these calculations is provided by the Intel^® oneAPI AI Toolkit, version 2024.0.1.3.

This dual-parallel design addresses the substantial computational cost of the GSADF test (approaching $O\left(T^4\right)$ for recursive sub-windowing), making the analysis of high-frequency, long-horizon data feasible. Without these optimizations, GSADF computations for series longer than 5000 data points would be practically infeasible.

Performance gains were particularly notable for long time series, where GPU batch workloads could be fully utilized. In contrast, for shorter datasets, the parallelization overhead often outweighed the benefits—highlighting the importance of dataset-specific tuning.

Although this implementation does not yet include low-level hardware optimizations such as NUMA-aware memory placement, such techniques remain a promising direction for further improving memory throughput and inter-device communication efficiency in future work.

For the cryptocurrency and spot index data, which were computationally light, all calculations were performed on a MacBook equipped with Apple M3 cores. Although GPU and MPI parallelization were not employed, thread-level parallelism was utilized via Python 3.12.11’s native multiprocessing to efficiently leverage multiple CPU cores.

3. Results

3.1. GSADF and MSM to S& 500 Futures in 2005–2019

Figure 1 presents the results of the GSADF test and the MSM applied to daily dollar-bar data of S&P 500 futures from March 2005 to December 2019. The 90% and 95% critical values were estimated via 1000 Monte Carlo simulations. The MSM was calculated with two regimes, and the exogenous variables included the lag = 1 value and the 20-day volatility.

The GSADF test is designed to detect episodes of transient explosiveness, rejecting the null hypothesis of a pure unit root process whenever the statistic exceeds the critical thresholds. This characteristic makes it useful for identifying structural breaks and speculative dynamics in financial markets.

Between May and July 2007, the GSADF statistic intermittently crossed the 90% critical threshold, coinciding with the escalation of the subprime mortgage crisis, following the February 2007 bankruptcy of New Century Financial (BLB&G., 2010). The GSADF signal intensified during the collapse of two Bear Stearns hedge funds in June and BNP Paribas’s suspension of three ABS funds in August (BNP Paribas, 2007), suggesting that the GSADF test captured early signs of systemic stress and an impending market regime shift.

A pronounced spike above the 95% critical threshold in October 2008 coincides precisely with the onset of the global financial crisis, triggered by the collapse of Lehman Brothers on 15 September 2008. This phase typifies a self-reinforcing crash, where feedback loops drive rapid, nonlinear price declines—an archetype of bubble bursts.

The GSADF statistic again exceeded the 95% threshold on 10 January 2018, shortly before the “VIX shock” in early February. This period was marked by record-breaking highs in U.S. equity indices and increased speculative activity. The subsequent volatility spike led to the unwinding of inverse VIX products like XIV and a cascade of algorithmic sell-offs, forming a feedback loop (Antoshin et al., 2018). The GSADF test appears to have preemptively detected the instability underlying this sharp market correction.

MSM results indicate that market instability began over a year before the GSADF statistic exceeded the 95% threshold, persisting until 2010. This suggests that MSM detected ongoing instability well ahead of the market’s acute stress points, providing a broader context for the instability observed in the GSADF statistic. Similarly, during the 2018 “VIX shock”, instability emerged after the GSADF statistic exceeded the 95% threshold, further illustrating the test’s ability to capture explosive market dynamics. MSM corroborates this finding, showing that the instability intensified following the identification of increased volatility by the GSADF.

These results affirm the GSADF test’s capacity to identify local structural breaks and speculative surges, even in mature markets like the S&P 500. Unlike conventional ADF tests, which may fail to detect periodically collapsing bubbles as highlighted by Evans (1991), GSADF offers a more robust framework for real-time market monitoring and risk assessment.

The 90th percentile zones in the figures are not critical regions but can be interpreted as precautionary zones. Movements within these regions may signal potential precursors to structural breaks, but they do not necessarily lead to confirmed breaks, as they can also be influenced by statistical noise. Therefore, when the market enters these zones, further analysis and careful observation are required. It is advisable to increase caution and monitoring as the likelihood of structural changes grows, although the transition to the 95% zone is not guaranteed.

While the 90th percentile zones provide valuable insights into potential market shifts, additional analysis and monitoring are recommended before drawing definitive conclusions. Future research may explore combining this approach with other models to improve the detection of early-stage market instability and confirm structural breaks.

In the following section, we apply this methodology to the cryptocurrency market, where price dynamics are more volatile and speculative episodes are more frequent. We examine whether GSADF can similarly uncover meaningful signals from this emerging asset class.

3.2. GSADF and MSM to BTC-USD in 2023–2025

We applied both the GSADF test and the Markov Switching Model (MSM) to the BTC-USD time series covering the period from October 2023 to April 2025 to assess the presence of explosive behavior and regime instability.

As shown in Figure 2, the GSADF statistic exceeded the 90% and 95% critical thresholds intermittently from late October 2023 through early April 2024, suggesting multiple episodes of transient explosiveness. These periods coincide with major macro-financial developments, including the Federal Reserve’s pause in rate hikes (Board of Governors of the Federal Reserve System, 2023), increased likelihood of spot Bitcoin ETF approval (Reuters, 2023), geopolitical uncertainty (International Monetary Fund, 2023), and shifting regulatory expectations following the 2024 U.S. presidential election.

In contrast, the MSM identified several distinct periods of heightened regime instability—specifically during July, August, and September 2024, as well as April 2025—when the smoothed probability of being in the high-volatility regime (Regime 1) exceeded 0.8. However, these periods did not align with peaks in the GSADF statistic. This divergence underscores the models’ differing sensitivities: while GSADF is calibrated to detect persistent deviations from the unit root null (e.g., speculative bubbles), MSM excels at identifying short-term regime shifts and volatility clustering.

Importantly, the MSM revealed that the BTC-USD market exhibited regime switching dynamics more frequently and more sharply than conventional equity markets such as the S&P 500 (see Figure 4). Yet the absence of GSADF peaks during these high-volatility MSM regimes suggests that short-term instability alone does not constitute a structural break or bubble episode.

These findings highlight the complementary nature of the two approaches: GSADF is well suited for identifying structural market transitions driven by sustained speculative pressures, whereas MSM captures rapid regime changes often driven by noise or sentiment shocks. A joint application of both methods offers a richer understanding of crypto market dynamics, distinguishing between transient turbulence and deeper structural shifts.

Due to the lack of reliable exogenous predictors and the limited availability of return-based regime indicators, MSM was not applied to the altcoin series. In those cases, GSADF alone was used to identify potential bubble episodes.

3.3. GSADF to Altcoins in 2023–2025

In this section, we apply the GSADF test to a selection of major crypto asset pairs: ETH-USD, SOL-USD, BCH-USD, and AAVE-USD.

As shown in Figure 3a, ETH-USD experienced short-lived structural changes in November and December 2023, and a sharp spike in March 2024 that was temporally aligned with BTC. However, unlike BTC, which exhibited a prolonged elevation of the GSADF statistic following the peak, ETH’s spike declined rapidly below the critical threshold. This suggests that while both assets reacted to the same macroeconomic event, their structural responses differed in duration and intensity. Notably, no significant change was detected after the 2024 U.S. election, further indicating that Ethereum may follow a distinct price formation mechanism compared to Bitcoin. In February 2025, ETH recorded a 36% decline from its 7-week local high (Santiment, 2025), likely driven by a combination of deteriorating market sentiment and panic-induced retail selling amid widespread fear, uncertainty, and doubt (FUD).

As shown in Figure 3b, SOL-USD exhibited a more prolonged period of structural change between November 2023 and January 2024. This shift was clearly linked to rapid expansion in Solana’s DeFi ecosystem. In November 2023, the monthly trading volume on decentralized exchanges (DEXs) built on Solana surged past USD 100 billion and reached USD 129 billion in November 2024—surpassing Ethereum’s previous all-time high of USD 117 billion set in May 2021 (SolanaFloor, 2024). This expansion was driven by Solana’s low fees and fast processing capabilities, combined with a retail influx fueled by a meme coin boom.

Another wave of structural change was observed in March–April 2024, synchronized with BTC and ETH. However, after the summer of 2024, no significant GSADF spikes were detected, and the market structure appeared relatively stable.

For AAVE-USD, Figure 3c, short-term structural changes were observed in November 2023, April 2024, and November 2025, though these episodes also lacked persistence.

As illustrated by these short-lived and idiosyncratic structural shifts—particularly in AAVE—there appears to be a broader pattern of divergence among related crypto assets. Unlike traditional markets, the cryptocurrency market exhibits a low overlap in structural breaks, even among tokens within the same ecosystem, such as ETH and AAVE.

According to Morales et al. (2023), this idiosyncratic behavior is especially evident in the ERC20 token network, where user behavior and transaction patterns vary significantly. For instance, a small number of users may create diverse portfolios and facilitate transactions between different tokens, while others focus on specific tokens. This variation in user behavior adds to the complexity and individuality of cryptocurrency market dynamics.

In our analysis, AAVE’s price movements were found to be strongly influenced by DeFi market dynamics, including liquidity provision and lending activity, while ETH, despite being a key player in DeFi, is more affected by broader network upgrades, infrastructure changes, and its role as the foundational token in the Ethereum ecosystem. Consequently, the price movements of ETH and AAVE do not always align, despite both being part of the same ecosystem. This highlights the distinct roles and dynamics of these tokens within the broader cryptocurrency market.

In the case of BCH-USD, Figure 3d, structural changes began slightly later, from December 2023, possibly due to its genealogical relationship with BTC. However, these changes were short-lived, and no significant shifts were detected in November 2023 or April 2025.

These results indicate that BTC and BCH are more sensitive to macroeconomic and institutional developments, while DeFi-related altcoins such as ETH, SOL, and AAVE exhibit structural changes tied to ecosystem-specific drivers.

In particular, ETH and SOL were influenced by internal factors such as upgrade schedules, Layer-2 migration plans, and developer activity. Their structural dynamics showed limited synchrony with BTC. A notable example is the structural change observed in ETH in April 2024, which likely stemmed from Ethereum-specific fundamental shifts—including migration pressure to Layer-2 networks, declining transaction fees, and delays in major upgrades Cointelegraph (2024).

In summary, the timing and persistence of structural breaks vary depending on the asset’s role and its sensitivity to external or internal market conditions. The GSADF test appears to capture not merely price fluctuations, but structurally significant episodes of speculative behavior consistent with macroeconomic, policy, or geopolitical catalysts. These findings support the utility of the GSADF procedure as a robust statistical tool for detecting speculative bubbles in cryptocurrency markets.

3.4. GSADF and MSM to the S&P 500 in 2023–2025

To compare structural dynamics between cryptocurrency and traditional equity markets, we applied the GSADF test to the S&P 500 index over the same two-year period using an identical methodology.

As shown in Figure 4, the GSADF statistics for the S&P 500 remained well below the critical thresholds throughout most of the period, and no significant structural break was observed. The brief fluctuation around April 2025 did not evolve into a persistent regime shift and is interpreted as a short-term reaction to transient macroeconomic shocks.

These results suggest that the time series structure of traditional equity indices, such as the S&P 500, is considerably more stable than that of crypto assets. Even in the presence of major events—such as presidential elections or temporary macroeconomic disturbances—the underlying structure appears resistant to abrupt changes, tending instead to absorb such events as statistical noise.

To compare structural dynamics between cryptocurrency and traditional equity markets, we applied both the GSADF test and MSM to the S&P 500 index over the same two-year period using an identical methodology.

As shown in Figure 4, the GSADF statistics for the S&P 500 remained well below the critical thresholds throughout most of the observation window, and no significant structural break was detected. A minor uptick around April 2025 was observed but did not culminate in a persistent regime shift, suggesting a transitory market response to temporary macroeconomic shocks.

Interestingly, the MSM transitioned into a high-probability unstable regime shortly after the GSADF statistic approached the 95th percentile in August 2024. Furthermore, in March–April 2025, MSM entered and remained in an unstable state even before GSADF crossed the 90% threshold. These results suggest that MSM may be more sensitive to early-stage market tremors and residual volatility stemming from broader macro-financial conditions.

Taken together, these findings highlight a complementary relationship between the two models. The GSADF test acts as a robust detector of structural breaks and speculative surges, while MSM functions as a volatility-sensitive alarm system—capturing both the precursors and the aftermath of potential regime transitions. Compared to cryptocurrency markets, the time series structure of traditional equity indices such as the S&P 500 appears considerably more resilient to such structural shifts, often absorbing shocks as statistical noise rather than entering sustained periods of instability.

Interestingly, the MSM transitioned into a high-probability unstable regime shortly after the GSADF statistic approached the 95th percentile in August 2024. Furthermore, in March–April 2025, MSM entered and remained in an unstable state even before GSADF crossed the 90th percentile. These results suggest that MSM may be more sensitive to early-stage market tremors and residual volatility stemming from broader macro-financial conditions.

Taken together, these findings highlight a complementary relationship between the two models. The GSADF test acts as a robust detector of structural breaks and speculative surges, while MSM functions as a volatility-sensitive alarm system—capturing both the precursors and the aftermath of potential regime transitions.

Compared to cryptocurrency markets, the time series structure of traditional equity indices such as the S&P 500 appears considerably more resilient to such structural shifts, often absorbing shocks as statistical noise rather than entering sustained periods of instability.

In sharp contrast to the structural stability observed in the S&P 500, the Bitcoin market exhibited repeated signs of structural change and regime instability during the same period.

Compared to DeFi-related altcoins such as ETH, SOL, and AAVE, Bitcoin’s regime shifts were more temporally aligned and persistent, suggesting its emerging role as a relatively “safe-haven” asset within the crypto ecosystem.

This contrast supports the interpretation that Bitcoin’s market value is no longer driven solely by speculative sentiment but increasingly reflects fundamental valuation factors. It may therefore function as a leading indicator of macro-level risk perception and economic shifts within the digital asset landscape.

3.5. Structural Co-Explosivity Across Crypto Assets and Equity Markets

To examine the interdependence of structural breaks across crypto assets, we constructed a co-explosivity matrix based on the co-occurrence of GSADF test exceedances.

Specifically, we identified time points at which the GSADF statistic exceeded the 95th percentile critical threshold, treating these as episodes of explosive dynamics. For each asset pair, we counted the number of days on which both assets simultaneously exceeded this threshold. The resulting symmetric matrix was then row-normalized by dividing each entry by the corresponding diagonal element, yielding a conditional co-explosivity rate—that is, the frequency with which asset B exhibited explosive behavior conditional on asset A being in such a state (see Figure 5).

The heatmap reveals that the S&P 500 index exhibits behavior that is distinctly different from that of all examined crypto assets. Aside from a marginal co-occurrence with ETH, no significant structural break overlap was detected between the S&P 500 and the crypto assets. This finding suggests a structural decoupling between traditional financial markets—driven primarily by macroeconomic and policy factors—and the highly volatile, speculation-driven nature of crypto markets.

In contrast, Bitcoin (BTC) demonstrated a high co-explosivity rate with nearly all crypto assets, particularly with ETH and BCH, exceeding those with AAVE and SOL. This suggests that BTC, as the dominant benchmark asset in the crypto ecosystem, plays a central role in transmitting market-wide speculative dynamics. The strong co-explosivity between BTC and BCH is also intuitive, given their historical and genealogical relationship.

Interestingly, ETH exhibited its highest co-explosivity with BTC rather than AAVE, despite both tokens being part of the same Ethereum ecosystem. Co-explosivity between ETH and AAVE was lower than that between ETH and BCH or SOL, highlighting the sectoral and functional heterogeneity within the ecosystem. This implies that while ETH functions as an infrastructural Layer-1 asset, AAVE behaves more idiosyncratically, reacting to DeFi-specific factors such as lending rates, regulatory announcements, and liquidity shocks.

Moreover, AAVE’s highest co-explosivity was observed with BTC rather than ETH, suggesting that even DeFi tokens are susceptible to broader market overheating, especially in BTC-led speculative phases. In such conditions, rising risk appetite may lead to a spillover of capital into smaller-cap tokens, amplifying co-explosivity.

To capture the temporal dynamics of these relationships, we constructed a rolling co-explosivity matrix using a 30-day window and plotted the time evolution (see Figure 6). This dynamic visualization reveals a market-wide speculative phase during March–April 2024, characterized by a sequential rise in co-explosivity, initially between BTC and ETH, followed by increased alignment with BCH and SOL. This pattern suggests a cascading capital inflow starting with major assets and diffusing toward riskier altcoins.

Throughout this period, the S&P 500 showed no signs of co-explosive behavior, reaffirming the structural divergence between traditional and crypto asset classes. The only notable co-explosive episode between ETH and the S&P 500 occurred briefly in April 2025, but was transient and non-persistent. This likely reflects a common reaction to a macroeconomic shock, rather than any fundamental coupling between the two markets.

Overall, the strength and timing of co-explosive episodes appear closely tied to each asset’s market role and sectoral function. The early rise of BTC’s co-explosivity confirms its structural centrality, while the delayed response of assets such as AAVE and SOL supports the interpretation that DeFi tokens act as secondary amplifiers of speculative market conditions.

4. Discussion and Conclusions

Speculative bubbles in financial markets can offer short-term profit opportunities but also pose substantial systemic risks. Detecting such nonstationary structural changes at an early stage is thus essential for financial engineering, asset management, and macroprudential oversight.

This study applied the GSADF test to high-frequency data for five major crypto assets—BTC, ETH, SOL, AAVE, and BCH—spanning April 2023 to April 2025. The test identified multiple explosive episodes, particularly in BTC, ETH, and SOL. These structural breaks coincided with macroeconomic shifts, institutional actions, and ecosystem-specific developments, highlighting the heterogeneous and episodic nature of regime transitions across the crypto landscape.

Applying the same methodology to the S&P 500 index revealed a stark contrast: despite observable short-term volatility, no statistically significant structural breaks were detected. This suggests that traditional equity markets exhibit greater structural resilience and are more capable of absorbing transient macroeconomic shocks compared to crypto markets.

Among crypto assets, Bitcoin displayed the most temporally persistent regime shifts, potentially reflecting its growing role as a macro-sensitive digital reserve asset. In contrast, Ethereum and DeFi-related tokens (e.g., AAVE, SOL) experienced shorter-lived and more idiosyncratic shifts, often triggered by internal technical factors such as protocol upgrades or liquidity shocks. Notably, even within the same ecosystem, ETH and AAVE showed weak co-explosivity, indicating that DeFi tokens respond more to sector-specific fundamentals than ecosystem-wide trends.

To further examine the temporal resolution of these shifts, we compared the GSADF results with regime probabilities from an MSM. While MSM helped identify rapid or transient instability not captured by GSADF, the model’s assumptions and limited interpretability raise concerns regarding its suitability for highly volatile assets like cryptocurrencies. Future work may explore more flexible or data-driven alternatives for capturing short-term nonlinear dynamics.

We also emphasize that while GSADF is effective in detecting the onset of explosive episodes, it does not predict their termination. In principle, the LPPLS model could address this gap. However, the large number of nonlinear parameters and fitting instability in LPPLS limit its practical utility—particularly for assets with short or noisy explosive phases. Advances in regularized optimization or physics-informed priors may help stabilize such models in future work.

Finally, our co-occurrence matrix analysis revealed a layered structure within the crypto market. Bitcoin emerged as a structurally central asset during explosive regimes, while DeFi tokens such as AAVE showed delayed and less consistent co-explosivity. The S&P 500 remained largely isolated, underscoring the structural disconnect between traditional equity markets and the crypto ecosystem. Dynamic co-explosivity plots further revealed wave-like propagation of speculative dynamics from BTC to other tokens, highlighting crypto’s endogenous synchronization structure.

In sum, this study demonstrates the utility of GSADF-based tests for identifying market regime shifts and episodic explosiveness in crypto markets. By integrating statistical detection with network-based co-occurrence analysis, we provide a framework for diagnosing both individual and systemic bubble-like behavior. Future research should aim to refine these tools—both methodologically and computationally—while exploring alternative models for bubble forecasting, short-term regime detection, and cross-asset contagion.