The logarithmic vector multiplicative error model (log-vMEM) provides a methodology for modeling and forecasting multivariate positive-valued time series. However, the number of autoregressive parameters increases rapidly with the system dimension and lag order, posing challenges for estimation and model selection. This paper develops a regularized estimation framework for the log-vMEM with multivariate Gamma errors by incorporating hierarchical lag structures to induce sparsity and accommodate heterogeneous lag dynamics. We consider three hierarchical lag structures: componentwise, own-other, and elementwise, in combination with group lasso, group MCP, and group SCAD, adaptive group lasso penalties. Parameter estimation is carried out through an alternating block-coordinate optimization algorithm with sequential Gauss-Seidel updates, while GPU-accelerated numerical quadrature is employed to address the computational burden of evaluating the multivariate Gamma likelihood. Simulation experiments investigate parameter recovery and lag selection under different sparsity structures. The real data analysis is carried out on the realized volatility measures for Microsoft (NASDAQ:MSFT) which allows a comparison of the competing regularization approaches in an empirical financial setting. Together, these developments provide a computational framework for sparse multivariate volatility modeling with flexible lag structures and dependent Gamma errors.

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