This paper investigates the maximum likelihood estimator (MLE) for unstable autoregressive moving-average (ARMA) time series with the noise sequence satisfying a general autoregressive heteroscedastic (GARCH) process. Under some mild conditions, it is shown that the MLE satisfying the likelihood equation exists and is consistent. The limiting distribution of the MLE is derived in a unified manner for all types of characteristic roots on or outside the unit circle and is expressed as a functional of stochastic integrals in terms of Brownian motions. For various types of unit roots, the limiting distribution of the MLE does not depend on the parameters in the moving-average component and hence, when the GARCH innovations reduce to usual white noises with a constant conditional variance, they are the same as those for the least squares estimators (LSE) for unstable autoregressive models given by Chan and Wei (1988). In the presence of the GARCH innovations, the limiting distribution will involve a sequence of independent bivariate Brownian motions with correlated components. These results are different from those already known in the literature and, in this case, the MLE of unit roots will be much more efficient than the ordinary least squares estimation.
References
AITCHISON, J. and SILVEY, S. D. 1958 . Maximum-likelihood estimation of parameters subject to restraints. Ann. Math. Statist. 29 813 828. Z .
BAI, J. 1993 . On the partial sums of residuals in autoregressive and moving average models. J. Time Ser. Anal. 14 247 260. Z .
BILLINGSLEY, P. 1968 . Convergence of Probability Measures. Wiley, New York. Z .
BOLLERSLEV, T. 1986 . Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307 327. Z .
BOLLERSLEV, T., CHOU, R. Y. and KRONER, K. F. 1992 . ARCH modeling in finance. J. Econometrics 52 5 59. Z .
BOLLERSLEV, T., ENGLE, R. F. and NELSON, D. B. 1994 . ARCH model. In Handbook of EconoZ . metrics IV R. F. Engle and D. C. McFadden, eds. 2959 3038. North-Holland, Amsterdam. Z . Z .
CHAN, N. H. and WEI, C. Z. 1987 . Asy mptotic inference for nearly nonstationary AR 1 processes. Ann. Statist. 15 1050 1063. Z .
CHAN, N. H. and WEI, C. Z. 1988 . Limiting distributions of least squares estimates of unstable autoregressive processes. Ann. Statist. 16 367 401. Z .
CHUNG, K. L. 1968 . A Course in Probability Theory. Harcourt, Brace and World, New York. Z .
CHUNG, K. L. and WILLIAMS, R. J. 1990 . Introduction of Stochastic Integration, 2nd ed. Birkhauser, Boston. ¨ Z .
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.
DICKEY, D. A., HASZA, D. P. and FULLER, W. A. 1984 . Testing for units roots in seasonal time series. J. Amer. Statist. Assoc. 79 355 367. Z .
ENGLE, R. F. 1982 . Autoregressive conditional heteroskedasticity with estimates of variance of U.K. inflation. Econometrica 50 987 1008. Z .
FULLER, W. A. 1976 . Introduction to Statistical Time Series. Wiley, New York. Z .
HASZA, D. P. and FULLER, W. A. 1979 . Estimation for autoregressive processes with unit roots. Ann. Statist. 7 1106 1120. Z .
JEGANATHAN, P. 1991 . On the asy mptotic behavior of least squares estimators in AR time series with roots near the unit circle. Econometric Theory 7 269 306. Z .
KIM, K. and SCHMIDT, P. 1993 . Unit root tests with conditional heteroskedasticity. J. Econometrics 59 287 300. Z .
KURTZ, T. G. and PROTTER, P. 1991 . Weak limit theorems to stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035 1070. Z . Z .
LEE, S.-W. and HANSEN, B. E. 1994 . Asy mptotic theory for GARCH 1, 1 quasi-maximum likelihood estimator. Econometric Theory 10 29 52. Z .
LING, S. 1995 . On probability properties of a double threshold ARMA conditional heteroskedasticity models. Technical report, Dept. Statistics, Univ. Hong Kong. Z .
LING, S. and LI, W. K. 1996 . Limiting distributions of maximum likelihood estimators for unstable ARMA time series with GARCH errors. Technical report, Dept. Statistics, Univ. Hong Kong. Z .
LING, S. and LI, W. K. 1997a . On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity. J. Amer. Statist. Assoc. To appear. Z . Z .
LING, S. and LI, W. K. 1997b . Estimating and testing for unit root processes with GARCH 1, 1 errors. Technical report, Dept. Statistics, Univ. Hong Kong. Z . Z .
NELSON, D. B. 1990 . Stationarity and persistence in the GARCH 1, 1 model. Econometric Theory 6 318 334. Z .
PANTULA, S. G. 1989 . Estimation of autoregressive models with ARCH errors. Sankhy a Ser. B 50 119 138. Z .
PANTULA, S. G. and HALL, A. 1991 . Testing for unit roots in autoregressive moving average models. J. Econometrics 48 325 353. Z .
PETERS, T. A. and VELOCE, W. 1988 . Robustness of unit root tests in ARMA models with generalized ARCH errors. Unpublished manuscript. Z .
PHILLIPS, P. C. B. 1987 . Time series regression with a unit root. Econometrica 55 277 301. Z .
PHILLIPS, P. C. B. and DURLAUF, S. N. 1986 . Multiple time series regression with integrated processes. Rev. Econom. Stud. LIII 473 495. Z .
TSAY, R. S. and TIAO, G. C. 1984 . Consistent estimates of autoregressive parameters and the extended sample autocorrelation function of nonstationary and stationary ARMA models. J. Amer. Statist. Assoc. 79 84 96. Z .
TSAY, R. S. and TIAO, G. C. 1990 . Asy mptotic properties of multivariate nonstationary processes with applications to autoregressions. Ann. Statist. 18 220 250. Z .
WEISS, A. A. 1986 . Asy mptotic theory for ARCH models: estimation and testing. Econometric Theory 2 107 131. Z .
WOOLDRIDGE, J. M. and WHITE, H. 1988 . Some invariance principles and central limit theorems for dependent heterogeneous processes. Econometric Theory 4 210 230. Z .
YAP, S. F. and REINSEL, G. C. 1995a . Estimation and testing for unit roots in a partially nonstationary vector autoregressive moving average model. J. Amer. Statist. Assoc. 90 253 267. Z .
YAP, S. F. and REINSEL, G. C. 1995b . Results on estimation and testing for unit roots in the nonstationary autoregressive moving average model. J. Time Ser. Anal. 16 339 353.
