## Semiparametric robust tests on seasonal or cyclical long memory time series.(English. English summary)Zbl 1023.62093

Let $$f$$ be the spectral density of a time series and let at some frequencies $$\omega_i\in[0,\pi]$$, $$i=1,\dots,H$$, $$f(\omega_i\pm\lambda)\sim C\lambda^{-2d_i^{\pm}}$$ as $$\lambda\to 0$$. The author considers a hypothesis of the form $$H_0$$: $$d_1^{+}=d_1^{-}=\dots=d_H^{-}$$ or $$H_0$$: $$d_i^{pm}=c_i^{\pm}$$, $$i=1,\dots, d$$, for some fixed $$c_i^{\pm}$$. Lagrange multiplier (LM) tests are constructed for these hypotheses basing on the Whittle quasi-likelihood function. It is shown that the LM statistic is asymptotically $$\chi^2$$ distributed. The test power is investigated under local alternatives. The performance of the test is compared with that of the Robinson test for some parametric models. The author’s conclusion is that the Robinson test performance is significantly worse. Results of simulations and an empirical application to a UK inflation series are presented.
Reviewer: R.E.Maiboroda

### MSC:

 62M15 Inference from stochastic processes and spectral analysis 62H15 Hypothesis testing in multivariate analysis 91B84 Economic time series analysis
Full Text:

### References:

 [1] 1J. ARTECHE, and P. M. ROBINSON(1999 ) Seasonal and cylical long memory . InAsymptotics, Nonparametrics and Time Series(ed. S. Ghosh). New York: Marcel Dekker, Inc., 115-48. [2] DOI: 10.1111/1467-9892.00170 · Zbl 0974.62079 [3] CHAN N. H., Ann. Statist 23 pp 1662– (1995) [4] WEI C. Z., Ann. Statist. 16 pp 367– (1988) [5] DAVIES R. B., Biometrika 74 pp 95– (1987) [6] GEWEKE J., J. Time Ser. Anal 4 pp 221– (1983) [7] GIRAITIS L., Liet. Matem. Rink 35 pp 65– (1995) [8] 8P. HALL, and C. C. HEYDE(1980 ) Martingale limit theory and its application .Probability and Mathematical Statistics(eds Z. W. Bimbaum and E. Lukacs). New York: Academic Press, 52-65. [9] HASSLER U., J. Bus. Econ. Statist. 13 pp 37– (1995) [10] 10M. HENRY, and P. M. ROBINSON(1996 ) Bandwidth choice in Gaussian semiparametric estimation of long range dependence . InAthens Conference on Applied Probability and Time Series, Volume II: Time Series Analysis, in memory of E. J. Hannan (eds P. M. Robinson and M. Rosenblatt). New York: Springer Verlag, 220-32. [11] HEYDE C. C., J. Appl. Probab. 9 pp 235– (1972) [12] HOSKING J. R. M., Water Resour. Res. 20 pp 1898– (1984) [13] 13H. R. KUNSCH(1987 ) Statistical aspects of self-similar processes . InProceedings of the First World Congress of the Bernoulli Society, 1, (eds Yu. Prohorov and V. V. Sazanov) Utrecht: VNU Science Press, 67-74. [14] LOBATO I. N., Rev. Econ. Stud. 65 pp 475– (1998) [15] NERLOVE M., Econometrica 32 pp 241– (1964) [16] PORTER-HUDAK S., J. Am. Statist. Assoc. 85 pp 338– (1990) [17] DOI: 10.1016/0169-2070(93)90009-C [18] ROBINSON P. M., J. Am. Statist. Assoc. 89 pp 1420– (1994) [19] Ann. Statist. 23 pp 1048– (1995) [20] Ann. Statist. 23 pp 1630– (1995) [21] DOI: 10.1017/S0266466699153027 · Zbl 1054.62584
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.