Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T10:49:43.969Z Has data issue: false hasContentIssue false
  • English
  • Français

TESTING AND INFERENCE IN NONLINEAR COINTEGRATING VECTOR ERROR CORRECTION MODELS

Published online by Cambridge University Press:  06 August 2013

Dennis Kristensen  and
Anders Rahbek
Dennis Kristensen*
Affiliation:
University College London, and, Institute of Fiscal Studies
Anders Rahbek
Affiliation:
University of Copenhagen
*
*Address correspondence to Dennis Kristensen, University College London, Gower Street, London WC1E 6BT, United Kingdom; e-mail: d.kristensen@ucl.ac.uk.
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyze estimators and tests for a general class of vector error correction models that allows for asymmetric and nonlinear error correction. For a given number of cointegration relationships, general hypothesis testing is considered, where testing for linearity is of particular interest. Under the null of linearity, parameters of nonlinear components vanish, leading to a nonstandard testing problem. We apply so-called sup-tests to resolve this issue, which requires development of new(uniform) functional central limit theory and results for convergence of stochastic integrals. We provide a full asymptotic theory for estimators and test statistics. The derived asymptotic results prove to be nonstandard compared to results found elsewhere in the literature due to the impact of the estimated cointegration relations. This complicates implementation of tests motivating the introduction of bootstrap versions that are simple to compute. A simulation study shows that the finite-sample properties of the bootstrapped tests are satisfactory with good size and power properties for reasonable sample sizes.

Type
ARTICLES
Information
Econometric Theory , Volume 29 , Issue 6 , December 2013 , pp. 1238 - 1288
Copyright
Copyright © Cambridge University Press 2013 

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.)

Footnotes

We thank the co-editor Michael Jansson and editor Peter C.B. Phillips and the anonymous referees whose comments and suggestions much improved the paper. We also thank Juan Carlos Escanciano and Myung Hwan Seo for very helpful discussions and suggestions. Both authors are affiliated with CREATES, funded by the DanishNational Research Foundation. The Velux Foundation funded a longer research visit for Kristensen at the University of Copenhagen, where part of this research was conducted. Part of the research was also conducted while Kristensen visited Princeton University, whose hospitality is gratefully acknowledged. Kristensen received research support from the National Science Foundation (grant SES-0961596). Rahbek gratefully acknowledges funding from the Danish Council for Independent Research, Social Sciences (grant 10-079774). We also thank participants at the 18th Annual Symposium of the Society for Nonlinear Dynamics and Econometrics in Novara, Italy, and at the Montreal Econometrics Seminar and Oxbridge Time Series Group Workshop 2010, Cambridge.

References

REFERENCES

Andrews, D.W.K. (1994) Empirical process methods in econometrics. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. IV., pp. 2247–2294. Elsevier.Google Scholar
Andrews, D.W.K. & Buchinsky, M. (2001) A three-step method for choosing the number of bootstrap repetitions. Econometrica 68, 23–51.CrossRefGoogle Scholar
Baghli, M. (2005) Nonlinear error-correction models for the FF/DM rate. Studies in Nonlinear Dynamics & Econometrics 9, Issue 1, article 7.Google Scholar
Bec, F. & Rahbek, A. (2004) Vector equilibrium correction models with non-linear discontinuous adjustments. Econometrics Journal 7, 628–651.CrossRefGoogle Scholar
Brown, B.M. (1971) Martingale central limit theorems. Annals of Mathematical Statistics 42,59–66.CrossRefGoogle Scholar
Caner, M. & Hansen, B. (2001) Threshold autoregression with a unit root. Econometrica 69, 1555–1596.CrossRefGoogle Scholar
Cavaliere, G., Rahbek, A., & Taylor, R. (2010a) Testing for cointegration in vector autoregressions with non-stationary volatility. Journal of Econometrics 158, 7–24.Google Scholar
Cavaliere, G., Rahbek, A., & Taylor, R. (2010b) Co-integration rank tests under conditional heteroskedasticity. Econometric Theory 26, 1719–1760.CrossRefGoogle Scholar
Cavaliere, G., Rahbek, A., & Taylor, R. (2012) Bootstrap determination of the cointegration rank in VAR models. Econometrica 80, 1721–1740.Google Scholar
Choi, I. & Saikkonen, P. (2004) Testing linearity in cointegrating smooth transition regressions. Econometrics Journal 7, 341–365.CrossRefGoogle Scholar
Davies, R.B. (1987) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33–43.Google Scholar
de Jong, R.M. (2001) Nonlinear estimation using estimated cointegrating relations. Journal of Econometrics 101, 109–122.CrossRefGoogle Scholar
de Jong, R.M. (2002) Nonlinear minimization estimators in the presence of cointegrating relations. Journal of Econometrics 110, 241–259.Google Scholar
Escanciano, J.C. (2007) Model checks using residual marked empirical processes. Statistica Sinica 17, 115–138.Google Scholar
Escribano, A. (2004) Nonlinear Error Correction: The Case of Money Demand in the United Kingdom (1878-2000). Macroeconomic Dynamics 8, 76–116.Google Scholar
Franq, C., Horvatha, L., & Zakoïan, J.-M. (2010) Sup-tests for linearity in a general nonlinear AR(1) model. Econometric Theory 26, 965–993.CrossRefGoogle Scholar
Gao, J. & Phillips, P.C.B. (2012) Semiparametric Estimation in Multivariate Nonstationary Time Series Models. Working paper, Monash University.Google Scholar
Gonzalo, J. & Pitarakis, J.-Y. (2006) Threshold effects in multivariate error correction models. In Mills, T.C. & Patteson, K. (eds.), Palgrave Handbook of Econometrics: Econometric Theory vol. 1, pp. 578–609. Palgrave Macmillan.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Applications. Academic Press.Google Scholar
Hansen, B.E. (1992) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.CrossRefGoogle Scholar
Hansen, B.E. (1996a) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.CrossRefGoogle Scholar
Hansen, B.E. (1996b) Stochastic equicontinuity for unbounded dependent heterogeneous arrays. Econometric Theory 12, 347–359.CrossRefGoogle Scholar
Hansen, B.E. & Seo, B. (2002) Testing for two-regime threshold cointegration in vector error correction models. Journal of Econometrics 110, 293–318.CrossRefGoogle Scholar
Jensen, S.T. & Rahbek, A. (2007) A Note on the Law of Large Numbers for Functions of Geometrically Ergodic Time Series. Econometric Theory 23, 761–766.CrossRefGoogle Scholar
Johansen, S. (1992) Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 1551–1580.CrossRefGoogle Scholar
Johansen, S. (1996) Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, 2nd ed. Oxford University Press.Google Scholar
Johansen, S. (2010) Some identification problems in the cointegrated vector autoregressive model. Journal of Econometrics 158, 262–273.CrossRefGoogle Scholar
Kallenberg, O. (2002) Foundations of Modern Probability, 2nd edition. Springer.CrossRefGoogle Scholar
Kapetanios, G., Shin, Y. & Snell, A. (2006) Testing for cointegration in nonlinear smooth transition error correction models. Econometric Theory 22, 279–303.CrossRefGoogle Scholar
Karlsen, H.A., Mykelbust, T., & Tjøstheim, D. (2007) Nonparametric estimation in a nonlinear cointegration type model. Annals of Statistics 35, 252–299.CrossRefGoogle Scholar
Kilic, R. (2011) Testing for co-integration and nonlinear adjustment in a smooth transition error correction model. Journal of Time Series Analysis 32, 647–660.CrossRefGoogle Scholar
Kristensen, D. and Rahbek, A. (2005) Asymptotics of the QMLE for a class of ARCH(q) models. Econometric Theory 21, 946–961.CrossRefGoogle Scholar
Kristensen, D. & Rahbek, A. (2009) Asymptotics of the QMLE for non-linear ARCH models. Journal of Time Series Econometrics, Issue 1, article 2.CrossRefGoogle Scholar
Kristensen, D. & Rahbek, A. (2010) Likelihood-based inference for cointegration with nonlinear error-correction. Journal of Econometrics 158, 78–94.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
Linton, O.B. & Seo, M. (2007) A smoothed least squares estimator for threshold regression models. Journal of Econometrics 141, 704–735.Google Scholar
Luukkonen, R., Ripatti, A., & Saikkonen, P. (1996) Testing for a valid normalization of cointegrating vectors in vector autoregressive processes. Journal of Business & Economic Statistics 17, 195–204.Google Scholar
Magnus, J.R. & Neudecker, H. (1988) Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley.Google Scholar
Meitz, M. and Saikkonen, P. (2011) Parameter estimation in nonlinear AR-GARCH models. Econometric Theory 27, 1236–1278.CrossRefGoogle Scholar
Meyn, S.P. & Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. Springer-Verlag.Google Scholar
Nedeljkovic, M. (2009) Testing for Smooth Transition Nonlinearity in Adjustments of Cointegrating Systems. Warwick Economic Research papers 876, Department of Economics, University of Warwick.Google Scholar
Newey, W.K. & McFadden, D. (1994) Large sample estimation and hypothesis testing. In Engle, R.F. & McFadden, D. (eds.), Handbook of Econometrics, vol. 4, pp. 2111–2245. Elsevier.Google Scholar
Paparoditis, E. & Politis, D.N. (2005) Bootstrap hypothesis testing in regression models. Statistics & Probability Letters 74, 356–365.CrossRefGoogle Scholar
Park, J. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 69, 117–162.CrossRefGoogle Scholar
Pitarakis, J.-Y. (2008) Comment on Threshold autoregression with a unit root. Econometrica 76, 1207–1217.Google Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer-Verlag.CrossRefGoogle Scholar
Rahbek, A., Kongsted, H.C., & Jørgensen, C. (1999) Trend stationarity in the I(2) cointegration model. Journal of Econometrics 90, 265–289.CrossRefGoogle Scholar
Saikkonen, P. (2005) Stability results for nonlinear error correction models. Journal of Econometrics 127, 69–81.CrossRefGoogle Scholar
Saikkonen, P. (2008) Stability of regime switching error correction models under linear cointegration. Econometric Theory 24, 294–318.CrossRefGoogle Scholar
Seo, M.H. (2006) Bootstrap testing for the null of no cointegration in a threshold vector error correction model. Journal of Econometrics 134, 129–150.CrossRefGoogle Scholar
Seo, M.H. (2008) Unit root test in a threshold autoregression: Asymptotic theory and residual-based block bootstrap. Econometric Theory 24, 1699–1716.CrossRefGoogle Scholar
Seo, M.H. (2011) Estimation of nonlinear error correction models. Econometric Theory 27, 201–234.Google Scholar
Shi, X. & Phillips, P.C.B. (2012) Nonlinear cointegrating regression under weak identification. Econometric Theory 28, 509–547.Google Scholar
van der Vaart, W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes. Springer-Verlag.CrossRefGoogle Scholar