A Primer on Unit Root Testing

The immense literature and diversity of unit root tests can at times be confusing even to the specialist and presents a truly daunting prospect to the uninitiated. In consequence, much empirical work still makes use of the simplest testing procedures because it is unclear from the literature and from recent reviews which tests if any are superior. This paper presents a survey of unit root theory with an emphasis on testing principles and recent developments. The general framework adopted makes it possible to consider tests of stochastic trends against trend stationarity and trend breaks of a general type. The main tests are listed, and asymptotic distributions are given in a simple form that emphasizes commonalities in the theory. Some simulation results are reported, and an extensive list of references and all annotated bibliography are provided.

[1]  Katsuto Tanaka,et al.  THE NONSTATIONARY FRACTIONAL UNIT ROOT , 1999, Econometric Theory.

[2]  John C. Chao,et al.  Model selection in partially nonstationary vector autoregressive processes with reduced rank structure , 1999 .

[3]  É. Moulines,et al.  Log-Periodogram Regression Of Time Series With Long Range Dependence , 1999 .

[4]  Mahmoud Zarepour,et al.  Bootstrapping unstable first order autoregressive process with errors in the domain of attraction of stable law , 1999 .

[5]  Michael McAleer,et al.  Cointegration Analysis of Seasonal Time Series , 1998 .

[6]  Peter C. B. Phillips,et al.  New Tools for Understanding Spurious Regressions , 1998 .

[7]  Serena Ng,et al.  AN AUTOREGRESSIVE SPECTRAL DENSITY ESTIMATOR AT FREQUENCY ZERO FOR NONSTATIONARITY TESTS , 1998, Econometric Theory.

[8]  In Choi,et al.  TIME-SERIES-BASED ECONOMETRICS , 1998, Econometric Theory.

[9]  Peter C. B. Phillips,et al.  Impulse response and forecast error variance asymptotics in nonstationary VARs , 1998 .

[10]  Rohit S. Deo,et al.  The mean squared error of Geweke and Porter‐Hudak's estimator of the memory parameter of a long‐memory time series , 1998 .

[11]  Phoebus J. Dhrymes.  Time series, unit roots, and cointegration , 1998 .

[12]  G. N. Fouskitakis,et al.  On the estimation of long-memory time series models , 1997, 1997 European Control Conference (ECC).

[13]  J. Bai,et al.  Estimating Multiple Breaks One at a Time , 1997, Econometric Theory.

[14]  Robin L. Lumsdaine,et al.  Multiple Trend Breaks and the Unit-Root Hypothesis , 1997, Review of Economics and Statistics.

[15]  Mark W. Watson,et al.  Estimating Deterministic Trends in the Presence of Serially Correlated Errors , 1994, Review of Economics and Statistics.

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

[17]  P. Robinson,et al.  Testing of unit root and other nonstationary hypotheses in macroeconomic time series , 1996 .

[18]  Juan Romo,et al.  Unit root bootstrap tests for AR(1) models , 1996 .

[19]  Philip Hans Franses,et al.  Periodicity and Stochastic Trends in Economic Time Series , 1996 .

[20]  Jon Faust.  Near Observational Equivalence and Theoretical size Problems with Unit Root Tests , 1996, Econometric Theory.

[21]  R. Baillie,et al.  Fractionally integrated generalized autoregressive conditional heteroskedasticity , 1996 .

[22]  P. Phillips,et al.  Efficiency Gains from Quasi-Differencing under Nonstationarity , 1996 .

[23]  Katsuto Tanaka,et al.  Time Series Analysis: Nonstationary and Noninvertible Distribution Theory , 1996 .

[24]  Richard T. Baillie,et al.  Long memory processes and fractional integration in econometrics , 1996 .

[25]  Peter C. B. Phillips,et al.  Econometric Model Determination , 1996 .

[26]  N. E. Savin,et al.  The Level and Power of the Bootstrap t Test in the AR(1) Model With Trend , 1996 .

[27]  Richard A. Davis,et al.  Maximum Likelihood Estimation for MA(1) Processes with a Root on or near the Unit Circle , 1996, Econometric Theory.

[28]  Peter C. B. Phillips,et al.  An Asymptotic Theory of Bayesian Inference for Time Series , 1996 .

[29]  S. Johansen.  Likelihood-Based Inference in Cointegrated Vector Autoregressive Models , 1996 .

[30]  An Empirical Bayesian Approach to Cointegrating Rank Selection and Test of the Present Value Model for Stock Prices , 1996 .

[31]  P. Robinson.  Gaussian Semiparametric Estimation of Long Range Dependence , 1995 .

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

[33]  G. Reinsel.  Elements of Multivariate Time Series Analysis , 1995 .

[34]  P. Phillips.  Automated Forecasts of Asia-Pacific Economic Activity , 1995 .

[35]  M. Chinn,et al.  DETERMINISTIC, STOCHASTIC, AND SEGMENTED TRENDS IN AGGREGATE OUTPUT: A CROSS-COUNTRY ANALYSIS , 1995 .

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

[37]  Joon Y. Park,et al.  Testing for Unit Roots in Models with Structural Change , 1994, Econometric Theory.

[38]  P. Robinson.  Efficient Tests of Nonstationary Hypotheses , 1994 .

[39]  Peter C. B. Phillips,et al.  Posterior Odds Testing for a Unit Root with Data-Based Model Selection , 1994, Econometric Theory.

[40]  Jae-Young Kim,et al.  Bayesian Asymptotic Theory in a Time Series Model with a Possible Nonstationary Process , 1994, Econometric Theory.

[41]  Brendan McCabe,et al.  A Consistent Test for a Unit Root , 1994 .

[42]  P. M. Robinson.  Advances in Econometrics: Time series with strong dependence , 1994 .

[43]  Eric Zivot,et al.  A Bayesian Analysis of Trend Determination in Economic Time Series , 1994 .

[44]  Christopher A. Sims.  Advances in econometrics : Sixth World Congress , 1994 .

[45]  J. Davidson.  Stochastic Limit Theory: An Introduction for Econometricians , 1994 .

[46]  Pentti Saikkonen,et al.  Testing for a Moving Average Unit Root in Autoregressive Integrated Moving Average Models , 1993 .

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

[48]  J. Beran.  Statistical methods for data with long-range dependence , 1992 .

[49]  P. Phillips,et al.  Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? , 1992 .

[50]  Peter Schmidt,et al.  LM Tests for a Unit Root in the Presence of Deterministic Trends , 1992 .

[51]  D. Andrews,et al.  Further Evidence on the Great Crash, the Oil-Price Shock, and the Unit-Root Hypothesis , 1992 .

[52]  Donald W. K. Andrews,et al.  An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator , 1992 .

[53]  Stephen R. Blough.  The Relationship between Power and Level for Generic Unit Root Tests in Finite Samples , 1992 .

[54]  C. Whiteman,et al.  The power problems of unit root test in time series with autoregressive errors , 1992 .

[55]  Victor Solo,et al.  Asymptotics for Linear Processes , 1992 .

[56]  J. Stock.  Deciding between I(1) and I(0) , 1992 .

[57]  Sat Gupta,et al.  Estimation in long memory time series models , 1992 .

[58]  Peter C. B. Phillips,et al.  To Criticize the Critics: An Objective Bayesian Analysis of Stochastic Trends , 1991 .

[59]  P. Jeganathan.  On the Asymptotic Behavior of Least-Squares Estimators in AR Time Series with Roots Near the Unit Circle , 1991, Econometric Theory.

[60]  H. V. Dijk,et al.  A Bayesian analysis of the unit root in real exchange rates , 1991 .

[61]  Robert L. Taylor,et al.  Boot-strapping unstable rst-order autoregressive processes , 1991 .

[62]  D. Andrews.  Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation , 1991 .

[63]  C. Whiteman,et al.  The Temporal Stability of Dividends and Stock Prices: Evidence from the Likelihood Function , 1991 .

[64]  R. Taylor,et al.  Bootstrap test of significance and sequential bootstrap estimation for unstable first order autoregressive processes , 1991 .

[65]  J. Stock,et al.  Confidence Intervals for the Largest Autoregressive Root in U , 1991 .

[66]  Katsuto Tanaka.  Testing for a Moving Average Unit Root , 1990, Econometric Theory.

[67]  J. Stock,et al.  Recursive and Sequential Tests of the Unit Root and Trend Break Hypothesis: Theory and International Evidence , 1990 .

[68]  D. Duffie,et al.  Security markets : stochastic models , 1989 .

[69]  Juan J. Dolado,et al.  COINTEGRA TION AND UNIT ROOTS , 1990 .

[70]  S. Porter-Hudak.  An Application of the Seasonal Fractionally Differenced Model to the Monetary Aggregates , 1990 .

[71]  Byung Sam Yoo,et al.  Seasonal integration and cointegration , 1990 .

[72]  P. Phillips,et al.  Asymptotic Properties of Residual Based Tests for Cointegration , 1990 .

[73]  J. Stock,et al.  INFERENCE IN LINEAR TIME SERIES MODELS WITH SOME UNIT ROOTS , 1990 .

[74]  P. Perron,et al.  The Great Crash, The Oil Price Shock And The Unit Root Hypothesis , 1989 .

[75]  Sam Ouliaris,et al.  Testing for a Unit Root in the Presence of a Maintained Trend , 1989 .

[76]  D. Osborn,et al.  The Performance of Periodic Autoregressive Models in Forecasting Seasonal U.K. Consumption , 1989 .

[77]  B. Raj.  Advances in econometrics and modelling , 1989 .

[78]  Jean-Marie Dufour,et al.  Optimal Invariant Tests for the Autocorrelation Coefficient in Linear Regressions with Stationary and Nonstationary Ar(1) Errors , 1989 .

[79]  G. Schwert,et al.  Tests for Unit Roots: a Monte Carlo Investigation , 1988 .

[80]  A. Bose.  Edgeworth correction by bootstrap in autoregressions , 1988 .

[81]  P. Phillips,et al.  TRENDS VERSUS RANDOM WALKS IN TIME SERIES ANALYSIS , 1988 .

[82]  Lawrence J. Christiano,et al.  Searching for a Break in Gnp , 1988 .

[83]  J. Stock,et al.  Variable Trends in Economic Time Series , 1988 .

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

[85]  Seiji Nabeya,et al.  Asymptotic Theory of a Test for the Constancy of Regression Coefficients Against the Random Walk Alternative , 1988 .

[86]  Francis X. Diebold,et al.  Unit roots in economic time series: a selective survey , 1988 .

[87]  R. Kashyap,et al.  ESTIMATION IN LONG‐MEMORY TIME SERIES MODEL , 1988 .

[88]  P. Phillips.  Multiple Regression with Integrated Time Series , 1987 .

[89]  C. Z. Wei,et al.  Asymptotic Inference for Nearly Nonstationary AR(1) Processes , 1987 .

[90]  Peter C. B. Phillips,et al.  Towards a Unified Asymptotic Theory for Autoregression , 1987 .

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

[92]  C. Granger,et al.  Co-integration and error correction: representation, estimation and testing , 1987 .

[93]  Maxwell L. King,et al.  Towards a Theory of Point Optimal Testing , 1987 .

[94]  P. Perron,et al.  Trends and random walks in macroeconomic time series : Further evidence from a new approach , 1988 .

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

[96]  H. Künsch,et al.  Discrimination between monotonic trends and long-range dependence , 1986, Journal of Applied Probability.

[97]  P. Phillips.  Understanding spurious regressions in econometrics , 1986 .

[98]  J. Rissanen.  Stochastic Complexity and Modeling , 1986 .

[99]  Alok Bhargava,et al.  On the Theory of Testing for Unit Roots in Observed Time Series , 1986 .

[100]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

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

[102]  D. Dickey,et al.  Testing for unit roots in autoregressive-moving average models of unknown order , 1984 .

[103]  Victor Solo,et al.  The Order of Differencing in ARIMA Models , 1984 .

[104]  D. Freedman.  On Bootstrapping Two-Stage Least-Squares Estimates in Stationary Linear Models , 1984 .

[105]  David A. Dickey,et al.  Testing for Unit Roots in Seasonal Time Series , 1984 .

[106]  Tze Leung Lai,et al.  Fixed Accuracy Estimation of an Autoregressive Parameter , 1983 .

[107]  Alok Bhargava,et al.  Testing Residuals from Least Squares Regression for Being Generated by the Gaussian Random Walk , 1983 .

[108]  Kenneth J. Singleton,et al.  On Unit Roots and the Empirical Modeling of Exchange Rates , 1982 .

[109]  T. Lai,et al.  Least Squares Estimates in Stochastic Regression Models with Applications to Identification and Control of Dynamic Systems , 1982 .

[110]  C. Nelson,et al.  Trends and random walks in macroeconmic time series: Some evidence and implications , 1982 .

[111]  N. Savin,et al.  The Calculation of the Limiting Distribution of the Least Squares Estimator of the Parameter in a Random Walk Model , 1981 .

[112]  W. Fuller,et al.  LIKELIHOOD RATIO STATISTICS FOR AUTOREGRESSIVE TIME SERIES WITH A UNIT ROOT , 1981 .

[113]  C. Nelson,et al.  A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the ‘business cycle’☆ , 1981 .

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

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

[116]  I. Johnstone,et al.  On Asymptotic Posterior Normality for Stochastic Processes , 1979 .

[117]  R. Hall.  Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence , 1978, Journal of Political Economy.

[118]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[119]  J. Ware,et al.  Applications of Statistics , 1978 .

[120]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[121]  Ian B. MacNeill,et al.  Properties of Sequences of Partial Sums of Polynomial Regression Residuals with Applications to Tests for Change of Regression at Unknown Times , 1978 .

[122]  Hirotugu Akaike,et al.  On entropy maximization principle , 1977 .

[123]  Chris Chatfield,et al.  Introduction to Statistical Time Series. , 1976 .

[124]  K. Berk.  Consistent Autoregressive Spectral Estimates , 1974 .

[125]  John S. White.  THE LIMITING DISTRIBUTION OF THE SERIAL CORRELATION COEFFICIENT IN THE EXPLOSIVE CASE , 1958 .

[126]  U. Grenander,et al.  Statistical analysis of stationary time series , 1958 .