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TESTS FOR NONLINEAR COINTEGRATION

Published online by Cambridge University Press:  07 October 2009

In Choi  and
Pentti Saikkonen
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Abstract

This paper develops tests for the null hypothesis of cointegration in the nonlinear regression model with I(1) variables. The test statistics we use in this paper are Kwiatkowski, Phillips, Schmidt, and Shin’s (1992; KPSS hereafter) tests for the null of stationarity, though using other kinds of tests is also possible. The tests are shown to depend on the limiting distributions of the estimators and parameters of the nonlinear model when they use full-sample residuals from the nonlinear least squares and nonlinear leads-and-lags regressions. This feature makes it difficult to use them in practice. As a remedy, this paper develops tests using subsamples of the regression residuals. For these tests, first, the nonlinear least squares and nonlinear leads-and-lags regressions are run and residuals are calculated. Second, the KPSS tests are applied using subresiduals of size b. As long as b/T → 0 as T → ∞, where T is the sample size, the tests using the subresiduals have limiting distributions that are not affected by the limiting distributions of the full-sample estimators and the parameters of the model. Third, the Bonferroni procedure is used for a selected number of the subresidual-based tests. Monte Carlo simulation shows that the tests work reasonably well in finite samples for polynomial and smooth transition regression models when the block size is chosen by the minimum volatility rule. In particular, the subresidual-based tests using the leads-and-lags regression residuals appear to be promising for empirical work. An empirical example studying the U.S. money demand equation illustrates the use of the tests.

Type
Research Article
Information
Econometric Theory , Volume 26 , Issue 3 , June 2010 , pp. 682 - 709
Copyright
Copyright © Cambridge University Press 2009

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Footnotes

Choi acknowledges financial support from the RGC Competitive Earmarked Research Grant 2003–2004 under Project No. HKUST6223/03H. Saikkonen thanks the Research Unit of Economic Structures and Growth (RUESG) in the University of Helsinki and the Yrjö Jahnsson Foundation for financial support. The authors are grateful to Bruce Hansen, Peter Phillips and two anonymous referees for their comments on this paper.

References

REFERENCES

Abadir, K. (1990) The Limiting Distribution of a Functional of a Wiener Process. Mimeo, The American University of Cairo.Google Scholar
Anderson, T.W. & Darling, D.A. (1952) Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes. Annals of Mathematical Statistics 23, 193–212.CrossRefGoogle Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.10.2307/2938229CrossRefGoogle Scholar
Andrews, D.W.K. & McDermott, C.J. (1995) Nonlinear econometric models with deterministically trending variables. Review of Economic Studies 62, 343–360.CrossRefGoogle Scholar
Balke, N.S. & Fomby, T.B. (1997) Threshold cointegration. International Economic Review 38, 627–645.CrossRefGoogle Scholar
Bec, F. & Rahbek, A. (2004) Vector equilibrium correction models with non-linear discontinuous adjustments. Econometrics Journal 7, 628–651.CrossRefGoogle Scholar
Caner, M. & Kilian, L. (2001) Size distortions of the null hypothesis of stationarity: Evidence and implications for the PPP debate. Journal of International Money and Finance 20, 639–657.10.1016/S0261-5606(01)00011-0CrossRefGoogle Scholar
Chang, Y. & Park, J.Y. (2003) Index models with integrated time series. Journal of Econometrics 114, 73–106.10.1016/S0304-4076(02)00220-8CrossRefGoogle Scholar
Chang, Y., Park, J.Y., & Phillips, P.C.B. (2001) Nonlinear econometric models with cointegrated and deterministically trending regressors. Econometrics Journal 4, 1–36.CrossRefGoogle Scholar
Choi, I. (2005) Subsampling vector autoregressive tests of linear constraints. Journal of Econometrics 124, 55–89.CrossRefGoogle Scholar
Choi, I. & Ahn, B.C. (1995) Testing for cointegration in a system of equations. Econometric Theory 11, 952–983.CrossRefGoogle Scholar
Choi, I. & Saikkonen, P. (2004) Testing linearity in cointegrating smooth transition regressions. Econometrics Journal 7, 341–365.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.CrossRefGoogle Scholar
Engle, R. & Granger, C.W.J. (1987) Cointegration and error correction: Representation, estimation and testing. Econometrica 55, 251–276.CrossRefGoogle Scholar
Evans, G.B.A. & Savin, E. (1981) Testing for unit roots: 1. Econometrica 49, 753–779.10.2307/1911521CrossRefGoogle Scholar
Granger, C.W.J. (1981) Some properties of time series data and their use in econometric model specification. Journal of Econometrics 16, 121–130.CrossRefGoogle Scholar
Hansen, B.E. (1992) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.CrossRefGoogle Scholar
Hansen, B.E. & Seo, B. (2002) Testing for two-regime threshold cointegration in vector error correction models. Journal of Econometrics 110, 293–318.10.1016/S0304-4076(02)00097-0CrossRefGoogle Scholar
Hoffman, D.L. & Rasche, R.H. (1991) Long-run income and interest elasticities of money demand in the United States. Review of Economics and Statistics 78, 665–674.CrossRefGoogle Scholar
Hoffman, D.L., Rasche, R.H., & Tieslau, M.A. (1995) The stability of long-run money demand in five industrial countries. Journal of Monetary Economics 35, 317–339.CrossRefGoogle Scholar
Hong, S.H. & Phillips, P.C.B. (2004) Testing Linearity in Cointegrating Relations with an Application to PPP. Cowles Foundation Discussion Paper 1541, Yale University.Google Scholar
Jin, S. (2004) Discrete Choice Modeling with Nonstationary Panels Applied to Exchange Rate Regime Choice. Mimeo, Yale University.Google Scholar
Kasparis, I. (2008) Detection of functional form misspecification in cointegrating relations. Econometric Theory 24, 1373–1403.CrossRefGoogle Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992) 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? Journal of Econometrics 54, 159–178.CrossRefGoogle Scholar
Lütkepohl, H., Teräsvirta, T., & Wolters, J. (1999) Investigating stability and linearity of a German M1 demand function. Journal of Applied Econometrics 14, 511–525.10.1002/(SICI)1099-1255(199909/10)14:5<511::AID-JAE529>3.0.CO;2-C3.0.CO;2-C>CrossRefGoogle Scholar
Neftci, S.A. (1984) Are economic time series asymmetric over the business cycle? Journal of Political Economy 92, 307–328.CrossRefGoogle Scholar
Park, J. & Phillips, P.C.B. (1999) Asymptotics for nonlinear transformations of time series. Econometric Theory 15, 269–298.CrossRefGoogle Scholar
Park, J. & Phillips, P.C.B. (2001) Nonlinear regressions with integrated time series. Econometrica 69, 117–161.CrossRefGoogle Scholar
Patterson, K. (2000) An Introduction to Applied Econometrics: A Time Series Approach. MacMillan.Google Scholar
Politis, D.N., Romano, J.P., & Wolf, M. (1999) Subsampling. Springer-Verlag.CrossRefGoogle Scholar
Pötscher, B. & Prucha, I.R. (1997) Dynamic Nonlinear Econometric Models. Springer.CrossRefGoogle Scholar
Romano, J.P. & Wolf, M. (2001) Subsampling intervals in autoregressive models with linear time trend. Econometrica 69, 1283–1314.CrossRefGoogle Scholar
Saikkonen, P. (1991) Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21.CrossRefGoogle Scholar
Saikkonen, P. (2008) Stability of regime switching error correction models under linear cointegration. Econometric Theory 24, 294–318.CrossRefGoogle Scholar
Saikkonen, P. & Choi, I. (2004) Cointegrating smooth transition regressions. Econometric Theory 20, 301–340.CrossRefGoogle Scholar
Shin, Y. (1994) A residual-based test of the null of cointegration against the alternative of no cointegration. Econometric Theory 10, 91–115.CrossRefGoogle Scholar
Stock, J.H. & Watson, M.W. (1993) A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61, 783–820.CrossRefGoogle Scholar
White, J. (1958) The limiting distribution of the serial correlation in the explosive case. Annals of Mathematical Statistics 29, 1188–1197.10.1214/aoms/1177706450CrossRefGoogle Scholar