• Bernoulli
  • Volume 18, Number 4 (2012), 1448-1464.

Inference of seasonal long-memory aggregate time series

Kung-Sik Chan and Henghsiu Tsai

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Time-series data with regular and/or seasonal long-memory are often aggregated before analysis. Often, the aggregation scale is large enough to remove any short-memory components of the underlying process but too short to eliminate seasonal patterns of much longer periods. In this paper, we investigate the limiting correlation structure of aggregate time series within an intermediate asymptotic framework that attempts to capture the aforementioned sampling scheme. In particular, we study the autocorrelation structure and the spectral density function of aggregates from a discrete-time process. The underlying discrete-time process is assumed to be a stationary Seasonal AutoRegressive Fractionally Integrated Moving-Average (SARFIMA) process, after suitable number of differencing if necessary, and the seasonal periods of the underlying process are multiples of the aggregation size. We derive the limit of the normalized spectral density function of the aggregates, with increasing aggregation. The limiting aggregate (seasonal) long-memory model may then be useful for analyzing aggregate time-series data, which can be estimated by maximizing the Whittle likelihood. We prove that the maximum Whittle likelihood estimator (spectral maximum likelihood estimator) is consistent and asymptotically normal, and study its finite-sample properties through simulation. The efficacy of the proposed approach is illustrated by a real-life internet traffic example.

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Bernoulli, Volume 18, Number 4 (2012), 1448-1464.

First available in Project Euclid: 12 November 2012

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asymptotic normality consistency seasonal auto-regressive fractionally integrated moving-average models spectral density spectral maximum likelihood estimator Whittle likelihood


Chan, Kung-Sik; Tsai, Henghsiu. Inference of seasonal long-memory aggregate time series. Bernoulli 18 (2012), no. 4, 1448--1464. doi:10.3150/11-BEJ374.

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  • [1] Beran, J. (1994). Statistics for Long-Memory Processes. Monographs on Statistics and Applied Probability 61. New York: Chapman & Hall.
  • [2] Bisognin, C. and Lopes, S.R.C. (2007). Estimating and forecasting the long memory parameter in the presence of periodicity. J. Forecast. 26 405–427.
  • [3] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [4] Chambers, M.J. (1996). The estimation of continuous parameter long-memory time series models. Econometric Theory 12 374–390.
  • [5] Chan, K.S. and Tsai, H. (2008). Inference of seasonal long-memory aggregate time series. Technical Report 391, Dept. Statistics & Actuarial Science, Univ. Iowa.
  • [6] Granger, C.W.J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1 15–29.
  • [7] Hosking, J.R.M. (1981). Fractional differencing. Biometrika 68 165–176.
  • [8] Hosoya, Y. (1996). The quasi-likelihood approach to statistical inference on multiple time-series with long-range dependence. J. Econometrics 73 217–236.
  • [9] Hsu, N.J. and Tsai, H. (2009). Semiparametric estimation for seasonal long-memory time series using generalized exponential models. J. Statist. Plann. Inference 139 1992–2009.
  • [10] Lopes, S.R.C. (2008). Long-range dependence in mean and volatility: Models, estimation and forecasting. In In and Out of Equilibrium 2. Progress in Probability 60 497–525. Basel: Birkhäuser.
  • [11] Man, K.S. and Tiao, G.C. (2006). Aggregation effect and forecasting temporal aggregates of long memory processes. International Journal of Forecasting 22 267–281.
  • [12] Montanari, A., Rosso, R. and Taqqu, M. (2000). A seasonal fractional ARIMA model applied to Nile River monthly flows at Aswan. Water Resources Research 36 1249–1259.
  • [13] Ohanissian, A., Russell, J.R. and Tsay, R.S. (2008). True or spurious long memory? A new test. J. Bus. Econom. Statist. 26 161–175.
  • [14] Palma, W. and Bondon, P. (2003). On the eigenstructure of generalized fractional processes. Statist. Probab. Lett. 65 93–101.
  • [15] Palma, W. and Chan, N.H. (2005). Efficient estimation of seasonal long-range-dependent processes. J. Time Ser. Anal. 26 863–892.
  • [16] Porter-Hudak, S. (1990). An application of the seasonal fractionally differenced model to the monetary aggregates. J. Amer. Statist. Assoc. 85 338–344.
  • [17] Ray, B.K. (1993). Long-range forecasting of IBM product revenues using a seasonal fractionally differenced ARMA model. International Journal of Forecasting 9 255–269.
  • [18] Tsai, H. and Chan, K.S. (2005). Temporal aggregation of stationary and nonstationary discrete-time processes. J. Time Ser. Anal. 26 613–624.
  • [19] Wei, W.W.S. (1978). Some consequences of temporal aggregation seasonal time series models. In Seasonal Analysis of Economic Time Series (A. Zellner, ed.). Washington, DC: US Department of Commerce, Bureau of Census.