在本篇論文中，我們探討關於多元線性時間序列迴歸模型之參數估計，其中解釋變數與干
擾項皆為長期記憶過程。Robinson and Hidalgo(1997)和Hidalgo and Robinson(2002)提出
頻率領域加權最小平方法，估計迴歸模型之係數，他們證明出此估計量能夠達到高斯-馬可夫界
以及具有n^(1/2)的收斂速度。本文中我們提出時間領域廣義最小平方法，藉由對干擾項配適自我
迴歸模型，進而建構干擾項之自我共變異數矩陣，得到最佳線性不偏估計量。模擬結果顯示，
與Robinson and Hidalgo(1997)和Hidalgo and Robinson(2002)相較，當考慮的干擾項模
型為FARIMA過程，且AR或MA之參數的值較大的情況下，我們提出的方法有較大的相對效
率。此外，利用變異數縮減方法，考慮時間領域與頻率領域估計量之線性組合，提出一個變異
數縮減的估計量，其中線性組合的權重，我們利用拔靴法來估計。從模擬結果得知，新提出的
降低變異數的估計量可有效改善估計的效率性。

In this paper, we study the parameter estimation of the multiple linear time series
regression model with long memory stochastic regressors and innovations. Robinson and
Hidalgo (1997) and Hidalgo and Robinson (2002) proposed a class of frequency-domain
weighted least squares estimates. Their estimates are shown to achieve the Gauss-Markov
bound with standard convergence rate. In this study, we proposed a time-domain generalized LSE approach, in which the inverse autocovariance matrix of the innovations is estimated via autoregressive coefficients. Simulation studies are performed to compare the proposed estimates with Robinson and Hidalgo (1997) and Hidalgo and Robinson (2002). The results show the time-domain generalized LSE is comparable to Robinson and Hidalgo (1997) and Hidalgo and Robinson (2002) and attains higher efficiencies when the
autoregressive or moving average coefficients of the FARIMA models have larger values.
A variance reduction estimator, called TF estimator, based on linear combination of the
proposed estimator and Hidalgo and Robinson (2002)'s estimator is further proposed to
improve the efficiency. Bootstrap method is applied to estimate the weights of the linear combination. Simulation results show the TF estimator outperforms the frequency-domain as well as the time-domain approaches.

1 Introduction 1
2 Background 2
2.1 Long memory model 2
2.2 Model selection criterion 3
3 Frequency-domain generalized LSE 5
3.1 Robinson and Hidalgo (1997) method 5
3.2 Hidalgo and Robinson (2002) method 8
4 Newly proposed method 9
4.1 Time-domain generalized LSE 9
4.2 TF estimator 16
5 Simulation study 18
5.1 Simulation procedure 18
5.2 Comparison of the three methods 20
6 Conclusion 22
References 24
Appendix 25
A.1 Tables 25
A.1.1 Robinson and Hidalgo (1997) 25
A.1.2 Hidalgo and Robinson (2002) 27
A.1.3 Time-domain generalized LSE Method 33
A.1.4 TF Method 36
A.2 Figures 38
A.3 Summary of Tables and Figures 75

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