Nonparametric pseudo-Lagrange multiplier stationarity testing

Landajo, Manuel and Presno, María José (2010): Nonparametric pseudo-Lagrange multiplier stationarity testing.

Abstract

The framework of stationarity testing is extended to allow a generic smooth trend function estimated nonparametrically. The asymptotic behavior of the pseudo-Lagrange Multiplier test is analyzed in this setting. The proposed implementation delivers a consistent test whose limiting null distribution is standard normal. Theoretical analyses are complemented with simulation studies and some empirical applications.

• Item Type: MPRA Paper
• Language: English
• Keywords: Time series, stationarity testing, limiting distribution, nonparametric regression, nonparametric hypothesis testing
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Amsler, C., Schmidt, P. and Vogelsang, T.J., 2009. The KPSS test using fixed-b critical values: size and power in highly autocorrelated time series. Journal of Time Series Econometrics 1(1), Article 5. doi 10.2202/1941-1928 1027.

Anderson, T.W., 1971. The Statistical Analysis of Time Series. John Wiley & Sons, New York.

Becker, R., Enders, W. and Lee, J., 2006. A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis 27, 381-409.

Bhansali, R.J., Giraitis, L. and Kokoszka, P.S., 2007. Convergence of quadratic forms with nonvanishing diagonal. Statistics & Probability Letters 77, 726-734.

Bierens, H.J., 1997. Testing the unit root with drift hypothesis against nonlinear trend stationarity, with an application to the US price level and interest rate. Journal of Econometrics 81, 29-64.

Billingsley, P., 1968. Convergence of Probability Measures. John Wiley & Sons, New York.

Brooks, C., 2008. Introductory Econometrics for Finance. Cambridge University Press, Cambridge.

Busetti, F. and Harvey, A.C., 2001. Testing for the presence of a random walk in series with structural breaks. Journal of Time Series Analysis 22, 127-150. 639-657.

Caner, M. and Hansen, B.E., 2001. Threshold autoregressions with a unit root. Econometrica 69, 1555-1596.

DiCecio, R., Engemann, K.M., Owyang, M.T. and Wheeler, C.H., 2008. Changing trends in the labor force: a survey. Federal Reserve Bank of St. Louis Review 90, 47-62.

Diebold, F.X. and Kilian, L., 2000. Unit-root tests are useful for selecting forecasting models. Journal of Business and Economics Statistics 18, 265-273.

Dore, M.H.I. and Johnston, M., 2000. The carbon cycle and the value of forests as a carbon sink: a boreal case study. In Sustainable Forest Management and Global Climate Change: Selected Case Studies from the Americas. Edward Elgar Publishing, Cheltenham, UK.

Enders, W. and Granger, C.W.J., 1998. Unit-root tests and asymmetric adjustment with an example using the term structure of interest rates. Journal of Business and Economic Statistics 16, 304-311.

Enders, W. and Lee, J., 2004. Testing for a unit root with a nonlinear Fourier function. Mimeo, University of Alabama. Escanciano, J.C., 2006. Goodness-of-fit tests for linear and nonlinear time series models. Journal of the American Statistical Association 101, 531-541.

Gallant, A.R., 1987. Nonlinear Statistical Models. John Wiley & Sons, New York.

Gay-Garcia, C., Estrada, F. and Sánchez, A., 2009. Global and hemispheric temperatures revised. Climatic Change 94, 333-349.

Van Gelder, P.H.A.J.M., Wang, W. and Vrijling, J.K., 2007. Statistical estimation methods for extreme hydrological events. In Extreme Hydrological Events: New Concepts for Security. Springer Netherlands.

Grenander, U., 1981. Abstract inference. John Wiley & Sons, New York.

Gustavsson, M. and Österholm, P., 2006. The informational value of unemployment statistics: a note on the time series properties of participation rates. Economics Letters 92, 428-433.

Gustavsson, M. and Österholm, P., 2010. Labor-force participation rates and the informational value of unemployment rates: evidence from disaggregated US data. Working paper 2010:14. Departament of Economics, Uppsala Universitet. (avalaible at: http://www.nek.uu.se)

Harvey, D.I., Leybourne, S.J. and Taylor, A.M.R., 2009. Unit root testing in practice: dealing with uncertainty over the trend and initial conditions. Econometric Theory 25, 587-636.

Harvey, D.I. and Mills, T.C., 2004. Tests for stationarity in series with endogenously determined structural change. Oxford Bulletin of Economics and Statistics 66, 863-894.

Hashimzade, N. and Vogelsang, T.J., 2008. Fixed-b asymptotic approximation of the sampling behaviour of nonparametric spectral density estimators. Journal of Time Series Analysis 29, 142-162.

Hobijn, B., Franses, P.H. and Ooms, M., 2004. Generalizations of the KPSS-test for stationarity. Statistica Neerlandica 58, 483-502.

Hong, Y. and Lee Y.J., 2003. Generalized spectral tests for conditional mean models in time series with conditional heteroscedasticity of unknown form. Review of Economic Studies 72, 499-541.

Hong, Y. and White, H., 1995. Consistent specification testing via nonparametric series regression. Econometrica 63, 1133-1159.

Jewell, T., Lee, J., Tieslau, M. and Stracizich, M.C., 2003. Stationarity of health expenditures and GDP: evidence from panel unit root tests with heterogeneous structural breaks. Journal of Health Economics 22, 313-323.

Kapetanios, G., Shin, Y. and Snell, A., 2003. Testing for a unit root in the nonlinear STAR framework. Journal of Econometrics 112, 359-379.

Kiefer, N.M. and Vogelsang, T.J., 2005. A new asymptotic theory for heteroskedasticity and autocorrelation robust tests. Econometric Theory 21,1130-1164.

Kurozumi, E., 2002. Testing for stationarity with a break. Journal of Econometrics 108, 63-99.

Kwiatkowski, D., Phillips, P.C.B., Schmidt, P. and 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.

Landajo, M. and Presno, M.J., 2010. Stationarity testing under nonlinear models. Some asymptotic results. Journal of Time Series Analysis 31, 392-405.

Lee, J. and Strazicich, M., 2001. Testing the null of stationarity in the presence of a structural break. Applied Economics Letters 8, 377-382.

Lee, J. and Strazicich, M., 2003. Minimum Lagrange Multiplier unit root with two structural breaks. Review of Economics and Statistics 85, 1082-1089.

Leybourne, S.J., and McCabe, B.P.M., 1994. A consistent test for a unit root. Journal of Business and Economics Statistics 12, 157-166.

Leybourne, S., Newbold, P. and Vougas, D., 1998. Unit roots and smooth transition. Journal of Time Series Analysis 19, 83-97.

MacNeill, I.B., 1978. Properties of sequences of partial sums of polynomial regression residuals with applications to tests for change of regression at unknown times. Annals of Statistics 6 (2), 422-433.

Madsen, J., Mishra, V. and Smyth, R., 2008. Are labour force participation rates non-stationary? Evidence from 130 years for G7 countries. Australian Economic Papers 47, 166-189.

Mills, T.C. and Markellos, R.N., 2008. The Econometric Modelling of Financial Time Series. Cambridge University Press, Cambridge.

Nabeya, S. and Tanaka, K., 1988. Asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative. The Annals of Statistics 16, 218-235.

Nelson, C.R. and Plosser, C.I., 1982. Trends and random walks in macroeconomic time series: some evidence and implications. Journal of Monetary Economics 10, 139-162.

Perron, P., 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361-1401.

Perron, P., 2006. Dealing with Structural Breaks. In Palgrave Handbook of Econometrics, Vol. 1: Econometric Theory, Patterson, K. and T.C. Mills (eds.), Palgrave Macmillan, 278-352.

Phillips, P.C.B., 2005. HAC estimation by automated regression. Econometric Theory 21, 116-142.

Pötscher, B.M. and Prucha, I.R., 1991. Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part II: asymptotic normality. Econometric Reviews 10, 253-325.

Presno, M.J. and Landajo, M., 2010. Computation of limiting distributions in stationarity testing with a generic trend. Metrika 71, 165-183.

Rodrigues, P.M.M. and Taylor, A.M. R., 2009. The flexible Fourier form and local GLS de-trend unit root tests. Bank of Portugal Working Paper 2009-19.

Sollis, R., 2004. Asymmetric adjustment and smooth transitions: a combination of some unit root tests. Journal of Time Series Analysis 25, 409-417.

Sul, D., Phillips, P.C.B. and Choi, C., 2005. Prewhitening bias in HAC estimation. Oxford Bulletin of Economics and Statistics 67, 517-546.

Sun, Y.X., Phillips, P.C.B. and Jin, S., 2008. Optimal bandwidth selection in heteroskedasticity-autocorrelation robust testing. Econometrica 76, 175-194.

Tanaka, K., 1990. The Fredholm approach to asymptotic inference on nonstationary and noninvertible time series models. Econometric Theory 6, 411-432.

Tanaka, K., 1996. Time Series Analysis: Nonstationary and Noninvertible Distribution Theory. John Wiley & Sons, New York.

Teräsvirta, T., 2005. Forecasting economic variables with nonlinear models. SSE/EFI Working Paper Series in Economics and Finance, Nº 598. Stockholm School of Economics.

Wang, W., 2006. Stochasticity, Nonlinearity and Forecasting of Streamflow Processes. IOS Press, Amsterdam.