Bootstrap Unit Root Tests for Nonlinear Threshold Models

While a number of unit root testing procedures have been developed to account for nonlinearity under the alternative hypothesis of stationarity, almost all available tests assume a linear DGP under the unit root null hypothesis. This paper establishes some theoretical results relating to the inclusion of nonlinear terms in an ADF regression and proposes two new unit root tests that allow the process to be nonlinear under the null hypothesis. More specifically, block and model-based bootstrap procedures are developed for smooth transition threshold models. Simulations show that the latter is preferred and the model-based bootstrap test delivers a good size performance across all specifications, including linear and effectively abrupt transition models. The modelbased test also dominates the standard ADF test in terms of power and an application to the US unemployment rate shows that it can overturn conclusions based on an erroneous linearity assumption.

[1]  Mototsugu Shintani,et al.  Testing for a Unit Root Against Transitional Autoregressive Models , 2016 .

[2]  M. Pippenger,et al.  A Note on the Empirical Power of Unit Root Tests under Threshold Processes , 2009 .

[3]  Alain Guay,et al.  Adaptive consistent unit-root tests based on autoregressive threshold model , 2008 .

[4]  Rehim Kiliĉ.  Linearity Tests and Stationarity , 2004 .

[5]  Javier Ordóñez,et al.  Common smooth transition trend-stationarity in European unemployment , 2008 .

[6]  P. Newbold,et al.  Tests for Symmetric and Asymmetric Nonlinear Mean Reversion in Real Exchange Rates , 2002 .

[7]  Chi Young Choi,et al.  How Useful are Tests for Unit-Root in Distinguishing Unit-Root Processes from Stationary but Non-Linear Processes? , 2007 .

[8]  Bruce E. Hansen,et al.  THRESHOLD AUTOREGRESSION WITH A UNIT ROOT , 2001 .

[9]  James D. Hamilton.  Time Series Analysis , 1994 .

[10]  J. MacKinnon.  Bootstrap Methods in Econometrics , 2006 .

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

[12]  M. Seo,et al.  UNIT ROOT TEST IN A THRESHOLD AUTOREGRESSION: ASYMPTOTIC THEORY AND RESIDUAL-BASED BLOCK BOOTSTRAP , 2008, Econometric Theory.

[13]  Bruce E. Hansen,et al.  Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis , 1996 .

[14]  J. Horowitz.  The bootstrap in econometrics , 2003 .

[15]  M. Friedman.  THE ROLE OF MONETARY POLICY , 1995 .

[16]  Theodore Panagiotidis,et al.  Testing for non-linearity in labour markets: the case of Germany and the UK , 2003 .

[17]  Philipp Sibbertsen,et al.  Phillips-Perron-type unit root tests in the nonlinear ESTAR framework , 2006 .

[18]  The informational value of unemployment statistics: A note on the time series properties of participation rates , 2006 .

[19]  J. MacKinnon,et al.  Bootstrap Testing in Nonlinear Models , 1999 .

[20]  P. Bühlmann.  Bootstraps for Time Series , 2002 .

[21]  Veli Yılancı.  Are Unemployment Rates Nonstationary or Nonlinear? Evidence from 19 OECD Countries , 2008 .

[22]  Peter C. B. Phillips,et al.  Weak Convergence of Sample Covariance Matrices to Stochastic Integrals Via Martingale Approximations , 1988, Econometric Theory.

[23]  George Kapetanios,et al.  Testing for a unit root in the nonlinear STAR framework , 2003 .

[24]  D. Osborn,et al.  Nonlinearity and structural change in the interest rate reaction functions for the US, UK and Germany , 2006 .

[25]  R. Sollis.  A simple unit root test against asymmetric STAR nonlinearity with an application to real exchange rates in Nordic countries , 2009 .

[26]  George Kapetanios,et al.  Unit Root Tests in Three-Regime Setar Models , 2002 .

[27]  L. Summers,et al.  Hysteresis and the European Unemployment Problem , 1986, NBER Macroeconomics Annual.

[28]  Clive W. J. Granger,et al.  Unit Root Tests and Asymmetric Adjustment with an Example Using the Term Structure of Interest Rates , 1998 .

[29]  Marco Bianchi,et al.  Unemployment Persistence: Does the Size of the Shock Matter? , 1996 .

[30]  H. Künsch.  The Jackknife and the Bootstrap for General Stationary Observations , 1989 .

[31]  Timo Teräsvirta,et al.  MODELING ASYMMETRIES AND MOVING EQUILIBRIA IN UNEMPLOYMENT RATES , 1999, Macroeconomic Dynamics.

[32]  T. Teräsvirta.  Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models , 1994 .

[33]  A nonlinear alternative to the unit root hypothesis , 2003 .

[34]  Gary Koop,et al.  Dynamic Asymmetries in U.S. Unemployment , 1999 .

[35]  Marine Carrasco,et al.  Tests for Unit-Root versus Threshold Specification With an Application to the Purchasing Power Parity Relationship , 2004 .

[36]  David I. Harvey,et al.  Testing for Time Series Linearity , 2007 .

[37]  Paul Newbold,et al.  Unit roots and smooth transitions , 1998 .

[38]  Timo Teräsvirta,et al.  Testing linearity against smooth transition autoregressive models , 1988 .