Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T17:52:57.894Z Has data issue: false hasContentIssue false
  • English
  • Français

HETEROSKEDASTIC TIME SERIES WITH A UNIT ROOT

Published online by Cambridge University Press:  01 October 2009

Giuseppe Cavaliere  and
A.M. Robert Taylor
Giuseppe Cavaliere
Affiliation:
University of Bologna
A.M. Robert Taylor*
Affiliation:
University of Nottingham
*
*Address correspondence to Robert Taylor, School of Economics, The Sir Clive Granger Building, University of Nottingham, Nottingham NG7 2RD, United Kingdom; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we provide a unified theory, and associated invariance principle, for the large-sample distributions of the Dickey–Fuller class of statistics when applied to unit root processes driven by innovations displaying nonstationary stochastic volatility of a very general form. These distributions are shown to depend on both the spot volatility and the integrated variation associated with the innovation process. We propose a partial solution (requiring any leverage effects to be asymptotically negligible) to the identified inference problem using a wild bootstrap–based approach. Results are initially presented in the context of martingale differences and are later generalized to allow for weak dependence. Monte Carlo evidence is also provided that suggests that our proposed bootstrap tests perform very well in finite samples in the presence of a range of innovation processes displaying nonstationary volatility and/or weak dependence.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 5 , October 2009 , pp. 1228 - 1276
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, D.W.K. (1988) Laws of large numbers for dependent non-identically distributed random variables. Econometric Theory 4, 458–467.CrossRefGoogle Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Andrews, D.W.K. & Buchinsky, M. (2001) Evaluation of a three-step method for choosing the number of bootstrap repetitions. Journal of Econometrics 103, 345–386.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2001) Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics (with discussion). Journal of the Royal Statistical Society, Series B 63, 167–241.CrossRefGoogle Scholar
Basawa, I.V., Mallik, A.K., McCormick, W.P., Reeves, J.H., & Taylor, R.L. (1991) Bootstrapping unstable first-order autoregressive processes. Annals of Statistics 19, 1098–1101.CrossRefGoogle Scholar
Beare, B. (2005) Robustifying Unit Root Tests to Permanent Changes in Innovation Variance. Mimeo, Yale University.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Boswijk, H.P. (2001) Testing for a Unit Root with Near Integrated Volatility. Tinbergen Institute Discussion paper 2001-077/4.Google Scholar
Boswijk, H.P. (2005) Adaptive Testing for a Unit Root with Nonstationary Volatility. UvA- Econometrics Discussion paper 2005/07.Google Scholar
Burridge, P. & Taylor, A.M.R. (2001) On regression-based tests for seasonal unit roots in the presence of periodic heteroscedasticity. Journal of Econometrics 104, 91–117.CrossRefGoogle Scholar
Cavaliere, G. (2004) Unit root tests under time-varying variances. Econometric Reviews 23, 259–292.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2007) Testing for unit roots in time series models with non-stationary volatility. Journal of Econometrics 140, 919–947.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2008) Bootstrap unit root tests for time series with non-stationary volatility. Econometric Theory 24, 43–71.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2009) Bootstrap M unit root tests. Econometric Reviews 28, 393–421.CrossRefGoogle Scholar
Chan, N.H. & Tran, L.T. (1989) On the first order autoregressive process with infinite variance. Econometric Theory 5, 354–362.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2003) A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis 24, 379–400.CrossRefGoogle Scholar
Clark, P.K. (1973) A subordinated stockastic process model with finite variance for speculative prices. Econometrica 41, 135–155.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.CrossRefGoogle Scholar
Davidson, R. & Flachaire, E. (2008) The wild bootstrap, tamed at last. Journal of Econometrics 146, 162–169.CrossRefGoogle Scholar
de Jong, R. & Davidson, J. (2000) The functional central limit theorem and weak convergence to stochastic integrals, part 1. Econometric Theory 16, 621–642.CrossRefGoogle Scholar
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Dickey, D.A. & Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072.CrossRefGoogle Scholar
Duffie, D., Pan, J. & Singleton, K. (2000) Transformation analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376.CrossRefGoogle Scholar
Elliott, G., Rothenberg, T.J. & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.CrossRefGoogle Scholar
Ferretti, N. & Romo, J. (1996) Bootstrap tests for unit root AR(1) models. Biometrika 84, 849–860.CrossRefGoogle Scholar
Georgiev, I. (2008) Asymptotics for cointegrated processes with infrequent stochastic level shifts and outliers. Econometric Theory 24, 587–615.CrossRefGoogle Scholar
Giné, E. & Zinn, J. (1990) Bootstrapping general empirical measures. Annals of Probability 18, 851–869.CrossRefGoogle Scholar
Gonçalves, S. & Kilian, L. (2004) Bootstrapping autoregressions with conditional heteroskedasticity of unknown form. Journal of Econometrics 123, 89–120.CrossRefGoogle Scholar
Gonçalves, S. & Kilian, L. (2007) Asymptotic and bootstrap inference for AR(∞) processes with conditional heteroskedasticity. Econometric Reviews 26, 609–641.CrossRefGoogle Scholar
Hall, P. (1977) Martingale invariance principles. Annals of Probability 5, 875–887.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Applications. Academic Press.Google Scholar
Hansen, B.E. (1992a) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.CrossRefGoogle Scholar
Hansen, B.E. (1992b) Consistent covariance matrix estimation for dependent heterogeneous processes. Econometrica 60, 967–972.CrossRefGoogle Scholar
Hansen, B.E. (1995) Regression with nonstationary volatility. Econometrica 63, 1113–1132.CrossRefGoogle Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.CrossRefGoogle Scholar
Hansen, B.E. (2000) Testing for structural change in conditional models. Journal of Econometrics 97, 93–115.CrossRefGoogle Scholar
Inoue, A. & Kilian, L. (2002) Bootstrapping autoregressive processes with possible unit roots. Econometrica 70, 377–391.CrossRefGoogle Scholar
Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory 18, 1449–1459.CrossRefGoogle Scholar
Kim, C.-J. & Nelson, C.R. (1999) Has the US economy become more stable? A Bayesian approach based on a Markov-switching model of the business cycle. Review of Economics and Statistics 81, 608–616.CrossRefGoogle Scholar
Kurtz, T.G. & Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.CrossRefGoogle Scholar
Lee, C.C. & Phillips, P.C.B. (1994) An ARMA Pre-whitened Long-Run Variance Estimator. Manuscript, Yale University.Google Scholar
Ling, S., Li, W.K. & McAleer, M. (2003) Estimation and testing for unit root process with GARCH(1,1) errors: Theory and Monte Carlo evidence. Econometric Reviews 22, 179–202.CrossRefGoogle Scholar
Liu, R.Y. (1988) Bootstrap procedures under some non i.i.d. models. Annals of Statistics 16, 1696–1708.CrossRefGoogle Scholar
Loretan, M. & Phillips, P.C.B. (1994) Testing covariance stationarity under moment condition failure with an application to common stock returns. Journal of Empirical Finance 1, 211–248.CrossRefGoogle Scholar
MacKinnon, J. (2006) Bootstrap methods in econometrics. Economic Record 82, S2–S18.CrossRefGoogle Scholar
Mammen, E. (1993) Bootstrap and wild bootstrap for high dimensional linear models. Annals of Statistics 21, 255–285.CrossRefGoogle Scholar
McConnell, M.M. & Perez Quiros, G. (2000) Output fluctuations in the United States: What has changed since the early 1980s? American Economic Review 90, 1464–1476.CrossRefGoogle Scholar
McLeish, D.L. (1974) Dependent central limit theorems and invariance principles. Annals of Probability 2, 620–628.CrossRefGoogle Scholar
Nelson, C.R., Piger, J. & Zivot, E. (2001) Markov regime switching and unit-root tests. Journal of Business & Economic Statistics 19, 404–415.CrossRefGoogle Scholar
Nelson, D.B. (1990a) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Nelson, D.B. (1990b) ARCH models as diffusion approximations. Journal of Econometrics 45, 7–38.CrossRefGoogle Scholar
Ng, S. & Perron, P. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.CrossRefGoogle Scholar
Omori, Y., Chib, S., Shephard, N., & Nakajimaa, J. (2007) Stochastic volatility with leverage: Fast and efficient likelihood inference. Journal of Econometrics 140, 425–449.CrossRefGoogle Scholar
Paparoditis, E.P. & Politis, D.N. (2003) Residual-based block bootstrap for unit root testing. Econometrica 71, 813–855.CrossRefGoogle Scholar
Park, J.Y. (2002a) Nonstationary nonlinear heteroskedasticity. Journal of Econometrics 110, 383–415.CrossRefGoogle Scholar
Park, J.Y. (2002b) An invariance principle for sieve bootstrap in time series. Econometric Theory 18, 469–490.CrossRefGoogle Scholar
Park, J.Y. (2003) Bootstrap unit root tests. Econometrica 71, 1845–1895.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 69, 117–161.CrossRefGoogle Scholar
Perron, P. & Ng, S. (1996) Useful modifications to some unit root tests with dependent errors and their local asymptotic properties. Review of Economic Studies 63, 435–463.CrossRefGoogle Scholar
Phillips, P.C.B. (1987a) Toward a unified asymptotic theory for autoregression. Biometrika 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (1987b) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. (1990) Time series regression with a unit root and infinite variance errors. Econometric Theory 6, 44–62.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335–346.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Phillips, P.C.B. & Xiao, Z. (1998) A primer on unit root testing. Journal of Economic Surveys 12, 423–470.CrossRefGoogle Scholar
Phillips, P.C.B. & Xu, K.-L. (2006) Inference in autoregression under heteroskedasticity. Journal of Time Series Analysis 27, 289–308.CrossRefGoogle Scholar
Revuz, D. & Yor, M. (1991) Continuous Martingales and Brownian Motion. Springer-Verlag.CrossRefGoogle Scholar
Roberts, G.O., Papaspiliopoulos, O., & Dellaportas, P. (2004) Bayesian inference for non-Gaussian Ornstein–Uhlenbeck stochastic volatility processes. Journal of the Royal Statistical Society, Series B 66, 369–393.CrossRefGoogle Scholar
Schmidt, P. & Phillips, P.C.B. (1992) LM tests for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54, 257–287.CrossRefGoogle Scholar
Sensier, M. & van Dijk, D. (2004) Testing for volatility changes in U.S. macroeconomic time series. Review of Economics and Statistics 86, 833–839.CrossRefGoogle Scholar
Shephard, N. (2005) Stochastic Volatility: Selected Readings. Oxford University Press.CrossRefGoogle Scholar
Stock, J.H. (1994) Unit roots, structural breaks and trends. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. 4, pp. 2739–2840. Elsevier Science.Google Scholar
Stock, J.H. & Watson, M.W. (1999) A comparison of linear and nonlinear univariate models for forecasting macroeconomic time series. In Engle, R.F. & White, H. (eds.), Cointegration, Causality and Forecasting: A Festschrift in Honour of Clive W.J. Granger, pp. 1–44. Oxford University Press.Google Scholar
Wooldridge, J.M. & White, H. (1988) Some invariance principles and central limit theorems for dependent heterogeneous processes. Econometric Theory 4, 210–230.CrossRefGoogle Scholar
Wu, C.F.J. (1986) Jackknife, bootstrap, and other resampling methods. Annals of Statistics 14, 1261–1295.Google Scholar
Xu, K.-L. (2008) Bootstrapping autoregression under nonstationary volatility. Econometrics Journal 11, 1–26.CrossRefGoogle Scholar
Xu, K.-L. & Phillips, P.C.B. (2008) Adaptive estimation of autoregressive models with time-varying variances. Journal of Econometrics 142, 265–280.CrossRefGoogle Scholar