許多實務上的研究都顯示，股票波動與其交易量都具有長相關，且可用分數差分過程來描述。這個研究主要目的就是去探討它們之間的關係。這篇文章中我們採用四種估計波動的方法，包含對數報酬平方法、歷史資料估計法、疊代t的估計法以及GARCH模型估計法。結果顯示在四種估計波動方法中，對數報酬平方法通常具有最大的分數差分階次，以及最高比例的分數共整合現象。分數差分的階次有聯合估計與分別估計，而共整合參數分別可用簡單線性迴歸、頻譜方面的迴歸以及半參數估計法來得到。最後對無分數共整合的序列也會對它們建立模型關係。

Many empirical studies show that both equity volatility and its trading volume have long range dependence and can be modeled as fractional integrated processes. The objective of this study is to investigate relationship between volatility and volume.We adopt four estimators of volatility, which includes the squared log returns, historical volatility, iterative t estimators and $GARCH$ estimators. The results show that among the four estimators squared log returns usually have the largest integration orders and produce hightest ratios of fractional cointegration. The fractional integrated orders are estimated separately and jointly, and the cointegration parameters are estimated by ordinary least squares, a narrow band frequency domain least squares method and a semiparametric estimator of Whittle likelihood. Models are also established when volatility and volume are not fractional cointegrated.

1. Introduction 1
2. Literature review 5
3. Cross spectrum analysis 8
4. Semiparametric estimation in long memory models 11
5. Results 20
6. Conclusion 24

1. Andersen, T. G. (1996). Return volatility and trading volume: An information flow
interpretation of stochastic volatility, Journal of Finance 51, 169-204.
2. Andersen, T. G., Bollerslev, T., Diebold, F. X. and Ebens, H. (2001). The distribution
of realized stock return volatility, Journal of Financial Economics 61, 43-76.
3. Andersen, T. G., Bollerslev, T., Diebold, F. X. and Labys, P. (2001). The distribution
of exchange rate volatility, Journal of the American Statistical Association 96, 42-55.
4. Bollerslev, T. and Jubinski, D. (1999). Equity trading volume and volatility: Latent
information arrivals and common long-run dependencies, Journal of Business and
Economic Statistics 17, 9-21.
5. Breidt, F. J., Crato, N. and de Lima, P. (1998). On the detection and estimation of
long memory in stochastic volatility, Journal of Econometrics 83, 325-348.
6. Christensen, B. J. and Nielsen, M. O. (2001). Semiparametric analysis of stationary
fractional cointegration and the implied-realized volatility relation, Working Paper
Department of Economics, University of Aarhus.
7. Christensen, B. J. and Prabhala, N. R. (1998). The relation between implied and
realized volatility, Journal of Financial Economics 50, 125-150.
8. Clark, P. K. (1973). A subordinated stochastic process model with finite variance for
speculative prices, Econometrica 41,135-155.
9. Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from
incomplete data via the EM algorithm, Journal of the Royal Statistical Society, Series
B 39, 1-38.
10. Dzhaparidze, K. (1986). Parameter estimation and hypothesis testing in spectral
analysis of stationary time series. Springer-Verlag, 35-66, New York.
11. Fox, R. and Taqqu, M. S. (1986). Large-sample properties of parameter estimates for
strongly dependent stationary Gaussian time series, Annals of Statistics 14, 517-532
12. Fuller, W. A. (1996). Introduction to statistical time series (2nd ed.), Wiley, New
York.
13. 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.
14. Granger, C. W. J. (1981). Some properties of time series dada and their use in
econometric model specification, Journal of Econometrics 16, 121-130.
15. Hassler, U., mARMOL, F. and Velasco, C. (2000). Residual log-periodogram inference
for long-ren relationships, Forthcoming in Journal of Econometrics.
16. Henry, M. and Zaffaroni, P. (2003). The long range dependence paradigm for macroeconomics
and finance, in Doukhan P., Oppenheim G. and Tauua M. S., eds, ’Theory
and applications of long-range dependence’, Birkhauser, Boston, pp.417-438.
17. Johansen, S. (1988). Statistical analysis of cointegration vectors, Journal of Economic
Dynamics Control 12, 231-254.
18. Lo, A. W. and Wang, J. (1996). Trading volume, part 1: Definitions, data analysis
and implacations of portfolio theory, unpublished manuscript, Massachusetts Institute
of Technology, Sloan School of Management.
19. Lobato, I. N. (1999). A semiparametric two-step estimator in a multivariate long
memory model, Journal of Econometrics 90, 129-153.
20. Lobato, I. N. and Robinson, P. M. (1996). Averaged periodogram estimation of long
memory, Journal of Econometrics 73, 303-324.
21. Lobato, I. N. and Velasco, C. (2000). Long memory in stock-market trading volume,
Journal of Business and Economic Statistics 18, 410-427.
22. Marinucci, D. and Robinson, P. M. (2001). Semiparametric fractional cointegration
analysis, Journal of Econometrics 105, 225-247.
23. Nielsen, M. O. (2004). Local Whittle analysis of stationary fractional cointegration
and the implied-realied volatility relation, Working Paper Cornell University.
24. Randal, J. A., Thomson, P. J. and Lally, M. T.. Non-parametric estimation of
historical volatility, Working Paper.
25. Ray, B. K. and Tsay, R. S. (2000). Long-range dependence in daily stock volatilities,
Journal of Business and Ecnomic Statistics 18, 254-262.
26. Robinson, P.M. (1994). Semiparametric analysis of long-memory time series, Annals
of Statistics 22, 515-539.
27. Robinson, P.M. (1995a). Gaussian semiparametric estimation of long range dependence,
Annals of Statistics 23, 1630-1661.
28. Robinson, P.M. (1995b). Log-periodogram regression of time series with long range
dependence, Annals of Statistics 25, 1048-1072.
29. Robinson, P. M. and Marinucci, D. (2003). Semiparametric frequency domain analysis
of fractional cointegration, in P. M. Robinson, ed, ’Time series with long memory’,
Oxford University Press, Oxford, pp.334-373.
30. Robinson, P. M. and Yajima, Y. (2002). Determination of cointegrating rank in
fractional systems, Journal of Economics 106, 217-241.
31. Tsay, R. S.(2002). Analysis of financial time series. Wiley Inter-Science, 93-95,
United States of America.
32. Watson, M. W. (1994). Vector autoregressions and cointegration, in R. F. ENGLE
and D. L. McFadden, eds, ’Handbook of Econometrics, Vol. IV’, North-Holland,
Amsterdam, chapter 47, pp. 2843-2915.
33. Wei, W. S. (1997). Time series analysis. Addism Wesley, 316-322, New York.