Testing for Nonlinear Adjustment in Smooth Transition Vector Error Correction Models
The smooth transition autoregressive (STAR) model was proposed by Chan and Tong (1986) as a generalization of the threshold autoregressive (TAR) model, and since then it has attracted wide attention in the recent literature on the business cycles and the equilibrium parity relationships of commodity prices, exchange rates, and equity prices. Economic behavior is affected by asymmetric transaction costs and institutional rigidities, and thus a large number of studies - for example, Neftci (1984), Terasvirta and Anderson (1992), and Michael, Nobay, and Peel (1997) - have shown that many economic variables and relations display asymmetry and nonlinear adjustment. One of the most crucial issues in models of this kind is testing for the presence of nonlinear adjustment with the null of linearity. Luukkonen, Saikkonen, and Terasvirta (1988) expanded the transition function and proposed the variable addition tests as the tests of linearity against smooth transition nonlinearity, and the tests have been used in many empirical studies. However, the test statistics are based on the polynomial approximation, and the approximation errors may affect statistical inference depending on the parameter values of transition rate and location. Furthermore, the tests are not directly related to the smooth transition model, and thus we cannot retrace what causes the rejection of linearity. This paper considers the direct tests for nonlinear adjustment, which are based on the exact specification of smooth transition. The smooth transition model entails transition parameters, which cannot be identified under the null hypothesis. However, the optimality issue in the smooth transition model has not been treated extensively. The optimality issue regarding unidentified parameters has been developed by Davies (1987), Andrews (1993), and Hansen (1996). Hansen (1996) particularly considered the optimality issue in threshold models. The threshold parameter cannot be identified under the null hypothesis, and as a result the likelihood ratio statistic has the nonstandard distribution. The smooth transition model generalizes the threshold model, and thus this paper develops the appropriate tests and the associated distribution theory based on the optimality argument. Many empirical studies have found evidence on the presence of stochastic nonlinear dependence in equilibrium relations such as purchasing power parity. For example, Michael, Nobay, and Peel (1997), considering the equilibrium model of real exchange rate in the presence of transaction costs, found strong evidence of nonlinear adjustment, which conforms to the exponential smooth transition model. There exists a huge literature, and it is growing in this area. However, the econometric methods and the formal theory have been limited. This paper proposes the tests for nonlinear adjustment in the smooth transition vector error correction models, and thereby fills the deficiency in the literature. One technical difficulty is to estimate the smooth transition model. As noted by Haggan and Ozaki (1981) and Terasvirta (1994), it is difficult to estimate the smooth transition parameters jointly with the other slope parameters. The gradient of the transition parameter forces its estimate to blow up to infinity; thus, we cannot depend on the standard estimation algorithm. Our tests are based on the Lagrange multiplier statistic, which can be calculated under the null hypothesis. Therefore, our tests are easy to implement and thus useful. This paper finds that our tests have the asymptotic distribution, which is based on the Gaussian process. However, the asymptotic distribution depends on the nuisance parameters and the covariances are data-dependent; thus, the tabulation of asymptotic distribution is not feasible. This paper suggests the bootstrap inference to approximate the sampling distribution of the test statistics. Simulation evidence shows that the bootstrap inference generates moderate size and power performances
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- Seo, Byeongseon, 1998. "Tests For Structural Change In Cointegrated Systems," Econometric Theory, Cambridge University Press, vol. 14(02), pages 222-259, April.
- Robert B. Davies, 2002. "Hypothesis testing when a nuisance parameter is present only under the alternative: Linear model case," Biometrika, Biometrika Trust, vol. 89(2), pages 484-489, June.
- Enders, Walter & Granger, Clive W J, 1998.
"Unit-Root Tests and Asymmetric Adjustment with an Example Using the Term Structure of Interest Rates,"
Journal of Business & Economic Statistics,
American Statistical Association, vol. 16(3), pages 304-11, July.
- Enders, Walter & Granger, C. W. J., 1998. "Unit Root Tests and Asymmetric Adjustment with an Example Using the Term Structure of Interest Rates," Staff General Research Papers 1388, Iowa State University, Department of Economics.
- Andrews, Donald W K & Ploberger, Werner, 1994.
"Optimal Tests When a Nuisance Parameter Is Present Only under the Alternative,"
Econometric Society, vol. 62(6), pages 1383-1414, November.
- Tom Doan, . "REGHBREAK: RATS procedure to perform structural break test with bootstrapped p-values," Statistical Software Components RTS00176, Boston College Department of Economics.
- Tom Doan, . "APGRADIENTTEST: RATS procedure to perform Andrews-Ploberger Structural Break Test for GARCH/Maximum Likelihood," Statistical Software Components RTS00007, Boston College Department of Economics.
- Donald W.K. Andrews & Werner Ploberger, 1992. "Optimal Tests When a Nuisance Parameter Is Present Only Under the Alternative," Cowles Foundation Discussion Papers 1015, Cowles Foundation for Research in Economics, Yale University.
- Tom Doan, . "APBREAKTEST: RATS procedure to implement Andrews-Ploberger Structural Break Test," Statistical Software Components RTS00006, Boston College Department of Economics.
- Bruce E. Hansen & Mehmet Caner, 1997.
"Threshold Autoregressions with a Unit Root,"
Boston College Working Papers in Economics
381, Boston College Department of Economics.
- Dirk Te Velde, 2001. "Balance of payments prospects in EMU," NIESR Discussion Papers 178, National Institute of Economic and Social Research.
- Dahl, Christian M. & Gonzalez-Rivera, Gloria, 2003. "Testing for neglected nonlinearity in regression models based on the theory of random fields," Journal of Econometrics, Elsevier, vol. 114(1), pages 141-164, May.
- Forbes, C.S. & Kalb, G.R.J. & Kofman, P., 1997.
"Bayesian Arbitrage Threshold Analysis,"
Monash Econometrics and Business Statistics Working Papers
3/97, Monash University, Department of Econometrics and Business Statistics.
- Donald W.K. Andrews, 1990.
"Tests for Parameter Instability and Structural Change with Unknown Change Point,"
Cowles Foundation Discussion Papers
943, Cowles Foundation for Research in Economics, Yale University.
- Andrews, Donald W K, 1993. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Econometrica, Econometric Society, vol. 61(4), pages 821-56, July.
- Enders, Walter & Siklos, Pierre L, 2001.
"Cointegration and Threshold Adjustment,"
Journal of Business & Economic Statistics,
American Statistical Association, vol. 19(2), pages 166-76, April.
- Tom Doan, . "RATS programs to replicate Enders-Siklos(2001) JBES paper on threshold cointegration," Statistical Software Components RTZ00053, Boston College Department of Economics.
- Tom Doan, . "ENDERSIKLOS: RATS procedure to perform Enders-Siklos test for cointegration with threshold effect," Statistical Software Components RTS00064, Boston College Department of Economics.
- Chang, Yoosoon & Park, Joon Y., 2003. "Index models with integrated time series," Journal of Econometrics, Elsevier, vol. 114(1), pages 73-106, May.
- Michael, Panos & Nobay, A Robert & Peel, David A, 1997.
"Transactions Costs and Nonlinear Adjustment in Real Exchange Rates: An Empirical Investigation,"
Journal of Political Economy,
University of Chicago Press, vol. 105(4), pages 862-79, August.
- Tom Doan, . "RATS programs to replicate Michael-Nobay-Peel ESTAR models," Statistical Software Components RTZ00113, Boston College Department of Economics.
- Neftci, Salih N, 1984. "Are Economic Time Series Asymmetric over the Business Cycle?," Journal of Political Economy, University of Chicago Press, vol. 92(2), pages 307-28, April.
- White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-38, May.
- Bruce E. Hansen, 1996.
"Sample Splitting and Threshold Estimation,"
Boston College Working Papers in Economics
319., Boston College Department of Economics, revised 12 May 1998.
- Hansen, B.E., 1991.
"Inference when a Nuisance Parameter is Not Identified Under the Null Hypothesis,"
RCER Working Papers
296, University of Rochester - Center for Economic Research (RCER).
- Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, vol. 64(2), pages 413-30, March.
- Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
- Engle, Robert F & Granger, Clive W J, 1987.
"Co-integration and Error Correction: Representation, Estimation, and Testing,"
Econometric Society, vol. 55(2), pages 251-76, March.
- Engle, Robert & Granger, Clive, 2015. "Co-integration and error correction: Representation, estimation, and testing," Applied Econometrics, Publishing House "SINERGIA PRESS", vol. 39(3), pages 106-135.
- Balke, Nathan S & Fomby, Thomas B, 1997.
International Economic Review,
Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 38(3), pages 627-45, August.
- Nathan S. Balke & Thomas B. Fomby, 1992. "Threshold cointegration," Research Paper 9209, Federal Reserve Bank of Dallas.
- Tom Doan, . "RATS programs to replicate Balke-Fomby threshold cointegration," Statistical Software Components RTZ00010, Boston College Department of Economics.
- Hansen, Bruce E. & Seo, Byeongseon, 2002.
"Testing for two-regime threshold cointegration in vector error-correction models,"
Journal of Econometrics,
Elsevier, vol. 110(2), pages 293-318, October.
- Tom Doan, . "RATS programs to replicate Hansen/Seo paper on threshold cointegration," Statistical Software Components RTZ00092, Boston College Department of Economics.
- Terasvirta, T & Anderson, H M, 1992. "Characterizing Nonlinearities in Business Cycles Using Smooth Transition Autoregressive Models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 7(S), pages S119-36, Suppl. De.
- Kapetanios, George & Shin, Yongcheol & Snell, Andy, 2003. "Testing for a unit root in the nonlinear STAR framework," Journal of Econometrics, Elsevier, vol. 112(2), pages 359-379, February.
- Anderson, Heather M, 1997. "Transaction Costs and Non-linear Adjustment towards Equilibrium in the US Treasury Bill Market," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 59(4), pages 465-84, November.
- Tse, Yiuman, 2001. "Index arbitrage with heterogeneous investors: A smooth transition error correction analysis," Journal of Banking & Finance, Elsevier, vol. 25(10), pages 1829-1855, October.
- Saikkonen, Pentti & Choi, In, 2004. "Cointegrating Smooth Transition Regressions," Econometric Theory, Cambridge University Press, vol. 20(02), pages 301-340, April.
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