Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T19:41:26.642Z Has data issue: false hasContentIssue false
  • English
  • Français

COINTEGRATING SMOOTH TRANSITION REGRESSIONS

Published online by Cambridge University Press:  10 February 2004

Pentti Saikkonen  and
In Choi
Pentti Saikkonen
Affiliation:
University of Helsinki
In Choi
Affiliation:
Hong Kong University of Science and Technology and Cowles Foundation, Yale University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the smooth transition regression model where regressors are I(1) and errors are I(0). The regressors and errors are assumed to be dependent both serially and contemporaneously. Using the triangular array asymptotics, the nonlinear least squares estimator is shown to be consistent, and its asymptotic distribution is derived. It is found that the asymptotic distribution involves a bias under the regressor-error dependence, which implies that the nonlinear least squares estimator is inefficient and unsuitable for use in hypothesis testing. Thus, this paper proposes a Gauss–Newton type estimator that uses the nonlinear least squares estimator as an initial estimator and is based on regressions augmented by leads and lags. Using leads and lags enables the Gauss–Newton estimator to eliminate the bias and have a mixture normal distribution in the limit, which makes it more efficient than the nonlinear least squares estimator and suitable for use in hypothesis testing. Simulation results indicate that the results obtained from the triangular array asymptotics provide reasonable approximations for the finite-sample properties of the estimators and t-tests when sample sizes are moderately large. The cointegrating smooth transition regression model is applied to the Korean and Indonesian data from the Asian currency crisis of 1997. The estimation results partially support the interest Laffer curve hypothesis. But overall the effects of interest rate on spot exchange rate are shown to be quite negligible in both nations.This paper was partly written while the first author was visiting the Institute of Statistics and Econometrics at Humboldt University, Berlin. This author acknowledges financial support from the Alexander von Humboldt Foundation under a Humboldt Research Award and from the Yrjö Jahnsson Foundation. The second author wrote this paper while visiting the Cowles Foundation for Research in Economics, Yale University. This author thanks the faculty and staff of the Cowles Foundation, especially Don Andrews, John Geanakoplos, David Pearce, Peter Phillips, and Nora Wiedenbach, for their support and hospitality. The second author was financially supported for the research in this paper by Kookmin University. The authors thank Don Andrews, Helmut Lütkepohl, Peter Phillips, Bruce Hansen, and two referees for their valuable comments on this paper. Part of the data studied in this paper was provided by Chi-Young Song, whom we thank.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 2 , April 2004 , pp. 301 - 340
Copyright
© 2004 Cambridge University Press

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. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.Google Scholar
Andrews, D.W.K. & C.J. McDermott (1995) Nonlinear econometric models with deterministically trending variables. Review of Economic Studies 62, 343–360.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.
Brillinger, D.R. (1975) Time Series: Data Analysis and Theory. Holt, Rinehart, and Winston.
Chang, Y. & J.Y. Park (1998) Nonlinear Index Models. Mimeo, Rice University.
Chang, Y., J.Y. Park, & P.C.B. Phillips (1999) Nonlinear Econometric Models with Cointegrated and Deterministically Trending Regressors. Mimeo, Rice University.
Choi, I. (1994) Residual based tests for the null of stationarity with applications to U.S. macroeconomic time series. Econometric Theory 19, 720–746.Google Scholar
Choi, I. & D.K. Park (2000) Causal Relation between Interest and Exchange Rates in the Asian Currency Crisis. Mimeo, University of Southern California.
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.
Dekle, R., C. Hsiao, & S. Wang (1999) Interest Rate Stabilization of Exchange Rates and Contagion in the Asian Crisis Countries. Mimeo, University of Southern California.
Elliott, G., T.J. Rothenberg, & J.H. Stock. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.Google Scholar
Feldstein, M. (1998) Refocusing the IMF. Foreign Affairs 77, 20–33.Google Scholar
Franses, P.H. & D. van Dijk (2000) Nonlinear Time Series Models in Empirical Finance. Cambridge University Press.
Furman, J. & J.E. Stiglitz (1998) Economic crises: Evidence and insights from East Asia. Brookings Papers on Economic Activity 2, 1–135.Google Scholar
Ghosh, A. & S. Phillips (1998) Interest Rates, Stock Market Prices, and Exchange Rates in East Asia. Mimeo, International Monetary Fund.
Goldfajn, I. & T. Baig (1998) Monetary Policy in the Aftermath of Currency Crises: The Case of Asia. IMF Working paper 98-170.
Goldfeld, S.M. & R. Quandt (1973) The estimation of structural shifts by switching regressions. Annals of Economic and Social Measurement 2, 475–485.Google Scholar
Granger, C.W.J. & T. Teräsvirta (1993) Modelling Non-linear Economic Relationships. Oxford University Press.
Granger, C.W.J., T. Teräsvirta, & H. Anderson (1993) Modelling non-linearity over the business cycle. In J.H. Stock and M.W. Watson (eds.), New Research on Business Cycles, Indicators and Forecasting, pp. 311–325. University of Chicago Press.
Hansen, B.E. (1992) Convergence to stochastic integrals for dependent heterogeneous processes. Econometric Theory 8, 489–500.Google Scholar
Jansen, E.S. & T. Teräsvirta (1996) Testing parameter constancy and super exogeneity in econometric equations. Oxford Bulletin of Economics and Statistics 58, 735–763.Google Scholar
Kaminsky, G. & S. Schmukler (1998) The Relationship between Interest Rates and Exchange Rates in Six Asian Countries. Mimeo, World Bank.
Kraay, A. (1998) Do High Interest Rates Defend Currencies During Speculative Attacks? Mimeo, World Bank.
Lütkepohl, H., T. Teräsvirta, & J. Wolters (1999) Investigating stability and linearity of a German M1 demand function. Journal of Applied Econometrics 14, 511–525.Google Scholar
Luukkonen, R., P. Saikkonen, & T. Teräsvirta (1988) Testing linearity against smooth transition autoregressive models. Biometrika 75, 491–499.Google Scholar
Michael, P., A.R. Nobay, & D.A. Peel (1997) Transaction costs and nonlinear adjustment in real exchange rates: An empirical investigation. Journal of Political Economy 105, 862–879.Google Scholar
Newey, W. & D.L. McFadden (1994) Large sample estimation and hypothesis testing. In R.F. Engle and D.L. McFadden (eds.), Handbook of Econometrics, vol. 4, pp. 2113–2445. Elsevier.
Park, J. & P.C.B. Phillips (1999) Asymptotics for nonlinear transformations of time series. Econometric Theory 15, 269–298.Google Scholar
Park, J. & P.C.B. Phillips (2001) Nonlinear regressions with integrated time series. Econometrica 69, 117–161.Google Scholar
Park, Y.C., Y. Wang, & C.-S. Chung (1999) Exchange Rate Policies in Korea: Has Exchange Rate Volatility Increased after the Crisis? Mimeo, Korea Institute for Economic Policy.
Phillips, P.C.B. & M. Loretan (1991) Estimating long run economic equilibria. Review of Economic Studies 58, 407–436.Google Scholar
Phillips, P.C.B. & V. Solo (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.Google Scholar
Pötscher, B. & I.R. Prucha (1997) Dynamic Nonlinear Econometric Models. Springer-Verlag.
Radelet, S. & J. Sachs (1998a) The East Asian financial crisis: Diagnosis, remedies, prospects. Brookings Papers on Economic Activity, 1–74.
Radelet, S. & J. Sachs (1998b) The Onset of the East Asian Financial Crises. Mimeo, Harvard Institute for International Development.
Saikkonen, P. (1991) Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21.Google Scholar
Saikkonen, P. (1993) Continuous weak convergence and stochastic equicontinuity results for integrated processes with an application to the estimation of a regression model. Econometric Theory 9, 155–188.Google Scholar
Saikkonen, P. (2001) Consistent estimation in cointegrated vector autoregressive processes with nonlinear time trends in cointegrating relations. Econometric Theory 17, 296–326.Google Scholar
Saikkonen, P. & H. Lütkepohl (1996) Infinite order cointegrated vector autoregressive processes: Estimation and inference. Econometric Theory 12, 814–844.Google Scholar
Sarno, L. (1999) Adjustment costs and nonlinear dynamics in the demand for money: Italy, 1861–1991. International Journal of Finance and Economics 4, 155–177.Google Scholar
Stock, J.H. & M.W. Watson (1993) A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61, 783–820.Google Scholar
Stout, W.F. (1974) Almost Sure Convergence. Academic Press.
Taylor, M.P., D.A. Peel, & L. Sarno (2001) Nonlinear mean-reversion in exchange rates: Towards a solution to the purchasing power parity puzzle. International Economic Review 42, 1015–1042.Google Scholar
Teräsvirta, T. (1998) Modeling economic relationships with smooth transition regressions. In A. Ullah and D.E.A. Giles (eds.), Handbook of Applied Economic Statistics, pp. 507–552. Dekker.
Teräsvirta, T. & H.M. Anderson (1992) Characterizing nonlinearities in business cycles using smooth transition autoregressive models. Journal of Applied Econometrics 7, S119–S136.Google Scholar
Tong, H. (1983) Threshold Models in Non-Linear Time Series Analysis. Springer-Verlag.
van Dijk, D., T. Teräsvirta, & P.H. Franses (2002) Smooth transition autoregressive models: A survey of recent developments. Econometric Review 21, 1–47.Google Scholar
White, H. (1984) Asymptotic Theory for Econometricians. Academic Press.