Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-08-01T20:29:49.877Z Has data issue: false hasContentIssue false
  • English
  • Français

ASYMPTOTIC THEORY FOR MAXIMUM LIKELIHOOD ESTIMATION OF THE MEMORY PARAMETER IN STATIONARY GAUSSIAN PROCESSES

Published online by Cambridge University Press:  13 September 2011

Offer Lieberman ,
Roy Rosemarin  and
Judith Rousseau
Offer Lieberman*
Affiliation:
University of Haifa
Roy Rosemarin
Affiliation:
London School of Economics
Judith Rousseau
Affiliation:
CEREMADE, University Paris Dauphine
*
*Address correspondence to Offer Lieberman, Department of Economics, University of Haifa, Haifa 31905, Israel; e-mail: offerl@econ.haifa.ac.il.
Get access Rights & Permissions [Opens in a new window]

Abstract

Consistency, asymptotic normality, and efficiency of the maximum likelihood estimator for stationary Gaussian time series were shown to hold in the short memory case by Hannan (1973, Journal of Applied Probability 10, 130–145) and in the long memory case by Dahlhaus (1989, Annals of Statistics 34, 1045–1047). In this paper we extend these results to the entire stationarity region, including the case of antipersistence and noninvertibility.

Type
MISCELLANEA
Information
Econometric Theory , Volume 28 , Issue 2 , April 2012 , pp. 457 - 470
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abadir, K.M., Distaso, W., & Giraitis, L. (2007) Nonstationarity-extended local Whittle estimation. Journal of Econometrics 141, 1353–1384.CrossRefGoogle Scholar
Adenstedt, R.K. (1974) On large-sample estimation for the mean of a stationary random sequence. Annals of Statistics 2, 1095–1107.CrossRefGoogle Scholar
Beran, J. (1991) M-estimator of location for data with slowly decaying serial correlations. Journal of the American Statistical Association 86, 704–708.Google Scholar
Beran, J., Feng, Y., Franke, G., Hess, D., & Ocker, D. (2003) Semiparametric modeling of stochastic and deterministic trends and fractional stationarity. In Rangarajan, G. & Ding, M., (eds.) Processes with Long Range Correlations: Theory and Applications, pp. 225–250. Springer.CrossRefGoogle Scholar
Cheung, Y.W. & Diebold, F.X. (1994) On maximum likelihood estimation of the differencing parameter of fractionally integrated noise with unknown mean. Journal of Econometrics 62, 301–316.CrossRefGoogle Scholar
Dahlhaus, R. (1989) Efficient parameter estimation for self-similar processes. Annals of Statistics 17, 1749–1766.CrossRefGoogle Scholar
Dahlhaus, R. (2006) Correction note: Efficient parameter estimation for self-similar processes. Annals of Statistics 34, 1045–1047.CrossRefGoogle Scholar
Fox, R. & Taqqu, M.S. (1986) Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Annals of Statistics 14, 157–532.CrossRefGoogle Scholar
Fox, R. & Taqqu, M.S. (1987) Central limit theorems for quadratic forms in random variables having long-range dependence. Probability Theory and Related Fields 74, 213–240.CrossRefGoogle Scholar
Granger, C.W.J. & Joyeux, R. (1980) An introduction to long-range time series models and fractional differencing. Journal of Time Series Analysis 1, 15–31.CrossRefGoogle Scholar
Hannan, E.J. (1973) The asymptotic theory of linear time series models. Journal of Applied Probability 10, 130–145.CrossRefGoogle Scholar
Hauser, M.A. (1999) Maximum likelihood estimators for ARMA and ARFIMA models: A Monte Carlo study. Journal of Statistical Planning and Inference 80, 229–255.CrossRefGoogle Scholar
Hosking, J.R.M. (1981) Fractional differencing. Biometrika 68, 165–176.CrossRefGoogle Scholar
Karuppiah, J. & Los, C.A. (2005) Wavelet multiresolution analysis of high-frequency Asian FX rates, summer 1997. International Review of Financial Analysis 14, 211–246.CrossRefGoogle Scholar
Lieberman, O. & Phillips, P.C.B. (2004) Expansions for the distribution of the maximum likelihood estimator of the fractional difference parameter. Econometric Theory 20, 464–484.CrossRefGoogle Scholar
Lieberman, O., Rosemarin, R., & Rousseau, J. (2010) Asymptotic Theory for Maximum Likelihood Estimation of the Memory Parameter in Stationary Gaussian Processess (full version). Available athttp://www.ceremade.dauphine.fr/~rousseau/LRRlongmemo.html.CrossRefGoogle Scholar
Mandelbrot, B.B. & Van Ness, J.W. (1968) Fractional Brownian motions, fractional noises and applications. SIAM Review 10, 422–37.CrossRefGoogle Scholar
Nielsen, M. & Frederiksen, P.H. (2005) Finite sample comparison of parametric, semiparametric, and wavelet estimators of fractional integration. Econometric Reviews 24, 405–443.CrossRefGoogle Scholar
Peters, E.E. (1994) Fractal Market Analysis: Applying Chaos Theory to Investment and Economics. Wiley.Google Scholar
Polard, D. (1984) Convergence of Stochastic Processes. Wiley.CrossRefGoogle Scholar
Robinson, P.M. (1995a) Gaussian semiparametric estimation of long-range dependence. Annals of Statistics 23, 1630–1661.CrossRefGoogle Scholar
Robinson, P.M. (1995b) Log-periodogram regression of time series with long range dependence. Annals of Statistics 23, 1048–1072.CrossRefGoogle Scholar
Samarov, A. & Taqqu, M.S. (1988) On the efficiency of the sample mean in long-memory noise. Journal of Time Series Analysis 9, 191–200.CrossRefGoogle Scholar
Shao, X. (2010) Nonstationarity-extended Whittle estimation. Econometric Theory 26, 1060–1087.CrossRefGoogle Scholar
Shimotsu, K. (2010) Exact local Whittle estimation of fractional integration with unknown mean and time trend. Econometric Theory 26, 501–540.CrossRefGoogle Scholar
Shimotsu, K. & Phillips, P.C.B. (2004) Exact local Whittle estimation of fractional integration. Annals of Statistics 33, 1890–1933.Google Scholar
Shiryaev, A.N. (1999) Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific.CrossRefGoogle Scholar
Sowell, F.B. (1992) Maximum likelihood estimation of stationary univariate fractionally integrated time series models. Journal of Econometrics 53, 165–188.CrossRefGoogle Scholar
Terrin, N. & Taqqu, M.S. (1990) A noncentral limit theorem for quadratic forms of Gaussian stationary sequences. Journal of Theoretical Probability 3, 449–475.CrossRefGoogle Scholar
Tsai, H. (2009) On continuous-time autoregressive fractionally integrated moving average processes. Bernoulli 15, 178–194.CrossRefGoogle Scholar
Velasco, C. (1999a) Gaussian semiparametric estimation of nonstationary time series. Journal of Time Series Analysis 20, 87–127.CrossRefGoogle Scholar
Velasco, C. (1999b) Non-stationary log-periodogram regression. Journal of Econometrics 91, 325–371.CrossRefGoogle Scholar
Velasco, C. & Robinson, P.M. (2000) Whittle pseudo-maximum likelihood estimation for nonstationary time series. Journal of the American Statistical Association 95, 1229–1243.CrossRefGoogle Scholar
Yajima, Y. (1985) On estimation of long-memory time series models. Australian Journal of Statistics 27, 303–20.CrossRefGoogle Scholar