SELF-WEIGHTED ESTIMATION FOR LOCAL UNIT ROOT REGRESSIONS WITH APPLICATIONS

A new self-weighted (SW) least squares (LS) estimation approach is developed for local unit root (LUR) autoregression with conditionally heteroskedastic innovations. Asymptotically controlled trimming of the most extreme observations in LS regression is shown to lead to mixed Gaussian limit theory and studentization produces standard normality free of the unknown LUR parameter. The estimator is super consistent with a convergence rate slightly below the <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>P</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> $O_P(n)$ upper O Subscript upper P Baseline left parenthesis n right parenthesis rate of LS estimation. The asymptotic theory relies on a new framework of convergence to the local time of a Gaussian process, allowing for the sample moments generated from martingales and many other integrated dependent sequences. A new LUR test in augmented autoregression is developed using SW estimation, providing an alternative approach that employs Gaussian limit theory in contrast to the augmented Dickey–Fuller and other tests. Simulation results show that the test has good size control and power performance against mildly integrated alternatives. An empirical illustration is conducted evaluating the persistence of several market fundamentals as predictors of S&amp;P 500 excess returns.