Adaptive LASSO estimation for ARDL models with GARCH innovations

ABSTRACT

In this paper, we show the validity of the adaptive least absolute shrinkage and selection operator (LASSO) procedure in estimating stationary autoregressive distributed lag(p,q) models with innovations in a broad class of conditionally heteroskedastic models. We show that the adaptive LASSO selects the relevant variables with probability converging to one and that the estimator is oracle efficient, meaning that its distribution converges to the same distribution of the oracle-assisted least squares, i.e., the least square estimator calculated as if we knew the set of relevant variables beforehand. Finally, we show that the LASSO estimator can be used to construct the initial weights. The performance of the method in finite samples is illustrated using Monte Carlo simulation.

KEYWORDS:

• adaLASSO
• ARDL
• GARCH
• LASSO
• shrinkage
• sparse models
• time series

JEL CLASSIFICATION:

• C22

Acknowledgments

M. C. Medeiros acknowledges partial support from CNPq/Brazil. E. F. Mendes was partially supported by the Australian Center of Excellence Grant CE140100049. We are in debt to Gabriel Vasconcelos for superb research assistance. We are very grateful to two anonymous referees and the Guest Editors, Aman Ullah and Peter C.B. Phillips, for valuable comments and editorial guidance.

Notes

¹We write $\parallel b^{\prime }v{\parallel }_{2d}={\left(𝔼\right|b^{\prime }v{|}^{2d})}^{1∕2d}={\left(𝔼\right|b^{\prime }{vv}^{\prime }b{|}^d)}^{1∕2d}$, and ∥⋅∥ = ∥⋅∥₂.

²We use the notation [A]_i,j to denote the (i,j)^th element of matrix A.