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Semiparametrically point-optimal hybrid rank tests for unit roots. (English) Zbl 07114923
The main contribution of this paper is twofold. First, the authors derived the semiparametric power envelopes of unit root tests with serially correlated errors for two cases: symmetric or possibly non-symmetric innovation distributions . Their method of derivation seems to be novel and exploits the invariance structures embedded in the semiparametric unit root model. They use an application of the Asymptotic Representation Theorem (see, e.g., Theorem 15.1 in [A. W. van der Vaart, Asymptotic statistics. Cambridge: Cambridge Univ. Press (1998; Zbl 0910.62001)]) that subsequently yields the local asymptotic power envelope (Theorem 3.3).
As a second contribution, they provided two new classes of easy-to-implement unit root tests that are semiparametrically optimal in the sense that their asymptotic power curves are tangent to the associated semiparametric power envelopes.
Finally, the authors introduced a simplified version of the proposed test and they showed, in a Monte-Carlo study, that their theoretical results carry over to finite samples.
MSC:
62M07 Non-Markovian processes: hypothesis testing
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
Software:
itsmr; ITSM2000
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References:
[1] Ahn, S. K., Fotopoulos, S. B. and He, L. (2001). Unit root tests with infinite variance errors. Econometric Rev.20 461-483. · Zbl 1044.62090
[2] Bickel, P. J. (1982). On adaptive estimation. Ann. Statist.10 647-671. · Zbl 0489.62033
[3] Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1998). Efficient and Adaptive Estimation for Semiparametric Models. Springer, New York. Reprint of the 1993 original. · Zbl 0894.62005
[4] Brockwell, P. J. and Davis, R. A. (2016). Introduction to Time Series and Forecasting, 3rd ed. Springer Texts in Statistics. Springer, Cham. · Zbl 1355.62001
[5] Callegari, F., Cappuccio, N. and Lubian, D. (2003). Asymptotic inference in time series regressions with a unit root and infinite variance errors. J. Statist. Plann. Inference116 277-303. · Zbl 1020.62076
[6] Cassart, D., Hallin, M. and Paindaveine, D. (2010). On the estimation of cross-information quantities in rank-based inference. In Nonparametrics and Robustness in Modern Statistical Inference and Time Series Analysis: A Festschrift in Honor of Professor Jana Jurečková. Inst. Math. Stat. (IMS) Collect.7 35-45. IMS, Beachwood, OH.
[7] Chan, N. H. and Wei, C. Z. (1988). Limiting distributions of least squares estimates of unstable autoregressive processes. Ann. Statist.16 367-401. · Zbl 0666.62019
[8] Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric test statistics. Ann. Math. Stat.29 972-994. · Zbl 0092.36501
[9] Choi, I. (2015). Almost All About Unit Roots. Themes in Modern Econometrics. Cambridge Univ. Press, New York.
[10] Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. J. Amer. Statist. Assoc.74 427-431. · Zbl 0413.62075
[11] Dickey, D. A. and Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica49 1057-1072. · Zbl 0471.62090
[12] Dufour, J.-M. and King, M. L. (1991). Optimal invariant tests for the autocorrelation coefficient in linear regressions with stationary or nonstationary \(\text{AR}(1)\) errors. J. Econometrics47 115-143. · Zbl 0729.62079
[13] Elliott, G. and Müller, U. K. (2006). Minimizing the impact of the initial condition on testing for unit roots. J. Econometrics135 285-310.
[14] Elliott, G., Rothenberg, T. J. and Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica64 813-836. · Zbl 0888.62088
[15] Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests. Academic Press, New York.
[16] Hallin, M. and Puri, M. L. (1988). Optimal rank-based procedures for time series analysis: Testing an ARMA model against other ARMA models. Ann. Statist.16 402-432. · Zbl 0659.62111
[17] Hallin, M. and Puri, M. L. (1994). Aligned rank tests for linear models with autocorrelated error terms. J. Multivariate Anal.50 175-237. · Zbl 0805.62050
[18] Hallin, M., van den Akker, R. and Werker, B. J. M. (2011). A class of simple distribution-free rank-based unit root tests. J. Econometrics163 200-214. · Zbl 1441.62718
[19] Hallin, M., van den Akker, R. and Werker, B. J. M. (2015). On quadratic expansions of log-likelihoods and a general asymptotic linearity result. In Mathematical Statistics and Limit Theorems 147-165. Springer, Cham. · Zbl 1320.62031
[20] Hasan, M. N. (2001). Rank tests of unit root hypothesis with infinite variance errors. J. Econometrics104 49-65. · Zbl 1026.62092
[21] Jansson, M. (2008). Semiparametric power envelopes for tests of the unit root hypothesis. Econometrica76 1103-1142. · Zbl 1152.91713
[22] Jeganathan, P. (1995). Some aspects of asymptotic theory with applications to time series models. Econometric Theory11 818-887.
[23] Jeganathan, P. (1997). On asymptotic inference in linear cointegrated time series systems. Econometric Theory13 692-745.
[24] Klaassen, C. A. J. (1987). Consistent estimation of the influence function of locally asymptotically linear estimators. Ann. Statist.15 1548-1562. · Zbl 0629.62041
[25] Kreiss, J.-P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist.15 112-133. · Zbl 0616.62042
[26] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer Series in Statistics. Springer, New York. · Zbl 0605.62002
[27] Müller, U. K. (2011). Efficient tests under a weak convergence assumption. Econometrica79 395-435.
[28] Müller, U. K. and Elliott, G. (2003). Tests for unit roots and the initial condition. Econometrica71 1269-1286. · Zbl 1152.62371
[29] Müller, U. K. and Watson, M. W. (2008). Testing models of low-frequency variability. Econometrica76 979-1016. · Zbl 1152.91717
[30] Patterson, K. (2011). Unit Root Tests in Time Series. Vol. 1. Palgrave Texts in Econometrics. Palgrave Macmillan, New York.
[31] Patterson, K. (2012). Unit Root Tests in Time Series. Vol. 2. Palgrave Texts in Econometrics. Palgrave Macmillan, New York.
[32] Phillips, P. C. B. (1987). Time series regression with a unit root. Econometrica55 277-301. · Zbl 0613.62109
[33] Phillips, P. C. B. and Perron, P. (1988). Testing for a unit root in time series regression. Biometrika75 335-346. · Zbl 0644.62094
[34] Rothenberg, T. J. and Stock, J. H. (1997). Inference in a nearly integrated autoregressive model with nonnormal innovations. J. Econometrics80 269-286. · Zbl 0888.62094
[35] Rudin, W. (1987). Real and Complex Analysis, 3rd ed. McGraw-Hill, New York. · Zbl 0925.00005
[36] Saikkonen, P. and Luukkonen, R. (1993). Point optimal tests for testing the order of differencing in ARIMA models. Econometric Theory9 343-362. · Zbl 0775.62240
[37] Schick, A. (1986). On asymptotically efficient estimation in semiparametric models. Ann. Statist.14 1139-1151. · Zbl 0612.62062
[38] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics3. Cambridge Univ. Press, Cambridge.
[39] White, J. S. (1958). The limiting distribution of the serial correlation coefficient in the explosive case. Ann. Math. Stat.29 1188-1197. · Zbl 0099.13004
[40] Zhou, B., van den Akker, R. and Werker, B. J. M. (2019). Supplement to “Semiparametrically point-optimal hybrid rank tests for unit roots.” DOI:10.1214/18-AOS1758SUPP.
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