Abstract
The possibility that a series can contain an unknown number of smooth breaks raises two distinct problems. First, even if the breaks are sharp, the number of breaks and the break dates themselves are generally unknown and need to be estimated along with the other parameters of the model. Second, even if the number of breaks is known, the possibility of a smooth break means that the functional form of the break is unknown to the researcher. A misspecification of the functional form of the breaks may be as problematic as ignoring the breaks altogether. Moreover, even if a series contains no breaks, it may be subject to other nonlinearities or parameter instabilities. We summarize a number of papers that use a variant of Gallant’s (J Economet 15:211−245, 1981) Flexible Fourier Form to control for the unknown number and form of the breaks. This chapter details and illustrates several unit root tests, stationarity tests, and tests for parameter instability that are based on a Fourier approximation.
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Notes
- 1.
The abbreviations stand for exponential smooth transition autoregressive, logistic smooth transition autoregressive, artificial neural network, generalized autoregressive, and multiple adaptive regressive splines, respectively.
- 2.
Note that these breaks were initially examined and originally analyzed in Clements and Hendry (1999).
- 3.
References
Amsler, C. and J. Lee (1995), An LM test for a unit root in the presence of a structural change, Econometric Theory 11, 359 −368.
Andrews DWK, Lee I., and W. Ploeberger (1996), Optimalchangepoint test for normal linear regression. Journal of Econometrics 70, 9 – 38.
Bai, J. and P. Perron (1998), Estimating and testing linear models with multiple structural changes. Econometrica 66, 47 – 78.
Bai, J. and P. Perron (2003), Computation and analysis of multiple structural change models. Journal of Applied Econometrics 18, 1 – 22.
Basu, S., J. Fernald, and M. Shapiro, (2001). Productivity growth in the 1990s: technology, utilization, or adjustment? Carnegie-Rochester Conference Series on Public Policy, 55(1), 117 − 165.
Becker, R., W. Enders, and J. Lee (2006), A stationary test with an unknown number of smooth breaks. Journal of Time Series Analysis 27,381 – 409.
Becker, R., W. Enders, and S. Hurn (2004), A general test for time dependence in parameters, Journal of Applied Econometrics19, 899-906.
Becker, R., W. Enders, and S. Hurn (2006), Modeling Inflation and Money Demand Using a Fourier Series Approximation. In C. Milas, P. Rothman, and D. van Dijk, eds. Nonlinear Time Series Analysis of Business Cycles. (Elvesier: Amsterdam) 2006. pp. 221–44.
Clements, M. and Hendry D (1999), On winning forecasting competitions. Spanish Economic Review 1, 123 – 160.
Davies, R. B. (1987), Hypothesis testing when a nuisance parameter is only identified under the alternative, Biometrika 47, 33–43.
Enders. W. and M. Holt (2012). Sharp Breaks or Smooth Shifts? An Investigation of the Evolution of Commodity Prices, American Journal of Agricultural Economics, 94, 659–673.
Enders, W. and J. Lee (2012a), A unit root test using a Fourier series to approximate smooth breaks. Oxford Bulletin of Economics and Statistics. 74, 574–599.
Enders, W. and J. Lee (2012b), The flexible Fourier form and Dickey-Fuller type unit root tests. Economics Letters 117, 196–199.
Gallant, A. R. (1981), On the bias in flexible functional forms and an essentially unbiased form: the flexible Fourier form, Journal of Econometrics 15, 211 −245.
Gallant, R. and G. Souza (1991), On the asymptotic normality of Fourier flexible form estimates, Journal of Econometrics 50, 329 − 353.
Kapetanios, G., Y. Shin and A. Snell (2003), Testing for a unit root in the nonlinear STAR framework, Journal of Econometrics 112, 359 − 379.
Kwiatkowski, D., P. Phillips, P. Schmidt, and Y. Shin (1992). Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root. Journal of Econometrics 54, 159 − 178.
Lee, J., and M. Strazicich, (2003), Minimum LM unit root tests with two structural breaks, Review of Economics and Statistics 85, 1082−1089.
Lee T., H. White and C. Granger (1993), Testing for neglected nonlinearity in time series models: a comparison of neutral network methods and alternative test. Journal of Econometrics 56, 269–290.
Leybourne, S., P. Newbold and D. Vougas (1998), Unit roots and smooth transitions, Journal of Time Series Analysis 19, 83−97.
Ludlow, J. and W. Enders (2000), Estimating Non-Linear ARMA Models Using Fourier Coefficients.. International Journal of Forecasting, 16. pp. 333–47.
Nyblom J. (1989), Testing for the constancy of parameters over time. Journal of the American Statistical Association 84, 223 – 230.
Patterson, D. M. and R. A. Ashley. (2000), A Nonlinear Time Series Workshop: A Toolkit for Detecting and Identifying Nonlinear Time Series Dependence. (Kluwer Academic Publishers: Norwell, Massachusetts).
Identifying Nonlinear Time Series Dependence. Kluwer Academic Publishers: Norwell, Massachusetts.
Perron, P. (1989), The great crash, the oil price shock, and the unit root hypothesis, Econometrica 57, 1361 − 1401.
Perron, P. (1997), Further evidence on breaking trend functions in macroeconomic variables, Journal of Econometrics,80, 355−385.
Terasvirta, T., D. Tjostheim and C.W.J. Granger, (2010). Modelling Nonlinear Economic Time Series. (Oxford University Press: New York).
Watson, M. and R. Engle (1985), Testing for regression coefficient stability with a stationary AR(1) alternative. Review of Economics and Statistics 67, 341 – 346.
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Jones, P.M., Enders, W. (2014). On the Use of the Flexible Fourier Form in Unit Root Tests, Endogenous Breaks, and Parameter Instability. In: Ma, J., Wohar, M. (eds) Recent Advances in Estimating Nonlinear Models. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8060-0_4
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