Coordinate gradient descent algorithm in adaptive LASSO for pure ARCH and pure GARCH models

This paper develops a coordinate gradient descent (CGD) algorithm, based on the work of Tseng and Yun (Math Program 117:387–423; 2009a; J Optim Theory Appl 140(3):513–535, 2009b), to optimize the constrained negative quasi maximum likelihood with adaptive LASSO penalization for pure autoregressive conditional heteroscedasticity (ARCH) model and its generalized form (GARCH). The strategy for choosing the appropriate values of the shrinkage parameter through information criteria (IC) is also discussed. We evaluate the numerical efficiency of the proposed algorithm through simulated data. Results of simulation studies show that for moderate sample sizes, the adaptive LASSO with the Bayesian variant of IC correctly estimates the ARCH structure at a high rate, even when model orders are over-specified. On the other hand, the adaptive LASSO has a low rate of correctly estimating true GARCH structure, especially when the model orders are over-specified regardless of the choice of IC. In our case study using daily ASX Ordinary log returns, the adaptive LASSO yields sparser ARCH and GARCH models while maintaining adequate fit for the volatility.