A New MARMA-GARCH Model with Applications to Cryptocurrency Volatility

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Abstract
This paper introduces a mixed causal-noncausal invertible-noninvertible autoregressive moving average generalized autoregressive conditional heteroskedasticity (MARMA-GARCH) model, which uniquely combines features to capture the nonlinear dynamics observed in financial markets. The MARMA component replicates explosive episodes observed during financial bubbles, while the GARCH model effectively captures time-varying conditional volatility. We propose two methods for parameter estimation: the Whittle function in the frequency domain and the maximum likelihood. Additionally, we suggest an identification scheme through high-order spectral densities and high-order dynamics of the residuals. Our findings demonstrate that an incorrect identification of the MARMA process leads to biased GARCH parameters, underestimating the effect of news and overestimating the volatility clustering. An empirical application to Bitcoin and Ethereum reveals the existence of noncausal dynamics in the mean, jointly with significant GARCH effects.
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