Penalized Convex Estimation in Dynamic Location Models

ABSTRACT This paper studies ‐penalized estimation for location models , where is defined by a possibly non‐Markovian recursion and is a martingale difference sequence with possibly time‐varying conditional variance. In such settings, standard LS/QML criteria are typically non‐convex. A two‐step plug‐in scheme is considered: a first‐step estimator (e.g., WLS or QMLE) is assumed to be strongly consistent, its fitted recursions are frozen, and a weighted least‐squares criterion with an penalty is minimized in a second step. The resulting objective is convex and compatible with standard LASSO algorithms. Under mild regularity conditions, the second‐step estimator is strongly consistent as soon as the penalties attached to the nonzero coordinates of the true parameter vanish. For penalties of order , its asymptotic distribution is derived, and adaptive penalties yield selection consistency and an oracle property. The second‐step estimator is unconstrained and, when combined with an unconstrained first‐step estimator, it yields a standard Gaussian limit with a tractable covariance matrix, in contrast with the non‐standard limits that typically arise for QMLE when some components of the true parameter are zero. The general results are specialized to several time‐series models and illustrated by Monte Carlo experiments and a real‐data application to interest rates.