在微結構理論的研究中，如何估計高頻資料中真實價格的累積波動是一個重
要的問題。Bandi 和 Russell (2006) 提出如何求最佳取樣頻率的方法，而Zhang
等人 (2005) 則提出了一個「兩個尺度」的統計量來解決這個問題。在這篇論文
中，我們提出了一個建立在訊干比統計量的累積波動估計值，其收斂速率是
Op (n^(−1/ 4) )。這個方法可以應用在常數及隨機波動模型，且改善了Zhang 等人 (2005)
估計值為Op (n^(−1/ 6) ) 的收斂速度。此外在常數波動模型下，我們證明了此統計量達
到與最大概似估計值相同的漸近效率。另外，我們分別對微結構噪音的變異數與
真實對數報酬率的四階動差提出了不偏估計量，其結果有助於累積波動的估計。
對本篇提出的估計值，我們將探討其理論的漸近性質與效率性，並藉由模擬的方
法進行驗證。

Estimating the integrated volatility of high frequency realized prices is an important
issue in microstructure literature. Bandi and Russell (2006) derived the optimal-sampling
frequency, and Zhang et al. (2005) proposed a "two-scales estimator" to solve the problem.
In this study, we propose a new estimator based on a signal to noise ratio statistic with
convergence rate of Op (n^(−1/ 4) ). The method is applicable to both constant and stochastic
volatility models and modi‾es the Op (n^(−1/ 6) ) convergence rate of Zhang et al. (2005). The
proposed estimator is shown to be asymptotic e±cient as the maximum likelihood estimate
for the constant volatility case. Furthermore, unbiased estimators of the two elements, the
variance of the microstructure noise and the fourth moment of the realized log returns, are
also proposed to facilitate the estimation of integrated volatility. The asymptotic prop-
erties and e®ectiveness of the proposed estimators are investigated both theoretically and
via simulation study.

1 Introduction 1
2 Literature review 3
3 An estimator based on Snr 5
3.1 Covariance matrix of the estimator . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Modified estimator of E . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Modified unbiased estimators of Q . . . . . . . . . . . . . . . . . . . . . . . 11
4 Approximate solutions and asymptotic efficiency 13
4.1 Approximate solutions to µi's . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Asymptotic efficiency of the proposed estimator . . . . . . . . . . . . . . . 14
5 Simulation results 14
5.1 Q and M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.2 Integrated volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Conclusion 16
Appendix A 17
Appendix B 22
Appendix C 27
References 36

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