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.

This article studies the first-order vector error correction (VEC(1)) model when its noise is a linear process of independent and identically distributed (i.i.d.) heavy-tailed random vectors with a tail index a ? (0,2). We show that the rate of convergence of the least squares estimator (LSE) related to the long-run parameters is n (sample size) and its limiting distribution is a stochastic integral in terms of two stable random processes, while the LSE related to the short-term parameters is not consistent. We further propose an automated approach via adaptive shrinkage techniques to determine the cointegrating rank in the VEC(1) model. It is demonstrated that the cointegration rank r(0) can be consistently selected despite the fact that the LSE related to the short-term parameters is not consistently estimable when the tail index a ? (1,2). Simulation studies are carried out to evaluate the performance of the proposed procedure in finite samples. Last, we use our techniques to explore the long-run and short-run behavior of the monthly prices of wheat, corn, and wheat flour in the United States.

This paper investigates the quasi-maximum likelihood estimator (QMLE) of the structure-changed and two-regime threshold double autoregressive model. It is shown that both the estimated threshold and change-point are n-consistent, and they converge weakly to the smallest minimizer of a compound Poisson process and the location of minima of a two-sided random walk, respectively. Other estimated parameters are n−consistent and asymptotically normal. The performance of the QMLE is assessed via simulation studies and a real example is given to illustrate our procedure.