Distribution free goodness-of-fit tests for linear processes. (English) Zbl 1084.62038
Summary: This article proposes a class of goodness-of-fit tests for the autocorrelation function of a time series process, including those exhibiting long-range dependence. Test statistics for composite hypotheses are functionals of a (approximated) martingale transformation of the Bartlett \(T_p\)-process with estimated parameters, which converge in distribution to the standard Brownian motion under the null hypothesis. We discuss tests of different natures such as omnibus, directional and Portmanteau-type tests. A Monte Carlo study illustrates the performance of the different tests in practice.
MSC:
| 62G10 | Nonparametric hypothesis testing |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 65C05 | Monte Carlo methods |
| 62M15 | Inference from stochastic processes and spectral analysis |
Keywords:
nonparametric model checking; spectral distribution; linear processes; martingale decomposition; local alternatives; omnibus; smooth tests; directional tests; long-range alternatives; Whittle estimatorReferences:
| [1] | Aki, S. (1986). Some test statistics based on the martingale term of the empirical distribution function. Ann. Inst. Statist. Math. 38 1–21. · Zbl 0659.62055 · doi:10.1007/BF02482496 |
| [2] | Anderson, T. W. (1997). Goodness-of-fit tests for autoregressive processes. J. Time Ser. Anal. 18 321–339. · Zbl 0885.62053 · doi:10.1111/1467-9892.00053 |
| [3] | Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201 |
| [4] | Bloomfield, P. (1973). An exponential model for the spectrum of a scalar time series. Biometrika 60 217–226. · Zbl 0261.62074 · doi:10.1093/biomet/60.2.217 |
| [5] | Box, G. E. P. and Pierce, D. A. (1970). Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. J. Amer. Statist. Assoc. 65 1509–1526. · Zbl 0224.62041 · doi:10.2307/2284333 |
| [6] | Brillinger, D. R. (1981). Time Series , Data Analysis and Theory , 2nd ed. Holden-Day, San Francisco. · Zbl 0486.62095 |
| [7] | Brockwell, P. J. and Davis, R. A. (1991). Time Series : Theory and Methods , 2nd ed. Springer, New York. · Zbl 0709.62080 |
| [8] | Brown, R. L., Durbin, J. and Evans, J. M. (1975). Techniques for testing constancy of regression relationships over time (with discussion). J. Roy. Statist. Soc. Ser. B 37 149–192. · Zbl 0321.62063 |
| [9] | Chen, H. and Romano J. P. (1999). Bootstrap-assisted goodness-of-fit tests in the frequency domain. J. Time Ser. Anal. 20 619–654. · Zbl 0940.62030 · doi:10.1111/1467-9892.00162 |
| [10] | Dahlhaus, R. (1985). On the asymptotic distribution of Bartlett’s \(U_p\)-statistic. J. Time Ser. Anal. 6 213–227. · Zbl 0602.62081 · doi:10.1111/j.1467-9892.1985.tb00411.x |
| [11] | Delgado, M. A. and Hidalgo, J. (2000). Bootstrap goodness-of-fit test for linear processes. Preprint, Universidad Carlos III de Madrid. · Zbl 0968.62041 · doi:10.1016/S0304-4076(99)00052-4 |
| [12] | Durbin, J., Knott, M. and Taylor, C. C. (1975). Components of Cramér–von Mises statistics. II. J. Roy. Statist. Soc. Ser. B 37 216–237. · Zbl 0335.62032 |
| [13] | Eubank, R. L. and LaRiccia, V. N. (1992). Asymptotic comparison of Cramér–von Mises and nonparametric function estimation techniques for testing goodness-of-fit. Ann. Statist. 20 2071–2086. · Zbl 0769.62033 · doi:10.1214/aos/1176348903 |
| [14] | Giraitis, L., Hidalgo, J. and Robinson, P. M. (2001). Gaussian estimation of parametric spectral density with unknown pole. Ann. Statist. 29 987–1023. · Zbl 1012.62098 · doi:10.1214/aos/1013699989 |
| [15] | Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle’s estimate. Probab. Theory Related Fields 86 87–104. · Zbl 0717.62015 · doi:10.1007/BF01207515 |
| [16] | Grenander, U. (1950). Stochastic processes and statistical inference. Ark. Mat. 1 195–277. · Zbl 0058.35501 · doi:10.1007/BF02590638 |
| [17] | Grenander, U. (1981). Abstract Inference. Wiley, New York. · Zbl 0505.62069 |
| [18] | Grenander, U. and Rosenblatt, M. (1957). Statistical Analysis of Stationary Time Series . Wiley, New York. · Zbl 0080.12904 |
| [19] | Hainz, G. and Dahlhaus, R. (2000). Spectral domain bootstrap tests for stationary time series. |
| [20] | Hannan, E. J. (1973). The asymptotic theory of linear time-series models. J. Appl. Probability 10 130–145. · Zbl 0261.62073 · doi:10.2307/3212501 |
| [21] | Hong, Y. (1996). Consistent testing for serial correlation of unknown form. Econometrica 64 837–864. · Zbl 0960.62559 · doi:10.2307/2171847 |
| [22] | Hosking, J. R. M. (1984). Modeling persistence in hydrological time series using fractional differencing. Water Resources Research 20 1898–1908. |
| [23] | Hosoya, Y. (1997). A limit theory for long-range dependence and statistical inference on related models. Ann. Statist. 25 105–137. · Zbl 0873.62096 · doi:10.1214/aos/1034276623 |
| [24] | Kac, M. and Siegert, A. J. F. (1947). An explicit representation of a stationary Gaussian process Ann. Math. Statist. 18 438–442. · Zbl 0033.38501 · doi:10.1214/aoms/1177730391 |
| [25] | Khmaladze, E. V. (1981). A martingale approach in the theory of goodness-of-fit tests. Theory Probab. Appl. 26 240–257. · Zbl 0481.60055 · doi:10.1137/1126027 |
| [26] | Khmaladze, E. V. and Koul, H. (2004). Martingale transforms goodness-of-fit tests in regression models. Ann. Statist. 32 995–1034. · Zbl 1092.62052 · doi:10.1214/009053604000000274 |
| [27] | Klüppelberg, C. and Mikosch, T. (1996). The integrated periodogram for stable processes. Ann. Statist. 24 1855–1879. · Zbl 0898.62116 · doi:10.1214/aos/1069362301 |
| [28] | Koul, H. and Stute, W. (1998). Regression model fitting with long memory errors. J. Statist. Plann. Inference 71 35–56. · Zbl 0931.62015 · doi:10.1016/S0378-3758(98)00016-0 |
| [29] | Koul, H. and Stute, W. (1999). Nonparametric model checks for time series. Ann. Statist. 27 204–236. · Zbl 0955.62089 · doi:10.1214/aos/1018031108 |
| [30] | Ljung, G. M. and Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika 65 297–303. · Zbl 0386.62079 · doi:10.1093/biomet/65.2.297 |
| [31] | Neyman, J. (1937). “Smooth” test for goodness of fit. Skand. Aktuarietidskr. 20 150–199. · Zbl 0018.03403 |
| [32] | Nikabadze, A. and Stute, W. (1997). Model checks under random censorship. Statist. Probab. Lett. 32 249–259. · Zbl 1003.62540 · doi:10.1016/S0167-7152(96)00081-8 |
| [33] | Paparoditis, E. (2000). Spectral density based goodness-of-fit tests for time series models. Scand. J. Statist. 27 143–176. · Zbl 0940.62084 · doi:10.1111/1467-9469.00184 |
| [34] | Prewitt, K. (1998). Goodness-of-fit test in parametric time series models. J. Time Ser. Anal. 19 549–574. · Zbl 0919.62109 · doi:10.1111/1467-9892.00108 |
| [35] | Robinson, P. M. (1994). Time series with strong dependence. In Advances in Econometrics : Sixth World Congress 1 (C. A. Sims, ed.) 47–95. Cambridge Univ. Press. |
| [36] | Robinson, P. M. (1995). Log-periodogram regression of time series with long range dependence. Ann. Statist. 23 1048–1072. · Zbl 0838.62085 · doi:10.1214/aos/1176324636 |
| [37] | Robinson, P. M. (1995). Gaussian semiparametric estimation of long-range dependence. Ann. Statist. 23 1630–1661. · Zbl 0843.62092 · doi:10.1214/aos/1176324317 |
| [38] | Schoenfeld, D. A. (1977). Asymptotic properties of tests based on linear combinations of the orthogonal components of the Cramér–von Mises statistic. Ann. Statist. 5 1017–1026. · Zbl 0369.62045 · doi:10.1214/aos/1176343956 |
| [39] | Sen, P. K. (1982). Invariance principles for recursive residuals. Ann. Statist. 10 307–312. · Zbl 0484.62098 · doi:10.1214/aos/1176345714 |
| [40] | Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics . Wiley, New York. · Zbl 1170.62365 |
| [41] | Stute, W. (1997). Nonparametric model checks for regression. Ann. Statist. 25 613–641. · Zbl 0926.62035 · doi:10.1214/aos/1031833666 |
| [42] | Stute, W., Thies, S. and Zhu, L. (1998). Model checks for regression: An innovation process approach. Ann. Statist. 26 1916–1934. · Zbl 0930.62044 · doi:10.1214/aos/1024691363 |
| [43] | Stute, W. and Zhu, L. (2002). Model checks for generalized linear models. Scand. J. Statist. 29 535–545. · Zbl 1035.62073 · doi:10.1111/1467-9469.00304 |
| [44] | Velasco, C. (1999). Non-stationary log-periodogram regression J. Econometrics 91 325–371. · Zbl 1041.62533 · doi:10.1016/S0304-4076(98)00080-3 |
| [45] | Velasco, C. and Robinson, P. M. (2000). Whittle pseudo-maximum likelihood estimation for nonstationary time series. J. Amer. Statist. Assoc. 95 1229–1243. · Zbl 1008.62087 · doi:10.2307/2669763 |
| [46] | Velilla, S. (1994). A goodness-of-fit test for autoregressive-moving-average models based on the standardized sample spectral distribution of the residuals J. Time Ser. Anal. 15 637–647. · Zbl 0807.62072 · doi:10.1111/j.1467-9892.1994.tb00218.x |
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