×

Fractional unit-root tests allowing for a fractional frequency flexible Fourier form trend: predictability of Covid-19. (English) Zbl 1494.62018

Summary: In this study we propose a fractional frequency flexible Fourier form fractionally integrated ADF unit-root test, which combines the fractional integration and nonlinear trend as a form of the Fourier function. We provide the asymptotics of the newly proposed test and investigate its small-sample properties. Moreover, we show the best estimators for both fractional frequency and fractional difference operator for our newly proposed test. Finally, an empirical study demonstrates that not considering the structural break and fractional integration simultaneously in the testing process may lead to misleading results about the stochastic behavior of the Covid-19 pandemic.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M07 Non-Markovian processes: hypothesis testing
62M15 Inference from stochastic processes and spectral analysis
62P05 Applications of statistics to actuarial sciences and financial mathematics
92D30 Epidemiology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dickey, D. A.; Fuller, W. A., Distribution of the estimates for autoregressive time series with a unit root, J. Am. Stat. Assoc., 74, 427-431 (1979) · Zbl 0413.62075
[2] Mayoral, L., Further evidence on the statistical properties of real GNP, Oxf. Bull. Econ. Stat., 68, 901-920 (2006)
[3] Mayoral, L., Testing for fractional integration versus short memory with structural breaks, Oxf. Bull. Econ. Stat., 74, 2, 278-305 (2011)
[4] Hamilton, J. D., Time Series Analysis (1994), Princeton: Princeton University Press, Princeton · Zbl 0831.62061
[5] Perron, P., The great crash, the oil price shock, and the unit root hypothesis, Econometrica, 57, 1361-1401 (1989) · Zbl 0683.62066
[6] Perron, P., Testing for a unit root in a time series with a changing mean, J. Bus. Econ. Stat., 8, 153-162 (1990)
[7] Rappaport, P.; Reichlin, L., Segmented trends and non-stationary time series, Econ. J., 99, 168-177 (1989)
[8] Zivot, E.; Andrews, K., Further evidence on the great crash, the oil price shock, and the unit root hypothesis, J. Bus. Econ. Stat., 10, 251-270 (1992)
[9] Lumsdaine, R. L.; Papell, D. H., Multiple trend breaks and the unit root hypothesis, Rev. Econ. Stat., 79, 2, 212-218 (1997)
[10] Omay, T.; Emirmahmutoğlu, F., The comparison of power and optimization algorithms on unit root testing with smooth transition, Comput. Econ., 49, 4, 623-651 (2017)
[11] Leybourne, S.; Newbold, P.; Vougas, D., Unit roots and smooth transitions, J. Time Ser. Anal., 19, 83-97 (1998) · Zbl 0902.62132
[12] Omay, T.; Emirmahmutoglu, F.; Hasanov, M., Structural break, nonlinearity, and asymmetry: a re-examination of PPP proposition, Appl. Econ., 50, 12, 1289-1308 (2018)
[13] Bierens, H. J., Testing the unit root with drift hypothesis against nonlinear trend stationarity, with an application to the US price level and interest rate, J. Econom., 81, 29-64 (1997) · Zbl 0944.62116
[14] Becker, R.; Enders, W.; Lee, J., A stationarity test in the presence of an unknown number of smooth breaks, J. Time Ser. Anal., 27, 381-409 (2006) · Zbl 1126.62076
[15] Enders, W.; Lee, J., A unit root test using a Fourier series to approximate smooth breaks, Oxf. Bull. Econ. Stat., 74, 574-599 (2012)
[16] Enders, W.; Lee, J., The flexible Fourier form and Dickey-Fuller type unit root tests, Econ. Lett., 117, 196-208 (2012) · Zbl 1255.62243
[17] Omay, T., Fractional frequency flexible Fourier form to approximate smooth breaks in unit root testing, Econ. Lett., 134, 123-126 (2015) · Zbl 1364.62278
[18] Dolado, J.; Gonzalo, J.; Mayoral, L., A fractional Dickey-Fuller test for unit roots, Econometrica, 70, 1963-2006 (2002) · Zbl 1132.91590
[19] Baillie, R. T., Long memory processes and fractional integration in econometrics, J. Econom., 73, 1, 5-59 (1996) · Zbl 0854.62099
[20] Beran, J., Statistics for Long Memory Processes (1994), New York: Chapman & Hall, New York · Zbl 0869.60045
[21] Granger, C. W.J., Long memory relationships and the aggregation of dynamic models, J. Econom., 14, 227-238 (1980) · Zbl 0466.62108
[22] Cuestas, J. C.; Gil-Alana, L. A., Testing for long memory in the presence of nonlinear deterministic trends with Chebyshev polynomials, Stud. Nonlinear Dyn. Econom., 20, 1, 57-74 (2016) · Zbl 1507.62375
[23] Chang, S. Y.; Perron, P., Fractional unit root tests allowing for a structural change in trend under both the null and alternative hypotheses, Econometrics, 5 (2017)
[24] Robinson, P. M., Efficient tests of nonstationary hypotheses, J. Am. Stat. Assoc., 89, 1420-1437 (1994) · Zbl 0813.62016
[25] Tanaka, K., The nonstationary fractional unit root, Econom. Theory, 15, 549-582 (1999) · Zbl 0985.62073
[26] Dolado, J., Gonzalo, J., Mayoral, L.: Testing \(\text{I}(1)\) against \(\text{I}(0)\) alternatives in the presence of deterministic components. Mimeo (2005)
[27] Gallant, A. R., On the bias in flexible functional forms and an essentially unbiased form: the Fourier flexible form, J. Econom., 15, 211-245 (1981) · Zbl 0454.62096
[28] Dolado, J., Gonzalo, J., Mayoral, L.: Structural breaks vs. long memory: what is what? Mimeo, Universidad Carlos III (2007)
[29] Omay, T.; Emirmahmutoğlu, F.; Shahzad, S. J.H., The comparison of optimization algorithms for selecting the fractional frequency in Fourier form unit root tests, Appl. Econ., 53, 7, 761-780 (2020)
[30] Andrews, D. W.K.; Guggenberger, P., A bias-reduced log-periodogram regression estimator for the long-memory parameter, Econometrica, 71, 2, 675-712 (2003) · Zbl 1153.62354
[31] Geweke, J.; Porter-Hudak, S., The estimation and application of long-memory time series models, J. Time Ser. Anal., 4, 221-237 (1983) · Zbl 0534.62062
[32] Gil-Alana, L. A., Testing the existence of multiple cycles in financial and economic time series, Ann. Econ. Financ., 8, 1, 1-20 (2007)
[33] Ahmed, A.; Salam, B.; Mohammad, M.; Akgul, A.; Khoshnaw, S. H., Analysis coronavirus disease (COVID-19) model using numerical approaches and logistic model, AIMS Bioeng., 7, 3, 130-146 (2020)
[34] Atangana, E.; Atangana, A., Facemasks simple but powerful weapons to protect against COVID-19 spread: can they have sides effects?, Results Phys., 19 (2020)
[35] Naik, P. A.; Yavuz, M.; Qureshi, S.; Zu, J.; Townley, S., Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan, Eur. Phys. J. Plus, 135, 10, 1-42 (2020)
[36] Akgül, A.; Ahmed, N.; Raza, A.; Iqbal, Z.; Rafiq, M.; Baleanu, D.; Rehman, M. A.U., New applications related to Covid-19, Results Phys., 20 (2021) · Zbl 1489.37103
[37] Memon, Z.; Qureshi, S.; Memon, B. R., Assessing the role of quarantine and isolation as control strategies for COVID-19 outbreak: a case study, Chaos Solitons Fractals, 144 (2020)
[38] Tuan, N. H.; Tri, V. V.; Baleanu, D., Analysis of the fractional corona virus pandemic via deterministic modeling, Math. Methods Appl. Sci., 44, 1, 1086-1102 (2021) · Zbl 1472.34097
[39] Atangana, A.; Akgül, A., Can transfer function and Bode diagram be obtained from Sumudu transform, Alex. Eng. J., 59, 4, 1971-1984 (2020)
[40] Atangana, A.; Akgül, A.; Owolabi, K. M., Analysis of fractal fractional differential equations, Alex. Eng. J., 59, 3, 1117-1134 (2020)
[41] Akram, T.; Abbas, M.; Riaz, M. B.; Ismail, A. I.; Ali, N. M., An efficient numerical technique for solving time fractional Burgers equation, Alex. Eng. J., 59, 4, 2201-2220 (2020)
[42] Akram, T.; Abbas, M.; Iqbal, A.; Baleanu, D.; Asad, J. H., Novel numerical approach based on modified extended cubic B-spline functions for solving non-linear time-fractional telegraph equation, Symmetry, 12, 7 (2020)
[43] Akram, T.; Abbas, M.; Ali, A.; Iqbal, A.; Baleanu, D., A numerical approach of a time fractional reaction-diffusion model with a non-singular kernel, Symmetry, 12, 10 (2020)
[44] Amin, M.; Abbas, M.; Iqbal, M. K.; Baleanu, D., Numerical treatment of time-fractional Klein-Gordon equation using redefined extended cubic B-spline functions, Front. Phys., 8 (2020)
[45] Amin, M.; Abbas, M.; Iqbal, M. K.; Ismail, A. I.M.; Baleanu, D., A fourth order non-polynomial quintic spline collocation technique for solving time fractional superdiffusion equations, Adv. Differ. Equ., 2019 (2019) · Zbl 1487.65163
[46] Iqbal, A.; Siddiqui, M. J.; Muhi, I.; Abbas, M.; Akram, T., Nonlinear waves propagation and stability analysis for planar waves at far field using quintic B-spline collocation method, Alex. Eng. J., 59, 4, 2695-2703 (2020)
[47] Khalid, N.; Abbas, M.; Iqbal, M. K.; Singh, J.; Ismail, A. I.M., A computational approach for solving time fractional differential equation via spline functions, Alex. Eng. J., 59, 5, 3061-3078 (2020)
[48] Khalid, N.; Abbas, M.; Iqbal, M. K., Non-polynomial quintic spline for solving fourth-order fractional boundary value problems involving product terms, Appl. Math. Comput., 349, 393-407 (2020) · Zbl 1429.65161
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.