On robust estimation for moderate deviations from a unit root

doi:10.1007/s11749-026-01020-z

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On robust estimation for moderate deviations from a unit root

2026-01-01

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Wang, Tao *

DOI：10.1007/s11749-026-01020-z

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Abstract

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We in this paper propose a robust estimation technique for a first-order autoregressiveprocess characterized by the root rho(n )= 1 + c/k(n), employing a kernel mode-basedobjective function construed on the mode value. Compared to traditional least squaresor maximum likelihood approaches, the newly developed method exhibits enhancedrobustness against outliers and non-normal errors. We suggest a computationallyefficient mode expectation-maximization algorithm leveraging a Gaussian kernel tonumerically estimate the coefficient. Under mild assumptions, we derive the asymp-totic distributions of the resulting kernel mode-based estimator, assuming thatknincreases to infinity at a slower rate thann. Specifically, for c < 0, we establish aconvergence rate of root nk(n )with a normal limit distribution, while for c > 0, the con-vergence rate is k(n)rho(n )(n)with a Cauchy limit distribution. Monte Carlo simulations arepresented to illustrate the favorable finite sample performance of the proposed esti-mation procedure. Furthermore, we extend these results to the general autoregressiveprocess with a coefficient satisfying n|1-rho(n)|-> infinity under weaker initial conditions.The convergence rates are demonstrated to be[n(1-rho(2)(n))(-1)](1/2 )and rho(n)(n)/(rho(2)(n )- 1)for nearly stationary and mildly explosive cases, respectively.

Keywords:

Kernel

Mildly explosive

Mode

Moderate deviations

Nearly stationary

Unit root