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We examine the relationship between the top five cryptos and the U.S. S&P500 index from January 2018 to December 2021. We use the novel General-to-specific Vector Autoregression (GETS VAR) and traditional Vector Autoregression (VAR) model to analyze the short- and long-run, cumulative impulse-response, and Granger causality test between S&P500 returns and the returns of Bitcoin, Ethereum, Ripple, Binance and Tether. Additionally, we used the Diebold and Yilmaz (DY) spillover index of variance decomposition to validate our findings. Evidence from the analysis suggests positive short- and long-run effects of historical S&P500 returns on Bitcoin, Ethereum, Ripple, and Tether returns--and negative short- and long-run effects of the historical returns of Bitcoin, Ethereum, Ripple, Binance, and Tether on S&P500 returns. Alternatively, evidence suggests a negative short- and long-run effect of historical S&P500 returns on Binance returns. The cumulative test of impulse-response indicates a shock in historical S&P500 returns stimulates a positive response from cryptocurrency returns while a shock in historical crypto returns triggers a negative response from S&P500 returns. Empirical evidence of bi-directional causality between S&P500 returns and crypto returns suggest the mutual coupling of these market. Although, S&P500 returns have high-intensity spillover effects on crypto returns than crypto returns have on S&P500. This contradicts the fundamental attribute of cryptocurrencies for hedging and diversification of assets to reduce risk exposure. Our findings demonstrate the need to monitor and implement appropriate regulatory policies in the crypto market to mitigate the potential risks of financial contagion., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2023 The Authors.)

Despite all the criticism, the improvement of the static CAPM, which has generated new dynamic models, provided investors with stronger guarantee through financial transactions. The CAPM and its static version were and are still very important in the financial scene. Nowadays, more sophisticated adaptations of the CAPM are found, which allow us to explain some matters in finance that had remained unqualified for a couple of time. Considering such discussion about the CAPM validity, this study intends creates a basis for reflection upon the conditional model comparing it with the static one. In order to verify such facts, tests of conditional models are examined (with beta varying throughout the exercise), something uncommonly studied in the literature. Such tests are suitable to incorporate variances and covariances that change throughout the time. Methodological wise, the study tested the conditional CAPM model by Jagannathan and Wang (1996).

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