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TESTING FOR GENERAL FRACTIONAL INTEGRATION IN THE TIME DOMAIN

Published online by Cambridge University Press:  01 December 2009

Uwe Hassler ,
Paulo M.M. Rodrigues  and
Antonio Rubia
Uwe Hassler
Affiliation:
Goethe University Frankfurt
Paulo M.M. Rodrigues*
Affiliation:
Banco de Portugal, Universidade Nova de Lisboa, and CASEE
Antonio Rubia
Affiliation:
University of Alicante
*
*Address correspondence to Paulo M.M. Rodrigues, Banco de Portugal, Economic Research Department, Av. Almirante Reis, 71-6th floor, 1150-012 Lisbon, Portugal; e-mail: pmrodrigues@bportugal.pt.
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Abstract

We propose a family of least-squares–based testing procedures that look to detect general forms of fractional integration at the long-run and/or the cyclical component of a time series, and that are asymptotically equivalent to Lagrange multiplier tests. Our setting extends Robinson’s (1994) results to allow for short memory in a regression framework and generalizes the procedures in Agiakloglou and Newbold (1994), Tanaka (1999), and Breitung and Hassler (2002) by allowing for single or multiple fractional unit roots at any frequency in [0, π]. Our testing procedure can be easily implemented in practical settings and is flexible enough to account for a broad family of long- and short-memory specifications, including ARMA and/or GARCH-type dynamics, among others. Furthermore, these tests have power against different types of alternative hypotheses and enable inference to be conducted under critical values drawn from a standard chi-square distribution, irrespective of the long-memory parameters.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 6: Newbold Conference Special Issue , December 2009 , pp. 1793 - 1828
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Agiakloglou, C. & Newbold, P. (1994) Lagrange multiplier tests for fractional difference. Journal of Time Series Analysis 14, 253–262.Google Scholar
Bouette, J.C., Chassagneux, J.F., Sibai, D., Terron, R., & Charpentier, A. (2006) Wind in Ireland: Long memory or seasonal effect? Stochastic Environmental Research and Risk Assessment 20, 141–151.CrossRefGoogle Scholar
Breitung, J. & Hassler, U. (2002) Inference on the cointegration rank in fractionally integrated processes. Journal of Econometrics 110(2), 167–185.Google Scholar
Brillinger, D.R. (1981) Time Series Data Analysis and Theory. Holden Day.Google Scholar
Chung, C.F. (1996) Estimating a generalized long memory process. Journal of Econometrics 73, 237–259.CrossRefGoogle Scholar
Dalla, V. & Hidalgo, J. (2005) A parametric bootstrap test for cycles. Journal of Econometrics 129, 219–261.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.Google Scholar
Davidson, J. (2000) Econometric Theory. Blackwell.Google Scholar
Demetrescu, M., Kuzin, V., & Hassler, U. (2008) Long memory testing in the time domain. Econometric Theory 24, 176–215.CrossRefGoogle Scholar
Dolado, J.J., Gonzalo, J., & Mayoral, L. (2002) A fractional Dickey-Fuller test for unit roots. Econometrica 70, 1963–2006.CrossRefGoogle Scholar
Gil-Alana, L.A. (2005) Modelling US monthly inflation in terms of a jointly seasonal and nonseasonal long memory process. Applied Stochastic Models in Business and Industry 21, 83–94.Google Scholar
Gil-Alana, L.A. & Robinson, P.M. (2001) Testing of seasonal fractional integration in UK and Japanese consumption and income. Journal of Applied Econometrics 16, 95–114.CrossRefGoogle Scholar
Giraitis, L., Hidalgo, J., & Robinson, P.M. (2001) Gaussian estimation of parametric spectral density with unknown pole. Annals of Statistics 29, 987–1023.CrossRefGoogle Scholar
Gonçalves, S. & Kilian, L. (2004) Bootstrapping autoregressions with conditional heteroskedasticity of unknown form. Journal of Econometrics 123, 89–120.Google Scholar
Gonçalves, S. & Kilian, L. (2007) Asymptotic and bootstrap inference for AR(∞) processes with conditional heteroskedasticity. Econometric Reviews 26, 609–641.Google Scholar
Gradshteyn, I.S. & Ryzhik, I.M. (2000) Table of Integrals, Series, and Products. Academic Press.Google Scholar
Gray, H.L., Zhang, N.F., & Woodward, W. (1989) On generalized fractional processes. Journal of Time Series Analysis 10, 233–57.Google Scholar
Hassler, U. (1994) (Mis)specification of long memory in seasonal time series. Journal of Time Series Analysis 15, 19–30.Google Scholar
Hassler, U. & Breitung, J. (2006) A residual-based LM type test against fractional cointegration. Econometric Theory 22, 1091–1111.Google Scholar
Hassler, U., Rodrigues, P.M.M., & Rubia, A. (2008) Testing for the General Unit Root Hypothesis in the Time Domain. Working paper 380, Fundacion de Cajas de Ahorros (FUNCAS).Google Scholar
Hidalgo, J. (2005) Semiparametric estimation for stationary processes whose spectra have an unknown pole. Annals of Statistics 33, 1843–1889.Google Scholar
Hidalgo, J. (2007) A nonparametric test for weak dependence against strong cycles and its bootstrap analougue. Journal of Time Series Analysis 28, 307–349.Google Scholar
Hidalgo, J. & Soulier, P. (2004) Estimation of the location and the exponent of the spectral singularity of a long memory process. Journal of Time Series Analysis 25, 55–81.Google Scholar
Hylleberg, S., Engle, R.F., Granger, C.W.J., & Yoo, B.S. (1990) Seasonal integration and cointegration. Journal of Econometrics 44, 215–238.Google Scholar
Koopman, S.J., Ooms, M., & Carnero, M.A. (2007) Periodic seasonal reg-ARFIMA-GARCH models for daily electricity spot prices. Journal of the American Statistical Association 102, 16–27.Google Scholar
Marinucci, D. & Robinson, P.M. (1999). Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 80, 111–122.CrossRefGoogle Scholar
Nielsen, M.Ø. (2004) Efficient likelihood inference in nonstationary univariate models. Econometric Theory 20, 116–146.Google Scholar
Nielsen, M.Ø. (2005) Multivariate Lagrange multiplier tests for fractional integration. Journal of Financial Econometrics 3, 372–398.Google Scholar
Porter-Hudak, S. (1990) An application of the seasonal fractionally differenced model to the monetary aggregates. Journal of the American Statistical Association 85, 338–344.CrossRefGoogle Scholar
Ramachandran, R. & Beaumont, P. (2001) Robust estimation of GARMA model parameters with application to cointegration among interest rates of industrialized countries. Computational Economics 17, 179–201.CrossRefGoogle Scholar
Robinson, P.M. (1991). Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 67–84.Google Scholar
Robinson, P.M. (1994). Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association 89, 1420–1437.Google Scholar
Robinson, P.M. (2005). The distance between rival nonstationary fractional processes. Journal of Econometrics 128, 283–300.Google Scholar
Schwert, G.W. (1989) Tests for unit roots: A Monte Carlo investigation. Journal of Business and Economic Statistics 7, 147–59.Google Scholar
Smallwood, A.D. & Norrbin, S.C. (2006) Generalized long memory processes, failure of cointegration tests and exchange rate dynamics. Journal of Applied Econometrics 21, 409–417.Google Scholar
Soares, L.J. & Souza, L.R., (2006) Forecasting electricity demand using generalized long memory. International Journal of Forecasting 22, 17–28.CrossRefGoogle Scholar
Tanaka, K. (1999) The nonstationary fractional unit root. Econometric Theory 15, 549–582.Google Scholar
Yajima, Y. (1996). Estimation of the frequency of the unbounded spectral density. In Proceedings of the Business and Economic Statistical Section. American Statistical Association.Google Scholar
White, H. (2001). Asymptotic Theory for Econometricians. Academic Press.Google Scholar
Woodward, W.A., Cheng, Q.C., & Gray, H.L. (1998). A k-factor GARMA long-memory model. Journal of Time Series Analysis 19, 485–504.CrossRefGoogle Scholar