Two-step estimations via the Dantzig selector for models of stochastic processes with high-dimensional parameters

We proposeatwo-step estimation procedure for stochastic process models with high-dimensional parameters of interest under heteroskedasticity. In low-dimensional settings, when a consistent estimator for a nuisance parameter that characterizes the conditional variance is available, one can construct an asymptotically normal estimator for the parameter of interest under appropriate conditions. Motivated by this fact, we extend the idea to high-dimensional settings. We first establish variable selection via the Dantzig selector, and then combine this with consistent estimation of the nuisance parameter to develop a two-step procedure that yields an asymptotically normal estimator. Our framework accommodates infinite-dimensional nuisance parameters in the conditional variance term. Therefore, this study extends sparse estimation methods to a broader class of stochastic process models. Applications to ergodic time series models, including integer-valued autoregressive models and ergodic diffusion processes, are presented.