Inference on nonstationary heavy-tailed AR processes via model selection

This article develops a series of Lasso-based procedures for statistical inference in the autoregressive (AR) process with heavy-tailed heteroscedastic noise. We first study the classical adaptive Lasso estimator for the nonstationary coefficient and show that it can distinguish between stationary and nonstationary autoregressions by detecting whether the coefficient is zero with high probability. After conducting unit-root tests, we proceed to consider the self-weighted least absolute deviation (SWLAD) estimator with an adaptive Lasso penalty applied to the stationary coefficients. As anticipated, the penalized SWLAD estimator exhibits the oracle properties. This implies that the penalized SWLAD estimator is consistent, accurately selects the correct sparsity pattern, and estimates the coefficients of the relevant variables with the same asymptotic efficiency as if only these variables had been included in the model. In particular, by assigning data-dependent weights to different coefficients in the penalty term, our method can effectively circumvent the issue of non-pivotal distributional ignorance in heavy-tailed time series analysis, thereby enabling effective unit-root tests and order selection without any prior information on the nuisance parameters, making it appealing in practice with wide applications in finance and econometrics. A simulation study is conducted to evaluate the performance of the test and estimator, and two real examples are presented to demonstrate their applicability.