Testing for Autocorrelation and Unit Roots in the Presence of Conditional Heteroskedasticity of Unknown Form

The presence of conditional heteroskedasticity invalidates standard autocorrelation tests such as the Durbin-Watson statistic and its many variants, and reduces the power of standard unit root tests like the Dickey-Fuller test. This paper addresses the problem of testing for AR(1) and AR(p) alternatives in the presence of conditional heteroskedasticity, constructs new autocorrelation tests (called g- and gl- tests), and derives an asymptotic theory for the new statistics. The g test is shown to be robust to any form of conditional heteroskedasticity, while the gl test is robust to the joint presence of conditional heteroskedasticity and lagged dependent variables. The paper also develops a GLS-type unit root test that allows for conditional heteroskedasticity. Its limit distribution turns out to be a convex mixture of the Dickey-Fuller distribution and the standard normal distribution. It is shown that the GLS statistic improves the power of the Dickey-Fuller test under condictional heteroskedasticity. Some Monte Carlo simulations with the new tests are reported.

[1]  G. Box,et al.  Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models , 1970 .

[2]  R. Chou,et al.  ARCH modeling in finance: A review of the theory and empirical evidence , 1992 .

[3]  Thomas J. Smith,et al.  A Unified Approach to Testing for Serial Correlation in Stock Returns , 1994 .

[4]  T. Breurch,et al.  A simple test for heteroscedasticity and random coefficient variation (econometrica vol 47 , 1979 .

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

[6]  P. Robinson,et al.  Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression , 1991 .

[7]  James Durbin,et al.  Testing for Serial Correlation in Least-Squares Regression When Some of the Regressors are Lagged Dependent Variables , 1970 .

[8]  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 .

[9]  Adrian Pagan,et al.  The Lagrange Multiplier Test and its Applications to Model Specification in Econometrics , 1980 .

[10]  Wolfgang Härdle,et al.  Nonparametric Curve Estimation from Time Series , 1989 .

[11]  W. Li,et al.  Limiting Distributions of Maximum Likelihood Estimators for Unstable Autoregressive Moving-Average Time Series with General Autoregressive Heteroscedastic Errors , 1998 .

[12]  P. Phillips,et al.  The Durbin-Watson ratio under infinite-variance errors , 1991 .

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

[14]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[15]  A. Gallant,et al.  Seminonparametric Estimation Of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications , 1989 .

[16]  K. French,et al.  Expected stock returns and volatility , 1987 .

[17]  J. Poterba,et al.  Mean Reversion in Stock Prices: Evidence and Implications , 1987 .

[18]  Daniel B. Nelson.  CONDITIONAL HETEROSKEDASTICITY IN ASSET RETURNS: A NEW APPROACH , 1991 .

[19]  Anil K. Bera,et al.  A Class of Nonlinear ARCH Models , 1992 .

[20]  Guido M. Kuersteiner,et al.  Efficiency IV Estimation for Autoregressive Models with Conditional Heterogeneity , 2000 .

[21]  Andre Lucas,et al.  Unit Root Tests Based on M Estimators , 1995, Econometric Theory.

[22]  Andrew A. Weiss,et al.  Asymptotic Theory for ARCH Models: Estimation and Testing , 1986, Econometric Theory.

[23]  W. Härdle.  Applied Nonparametric Regression , 1991 .

[24]  Y. Hong,et al.  Non-Parametric Estimation and the Risk Premium , 1988 .

[25]  H. Iemoto.  Modelling the persistence of conditional variances , 1986 .

[26]  E. Fama,et al.  Permanent and Temporary Components of Stock Prices , 1988, Journal of Political Economy.

[27]  M. Evans,et al.  Locally Optimal Properties of the Durbin-Watson Test , 1988, Econometric Theory.

[28]  R. Engle,et al.  Semiparametric ARCH Models , 1991 .

[29]  Adrian Pagan,et al.  The Econometric Analysis of Models with Risk Terms , 1988 .

[30]  G. Schwert,et al.  Heteroskedasticity in Stock Returns , 1989 .

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

[32]  L. Godfrey,et al.  REGRESSION EQUATIONS WHEN THE REGRESSORS INCLUDE LAGGED DEPENDENT VARIABLES , 1978 .

[33]  P. Hall,et al.  Martingale Limit Theory and its Application. , 1984 .

[34]  Wolfgang Härdle,et al.  APPLIED NONPARAMETRIC METHODS , 1992 .

[35]  D. Shin,et al.  UNIT ROOT TESTS BASED ON ADAPTIVE MAXIMUM LIKELIHOOD ESTIMATION , 1999, Econometric Theory.

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

[37]  T. Bollerslev,et al.  A CONDITIONALLY HETEROSKEDASTIC TIME SERIES MODEL FOR SPECULATIVE PRICES AND RATES OF RETURN , 1987 .

[38]  J. Poterba,et al.  Mean Reversion in Stock Prices: Evidence and Implications , 1987 .

[39]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[40]  Peter Schmidt,et al.  Unit root tests with conditional heteroskedasticity , 1993 .

[41]  W. Härdle.  Applied Nonparametric Regression , 1992 .

[42]  John R. Wingender,et al.  Skewness Persistence in Common Stock Returns , 1986, Journal of Financial and Quantitative Analysis.

[43]  Peter E. Rossi,et al.  Nonlinear dynamic structures , 1993 .

[44]  Robert F. Engle,et al.  Estimates of the Variance of U. S. Inflation Based upon the ARCH Model , 1983 .

[45]  J. Durbin,et al.  Testing for serial correlation in least squares regression. II. , 1950, Biometrika.

[46]  Maxwell L. King,et al.  Testing for autocorrelation in linear regression models: a survey , 1987 .

[47]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

[48]  Adrian Pagan,et al.  Alternative Models for Conditional Stock Volatility , 1989 .

[49]  Peter C. B. Phillips,et al.  Time Series Regression With a Unit Root and Infinite-Variance Errors , 1990, Econometric Theory.

[50]  P. Phillips,et al.  Efficient Estimation of Second Moment Parameters in Arch Models , 2001 .

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

[52]  H. White.  A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity , 1980 .

[53]  R. Litzenberger,et al.  SKEWNESS PREFERENCE AND THE VALUATION OF RISK ASSETS , 1976 .

[54]  J. Durbin,et al.  Testing for serial correlation in least squares regression. I. , 1950, Biometrika.

[55]  L. Godfrey.  TESTING AGAINST GENERAL AUTOREGRESSIVE AND MOVING AVERAGE ERROR MODELS WHEN THE REGRESSORS INCLUDE LAGGED DEPENDENT VARIABLES , 1978 .

[56]  A. Milhøj.  The moment structure of ARCH processes , 1985 .

[57]  D. Andrews,et al.  Testing for Serial Correlation against an ARMA(1,1) Process , 1996 .