The dance of the markets: unveiling bitcoin’s time-varying financial correlations using a GAS-based approach

Diversification allows investors to minimise risks by selecting assets that are negatively or not correlated with each other for their portfolios. However, changes in market dynamics can lead to variations in these relationships. Therefore, accurately modelling the correlation between financial assets enhances portfolio performance and risk management, and reduces losses (Chen et al., 2025; Feder-Sempach et al., 2024; Hull and White, 2004; Longin and Solnik, 2001). When a financial instrument is uncorrelated or negatively correlated with another asset, one speaks of a hedge. When a hedge negatively correlates with the other instrument during times of distress and economic downturns, it is defined as a safe-haven (Bedowska-Sójka and Kliber, 2021; Baur and Lucey, 2010). Traditional correlation models, based on Pearson’s coefficient, often fail to capture the dynamic and non-linear relationships between different securities, ignoring factors like volatility clustering and tail dependence (Algieri et al., 2021; Cont, 2001). Previous research indicates that financial time-series exhibit changing dependence structures, with asset interdependence intensifying during crises (Lawuobahsumo et al., 2024; Das and Uppal, 2004; Patton, 2004). This effect, known as asymmetric dependence (Ang and Bekaert, 2002), is prevalent in the cryptocurrency markets, which are particularly sensitive to economic changes and different types of news (Telli and Chen, 2020). Incorporating exogenous information such as macroeconomic indicators into correlation models is crucial, as it can impact asset co-movements (Bauwens et al., 2006) and enhance prediction accuracy (Diebold and Yilmaz, 2009).

To address these challenges, we focus on Bitcoin and its linkages with the key financial indicators (S&P 500, NASDAQ, VIX, and WTI crude oil), and propose a novel framework based on an asymmetric Student-t distribution (Azzalini, 2005), with correlations governed by Generalised Autoregressive Score (GAS, Creal et al., 2013) dynamics. This advanced approach overcomes the limitations of traditional models by offering a more flexible and accurate representation of financial returns. In particular, GAS models represent an easy and efficient way to make the parameters dynamic, as they are updated according to the maximum directional increase of the score function.

Hence, we contribute to the existing literature by providing an effective way to choose time-varying parameters with the introduction of a score function as a driving mechanism. This approach enables us to capture not only the first or second-order moments but also the broad density structure, and it further allows us to examine tail dependence, which is relevant in understanding extreme events such as market crashes or surges. By modelling extreme events jointly, one can assess how cryptocurrencies and stock indices behave under stress conditions, and understanding the complex relationship between cryptocurrencies and traditional financial assets offers insights for portfolio management and risk assessment. Our approach is further designed to solve the shortcomings of conventional models, providing a more adaptable and precise characterisation of financial data, as it encompasses time-varying skewness, heavy tails, and volatility clustering, features commonly observed in financial markets. For instance, Multivariate GARCH models, although effective at handling volatility clustering, fail to adequately capture skewness and tail risk. In contrast, our proposed GAS-based models offer superior adaptability and performance while maintaining reduced computational complexity.

We also introduce explanatory variables into the model specification to examine their impact on the time-varying correlation between the pairs under consideration. The correlations between cryptocurrencies and other financial variables can have implications for monetary policy and financial stability; therefore, our GAS-based approach can help policymakers to better understand the interconnectedness of these markets and formulate appropriate policies. Our results underscore the significance of exogenous variables in tracking time-varying correlations.

The remainder of the paper is organised as follows. In Section 2, we review the related literature. Section 3 introduces the methodology, covering the asymmetric Student-t case first and then exploring the Student-t, normal, and asymmetric normal cases as special instances. Section 4 shows an empirical analysis where the model is employed on real data (Bitcoin, S&P 500 index, NASDAQ index, VIX index, and WTI crude oil). Section 5 concludes the paper.

Attention to the cryptocurrency market has increased over the years due to its rapid growth, raising considerable research into risk management, portfolio optimisation, and price predictions. These digital currencies are characterised by high volatility and unique risk profiles, requiring robust analytical and forecasting methods. The growing body of literature on cryptocurrencies ^[1] focuses on applying traditional statistical models, machine learning techniques, and hybrid frameworks to enhance understanding and address these concerns. The following literature review summarises advancements in the field of cryptocurrency, highlighting key methodologies and their applications.

Due to the complex and evolving nature of cryptocurrencies, research efforts have recently converged on improving the analysis of volatility, risk, and market dynamics (Chen et al., 2025). Dudek et al. (2024) compares traditional statistical models with machine learning techniques for the forecast of cryptocurrency volatility. The authors find that no single model is the best method for predicting the volatility of each cryptocurrency, given that different models perform better for each cryptocurrency (Bitcoin, Ethereum, Litecoin, and Moreno), choice of the error metric, and forecast horizon. Chen et al. (2024) emphasise the importance of selecting the appropriate distribution to improve forecasting precision. Using alternative GARCH specifications and distributions for the innovation term, the authors show that the skewed and heavy-tailed distributions yield better volatility and Value at Risk (VaR) predictions, accurately reflecting the heavy tails in cryptocurrency markets. Furthermore, Huang et al. (2024) employ an ARMA-GARCH-VaR framework to assess market risk for Bitcoin, Ethereum, and Binance Coin and document that their models effectively capture volatility clustering and extreme tail risk.

To improve cryptocurrency volatility forecasting, Feng et al. (2024) introduce a daily dynamic tuning strategy capable of adapting models in real-time to explore factors influencing the performance of cryptocurrency market volatility prediction. Their results show that the tuning strategy significantly reduces forecasting errors, especially during market turbulence. García-Medina and Aguayo-Moreno (2024) explore cryptocurrency volatility forecasting using GARCH models, multilayer perceptron (MLP), long short-term memory (LSTM), and hybrid LSTM-GARCH models, focusing on the period surrounding the March 2020 pandemic declaration with hourly data. The findings indicate that deep neural networks, particularly MLP models, outperform GARCH models in terms of accuracy and computational efficiency, with no significant differences in predictive accuracy among MLP, LSTM, and LSTM-GARCH models. Portfolio analysis reveals that volatility forecasting improves Sharpe ratios for a uniform portfolio over long horizons. Ahmed et al. (2024) provide a comprehensive review of the current state of the literature concerned with the volatility of cryptocurrencies, identifying gaps and setting a future research agenda.

Other studies consider various aspects of cryptocurrency risk modelling. Bulut et al. (2024) use a fuzzy decision-making model to evaluate multiple cryptocurrency risk factors, identifying liquidity risk and regulatory uncertainty as the most significant. Boubaker et al. (2024) assess systemic risk tolerance in cryptocurrencies using machine learning and econometric models, highlighting the interconnectedness and extreme events that shape systemic vulnerabilities in crypto markets. They observe a common systemic risk tolerance trend among cryptocurrencies, with Bitcoin and Ethereum being less tolerant during systemic crises. Barson and Junior (2024) investigate the effectiveness of various model specifications and distributional innovations in capturing tail risk of cryptocurrencies (Bitcoin, Ethereum, and Litecoin), non-fungible tokens, stocks (FTSE 100 and S&P 500), and Gold. Results reveal that no single model or metric is universally superior; non-Gaussian distributions best capture the asymmetry and fat tails in asset returns, with Gold exhibiting greater homogeneity in its distributional assumptions, emphasising the importance of robust internal risk modelling to enhance investor confidence. Gkillas et al. (2024) explore the discontinuities and asymmetries in cryptocurrency price movements using a jump-diffusion model. They observe asymmetries with losses being more pronounced than gains during market downturns.

Some studies compare several models to identify the best method for predicting cryptocurrency volatility, tail risk, and price. Ünvan and Ergenc (2024) consider various machine learning models for a number of cryptocurrencies, assess the performance of competing models based on different measures of forecasting accuracy and find that no single model consistently outperforms others for all cryptocurrencies. AlMadany et al. (2024) utilise deep learning techniques alongside classical methods to forecast cryptocurrency returns and show that the hybrid EGARCH-LSTM or GARCH-LSTM models demonstrate slightly better accuracy when compared to other models. Muchtadi-Alamsyah et al. (2024) find that the combination of SVR with AR and GARCH models, and using polynomial or radial basis kernels, improved volatility, VaR, and expected shortfall forecasting compared to other distribution assumptions, highlighting an asymmetric non-linear relationship between returns and volatility for a combined Support Vector Regression (SVR) and an asymmetric GARCH model.

Another strand of the literature tries to investigate and understand the interactions between cryptocurrencies and other asset classes, with the aim of providing investors with valuable insights to make informed decisions in financial markets. While some research suggests that cryptocurrencies offer diversification and hedging benefits (Terraza et al., 2024; Colon et al., 2021; Guesmi et al., 2019; Urquhart and Zhang, 2019; Dyhrberg, 2016), others indicate their limited effectiveness as safe-havens or hedging instruments (Feder-Sempach et al., 2024; Bedowska-Sójka and Kliber, 2021; Smales, 2019; Wang et al., 2019; Klein et al., 2018). For instance, Bedowska-Sójka and Kliber (2021) document that Bitcoin and Ether can be occasionally considered as weak safe-haven against stock indices. Conversely, Gold could be regarded as a strong safe-haven within the years 2015–2019, but not during the coronavirus outbreak. Bouri et al. (2017a, b) and Demir et al. (2018) suggest that Bitcoin can serve as a short-term hedge in extreme market conditions. Osman et al. (2023) also find that cryptocurrencies can provide short-term diversification benefits by being disconnected from traditional markets.

Table 1 provides a summary of the literature on Bitcoin and its safe-haven properties.

Klein et al. (2018) employ multivariate BEKK-GARCH volatility models to analyse the dynamic correlation between Bitcoin and Gold. They report that while Gold acts as a safe-haven during market distress, Bitcoin behaves in the opposite way, positively correlating with downward market movements. Some studies have focused on the relationship between Bitcoin and commodities such as crude oil. Selmi et al. (2018) use the conditional and unconditional Quantile-on-Quantile Regression (QQR) technique to examine the impact of Gold and Bitcoin on the crude oil market and find that both can hedge oil market crises. Guesmi et al. (2019) employ the VARMA-DCC-GARCH framework and report that Bitcoin can reduce portfolio risk when added to a portfolio alongside crude oil. Moussa et al. (2021) use various regression techniques to analyse Bitcoin’s relationship with commodities such as oil, Gold, and coal, emphasising the importance of dynamic modelling for understanding their long-term interactions. Kumah and Odei-Mensah (2022) employ wavelet techniques and quantile regressions to investigate the hedging properties of various cryptocurrencies against oil market volatility and highlighted Ethereum, Stellar, Ripple, and Monero as potential hedges. Conlon and McGee (2020) question Bitcoin’s diversification potential during the COVID-19 pandemic, observing an increased correlation with the S&P 500. They conclude that Bitcoin did not act as a safe-haven during the crisis, potentially increasing portfolio downside risk when held alongside equities, consistent with what Klein et al. (2018) observed for commodity markets. Kim et al. (2020) also note significant correlations between Bitcoin and traditional assets during the cryptocurrency crash and the COVID-19 pandemic, indicating potential correlations during market crises. Shen (2022) apply the BEKK-GARCH model to assess time-varying correlations among Bitcoin, Gold, and major market indexes, revealing that Bitcoin is unable to hedge market risks during crashes. Ghorbel and Jeribi (2021) identify a weak association between Bitcoin and conventional assets, suggesting Bitcoin’s relative isolation from financial markets. Cortese et al. (2023) show a similar result, using a sparse statistical jump model (Nystrup et al., 2021) to identify the main drivers of crypto-markets, finding that momentum is the primary driver (Liu et al., 2022). Oosterlinck et al. (2023) provide evidence from the 2022 invasion of Ukraine, indicating that Gold and Bitcoin complement each other rather than serving as substitutes during crises, with Bitcoin diversifying oil risk more effectively than Gold. In a nutshell, these studies highlight the complex and evolving relationship between cryptocurrencies and traditional financial assets, emphasising the importance of dynamic modelling of correlation.

The literature on correlation models highlights the importance of dynamic distributions (e.g. Bedowska-Sójka and Kliber, 2021) and non-linear methods, such as Dynamic Conditional Correlation models (DCC) initially proposed by Engle (2002) and copulas (Cherubini et al., 2004), to accurately capture the asymmetries in traditional financial assets and cryptocurrency market dependencies. The main drawback of DCC is that it often relies on normality, which is a questionable assumption in a financial context.

Copula models (Patton, 2013) provide a flexible framework for capturing complex dependence structures, particularly through the Student-t copula of Demarta and McNeil (2005), which is often preferred in financial time-series modelling for its ability to represent tail dependence, excess kurtosis (Fischer et al., 2009; Huang et al., 2009; Breymann et al., 2003), and skewness (Smith et al., 2012). Parameters of the copula can be made time-varying, considering the autoregressive approach of Patton (2006), further extended by Oh and Patton (2018) that use a GAS equation term for the copula parameters, or through a regime-switching approach (Hamilton, 1989), specifically assuming a Markovian dynamics for the copula parameters (Cortese et al., 2024; Nasri and Rémillard, 2019; Härdle et al., 2015). However, their complexity, the computational effort required for feasible estimation, and the difficulty in selecting an appropriate copula function can be challenging and potentially lead to model misspecification. For instance, the skew−t copula of Demarta and McNeil (2005) is computationally inefficient as the marginal densities are not available in closed form. This forces the use of slow numerical methods, such as Monte Carlo simulation, for likelihood evaluation, which is impractical for the recursive updates of a dynamic GAS model.

Our method, instead, is closely related to the skew−t model of Azzalini (2005). The reason for adopting this approach is that, among the various skew−t methods available, it presents particular advantages in modelling asymmetric dependence structures in financial data. Recent comparative studies reveal that the Azzalini method allows for greater asymmetric tail dependence, which is crucial for accurately representing extreme co-movements in financial time-series (Deng et al., 2025). Indeed, compared to the skew−t copula of Demarta and McNeil (2005) and the one of Smith et al. (2012), the Azzalini skew−t method exhibits superior flexibility in capturing higher levels of asymmetric and tail dependence. ^[2].

Our focus on the Azzalini variant of the skew−t copula, therefore, reflects a deliberate methodological choice to achieve an optimal balance between empirical expressive power and computational feasibility for dynamic modelling. While alternative skew−t copulas provide additional structural flexibility, their integration within the GAS framework is constrained by practical considerations. The GAS model relies on the analytic score for recursive, time-varying updates, and more complex copulas typically lack closed-form marginal distributions. Consequently, they require numerically intensive approximations that are often impractical for high-dimensional or dynamic estimation. Furthermore, adopting more parameter-rich copulas is prone to overfitting and exacerbates challenges in parameter identification, particularly when the GAS framework already involves a substantial number of time-varying parameters.

Several advancements in modelling dynamic correlation matrices and volatility have emerged in the recent literature. Hafner and Wang (2023) introduced a model where the dynamics of correlation matrices are driven by the likelihood score corresponding to the matrix logarithm of the correlation matrix, maintaining positive-definiteness through a transformation akin to exponential GARCH for volatility. They employed a Student−t copula framework for the conditional dependence structure, facilitating separation of volatility and correlation for high-dimensional estimation. In a similar pursuit of scalability, D’Innocenzo and Lucas (2024) developed a recursive framework of bivariate partial correlation models, which uses stochastic recurrence equations to estimate partial correlations from bivariate slices, allowing for flexible restrictions. Finally, Zheng and Ye (2024) introduced a Cholesky-based generalised autoregressive score model for large-dimensional covariance matrices, leveraging score-driven updates resembling GARCH and stochastic volatility models to ensure computational feasibility. Their method also incorporates dynamic model averaging to handle model and parameter uncertainty in high-dimensional cases. Together, these contributions enhance our understanding of dynamic dependencies, risk modelling, and high-dimensional estimation in financial econometrics.

With an attention to cryptocurrency portfolio optimisation, Jeleskovic et al. (2024) integrate a GARCH-copula model within the Markowitz framework for optimising cryptocurrency portfolios, highlighting the benefits of incorporating time-varying dependence structures for enhancing risk-return for investors in volatile cryptocurrency markets. In a similar study, Dobrynskaya (2024) examines whether downside risk is priced in cryptocurrency markets using downside risk metrics and finds that downward market movements disproportionately influence risk premiums, offering insights for portfolio management and risk assessment.

GAS-based models are gaining attention in financial econometrics literature due to their flexibility in modelling time-varying parameters ^[3]. Researchers in cryptocurrency markets are considering GAS-based approaches as a new avenue for understanding the unique characteristics of this asset class, including extreme volatility, clustering, and asymmetric responses to shocks. For instance, Troster et al. (2019) analyse Bitcoin returns and risk forecast using GARCH and GAS approaches. Considering heavy-tailed GARCH specifications, they compare out-of-sample VaR forecasts under different specifications, highlighting the superiority of GAS models with heavy-tailed distributions. Liu et al. (2020) examine whether reasonable and economical methods for VaR forecasting can be relied upon by investors in cryptocurrency markets. The study concludes that the Laplace GAS specification performs well across different levels by accounting for time variation in both scale (volatility) and skewness (asymmetric responses to positive and negative volatility). Vieira and Laurini (2023) model time-varying higher-order (scale, skewness, and kurtosis) moments structure for Bitcoin return under non-traditional distributions innovations comparing predictive performance using a loss function based on VaR performance. Mili and Bouteska (2023) adopt the GAS method to forecast Bitcoin and fiat currencies correlations and show that the GAS copulas model better forecasts the volatility and dependency of multiple tail dependence regimes and produces a more accurate VaR forecast. Investigating the co-dependence and portfolio VaR of Bitcoin, Ethereum, Litecoin, and Ripple, Cheng (2023) use a GAS model to demonstrate its ability to effectively handle dynamic volatility and correlation during periods of volatility. Ivanovski and Hailemariam (2023) examine time-varying dependence among S&P500, NASDAQ, Dow Jones Industrial, Bitcoin, and Ethereum and show that the GAS framework outperforms the traditional DCC-GARCH model in capturing volatility persistence and nonlinearity between stocks and cryptocurrencies.

The existing literature within the cryptocurrency spectrum signifies advancements in the methodologies used for cryptocurrency market risk and volatility forecasting. Researchers are employing diverse approaches, ranging from traditional econometric models to advanced machine learning and artificial intelligence techniques, to address challenges in the cryptocurrency market. These advancements are driven by the market’s inherent characteristics, such as extreme price volatility, irregular trading patterns, and the influence of macroeconomic indicators and sentiment data. As this field continues to evolve, the intersection of traditional finance theories and advanced computational techniques offers promising avenues for more robust and accurate risk and volatility forecasting in the cryptocurrency market.

This study proposes a novel methodology for modelling time-varying pairwise correlations, which allows for a simple integration of exogenous variables as explanatory factors. The proposed method is then applied to Bitcoin returns. Investigating the correlation between Bitcoin and financial indicators such as VIX, S&P 500, WTI crude oil, and NASDAQ is critical for assessing Bitcoin’s nature as an asset class within the global financial system. For instance, the relationship with VIX could reveal how Bitcoin behaves during periods of market stress; correlation with the S&P 500 and NASDAQ could measure its integration with equity risk, determining if the cryptocurrency still offers diversification benefits; and its link with WTI crude oil could indicate its sensitivity to global economic cycles and geopolitical events.

The correlations are modelled using a Generalised Autoregressive Score dynamics where the time-t correlation depends on a constant, a term depending on its past value, one depending on the score of the log-likelihood function at time t − 1, and, lastly, a term depending on lagged exogenous factors. GAS models are a valuable tool for time-series analysis, offering flexibility and improved accuracy. They efficiently incorporate higher-order moments—such as skewness and heavy tails—and handle time-varying parameters within a unified framework, leading to better forecasting and risk management. The score-driven updating mechanism offers two key advantages: it allows econometricians to incorporate new information more efficiently than GARCH or DCC models, and it improves robustness to outliers by down-weighting the impact of extreme observations. Additionally, contrary to DCC models, GAS models easily allow for the inclusion of external factors in the correlation dynamics in a straightforward manner. Therefore, our GAS models, with their ability to naturally incorporate these external explanatory factors, enable us to establish why the correlation between Bitcoin and mainstream assets (S&P 500, NASDAQ, VIX and WTI crude oil) changes. The proposed methodology is hence an effective mechanism for both investors and regulators to seize market developments, inform trading strategies, and facilitate effective portfolio management.

Let y_1,t and y_2,t be the log returns, i.e. changes in the log prices, at time t, t = 1, …, T, of two assets (in our case, BTC and S&P 500; BTC and NASDAQ; BTC and VIX, and BTC and WTI). Building upon the DCC model proposed by Engle (2002), our approach involves a two-stage process. Initially, we model the mean and variance of asset returns separately using ARMA-GARCH models (Engle and Bollerslev, 1986). In the second stage, we focus on the marginal standardised residuals, namely:

Where $uˆi,t$ are the residuals in the mean equation and $σˆi,t$ are the volatilities from the estimated GARCH model for asset i.

Given a bivariate distribution f(η_1,t, η_2,t; ρ_t) for the pair (η_1,t, η_2,t), depending on a time-varying correlation parameter ρ_t, we assume that ρ_t is modelled via a GAS(1,1) model (Creal et al., 2013)

Where s_t = S_t∇_t is the scaled score of the log-likelihood. Here, $▿t=∂ℓt∂ρt$ is the score, i.e. the partial derivative of the log-likelihood function with respect to ρ_t, S_t is a scaling matrix, and X_t is a vector of explanatory variables. Given the information matrix $It=−Et−1∂2ℓt∂ρtρt′=Et−1▿t▿t′$⁠, the scaling matrix is typically constructed as $St=It−k$⁠, with the exponent k ∈ {0, 1/2, 1}. In our model, we consider only the correlation coefficient to be time-varying. In particular, to ensure that ρ_t belongs to (−1, 1), we consider the inverse hyperbolic tangent transformation δ_t = atanh(ρ_t). Hence, Eq. (1) becomes

Where X_t−1 denotes a vector of explanatory variables and $st=St×∂ℓt∂δt$⁠. Note that

In Appendix A we consider various analytical forms for the bivariate density functions, needed to derive the log-likelihood function and the score mentioned above, starting with the asymmetric Student-t distribution. The subsequent forms can be derived as special cases of this broader distribution. For each distribution, we give the scores and, only for models without asymmetries, Fisher’s information, as in this case, it is only available in closed form. Appendix B presents a simulation study aimed at evaluating the performance of the proposed statistical model.

For our analysis, we investigate the correlations between BTC ^[4] and each mainstream asset, namely S&P 500, NASDAQ, VIX, and WTI, and we use Gold (GOLD), Market Yield on US Treasury Securities at 10-Year Constant Maturity (DGS10), 10-Year Treasury Constant Maturity Minus 3-Month Treasury Constant Maturity (T10Y3M), and 10-Year Breakeven Inflation Rate (T10YIE) as explanatory variables.

In particular, by considering BTC, S&P 500, NASDAQ, VIX, and WTI crude oil, it is possible to assess how our proposed model captures correlation dynamics across different markets (cryptocurrency, equity, and commodity) ^[5]. We include the NASDAQ and S&P 500 indices to ensure that we account for sectoral differences within the equity market, given that NASDAQ is heavily weighted toward technology and growth-oriented companies, making it particularly sensitive to innovation-driven sectors, interest rate changes, and market sentiment surrounding high-growth stocks. At the same time, the S&P 500 represents a diversified basket of 500 large-cap US companies across various industries, providing a comprehensive measure of the overall equity market.

The considered explanatory variables provide an economically meaningful foundation for the observed time-varying correlations. For instance, Gold is the most popular safe-haven asset, considered a traditional inflation hedge and serving as a currency and store of value (Bedowska-Sójka and Kliber, 2021; Baur and Lucey, 2010). During periods of economic uncertainty, financial instability, or market stress, Gold protects market participants against volatility and currency devaluation. Because of its intrinsic value, it is also considered an effective hedge against inflation ^[6]. Including Gold as an exogenous factor directly tests whether the correlation between BTC and, for example, the S&P 500, is influenced by the simultaneous flow of capital into traditional safe-havens. If the BTC-equity correlation changes in the presence of strong demand for Gold, it helps determine if BTC is competing with or complementing Gold as a digital store of value.

The 10-Year Treasury yield (DGS10) serves as the benchmark for the “risk-free” rate and is a key measure of monetary policy expectations. Higher yields indicate tighter monetary policy, which may reduce the availability of cheap credit and slow economic growth, whereas lower yields can spur growth by encouraging investment and consumption. As a reflection of the opportunity cost of holding riskier assets versus risk-free government bonds, when interest rates rise, bond yields become more attractive, which can reduce demand for riskier assets. Conversely, falling yields make riskier investments more attractive as they promise higher returns, altering their correlation with bond prices. Hence, since the value of most financial assets (especially high-growth tech stocks like those in the NASDAQ) is sensitive to interest rates, the DGS10 acts as a fundamental lever for asset valuation. By including it, we can determine whether the dynamic correlation between BTC and equities is driven by macroeconomic shifts in interest rates.

The difference between the long-term (10-Year) and short-term (3-Month) Treasury yields is one of the most reliable predictors of economic recession. When this spread narrows or inverts, it signals heightened economic stress and gloomy economic perspectives (Algieri et al., 2025; Park, 2022; Estrella and Mishkin, 1998; Estrella and Hardouvelis, 1991). Put differently, a steepening yield curve suggests stronger economic growth expectations, while a flattening or inverted yield curve (short-term rates are higher than long-term rates) is a precursor to an economic slowdown or recession. An inverted yield curve causes correlations between risky assets (such as cryptocurrencies, equities, or commodities) to increase as all markets adjust to higher perceived risks. Conversely, during periods of steepening, correlations may shift as different assets react to positive growth expectations in varying ways (Bekaert and Engstrom, 2010; Ang et al., 2008). By adding T10Y3M as an exogenous variable, it is possible to test whether the correlations between BTC and other assets shift during different economic regimes, specifically whether BTC loses its diversification potential when a market downturn is looming.

The 10-Year Breakeven Inflation Rate measures what the market expects inflation to average over the next decade. With higher expectations of rising prices, investors look for assets that offer an alternative way to preserve purchasing power to anticipate future inflation (Liu and Valcarcel, 2024; Polizu et al., 2023; Orlowski and Soper, 2019; Ang et al., 2006; Estrella and Mishkin, 1996). Bitcoin is often touted as a hedge against fiat currency devaluation and inflation. The inclusion of T10YIE in the models allows us to test whether changes in inflation expectations independently affect the correlation between BTC and other assets, thereby assessing its viability as an inflation-hedging tool in investors’ eyes.

Our analysis covers the period from 7 August 2015 to 7 December 2023, for a total of 2050 observations per series. We source BTC, S&P 500, NASDAQ, and VIX data from Yahoo Finance ^[7], WTI, and Gold from Bloomberg ^[8], DGS10, T10Y3M, and T10YIE from the FRED ^[9] database.

For the included commodities, we consider their futures prices because they are key to many forward-looking decisions made by economic agents and serve as important price signals that help shape future spot prices (Ameur et al., 2022; Algieri and Leccadito, 2017). Futures prices reflect expectations of supply and demand for the respective commodities. For example, producers might determine their supply strategies based on futures contract prices, while investors could plan their asset allocation by analysing futures price trends. As futures contracts bind sellers and potential buyers to transact at a predetermined future date and price, they capture market sentiment around commodities. Additionally, futures contracts are commonly used for speculative purposes, making their connection with cryptocurrency markets more intuitive.

Table 2 shows the Pearson correlation coefficients among the variables. There is a weak positive correlation between BTC and the S&P 500, BTC and NASDAQ, and a weak negative correlation between BTC and VIX. We also notice low correlations between BTC and DGS10, BTC and T10Y3M, and BTC and T10YIE. Figure 1 displays log-returns of BTC, VIX, S&P 500, NASDAQ, and WTI, while Figure 2 provides time-series plots for the log-return for GOLD and the first differences of DGS10, T10Y3M, and T10YIE. Table 3 displays the descriptive statistics for the data, showing that returns are leptokurtic and negatively skewed, except for VIX. The null hypothesis of a Gaussian distribution is strongly rejected for all the series as indicated by the near-zero p-values of the Jarque-Bera test.

In our analysis, we scrutinise four pairs of variables and delve into the dynamic correlations between cryptocurrency and financial markets. Our focus lies in comparing how these correlations evolve across different pairs of cryptocurrency and financial market data. In particular, we consider pairs as given in Table 4.

We adopt the Log-Likelihood, the Akaike Information Criterion (AIC) (Akaike, 1974), and the Bayesian Information Criterion (BIC) (Schwarz, 1978) to ascertain our model’s performance ^[10]. Also, based on the significance of the estimated parameters, we can determine if our model captures relevant features regarding the data. Results in Table 5a, 5b, 5c, and 5d show that the asymmetric Student-t model, exhibiting the lowest AIC and BIC values across all considered pairs, is the preferred choice for modelling correlation dynamics. Accounting for asymmetry and heavy tails in modelling the relationship between cryptocurrency and financial market data yields the best fit, as our model effectively captures these features. In fact, the skewness and kurtosis parameters are statistically significant at the 1% confidence level.

Figure 3 shows the time-varying correlation between BTC and the other financial variables, estimated using the asymmetric Student-t model. We observe that since 2020, the correlations between BTC and the S&P 500, as well as BTC and NASDAQ, show a higher positive trend. The increase in correlation has started in the COVID-19 period (grey-shaded area in Figure 3). This increasing correlation may reflect the growing integration of Bitcoin into mainstream financial markets and its perception as a “risk-on” asset (or high-beta asset). The heightened positive relationship suggests that the diversification benefits of holding Bitcoin alongside major equity indices have diminished considerably since the pandemic, challenging the view of Bitcoin as an uncorrelated asset (Wang et al., 2019; Demir et al., 2018), especially during periods of high liquidity and risk-taking. Our results align with previous studies (Oosterlinck et al., 2023; Lawuobahsumo et al., 2022; Kumah and Odei-Mensah, 2022; Shen, 2022; Conlon and McGee, 2020; Kim et al., 2020; Guesmi et al., 2019; Klein et al., 2018; Selmi et al., 2018; Bouri et al., 2017a, b), confirming that BTC cannot serve as a hedge or safe-haven for the stock market.

In contrast, we observe a negative correlation between BTC and WTI, suggesting that the two assets play different roles in the economy. In addition, BTC and WTI are affected to varying extents by macroeconomic conditions and geopolitical developments, including the war. The negative correlation with WTI crude oil is relevant because it provides evidence of decoupling from traditional commodity price movements and offers a potentially valuable diversification opportunity for portfolios exposed to energy risk. As illustrated by the shaded regions in Figure 3, correlation levels show a sustained decline during the COVID-19 era, in contrast to the sharp escalation triggered by the invasion of Ukraine. The correlation between BTC and VIX remains relatively stable around zero likely due to their divergent volatility drivers. This near-zero correlation with VIX implies that Bitcoin’s own volatility and risk are often driven by idiosyncratic, crypto-specific factors (such as regulatory news or platform events) rather than being a direct amplifier or mitigator of generalised financial market fear.

To investigate the influence of external factors on the correlation between cryptocurrency markets and traditional assets, we include interest rates (DGS10), market expectations (T10Y3M), expected inflation (T10YIE) and Gold (GOLD), as explanatory variables. These variables enter the models since they have a significant economic role, and they likely influence the correlation structure between assets by altering market expectations, risk perceptions, and macroeconomic fundamentals (Liu and Valcarcel, 2024; Polizu et al., 2023; Şarkaya İçellioğlu and Öner, 2019). For instance, when interest rates change, the global economy and financial markets, including bonds, stocks, commodities, and crypto, are affected. Indeed, high interest rates lessen the appetite for higher-risk, higher-return assets, whereas low interest rates increase the appetite for the same types of risky assets (Polizu et al., 2023).

Estimation results for the four pairs are reported in Table 6. In particular, Table 6a shows the significance of each explanatory variable on the correlations between BTC and S&P500. The Student−t and asymmetric Student−t models outperform the Normal and asymmetric Normal models, both show lower AIC and BIC. This superior fit confirms the necessity of using specialised GAS models that explicitly account for the non-Normal characteristics of financial data, such as the heavy tails and asymmetry observed in the returns of the BTC and S&P 500 pair. The coefficient for GOLD is positive and significant at the 10% level for the pair BTC and S&P500, indicating that GOLD ties the two assets together and positively impacts their correlation. Specifically, a unit increase in GOLD returns leads to a 3.0496 unit increase in the inverse hyperbolic tangent correlation for the Student-t model and a 3.2294 unit increase for the asymmetric Student-t model. These results would suggest that in times of uncertainty, investors flock to Gold, and this sentiment could also affect their perception and trading of Bitcoin and securities linked to the S&P 500, leading to increased correlation. Economically, this finding is interesting because it implies that the correlation between Bitcoin and the broad equity market is not purely random but is mediated by a traditional systematic risk factor (the flight to Gold). Gold thus acts as an observable economic indicator that explains the co-movement of the two assets, highlighting that Bitcoin’s correlation with stocks is not completely idiosyncratic.

Table 6b evaluates the significance of each explanatory variable on BTC and NASDAQ correlations. We show that the Student−t and asymmetric Student−t models are the preferred choices. Again, the dominance of the Student-t specifications validates the model choice by accurately capturing the high volatility and extreme returns inherent in the BTC and NASDAQ relationship. However, none of the selected explanatory variables significantly impacts the correlation between BTC and NASDAQ. This striking difference from the S&P 500 result suggests that the strong positive co-movement between Bitcoin and the technology-heavy NASDAQ is likely driven by shared, high-frequency, or speculative sentiment factors that are intrinsic to the digital and tech investment ecosystem, rather than by the considered macroeconomic indicators (Gold, inflation expectations, or yield spreads). The correlation between these two “risk-on” tech assets appears more self-sustaining and less responsive to external systematic forces.

Table 6d shows the results evaluating the significance of the explanatory variables on the pairs BTC and WTI. The Student−t and asymmetric Student−t models fit the data better than the Normal and asymmetric Normal models, with lower AIC and BIC values and higher log-likelihoods. This sustained outperformance of the Student-t specifications across multiple pairs (S&P 500, NASDAQ, and WTI) firmly establishes the relevance of employing dynamic models that capture the non-Gaussian characteristics of cryptocurrency returns when examining financial market linkages. In Table 6d, we show that Gold is significant at a 5% level for all models and negatively impacts the BTC and WTI correlation. In particular, a unit increase in Gold causes a 2.6100, 2.9660, 3.5025, and 3.4686 units decrease in the inverse hyperbolic tangent correlation between BTC and the WTI for the Normal, asymmetric Normal, Student−t and asymmetric Student−t models, respectively. The negative coefficients for Gold indicate that when Gold prices rise, the correlation between BTC and crude oil tends to weaken. This result would suggest that Gold acts as a hedge or a diversifier. Specifically, as economic uncertainty increases, investors may shift funds into Gold, potentially decoupling BTC and WTI, which are more sensitive to economic growth and demand. This negative influence highlights Gold’s role as a systematic risk-separator; the flight to the traditional safe-haven (Gold) causes an observable push apart in the price movements of the digital asset (BTC) and the real-economy commodity (WTI), signalling two distinct risk regimes are at play. DGS10 is significant at a 5% level for the Normal model, and a 10% level for the asymmetric Normal, Student−t, and asymmetric Student−t models. DGS10 has a negative impact on BTC and WTI correlation, where a 1% increase in DGS10 causes a 36.70%, 56.21%, 45.70% and 67.16% decrease in the inverse hyperbolic tangent correlation between BTC and the WTI for the Normal, asymmetric Normal, Student−t and asymmetric Student−t models, respectively. With higher expectations of tighter monetary policy, investors tend to reduce their exposure to riskier assets like BTC and WTI, thus weakening their correlation. The negative impact of rising DGS10 (higher interest rates) demonstrates that the link between BTC and the real-economy commodity WTI is inversely responsive to the monetary environment. Tighter financial conditions—the cost of money—cause a broad-based reduction in risk exposure, thereby weakening the simple co-movement between these two fundamentally different risky assets.

Furthermore, the T10YIE is significant at 5% for the Normal and asymmetric models, showing that inflation expectations also play a role in the relationship between BTC and WTI. This result confirms that the evolving relationship between the digital asset and the physical commodity is sensitive to changes in the macroeconomic outlook regarding future purchasing power. Finally, we examine the impact of the explanatory variables on the correlation between Bitcoin and the VIX index and present the results in Table 6c. We do not observe a significant impact from the selected explanatory variables on the BTC-VIX correlation. Considering model performance, the asymmetric Student−t model best fits the data. The lack of significant influence from Gold, Treasury Yields, and Inflation Expectations on the BTC-VIX correlation suggests that the relationship between Bitcoin and generalised market fear (systematic risk) is largely idiosyncratic, meaning BTC’s risk dynamics are driven by factors—such as regulatory events or network security news—that are self-contained within the crypto ecosystem and are not directly predicted by the primary macroeconomic risk indicators of the traditional financial system.

Figure 4 plots the time-varying correlations between the BTC and S&P 500, NASDAQ, WTI and VIX, considering the impact of explanatory variables. Compared to the plot for the time-varying correlation without explanatory variables (Figure 3), they show higher correlation values, suggesting that these explanatory variables significantly impact the correlation between these asset classes. Consistent with the dynamics shown in Figure 3, the COVID-19 pandemic coincided with a pronounced increase in the correlation between Bitcoin and the two equity indices, as well as the VIX. In contrast, a descending trend is observed in the correlation between Bitcoin and WTI crude oil during the same period. Our results show that during the COVID-19 crash, Bitcoin’s correlation with the S&P 500 and Nasdaq experienced a sharp, synchronized spike, reaching peak levels that suggest a temporary loss of its “safe-haven” or “diversifier” status. In contrast, during the Ukraine invasion, the dynamics were more heterogeneous, reflecting Bitcoin’s complex role as both a “risk-on” asset and a potential alternative vehicle for value transfer during geopolitical instability.

We compare the performance of our models with the DCC model of Engle (2002) for all pairs using AIC and BIC. Based on AIC, the asymmetric Student−t models without explanatory variables perform better for all pairs except for BTC-VIX, where the asymmetric Student−t model with explanatory variables is the preferred choice. For BIC, the asymmetric Student−t model without explanatory variables performs better for all pairs except for BTC-S&P 500, where the Student-t is the best performing model. We summarise these results in Table 7.

Our findings offer market participants and practitioners valuable insights into how various market variables may impact the performance of their portfolios comprising these asset pairs.

We investigate our models’ capabilities in portfolio construction using the minimum variance portfolio approach. In particular, we compare our four proposals and the DCC model. We first compute the time-varying weights w_t of a two-component minimum-variance portfolio comprising a cryptocurrency and an asset. We optimise the weights for each point in time of our sample by solving the optimisation problem:

Where H_t is the covariance matrix and 1 is the unit vector. We use the global minimum variance portfolio (GMVP) technique, ignoring the expected return and focusing only on volatility. We obtain the weights of the minimum variance portfolio as:

The covariance matrix H_t has the form:

Where $σ1,t2$ and $σ2,t2$ are the variances of the assets 1 and 2 at time t, and σ_12,t is the covariance, where σ_12,t = ρ_tσ_1,tσ_2,t. We extract the conditional variances from the GARCH model fitted on univariate time-series data. We then use the time-varying correlations, ρ_t, from each fitted model (Normal, asymmetric Normal, Student-t, asymmetric Student-t, and the DCC models) to construct the covariances. The comparisons of the models’ portfolio performance are based on the Sharpe Ratio (Sharpe, 1966), which measures the investment risk-adjusted performance relative to its volatility. A portfolio with higher Sharpe ratios indicates better risk-adjusted performance.

In Table 8, we present the performance results for portfolios consisting of BTC and the selected asset (S&P 500, NASDAQ, VIX, and WTI). In each panel of the table, we test the difference between the Sharpe ratios obtained using the GAS model and the Sharpe ratios obtained using the DCC model. For this purpose, we use the robust version of the test of Ledoit and Wolf (2008) based on the studentised circular bootstrap approach. Panels A, B, and C show that the Sharpe Ratio of the Student-t model is the highest, with the highest annualised expected return and the lowest volatility. The Sharpe ratio of 0.0261 for BTC and the S&P 500 index reported in Panel A is the highest. This result suggests that, under this model, the portfolio provides the best risk-adjusted return among the models considered. In other words, the GAS framework appears to be better suited for modelling the volatility and correlation dynamics of these assets since for each unit of risk (volatility) taken, especially the GAS (1, 1) Student−t model offers the highest risk-adjusted return. In Panel D, we show that for the portfolio consisting of BTC and WTI, the asymmetric Normal model has the highest Sharpe Ratio. The Student-t and asymmetric Student-t, perform better than the DCC model for the majority of the constructed portfolio, with the only exception of BTC and WTI, where the asymmetric Normal model performs best.

Although the Student−t model frequently produces higher Sharpe ratios, particularly for BTC–S&P 500 and BTC–NASDAQ portfolios, this outcome does not reduce the significance of the Asymmetric Student−t model within our analytical framework. Our GAS(1,1) specification captures time-varying correlations through score-driven updates that incorporate past dependence, scaled likelihood scores, and lagged exogenous variables. The Asymmetric Student-t distribution is essential for modelling both heavy tails and skewness in the joint distribution of standardised residuals obtained from ARMA-GARCH-filtered returns. While Sharpe ratios focus on risk-adjusted returns and may favour more parsimonious models when asymmetry is limited, the Asymmetric Student−t model improves the predictive accuracy of correlation dynamics, especially in periods characterised by pronounced asymmetry. The flexibility of this distribution, together with the GAS framework’s capacity to integrate external information and reduce the influence of outliers, makes it a valuable approach for modelling evolving dependence structures in financial markets.

The consistent superiority of the GAS models, particularly those accounting for non-normality (Student−t and Asymmetric variants), over the traditional DCC model has important implications for portfolio management and financial stability analysis. The results fundamentally validate the need for models that are adaptive and sensitive to higher-order moments (skewness and kurtosis), moving beyond simplistic linear relationships.

The superior Sharpe Ratios achieved by the GAS models confirm that relying on static or less sophisticated dynamic models (such as DCC) misestimates true risk and limits the effectiveness of diversification. By accurately capturing time-varying correlations and heavy-tailed distributions, the GAS-based framework enables a more active and effective diversification strategy. Investors can use these models to dynamically adjust their crypto exposure during different market regimes, achieving the lowest possible volatility for a given level of return, which is the core goal of the Minimum Variance Portfolio.

The dominance of the Student−t distribution is an economic signal of Bitcoin’s integration. Since correlations tend to rise toward 1 during periods of high market stress (the “contagion effect”), the Student−t model’s success suggests it better anticipates the simultaneous collapse of assets during tail events. This prevents the “diversification illusion”, in which expected benefits vanish precisely when they are needed most, thereby making the resulting portfolio more resilient to systemic shocks.

The models indicate that Bitcoin is rapidly moving away from being a disconnected “safe-haven” or purely idiosyncratic asset. Instead, its time-varying correlation with equity markets (S&P 500, NASDAQ) is now highly dynamic and non-normal, suggesting it has become an integrated, albeit volatile, part of the broader global financial ecosystem, highly susceptible to shared systemic risk. Portfolio managers must model this dynamic linkage using advanced techniques, such as GAS, to avoid underestimating the downside risk arising from its co-movement with traditional assets.

Our analysis shows that the choice of the statistical model is crucial for accurately assessing risk and return in cryptocurrency portfolios. Our findings reinforce the importance of accounting for heavy tails and asymmetries when analysing cryptocurrency returns and, at the same time, suggest that the optimal portfolio model may vary depending on the specific cryptocurrency being considered. For portfolios containing Bitcoin, the GAS (1,1) Student−t model generally provides the highest Sharpe ratio and best risk-adjusted returns across most asset pairings (S&P500, NASDAQ, VIX).

In this study, we modelled the time-varying correlation between Bitcoin and some traditional financial assets (S&P 500 index, NASDAQ index, VIX index, and WTI). We proposed a novel framework that uses an asymmetric Student−t distribution for asset return correlations, enhanced with parameters governed by Generalised Autoregressive Score (GAS) dynamics. This advanced approach overcomes the shortcomings of conventional models, providing a more versatile and reliable representation of financial return correlations.

Results show that the asymmetric Student−t model consistently outperforms the alternatives (including DCC) for all the pairs considered. The model effectively captures the heavy-tailed and asymmetric nature of the data, which is crucial in cryptocurrency markets. Our framework also allows for the easy incorporation of explanatory variables into the model formulation to account for their impact on correlations, thereby significantly improving the understanding of correlation processes and market return dynamics.

The empirical findings offer several crucial economic insights regarding Bitcoin’s evolving role and its linkages to macroeconomic factors. The highly dynamic nature of the estimated correlations strongly suggests that Bitcoin is moving away from its historical status as a purely idiosyncratic or safe-haven asset. Instead, its correlation with equity indices (S&P 500 and NASDAQ) confirms its increasing integration into the broader financial system, thereby preventing the classic “diversification illusion” where expected low correlation vanishes and rapidly increases precisely during periods of market stress. Our findings further reveal that the inclusion of explanatory variables improves the understanding of correlation processes and the market return dynamics. In particular, Gold is relevant in several pair correlations: it positively impacts the correlation between Bitcoin and S&P 500 and negatively affects the correlation between Bitcoin and WTI, suggesting that Gold’s safe-haven properties are asset-specific rather than universal. The Market Yield on US Treasury Securities negatively impact Bitcoin and WTI correlations. This negative effect pinpoints the influence of broader economic factors and risk sentiment on asset correlations. Moreover, the significance of T10YIE shows that inflation expectations also play a role in the relationship between BTC and WTI.

Our portfolio analysis shows that accounting for heavy tails and asymmetries when managing a portfolio containing cryptocurrencies significantly improves its risk-adjusted returns. For financial professionals, our results emphasise the necessity of moving beyond traditional symmetric models. Investors, using conventional techniques, significantly underestimate downside portfolio risk because these models fail to account for the heightened co-movement of assets during bear markets. Our approach would also support policymakers in understanding risks more accurately and in having financial institutions conduct closer surveillance of those risks. The significant influence of US Treasury Yields and Inflation Expectations on Bitcoin’s correlation implies that monetary policy actions and forward guidance are now being transmitted to the cryptocurrency market. Therefore, central banks could integrate the crypto market’s dynamic response into their financial stability models to accurately forecast the overall impact of policy changes on capital flows. Detecting and monitoring correlations could thus enhance regulatory frameworks, strengthen market infrastructure and lessen market manipulation.

The present study is, however, not without limitations. A key methodological decision was the selection of the Azzalini skew−t model for the dependence structure. This choice was motivated by the optimal balance between its superior empirical power in capturing asymmetric tail dependence and its computational tractability for efficient estimation within the recursive GAS dynamic framework. We acknowledge the existence of other advanced skew−t formulations. Evaluating newer, more complex skew−t copula variants under the GAS dynamic framework remains a promising avenue for future research.

The authors would like to thank two anonymous referees and the Editor Prof. Wenfeng Wu for valuable comments and constructive feedback that greatly improved the manuscript.

The supplementary material for this article can be found online.