Semiparametric Power Envelopes for Tests of the Unit Root Hypothesis

This paper derives asymptotic power envelopes for tests of the unit root hypothesis in a zero-mean AR(1) model. The power envelopes are derived using the limits of experiments approach and are semiparametric in the sense that the underlying error distribution is treated as an unknown infinitedimensional nuisance parameter. Adaptation is shown to be possible when the error distribution is known to be symmetric and to be impossible when the error distribution is unrestricted. In the latter case, two conceptually distinct approaches to nuisance parameter elimination are employed in the derivation of the semiparametric power bounds. One of these bounds, derived under an invariance restriction, is shown by example to be sharp, while the other, derived under a similarity restriction, is conjectured not to be globally attainable.

[1]  S. Møller.  Habit persistence: Explaining cross-sectional variation in returns and time-varying expected returns , 2008 .

[2]  Richard K. Crump,et al.  Optimal Inference for Instrumental Variables Regression with Non-Gaussian Errors , 2007 .

[3]  B. Christensen,et al.  The Role of Implied Volatility in Forecasting Future Realized Volatility and Jumps in Foreign Exchange, Stock, and Bond Markets , 2007 .

[4]  J. Joensen,et al.  Are Economists More Likely to Hold Stocks , 2007 .

[5]  C. Christiansen.  Level-Arch Short Rate Models with Regime Switching: Bivariate Modeling of Us and European Short Rates , 2007 .

[6]  Dennis Kristensen,et al.  NONPARAMETRIC FILTERING OF THE REALIZED SPOT VOLATILITY: A KERNEL-BASED APPROACH , 2007, Econometric Theory.

[7]  Michael Jansson,et al.  Improving Size and Power in the Unit Root Testing , 2005 .

[8]  R. Koenker,et al.  Unit Root Quantile Autoregression Inference , 2004 .

[9]  S. B. Thompson.  ROBUST TESTS OF THE UNIT ROOT HYPOTHESIS SHOULD NOT BE “MODIFIED” , 2004, Econometric Theory.

[10]  Joon Y. Park.  Bootstrap Unit Root Tests , 2003 .

[11]  Graham Elliott,et al.  Tests for Unit Roots and the Initial Condition , 2003 .

[12]  Efstathios Paparoditis,et al.  Residual‐Based Block Bootstrap for Unit Root Testing , 2003 .

[13]  Michael McAleer,et al.  On Adaptive Estimation in Nonstationary Arma Models with Garch Errors , 2003 .

[14]  Victor Chernozhukov,et al.  Likelihood Estimation and Inference in a Class of Nonregular Econometric Models , 2003 .

[15]  K. Hirano,et al.  Asymptotic Efficiency in Parametric Structural Models with Parameter-Dependent Support , 2002 .

[16]  T. Rothenberg.  Testing for unit roots in AR and MA models , 2000 .

[17]  P. Phillips,et al.  A Primer on Unit Root Testing , 1998 .

[18]  Thomas J. Rothenberg,et al.  Inference in a nearly integrated autoregressive model with nonnormal innovations , 1997 .

[19]  Anton Schick,et al.  Efficient estimation in nonlinear autoregressive time-series models , 1997 .

[20]  Bas J. M. Werker,et al.  Adaptive estimation in time-series models , 1997 .

[21]  Serena Ng,et al.  Useful Modifications to some Unit Root Tests with Dependent Errors and their Local Asymptotic Properties , 1996 .

[22]  W. J. Hall,et al.  Asymptotically uniformly most powerful tests in parametric and semiparametric models , 1996 .

[23]  M. Herce.  Asymptotic Theory of LAD Estimation in a Unit Root Process with Finite Variance Errors , 1996, Econometric Theory.

[24]  P. Jeganathan.  Some Aspects of Asymptotic Theory with Applications to Time Series Models , 1995, Econometric Theory.

[25]  Serena Ng,et al.  Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag , 1995 .

[26]  Mark W. Watson,et al.  A SIMPLE ESTIMATOR OF COINTEGRATING VECTORS IN HIGHER ORDER INTEGRATED SYSTEMS , 1993 .

[27]  J. Stock,et al.  Efficient Tests for an Autoregressive Unit Root , 1992 .

[28]  Peter C. B. Phillips,et al.  Optimal Inference in Cointegrated Systems , 1991 .

[29]  W. Newey,et al.  Semiparametric Efficiency Bounds , 1990 .

[30]  P. Phillips.  Testing for a Unit Root in Time Series Regression , 1988 .

[31]  C. Z. Wei,et al.  Limiting Distributions of Least Squares Estimates of Unstable Autoregressive Processes , 1988 .

[32]  Chris A. J. Klaassen,et al.  Consistent Estimation of the Influence Function of Locally Asymptotically Linear Estimators , 1987 .

[33]  P. Phillips.  Time series regression with a unit root , 1987 .

[34]  A. Schick.  On Asymptotically Efficient Estimation in Semiparametric Models , 1986 .

[35]  D. Andrews.  Complete Consistency: A Testing Analogue of Estimator Consistency , 1986 .

[36]  R. Hogg,et al.  On adaptive estimation , 1984 .

[37]  W. J. Hall,et al.  Information and Asymptotic Efficiency in Parametric-Nonparametric Models , 1983 .

[38]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[39]  W. Fuller,et al.  Distribution of the Estimators for Autoregressive Time Series with a Unit Root , 1979 .

[40]  R. Beran.  An Efficient and Robust Adaptive Estimator of Location , 1978 .

[41]  C. J. Stone,et al.  Adaptive Maximum Likelihood Estimators of a Location Parameter , 1975 .

[42]  L. Lecam.  On the Assumptions Used to Prove Asymptotic Normality of Maximum Likelihood Estimates , 1970 .

[43]  Piotr Eliasz.  OPTIMAL MEDIAN UNBIASED ESTIMATION OF COEFFICIENTS ON HIGHLY PERSISTENT REGRESSORS , 2004 .

[44]  D. Pollard.  Another Look at Differentiability in Quadratic Mean , 1997 .

[45]  R. Koenker,et al.  Robust rank tests of the unit root hypothesis , 1997 .

[46]  A. .,et al.  SEMIPARAMETRIC LIKELIHOOD RATIO INFERENCE , 1996 .

[47]  Kreiss Jens-Peter,et al.  On Adaptive Estimation In Autoregressive Models When There Are Nuisance Functions , 1987 .

[48]  Anton Schick,et al.  A note on the construction of asymptotically linear estimators , 1987 .

[49]  J. Stock.  Unit roots, structural breaks and trends , 1986 .