Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T17:11:34.141Z Has data issue: false hasContentIssue false
  • English
  • Français

SUP-TESTS FOR LINEARITY IN A GENERAL NONLINEAR AR(1) MODEL

Published online by Cambridge University Press:  04 November 2009

Christian Francq ,
Lajos Horvath  and
Jean-Michel Zakoïan
Christian Francq
Affiliation:
Université Lille III, GREMARS-EQUIPPE
Lajos Horvath
Affiliation:
University of Utah
Jean-Michel Zakoïan*
Affiliation:
Université Lille III, GREMARS-EQUIPPE and CREST
*
*Address correspondence to Jean-Michel Zakoïan, CREST, 15 boulevard Gabriel Péri, 92245 Malakoff Cedex, France; e-mail: zakoian@ensae.fr.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider linearity testing in a general class of nonlinear time series models of order one, involving a nonnegative nuisance parameter that (a) is not identified under the null hypothesis and (b) gives the linear model when equal to zero. This paper studies the asymptotic distribution of the likelihood ratio test and asymptotically equivalent supremum tests. The asymptotic distribution is described as a functional of chi-square processes and is obtained without imposing a positive lower bound for the nuisance parameter. The finite-sample properties of the sup-tests are studied by simulations.

Type
ARTICLES
Information
Econometric Theory , Volume 26 , Issue 4 , August 2010 , pp. 965 - 993
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

Adler, R.J. (1990) An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Institute of Mathematical Statistics Lecture Notes, Monograph Series, 12. Hayward.CrossRefGoogle Scholar
Andrews, D.W.K. (1993) Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856.CrossRefGoogle Scholar
Andrews, D.W.K. & Ploberger, W. (1994) Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62, 1383–1414.Google Scholar
Andrews, D.W.K. & Ploberger, W. (1995) Admissibility of the likelihood ratio test when a nuisance parameter is present only under the alternative. Annals of Statistics 23, 1609–1629.CrossRefGoogle Scholar
Billingsley, P. (1961) The Lindeberg-Lévy theorem for martingales. Proceedings of the American Mathematical Society 12, 788–792.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Chan, K.S. & Tong, H. (1986) On estimating thresholds in autoregressive models. Journal of Time Series Analysis 7, 178–190.Google Scholar
Chesher, A. (1984) Testing for neglected heterogeneity. Econometrica 52, 865–872.CrossRefGoogle Scholar
Davies, R.B. (1977) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64, 247–254.CrossRefGoogle Scholar
Davies, R.B. (1987) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33–43.Google Scholar
Godfrey, L.G. (1988) Misspecification Tests in Econometrics. Cambridge University Press.Google Scholar
Gouriéroux, C. & Monfort, A. (1995) Statistics and Econometric Models, vol. II. Cambridge University Press.Google Scholar
Granger, C.W.J. & Teräsvirta, T. (1993) Modelling Nonlinear Economic Relationships. Oxford University Press.CrossRefGoogle Scholar
Haggan, V. & Ozaki, T. (1981) Modelling nonlinear random vibrations using an amplitude-dependent autoregressive time series model. Biometrika 68, 189–196.CrossRefGoogle Scholar
Hansen, B. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.CrossRefGoogle Scholar
King, M. & Shively, T.S. (1993) Locally optimal testing when a nuisance parameter is present only under the alternative. Review of Economics and Statistics 75, 1–7.CrossRefGoogle Scholar
Lee, L.-F. & Chesher, A.. (1986) Specification testing when score test statistics are identically zero. Journal of Econometrics 31, 121–149.Google Scholar
Luukkonen, R., Saikkonen, P., & Teräsvirta, T. (1988) Testing linearity against smooth transition autoregressive models. Biometrika 75, 491–499.CrossRefGoogle Scholar
Pötscher, B.M. & Prucha, I.R. (1989) A uniform law of large numbers for dependent and heterogeneous data processes. Econometrica 57, 675–683.CrossRefGoogle Scholar
Rotnitzky, A., Cox, D., Bottai, M., & Robins, J. (2000) Likelihood-based inference with singular information matrix. Bernoulli 6, 243–284.CrossRefGoogle Scholar
Stinchcombe, M.B. & White, H. (1998) Consistent specification testing with nuisance parameters present only under the alternative. Econometric Theory 14, 295–325.CrossRefGoogle Scholar
Stock, J. & Watson, M.W. (1999) A comparison of linear and nonlinear univariate models for forecasting macroeconomic times 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
Teräsvirta, T., van Dijk, D., & Medeiros, M.C. (2004) Linear models, smooth transition autoregressions, and neural networks for forecasting macroeconomic time series: A re-examination. International Journal of Forecasting 21, 755–774.Google Scholar
Tjøstheim, D. (1990) Non-linear time series and Markov chains. Advances in Applied Probability 22, 587–611.CrossRefGoogle Scholar
Tong, H. (1990) Non-linear Time Series. A Dynamical System Approach. Oxford University Press.CrossRefGoogle Scholar
Tong, H. & Lim, K.S. (1980) Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society, Series B 42, 245–292.Google Scholar