Voluntary carbon markets are at an early stage of development, characterized by low and irregular trading frequency. Such limited activity results in extended periods of unchanged prices and a high incidence of zero returns, making voluntary carbon markets a typical example of illiquid financial markets. To model dependence in such settings, we propose a multivariate zero-inflated GARCH-X model. The model extends the existing zero-inflated GARCH model to a multivariate setup, incorporating both a GARCH-X component with binary trading indicators as exogenous covariates and a time-step specification that updates only when trades occur.

The multivariate extension incorporates two different types of dependence. First, cross-dependence in trading activity is modeled using Markov networks applied to binary trading indicators. Second, cross-dependence in returns is analyzed using a copula-GARCH framework applied to residuals, with residuals corresponding to zero returns treated as undefined values. To make copula methods applicable to zero-inflated data, we introduce a joint probability integral transform approach. In this construction, the univariate marginals are defined conditional on the simultaneous trading activity of each asset, rather than conditioning only on each asset’s own trading activity. We prove that this method yields a consistent copula estimator when applied to the subset of observations where all assets have simultaneous non-zero trading activity. Dependence is quantified using both unconditional and conditional Kendall’s tau, estimated via kernel-based methods.

Theoretical results include consistency and asymptotic normality of the quasi-maximum likelihood estimator for the multivariate model under stationary covariates. The empirical study covers seven voluntary carbon credits and six conventional financial assets. We find no significant dependence between carbon credits and conventional assets, but observe strong correlation within the carbon market, especially among nature-based credits. Results suggest that voluntary carbon markets may operate independently of more liquid assets and are influenced by peer pricing due to a lack of standardization.