Bayesian Analysis

A Bayesian approach to estimating the long memory parameter

Sounak Chakraborty, Scott Holan, and Tucker McElroy

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We develop a Bayesian procedure for analyzing stationary long-range dependent processes. Specifically, we consider the fractional exponential model (FEXP) to estimate the memory parameter of a stationary long-memory Gaussian time series. In particular, we propose a hierarchical Bayesian model and make it fully adaptive by imposing a prior distribution on the model order. Further, we describe a reversible jump Markov chain Monte Carlo algorithm for variable dimension estimation and show that, in our context, the algorithm provides a reasonable method of model selection (within each repetition of the chain). Therefore, through an application of Bayesian model averaging, we incorporate all possible models from the FEXP class (up to a given finite order). As a result we reduce the underestimation of uncertainty at the model-selection stage as well as achieve better estimates of the long memory parameter. Additionally, we establish Bayesian consistency of the memory parameter under mild conditions on the data process. Finally, through simulation and the analysis of two data sets, we demonstrate the effectiveness of our approach.

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Bayesian Anal., Volume 4, Number 1 (2009), 159-190.

First available in Project Euclid: 22 June 2012

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Bayesian model averaging FEXP hierarchical Bayes long-range dependence reversible jump Markov chain Monte Carlo Spectral density


Holan, Scott; McElroy, Tucker; Chakraborty, Sounak. A Bayesian approach to estimating the long memory parameter. Bayesian Anal. 4 (2009), no. 1, 159--190. doi:10.1214/09-BA406.

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  • Beran, J. (1993). “Fitting long-memory models by generalized linear regression.” Biometrika, 80(4): 817–822.
  • — (1994). Statistics for Long Memory Processes. New York: Chapman & Hall.
  • Bertelli, S. and Caporin, M. (2002). “A Note on Calculating Autocovariances of Long-memory Processes.” Journal of Time Series Analysis, 23(5): 503–508.
  • Bloomfield, P. (1973). “An exponential model for the spectrum of a scalar time series.” Biometrika, 60: 217–226.
  • Bötcher, A. and Silberman, B. (1999). Introduction to Large Truncated Toeplitz Matrices. New York: Springer-Verlag.
  • Breidt, F. J. and Hsu, N.-J. (2002). “A Class of Nearly Long-memory Time Series Models.” International Journal of Forecasting, 18(2): 265–281.
  • Chen, W., Hurvich, C. M., and Lu, Y. (2006). “On the Correlation Matrix of the Discrete Fourier Transform and the Fast Solution of Large Toeplitz Systems for Long-memory Time Series.” Journal of the American Statistical Association, 101(474): 812–822.
  • Cheung, Y.-W. and 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.
  • Chib, S. and Greenberg, E. (1995). “Understanding the Metropolis-Hastings algorithm.” The American Statistician, 49: 327–335.
  • Chib, S. and Jeliazkov, I. (2001). “Marginal likelihood from the Metropolis-Hastings output.” Journal of the American Statistical Association, 96: 270–281.
  • Dahlhaus, R. (1989). “Efficient Parameter Estimation for Self-similar Processes.” The Annals of Statistics, 17: 1749–1766.
  • Davies, R. B. and Harte, D. S. (1987). “Tests for Hurst Effect.” Biometrika, 74: 95–101.
  • Denison, D. G. T., Holmes, C. C., Mallick, B. K., and Smith, A. F. M. (2002). Bayesian methods for nonlinear classification and regression. Wiley Series in Probability and Statistics. Chichester: John Wiley & Sons Ltd.
  • Denison, D. G. T., Mallick, B. K., and Smith, A. F. M. (1998). “A Bayesian CART Algorithm.” Biometrika, 85: 363–377.
  • Fox, R. and Taqqu, M. S. (1986). “Large-sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series.” The Annals of Statistics, 14: 517–532.
  • Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference Second Ed.. New York: Chapman & Hall/CRC.
  • Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2004). Bayesian data analysis. Texts in Statistical Science Series. Chapman & Hall/CRC, Boca Raton, FL, second edition.
  • Geweke, J. and Porter-Hudak, S. (1983). “The Estimation and Application of Long Memory Time Series Models.” Journal of Time Series Analysis, 4: 221–238.
  • 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.” Probability Theory and Related Fields, 86: 87–104.
  • Giraitis, L. and Taqqu, M. (1999). “Whittle Estimator for Finite-variance Non-Gaussian Time Series with Long Memory.” The Annals of Statistics, 27(1): 178–203.
  • Green, P. J. (1995). “Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination.” Biometrika, 82: 711–732.
  • Haslett, J. and Raftery, A. E. (1989). “Space-time Modelling with Long-memory Dependence: Assessing Ireland’s Wind Power Resource.” Applied Statistics, 38: 1–21.
  • Hoeting, J. A., Madigan, D., Raftery, A. E., and Volinsky, C. T. (1999). “Bayesian Model Averaging: a Tutorial.” Statistical Science, 14(4): 382–417.
  • Holan, S., McElroy, T., and Chakraborty, S. (2007). “A Bayesian approach to estimating the long memory parameter.” SRD Research Report No. 2007/13, US Census Bureau,
  • Hosking, J. R. M. (1981). “Fractional Differencing.” Biometrika, 68: 165–176.
  • Hurvich, C. M. (2001). “Model Selection for Broadband Semiparametric Estimation of Long Memory in Time Series.” Journal of Time Series Analysis, 22(6): 679–709.
  • — (2002). “Multistep Forecasting of Long Memory Series Using Fractional Exponential Models.” International Journal of Forecasting, 18(2): 167–179.
  • Hurvich, C. M. and Brodsky, J. (2001). “Broadband Semiparametric Estimation of the Memory Parameter of a Long-memory Time Series Using Fractional Exponential Models.” Journal of Time Series Analysis, 22(2): 221–249.
  • Janacek, G. J. (1982). “Determining the Degree of Differencing for Time Series Via the Log Spectrum.” Journal of Time Series Analysis, 3: 177–183.
  • Kass, R. and Raferty, A. (1995). “Bayes Factors.” Journal of the American Statistical Association, 90: 773–795.
  • Ko, K. and Vannucci, M. (2006a). “Bayesian wavelet analysis of autoregressive fractionally integrated moving-average processes.” Journal of Statistical Planning and Inference, 136: 3415–3434.
  • — (2006b). “Bayesian wavelet-based methods for the detection of multiple changes of the long memory parameter.” IEEE Transactions on Signal Processing, 54: 4461–4470.
  • Liseo, B., Marinucci, D., and Petrella, L. (2001). “Bayesian semiparametric inference on long-range dependence.” Biometrika, 88(4): 1089–1104.
  • Manley, G. (1974). “Central England temperatures: monthly means 1659 to 1973.” Q. J. R. Meteorology Soc., 100: 389–405.
  • Moulines, E. and Soulier, P. (2000). “Data Driven Order Selection for Projection Estimator of the Spectral Density of Time Series with Long Range Dependence.” Journal of Time Series Analysis, 21(2): 193–218.
  • Pai, J. S. and Ravishanker, N. (1998). “Bayesian Analysis of Autoregressive Fractionally Integrated Moving-average Processes.” Journal of Time Series Analysis, 19: 99–112.
  • Palma, W. (2007). Long-Memory Time Series. New York: John Wiley and Sons.
  • Petris, G. (1997). “Bayesian Analysis of Long Memory Time Series.” Ph.D. thesis, Duke University.
  • Pourahmadi, M. (1983). “Exact Factorization of the Spectral Density and Its Application to Forecasting and Time Series Analysis.” Communications in Statistics: Theory and Methods, 12: 2085–2094.
  • R Development Core Team (2007). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. URL
  • Ravishanker, N. and Ray, B. K. (1997). “Bayesian Analysis of Vector ARFIMA Processes.” The Australian & New Zealand Journal of Statistics, 39: 295–311.
  • Robert, C. (2001). Bayesian Choice. New York: Springer-Verlag.
  • Robert, C. P. and Casella, G. (2004). Monte Carlo statistical methods. Springer Texts in Statistics. New York: Springer-Verlag, second edition.
  • Robinson, P. M. (1994). “Efficient Tests of Nonstationary Hypotheses.” Journal of the American Statistical Association, 89: 1420–1437.
  • — (1995a). “Log-periodogram Regression of Time Series with Long Range Dependence.” The Annals of Statistics, 23: 1048–1072.
  • — (1995b). “Gaussian Semiparametric Estimation of Long Range Dependence.” The Annals of Statistics, 23: 1630–1661.
  • — (2003). Time Series With Long Memory. Oxford: Oxford University Press.
  • Rousseau, J. and Liseo, B. (2007). “Bayesian nonparametric estimation of the spectral density of a long memory Gaussian time series.”
  • Schwarz, G. (1978). “Estimating the Dimension of a Model.” The Annals of Statistics, 6: 461–464.
  • Sisson, S. A. (2005). “Transdimensional Markov Chains: A Decade of Progress and Future Perspectives.” Journal of the American Statistical Association, 100(471): 1077–1089.
  • Tierney, L. and Kadane, J. B. (1986). “Accurate Approximations for Posterior Moments and Marginal Densities.” Journal of the American Statistical Association, 81: 82–86.