本文提出一個用來描述長記憶季節相關的新模型，並稱之為generalized exponential model (GEXP) 模型。有關其統計推論，提供了兩種建構在frequency domain下的估計方法，一為OLS法、另一則為Lasso法。我們進一步用模擬的方法，探討這兩種方法的表現。結果顯示，用Lasso法來估計長距相關參數較不易受到Gegenbauer frequency估計不佳的影響。此外，當樣本數增加時，以Lasso法所選取的模式會與OLS法所選取的模式相近。當真實的GEXP模型有較少非零的參數時，利用Lasso法來選取變數的成功率亦會比使用OLS法來的高。在實證分析上，以太陽黑子資料作實例探討。

A new class of models, generalized exponential model (GEXP), is proposed for modeling seasonal long-memory processes which is a combination of a Gegenbauer model and a Bloomfield model. For inference, two estimation procedures are proposed in the frequency-domain; one is the traditional OLS approach the other is the Lasso approach. Due to different estimation methods, different model determination criteria are used. The performance of the Lasso estimator is investigated and compared with those obtained by the traditional OLS method in finite sample via simulation studies. Based on simulation results, we find that the long-memory estimate by the Lasso approach is less sensitive to bad performance of the estimator for the Gegenbauer frequency. We also find that, for the data with larger sample size, the model selected by the Lasso approach is similar to the OLS approach. But, for most of the cases, the OLS approach provides smaller MISE which measures the difference between the actual and the fitted spectral densities. For illustration, the methodology is applied to the sunspot data.

References
Baillie, R.T. and T. Bollerslev. (1993), Cointegration, fractional cointegration, and exchange rate dynamics, Journal of Finance, 49, 737-745.

Beran, J. (1993), Fitting long memory models by generalized regression, Biometrika, 80, 817-822.

Bloomfield, P. (1973), An exponential model for the spectrum of a scalar time series, Biometrika, 60, 217-226.

Chung, C.-F. (1994), A generalized fractionally integrated autoregressive moving-average process, Journal of Time Series Analysis, 17, 111-140.

Chung, C.-F. (1996), Estimating a generalized long memory process, Journal of Econometrics, 73, 237-259.

Efron, B., Tibshishirani, R. (1993), An introduction to the Bootstrap, Chapman and Hall, London.

Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004), Least angle regression, The Annals of Statistics, 32, 407-451.

Geweke, J. and Porter-Hudak, S. (1983), The estimation and application of long-memory time series models, Journal of Time Series Analysis, 4, 87-104.

Giraitis, L., and R. Leipus (1995) A generalized fractionally differencing approach in long-memory modeling, Lithuanian Mathematical Journal, 53-65

Gray, H.L., N.-F. Zhang, and W.A. Woodward. (1989), On generalized fractional process, Journal of Time Series Analysis, 10, 233-257.

Hurvish, C. M. (1987), Automatic selection of a linear predictor through frequency domain cross-validation, Communications in Statistics: Theory and Methods, 16, 3199-3234.

Hurvish, C. M. (2002), Multistep forcasting of long memory series using fractional exponential models, International Journal of Forecasting, 18, 167-179.

Montanari, A., R. Rosso, and M. S. Taqqu (2000), A seasonal fractional ARIMA model applied to hydrologic time series: identification, estimation and simulation, Water Resources Research, 33, 1035-1044.

Mouline E, Soulier P.H. (2000), Data driven order selection for projection estimator of the spectral density of time series with long range dependence, Journal of Time Series, 21, 193-218.

Mouline E, Soulier P.H. (1999), Broadband log-periodogram regression of time series with long range dependence, The Annals of Statistics, 27, 1415-1439.

Ooms, M. (1995) Flexible seasonal long memory and economic time series. Technique report, Erasmus University Rotterdam, Econometric Institute.

Ray, B.K. (1993), Long-range forecasting for IBM products revenues using a seasonal fractionally differenced ARMA model, International Journal of Forecasting, 9, 225-269.

Porter-Hudak, S. (1990), An application of seasonal fractionally differenced model to monetary aggregates, Journal of American Statistical Association, 85, 338-344.

Robinson, P. (1994), Time series with strong dependence, Advances in Econometrics, Sixth Worth Congress, Vol. 1, Cambridge University Press, 47-95.

Robinson, P. (1995), Log-periodogram regression of time-series with long range dependence, The annals of Statistics, 23, 1048-1072.

Tibshirani, R. (1996), Regression shrinkage and selection via the Lasso, Journal of the Royal Statistical Society Series B, 58, 267-288.

Walker, A. M. (1964), Asymptotic properties of least-squares estimates of parameters of the spectrum of a stationary non-deterministic time series, Journal of the Australian Mathematical Society, 4, 363-384.

Yajima, Y. (1996), Estimation of the frequency of unbounded spectral densities, Proceeding of the Business and Economic Statistical Section, ASA.