Abstract
This paper presents response surface estimates of finite sample critical values of the Augmented Fractional Dickey–Fuller test of Dolado et al. (Testing I(1) against I(d) alternatives in the presence of deterministic components. Universidad Carlos III, Madrid, mimeo, 2006). The null hypothesis that a series contains a unit root is tested against the alternative that it is fractionally integrated. It is shown that finite sample critical values depend critically on the lag order used to construct the test statistic as well as the hypothesized value of the fractional differencing parameter under the alternative. Non-linear non-parametric response surfaces provide a novel approach to constructing estimates of finite sample critical values.
Similar content being viewed by others
References
Attoh-Okine N, Mensah S and Nawaiseh N (2003). A new technique for using multivariate adaptive regression splines (MARS) in pavement roughness prediction. Transport 156: 51–56
Bai J and Perron P (2003). Critical values for multiple structural change tests. Econometr. J. 6: 72–78
Breiman L and Friedman J (1985). Estimating optimal transformations for multiple regression correlation. J Am Stat Assoc 80: 580–598
Briand L, Freimut B and Vollei F (2004). Using multiple adaptive regression splines to support decision making in code inspections. J Syst Softw 73: 205–217
Cheung Y and Lai K (1995a). Lag order and critical values of the augmented Dickey–Fuller test. J Bus Econ Stat 13: 277–280
Cheung Y and Lai K (1995b). Lag order and critical values of a modified Dickey–Fuller test. Oxf Bull Econ Stat 57: 411–419
Cook S (2001). Finite sample critical values of the augmented dickey–fuller statistic: a note on lag order. Econ Issues 6: 31–38
Craven P and Wahba G (1979). Smoothing noisy data with spline functions—estimating the correct degree of smoothing by the method of generalized cross-validation. Numer Math 31: 317–403
Deconinck E, Xu Q, Put R, Coomans D, Massart D and Vander Heyden Y (2005). Prediction of gastro-intestinal absorption using multivariate adaptive regression splines. J Pharm Biomed Anal 39: 1021–1030
De Gooijer J, Ray B and Kräger H (1999). Forecasting exchange rates using TSMARS. J Int Money Finance 17: 513–534
Dolado J, Gonzalo J and Mayoral L (2002). A fractional Dickey–Fuller test for unit roots. Econometrica 70: 1963–2006
Dolado J, Gonzalo J, Mayoral L (2005) Structural breaks versus long memory: what is what? mimeo. Universidad Carlos III, Madrid
Dolado J, Gonzalo J, Mayoral L (2006) Testing I(1) against I(d) alternatives in the presence of deterministic components. Mimeo, Universidad Carlos III, Madrid
Friedman J (1991a). Multivariate adaptive regression splines. Ann Stat 19: 1–67
Friedman J (1991b) Estimating functions of mixed ordinal and categorical variables using adaptive splines. Technical Report No. 108, Laboratory for Computational Statistics, Stanford University
Granger C and Hallman J (1991). Long memory series with attractors. Oxf Bull Econ Stat 53: 11–26
Granger C and Swanson N (1997). An introduction to stochastic unit-root processes. J Econ 80: 35–62
Hallman J (1990) Non-linear integrated series: cointegration and an application. PhD Dissertation, University of California, San Diego
Harvey D and Van Dijk D (2006). Sample size, lag order and critical values of seasonal unit root tests. Comp Stat Data Anal 50: 2734–2751
Le Blanc M and Crowley J (1999). Adaptive regression splines in the cox model. Biometrics 55: 204–213
Lewis P and Stevens J (1991). Nonlinear modeling of time series using multivariate adaptive regression splines (MARS). J Am Stat Assoc 86: 864–877
Lobato I and Velasco C (2007). Efficient wald tests for fractional unit roots. Econometrica 75: 575–589
Lopez C, Murray C and Papell D (2005). State of the art unit root tests and purchasing power parity. J Mon Cred Bank 37: 361–370
MacKinnon J (1991). Critical values for cointegration tests. In: Engle, R and Granger, C (eds) Long-run economic relationships: readings in cointegration, pp 267–276. Oxford University Press, Oxford
MacKinnon J (1994). Approximate asymptotic distribution functions for unit-root and cointegation tests. J Bus Econ Stat 12: 167–176
MacKinnon J (1996). Numerical distribution functions for unit root and cointegration tests. J Appl Econometr 11: 601–618
MacKinnon J (2000). Computing numerical distribution functions in econometrics. In: Pollard, A, Mewhort, D and Weaver, D (eds) High performance computing systems and applications, pp 455–470. Kluwer, Amsterdam
MacKinnon J, Haug A and Michelis L (1999). Numerical distribution functions of likelihood ratio tests for cointegration. J Appl Econometr 14: 563–577
Mukkamala S, Sung A, Abraham A and Ramos V (2005). Intrusion detection systems using adaptive regression splines. In: Seruca, I, Cordeiro, J, Hammoudi, S, and Filipe, J (eds) Enterprise informations systems VI. Springer, Heidelberg
Ng S and Perron P (2001). Lag length selection and the construction of unit root tests with good size and power. Econometrica 69: 1519–1554
Ng S and Perron P (2005). A note on the selection of time series models. Oxf Bull Econ Stat 67: 115–134
Presno M and Lopez A (2003). Response surface estimates of stationarity tests with a structural break. Econ Lett 78: 395–399
Put R, Xu Q, Massart D and Vander Heyden Y (2004). Multivariate adaptive regression splines (MARS) in chromatographic quantitative structure-retention relationship studies. J Chromatogr A 1055: 11–19
Said S and Dickey D (1984). Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 84: 599–607
Sephton P (1994). Cointegration tests on MARS. Comput Econ 7: 23–35
Sephton P (1995a). Response surface estimates of the KPSS stationarity test. Econ Lett 47: 255–261
Sephton P (1995b). A non-parametric view of money/income causality. Appl Fin Econ 5: 79–84
Sephton P (2001) Forecasting recessions: can we do better on MARS? Federal Reserve Bank of St Louis Review, pp 39–50
Sephton P (2005). Forecasting inflation using the term structure and MARS. Appl Econ Lett 12: 199–202
Sephton P (2006) Unit roots and purchasing power parity: another kick at the can. mimeo
York T and Eaves L (2001). Common disease analysis using multivariate adaptive regression splines (MARS): genetic analysis workshop 12 Simulated Sequence Data. Genet Epidemiol 21: S649–S654
York T, Eaves L and van den Oord E (2005). Multivariate adaptive regression splines: a powerful method for detecting disease-risk relationship differences among subgroups. Stat Med 25: 1355–1367
Zha W and Chan W (2005). Objective speech quality measurement using statistical data mining. EURASIP J Appl Signal Process 9: 1410–1424
Zhang H, Yu C, Zhu H and Shi J (2003). Identification of linear directions in multivariate adaptive spline models. J Am Stat Assoc 98: 369–376
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sephton, P.S. Critical values of the augmented fractional Dickey–Fuller test. Empir Econ 35, 437–450 (2008). https://doi.org/10.1007/s00181-007-0171-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00181-007-0171-0