Adaptive LASSO with coordinate gradient descent algorithm for M-BEKK-ARCH(q) model

This study proposes the adaptive LASSO estimator for the simultaneous parameter estimation and model selection of the multivariate Baba-Engle-Kroner-Kraft Autoregressive Conditional Heteroscedasticity (M-BEKK-ARCH) volatility model. A coordinate gradient descent (CGD) algorithm is developed to optimize the quasi-maximum likelihood (QML) with adaptive LASSO penalty. The strategy to select an appropriate value for the adaptive LASSO shrinkage parameter is also discussed. Under the condition where ARCH order q is known, we show the QML adaptive LASSO via CGD algorithm identifies correct models with reasonable percentages under moderate sample size in simulation studies. Furthermore, it also excludes irrelevant terms more often and has more stable parameter convergence compared to the existing modified shooting algorithm.