Fractional Dickey-Fuller test with or without prehistorical influence

BENSALMA, Ahmed (2021): Fractional Dickey-Fuller test with or without prehistorical influence.

Abstract

Recently the generalization of the standard Dickey-Fuller test to the fractional case has been proposed. The proposed test, called fractional Dickey-Fuller test can be applied to sample generated from a type I or a type II fractional process. Depending on whether the test is applied to sample generated from a type I or type II process, it is referred to as a test with or without prehistoric influence respectively. The main and the first objective of this paper is to study the impact induced by a pre-sample of the finite sample null distribution. In fact, the recently proposed test is built based on a composite null hypothesis rather than a sample one. The second objective is to highlight the theoretical justifications for the choice of the null composite hypothesis. All the theoretical results are illustrated with simulated and real data sets. Furthermore, to facilitate the reproducibility of our simulation data and figures we provide all the necessary supplementary material consisting of EViews programs.

• Item Type: MPRA Paper
• Language: English
• Keywords: ARFIMA; fractional integration, Dickey-Fuller test; Fractional Dickey-Fuller test; type I and type II fractional Brownian motion.
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C15 - Statistical Simulation Methods: General; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling

References

Akonom, J. and Gourieroux, C. (1987) `A functional central limit theorem for fractional processes', Discussion Paper #8801,CEPREMAP, Paris.

Bensalma, A. (2016) `A consistent test for unit root against fractional alternative', International Journal of Operational Research, Vol. 27, Nos. 1/2, pp.252--274.

Bensalma, A. (2018) `Testing the fractional integration parameter revisited: a fractional Dickey-Fuller test', International Journal of Mathematics in Operational Research, Vol. 12, No. 4, pp. 471-506.

Bensalma, A., (2021) `An Eviews program to perform the fractional Dickey-Fuller test'. Supplementary material 1

Bensalma, A., (2021)`A farctional Dickey-Fuller test: An Eviews program to evaluate the size and power of a type II fractional process based test'. Supplementary material 2

Bensalma, A.,(2021), 'Fractional Dickey-Fuller test: A simulation program with Eviews to compare the simple null distribution of the type I fractional process based test and the type II fractional process based test. Supplementary material 3.

Chan, N.H. and Terrin, N. (1995) `Inference for unstable long-memory processes with applications to fractional unit root autoregressions', The annals of Statistics, Vol. 23, No. 5, pp. 1662-1683.

Chan, N.H. and Wei, C.Z. (1988) `Limiting distributions of least squares estimates of unstable autoregressive processes', The annals of Statistics, Vol. 16, No. 1, pp. 367-401.

Davidson, J. and Hashimzade, N. (2009) `Type I and type II fractional Brownian motions: A reconsideration' Computational Statistics & Data Analysis, Vol. 53, No. 6, pp. 2089-2106.

Dickey, D.A. and Fuller, W.A. (1979) `Distribution of the estimators for autoregressive time series with a unit root', Journal of the American Statistical Association, Vol. 74, No. 366a, pp.427--431.

Dickey, D.A. and Fuller, W.A. (1981) `Likelihood ratio tests for autoregressive time series with a unit root', Econometrica, Vol. 49, No. 4, pp.1057--1072.

Dickey, D.A. and Pantula, S.G. (1987) `Determining the order of differencing in autoregressive processes', Journal of Business and Economic Statistics, Vol. 15, No. 4, pp.455--461.

Gourieroux, C., F. Maurel, and A. Monfort (1987), `Regression and nonstationarity', CREST document no. 8708.

Gourieroux, C. and A. Monfort (1997), `Times series and dynamic models', CAMBRIDGE UNIVERSITY PRESS.

Granger, C.W.J. and Joyeux, R. (1980) `An introduction to long memory time series models and fractional differencing', Journal of Time Series Analysis, Vol. 1, No. 1, pp.15--29.

Hamilton, J.D. (1994) `Time series analysis', Princeton University Press.

Hosking, J.R.M. (1981) `Fractional differencing', Biométrika, Vol. 68, No. 1, pp.165--176.

Liu, M. (1998) `Asymptotics of nonstationary fractional integrated series', Econometric Theory, Vol. 14, No. 5, pp.641--662.

Marinucci, D. and P.M. Robinson (1999) `Alternative forms of fractional Brownian motion', Journal of Statistical Inference and Planning, Vol. 80, pp.111-122.

Mandelbrot, B. and Van Ness, J. (1968).'Fractional Brownian motions, fractional noises and applications'. S.I.A.M. Review. Vol. 10,pp. 422-437.

Park, J.Y. and Phillips, C.B. (1988) 'Statistical inference in regressions with integrated processes: Part 1', Econometric Theory, Vol. 4, pp. 468-497.

Park, J.Y. and Phillips, C.B. (1989) 'Statistical inference in regressions with integrated processes: Part 2', Econometric Theory, Vol. 5, pp. 95-131.

Pantula, S.G. (1989) `Testing for unit root in time series data', Econometric Theory, Vol. 5,No 2, pp 256-271.

Schwert, W.S. (1989) `Tests for unit roots: A monte carlo investigation', Journal of business & Economic Statistics, Vol.7, No.2, pp. 147-159.

Silveira, G. (1991) `Contributions to strong approximations in time series with applications in nonparametric statistics and functional central limit theorems', PHD Thesis, University of London.

Sowell, F.B. (1990) `The fractional unit root distribution', Econometrica, Vol. 58, No. 2, pp.494--505.

Tanaka, K (1999) `The nonstationary fractional unit root', Econometric Theory, Vol. 15, No. 4, pp.549--582.

Tanaka, K (1996) `Time series analysis: Nonstationary and Noninvertible distribution Theory', New York : Wiley.

Wang, Q., Lin Y. & Gulati, C.M. (2003) `Asymptotic for general fractionally integrated processes with applications to unit root tests', Econometric Theory, Vol. 19, pp. 143-164.

Wang, Q., Lin Y. & Gulati, C.M. (2002) `Asymptotics for general nonstationary fractionally integrated processes without prehistoric influence', Journal of applied mathematics and decision sciences, Vol. 6, No. 4, pp. 255-269 pp.