過去二十多年已見證越來越多的非線性計量方法被應用在金融市場的實證研究。即使金融理論迄今還沒提供夠充分理由去解釋為何某些非線性實證模型可以使用，但由於非線性模型有時可捕捉到線性模型所無法解釋的現象，因此這些工具應該被研究人員適度的關注。本論文介紹三種最近發展的非線性計量方法，並呈現這些方法在金融市場之研究的應用。特別的是，這些非線性方法有助於分析變數間的動態因果關係。
本論文由三個篇文章所組成，每篇文章介紹一種非線性因果檢定方法之應用。第一篇文章是「美國與日本黃金期貨市場之線性與非線性動態」。這篇文章使用Diks與Panchenko (2006)所提出的無母數因果關係檢定法去探討世界前兩大黃金期貨之間的關聯。研究結果顯示，這兩個市場的報酬具有雙向的因果關係。但是，以FIGARCH模型控制市場波動後，僅發現美國對日本黃金期貨市場存在非線性因果關係。此結果表示，日本對美國的非線性因果關係幾乎純由市場波動所導致，但波動效果僅能解釋部分的反向非線性因果關係。
第二篇文章的題目是「台灣現貨與股價指數期貨之不對稱暨非線性動態關係」。這篇文章先利用Mackey-Glass時間數列模型去建構現貨與期貨的關係，進而執行基於此模型的因果關係檢定法，以便檢視這兩個市場變數的非線性動態關聯。研究結果顯示有雙向因果關係存在，而此非線性因果關幾乎係由市場波動所驅動。特別的是，進一步結果顯示，非線因果關係在股市多頭和股市空頭下呈現非常不對稱之情況。多頭市場下，現貨對期貨具有非線性因果關係，但在空頭市場下卻是期貨對現貨具有非線性因果關係。
第三篇文章的題目則是「台灣股價與交易量間狀態相依性之研究」。採用馬可夫轉換向量自我迴歸模型(Markov Switching Vector Autoregression model)去建構台灣股票市場的價量關係，我發現：2個狀態（高波動與低波動）的MS-VAR模型適用於描繪價量關係。另外，這篇文章分別運用狀態相依的Granger因果檢定法與衝擊反應函數，分析不同狀態下的價量因果關係，結果顯示：交易量變動對股價報酬的影響只發生在股市高波動的情況下，但不論股市處在高波動或低波動狀態，不論波動狀態為何，股價報酬對交易量都有影響。

The past twenty years have witnessed increasing nonlinear econometric methods used in the empirical study of financial markets. Despite financial theory commonly does not provide enough reason to support the use of the nonlinear models yet, the nonlinear tools sometimes can capture the phenomenon which cannot be explained by linear models. Therefore researchers should attach importance to these tools. This dissertation describes three types of recently-developed nonlinear econometric methods and display how these methods are applied to the studies of the financial markets. In particular, they can help researchers explore the dynamic causal relationship between financial variables. 
This dissertation consists of three essays. The first essay is "Linear and Nonlinear Dynamics between U.S. and Japanese gold futures market". Using Diks and Panchenko (2006) non-parametric causality test to explore the linkage between the world's top two gold futures markets, the study results show evidence of two-way causality. After controlling for market volatility with FIAGRCH models, only the nonlinear causality from U.S. to Japanese gold futures market is found. This suggests that the nonlinear causal from Japanese to U.S. gold futures market is almost a result of market volatility, but the volatility effect can only explain a part of nonlinear reverse causality. 
The second essay is "Asymmetric and Nonlinear Dynamic Relationship between Taiwan Spot and Stock Index Futures ". This article uses the Mackey-Glass (MG) time-series model to construct the relationship between spot and futures, and then performs the based MG causality tests. The study results show existence of the bidirectional nonlinear causality, and that the nonlinear causal almost is driven by market volatility. Further results demonstrate asymmetric nonlinear causality under good and bad news. Under good news, the spot returns nonlinearly causality index futures returns, but under the bad news the case is reversal.
The topic of the third essay is "Regime Dependence between Stock Prices and Trading Volume in Taiwan". This article uses the Markov Switching Vector Autoregression (MS-VAR) model to construct the price-volume relation in the Taiwan stock market. Results show that the two-regime MS-VAR model is suitable to describe the price-volume relationship. Moreover, the regime-dependent Granger causality tests and regime-independent impulse response functions are performed to analyze price-volume relationship under different regimes. Results show that the causality from volume changes to stock returns only exists in the high volatility regime, while stock returns cause volume changes irrespective of regimes.

Acknowledgements in Chinese	Ⅱ
Abstract in Chinese	Ⅶ
Abstract	Ⅷ
Chapter 	
1 Introduction to the Dissertation	1
2 Linear and Nonlinear Dynamics between the U.S. and Japanese Gold futures Markets	4
2.1 Introduction	6
2.2 Methodology	8
2.2.1 Linear Granger causality testing	9
2.2.2 Nonlinear Granger causality testing	10
2.3 Data	12
2.4 Results and Discussion	14
2.4.1 Unit root and Cointegration tests	14
2.4.2 Tests for Linear Granger Causality	17
2.4.3 Tests for Nonlinear Granger Causality	18
2.5 Conclusion	22
3 Asymmetrical and Nonlinear Dynamic Relationship between Spot and Stock Index Futures in Taiwan	24
3.1 Introduction	26
3.2 Methodology	29
3.2.1 Gregory-Hansen (GH) cointegration test	30
3.2.2 Tests for the nonlinear Granger causality	31
3.3 Data	32
3.4 Results and Discussion	32
3.5 Conclusion	40
4 The dynamics between stock prices and trading volume in Taiwan:
Evidence of regime dependence	42
4.1 Introduction	44
4.2 Literature Review	46
4.2.1 The theory	46
4.2.2 Empirical evidence	48
4.3 Methodology	50
4. 3.1 The Markov switching vector autoregression model	50
4. 3.2 Regime-independent and regime-dependent Granger causality tests	51
4. 3.3 Regime-dependent impulse response functions	52
4.4 The data	53
4.5 Empirical results and Discussion	53
4.6 Conclusion	61
5 Summary and Conclusion	63
Appendix	65
Reference	66

List of Figures
Figure 4.1 regime-dependent response of return to a shock in trading volume	62
Figure 4.2 regime-dependent response of trading volume to a shock in stock returns	63

List of Tables
Table 2.1 Preliminary statistics for daily returns of gold futures markets	13
Table 2.2 Unit Root Tests for daily gold futures prices		15
Table 2.3 Cointegration tests	17
Table 2.4 Results of Linear Granger causality test	18
Table 2.5 Results of Nonlinear Granger causality test	19
Table 2.6 Estimated FIGARCH Models	21
Table 2.7 Result of nonlinear causality with FIGARCH-filtered residuals	23
Table 3.1 Results of Unit Root Tests	34
Table 3.2 Results of the Gregory-Hansen cointegration tests	35
Table 3.3 Results of BDS test for spot and index futures return residuals	37
Table 3.4 Test for nonlinear causality	38
Table 3.5 Test for nonlinear causality under good and bad news	41
Table 4.1 Unit root test results	56
Table 4.2 Test for regime switching	57
Table 4.3 Estimated results of the MS-VAR model	60
Table 4.4 Regime-independent and regime-dependent Granger causality tests	61

Abhyankar, A. (1998). Linear and nonlinear Granger causality: evidence from the U.K. stock index futures market. Journal of Futures Markets, 18(5), 519-540. doi:10.1002/(SICI)1096-9934(199808)18:5<519::AID-FUT2>3.0.CO;2-U
Aruga, E. & Managi, S. (2011). Tests on price linkage between the U.S. and Japanese gold and silver futures markets. Economics Bulletin, 31(2), 1038-1046.
Baek, E., & Brock, W. (1992). A general test for nonlinear granger causality: Bivariate model. Unpublished manuscript.
Baillie, R. T., Han, Y., Myers, R. J., & Song, J. (2007). Long memory models for daily and high frequency commodity futures returns. Journal of Futures Markets, 27(7), 643–668. doi:10.1002/fut.20267
Beine, M., Capelle-Blancard, G., & Raymond, H. (2008). International nonlinear causality between stock markets. The European Journal of Finance, 14(8), 663. doi:10.1080/13518470802042112
Bekiros, S. D., & Diks, C. G. (2008). The relationship between crude oil spot and futures prices: Cointegration, linear and nonlinear causality. Energy Economics, 30(5), 2673-2685. doi:10.1016/j.eneco.2008.03.006
Blume, L., Easley, D., & O'hara, M. (1994). Market statistics and technical analysis: the role of volume. Journal of Finance, 49(1), 153-181.
Brock, W. A., Scheinkman, J. A., Dechert, W. D., & LeBaron, B. (1996). A test for independence based on the correlation dimension. Econometric Reviews, 15(3), 197-235. doi:10.1080/07474939608800353
Campbell, J. Y., Grossman, S. J., & Wang, J. (1993). Trading volume and serial Correlation in Stock Returns. Quarterly Journal of Economics, 108(4), 905-939. doi: 10.2307/2118454.
Chatrath, A., Adrangi, B., & Shank, T. (2001). Nonlinear dependence in gold and silver futures: is it chaos? The American Economist, 45(2), 25–32.
Chan, K. (1992). A further analysis of the lead-lag relationship between the cash market and stock index futures Market. Review of Financial Studies, 5(1), 123-152.
Chen, G., Firth, M., & Rui, O. M. (2001). The dynamic relation between stock returns, trading volume, and volatility. The Financial Review, 36(3), 153-174.
Ciner, C. (2002). The stock price-volume linkage on the Toronto stock exchange: before and after automation. Review of Quantitative Finance and Accounting, 19(4), 335–349. doi:10.1023/A:1021109325128
Clark, P. K. (1973). A subordinated stochastic process model with finite variance for speculative prices. Econometrica, 41(1), 135-155.
Copeland, T. E. (1976). A model of asset trading under the assumption of sequential information arrival. Journal of Finance, 31(4), 1149-1168.
Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 74(1), 33-43.
De Gooijer, J. G., & Sivarajasingham, S. (2008). Parametric and nonparametric granger causality testing: linkages between international stock markets. Physica A: Statistical Mechanics and its Applications, 387(11), 2547–2560. doi:10.1016/j.physa.2008.01.033
De Long, J. B., Shleifer, A., Summers, L. H., & Waldmann, R. J. (1990). Positive feedback investment strategies and destabilizing rational speculation. Journal of Finance, 45(2), 379-395.
DeJong, D. N., Nankervis, J. C., Savin, N. E., & Whiteman, C. H. (1992). The power problems of unit root tests in time series with autoregressive errors. Journal of Econometrics, 53(1-3), 323-344.
Diks, C., & Panchenko, V. (2005). A note on the hiemstra-jones test for granger non-causality. Studies in Nonlinear Dynamics and Econometrics, 9(2), 1–9.
Diks, C., & Panchenko, V. (2006). A new statistic and practical guidelines for nonparametric granger causality testing. Journal of Economic Dynamics and Control, 30(9-10), 1647–1669. doi:10.1016/j.jedc.2005.08.008
Ehrmann, M., Ellison, M., & Valla, N. (2003). Regime-dependent impulse response functions in a Markov-switching vector autoregression model. Economics Letters, 78(3), 295-299.
Elliott, G. (1999). Efficient tests for a unit root when the initial observation is drawn from its unconditional distribution. International Economic Review, 40(3), 767–783.
Elliott, G., Rothenberg, T. J., & Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813–836.
Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: representation, estimation, and testing. Econometrica, 55(2), 251–276.
Engle, R. F., & Kroner, K. F. (1995). Multivariate simultaneous generalized arch. Econometric Theory, 11(1), 122-150.
Epps, T. W. (1977). Security price changes and transaction volumes: some additional evidence. Journal of Financial and Quantitative Analysis, 12(1), 141-146.
Epps, T. W., & Epps, M. L. (1976). The stochastic dependence of security price changes and transaction volumes: implications for the mixture-of-distributions hypothesis. Econometrica, 44(2), 305-321.
Fleming, J., Ostdiek, B., & Whaley, R. E. (1996). Trading costs and the relative rates of price discovery in stock, futures, and option markets. Journal of Futures Markets, 16(4),353-387. doi:10.1002/(SICI)1096-9934(199606)16:4<353::AID-FUT1>3.0.CO;2-H
Fujihara, R. A., & Mougoue, M. (1997). An examination of linear and nonlinear causal relationships between price variability and volume in petroleum futures markets. Journal of Futures Markets, 17(4), 385–416. doi:10.1002/(SICI)1096-9934(199706) 17:4<385::AID-FUT2>3.0.CO;2-D
Gallant, A., Rossi, P., & Tauchen, G. (1992). Stock prices and volume. Review of Financial Studies, 5(2), 199-242.
Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica, 37(3), 424-438.
Granger, C. W. J., & Morgenstern, O. (1963). Spectral analysis of New York stock market prices. Kyklos, 16(1), 1-27. doi:10.1111/j.1467-6435.1963.tb00270.x
Gregory, A. W., & Hansen, B. E. (1996). Residual-based tests for cointegration in models with regime shifts. Journal of Econometrics, 70(1), 99-126. doi:10.1016/0304 -4076(69)41685-7
Grunbichler, A., Longstaff, F. A., & Schwartz, E. S. (1994). Electronic Screen Trading and the Transmission of Information: An Empirical Examination. Journal of Financial Intermediation, 3(2), 166-187. doi:10.1006/jfin.1994.1002
Gunduz, L., & Hatemi-J, A. (2005). Stock price and volume relation in emerging markets. Emerging Markets Finance and Trade, 41(1), 29-44.
Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business Cycle. Econometrica, 57(2), 357-384. doi:10.2307/1912559
He, H., & Wang, J. (1995). Differential information and dynamic behavior of stock trading volume. Review of Financial Studies, 8(4), 919-972. 
Hiemstra, C., & Jones, J. D. (1994). Testing for linear and nonlinear Granger causality in the Stock price-volume relation. Journal of Finance, 49(5), 1639-1664. doi: 10.2307/2329266. 
Hristu-Varsakelis, D., & Kyrtsou, C. (2008). Evidence for nonlinear asymmetric causality in US inflation, metal, and stock returns. Discrete Dynamics in Nature and Society, Volume 2008, Article ID 138547. doi:10.1155/2008/138547
Hsieh, D. A. (1991). Chaos and nonlinear dynamics: application to financial markets. Journal of Finance, 46(5), 1839-1877.
Jain, P. C., & Joh, G. (1988). The dependence between hourly prices and trading volume. Journal of Financial and Quantitative Analysis, 23(3), 269-283. doi: 10.2307/ 2331067. 
Jennings, R. H., Starks, L. T., & Fellingham, J. C. (1981). An equilibrium model of asset trading with sequential information arrival. Journal of Finance, 36(1), 143-161.
Johansen, S., Granger, C. W. J., & Mizon, G. E. (1995). Likelihood-based inference in cointegrated vector autoregressive models (Vol. 9). Cambridge Univ Press.
Johansen, Soren. (1991). Estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive models. Econometrica, 59(6), 1551–1580. doi: 10.2307/2938278
Kanas, A., & Ioannidis, C. (2010). Causality from real stock returns to real activity: evidence of regime-dependence. International Journal of Finance and Economics, 15(2), 180-197.
Kanas, A., & Kouretas, G. P. (2007). Regime dependence between the official and parallel foreign currency markets for US dollars in Greece. Journal of Macroeconomics, 29(2), 431-449. 
Karpoff, J. M. (1987). The relation between price changes and trading volume: a survey. Journal of Financial and Quantitative Analysis, 22(1), 109-126. doi: 10.2307/ 2330874. 
Kawaller, I. G., Koch, P. D., & Koch, T. W. (1987). The temporal price relationship between S&P 500 futures and the S&P 500 index. Journal of Finance, 42(5), 1309-1329.
Krolzig, H. M. (1997). Markov-switching vector autoregressions: modelling, statistical inference, and application to business cycle analysis. Lectures notes in economics and mathematical systems, 454. Springer-Verlag.
Krolzig, H. M., & Toro, J. (1999). A new approach to the analysis of shocks and the cycle in a model of output and employment. Economics Working Essays. Retrieved from http://ideas.repec.org/p/eui/euiwps/eco99-30.html
Kumar, A. (2008). A Markov-switching vector error correction model of the Indian stock prices and trading volume. Available at SSRN: http://ssrn.com/abstract=689661
Kwiatkowski, D., Phillips, P. C. B., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of econometrics, 54(1-3), 159–178.
Kyrtsou, C., & Labys, W. C. (2006). Evidence for chaotic dependence between US inflation and commodity prices. Journal of Macroeconomics, 28(1), 256-266. doi:10.1016/j.jmacro.2005.10.019
Kyrtsou, C., & Terraza, M. (2003). Is it possible to study chaotic and ARCH behaviour jointly? application of a noisy Mackey–Glass equation with heteroskedastic errors to the Paris stock exchange returns series. Computational Economics, 21(3), 257-276. doi:10.1023/A:1023939610962
Lamoureux, C. G., & Lastrapes, W. D. (1990). Heteroskedasticity in stock return data: volume versus GARCH effects. Journal of Finance, 45(1), 221-229.
Lin, H.-N., Chiang, S.-M., & Chen, K.-H. (2008). The dynamic relationships between gold futures markets: evidence from COMEX and TOCOM. Applied Financial Economics Letters, 4(1), 19. doi:10.1080/17446540701262868
MacKinnon, J. G. (1994). Approximate asymptotic distribution functions for unit-root and cointegration tests. Journal of Business and Economic Statistics, 12(2), 167-176.
MacKinnon, J. G. (1996). Numerical distribution functions for unit root and cointegration tests. Journal of Applied Econometrics, 11(6), 601–618. doi:10.1002/(SICI)1099-1255(199611)11:6<601::AID-JAE417>3.0.CO;2-T
MacKinnon, J. G. (1996). Numerical distribution functions for unit root and cointegration tests. Journal of Applied Econometrics, 11(6), 601–618. doi:10.1002/(SICI)1099-1255(199611)11:6<601::AID-JAE417>3.0.CO;2-T
MacKinnon, J. G., Haug, A. A., & Michelis, L. (1999). Numerical distribution functions of likelihood ratio tests for cointegration. Journal of Applied Econometrics, 14(5), 563–577.
Maheu, J. M., & McCurdy, T. H. (2000). Identifying bull and bear markets in stock reurns. Journal of Business and Economic Statistics, 18(1), 100-112.
Min, J. H., & Najand, M. (1999). A further investigation of the lead-lag relationship between the spot market and stock index futures: early evidence from Korea. Journal of Futures Markets, 19(2), 217-232. doi:10.1002/(SICI)1096-9934(199904)19:2 <217::AID-FUT5>3.0.CO;2-8
Newey, W. K., & West, K. D. (1994). Automatic lag selection in covariance matrix estimation. The Review of Economic Studies, 61(4), 631–653. doi:10.2307/2297912
Ozdemir, Z. A., & Cakan, E. (2007). Non-linear dynamic linkages in the international stock markets. Physica A: Statistical Mechanics and its Applications, 377(1), 173–180. doi:10.1016/j.physa.2006.11.013
Peguin-Feissolle, A., & Terasvirta, T. (1999). A general framework for testing the Granger noncausality hypothesis. SSE/EFI Working Essay Series in Economics and Finance, 343.
Perron, P. (1989). The great crash, the oil price shock, and the unit root hypothesis. Econometrica, 57(6), 1361-1401.
Psaradakis, Z., Sola, M., & Spagnolo, F. (2004). On Markov error-correction models, with an application to stock prices and dividends. Journal of Applied Econometrics, 19(1), 69-88. doi:10.1002/jae.729
Rashid, A. (2007). Stock prices and trading volume: An assessment for linear and nonlinear Granger causality. Journal of Asian Economics, 18(4), 595-612.
Robles-Fernandez, M. D., Nieto, L., & Fernandez, M. A. (2004). Nonlinear intraday dynamics in Eurostoxx50 index markets. Studies in Nonlinear Dynamics and Econometrics, 8(4), 1-28.
Rogalski, R. J. (1978). The dependence of prices and volume. Review of Economics and Statistics, 60(2), 268-274.
Saatcioglu, K., & Starks, L. T. (1998). The stock price–volume relationship in emerging stock markets: the case of Latin America. International Journal of Forecasting, 14(2), 215-225. 
Sarno, L., & Valente, G. (2000). The cost of carry model and regime shifts in stock index futures markets: An empirical investigation. Journal of Futures Markets, 20(7), 603-624. doi:10.1002/1096-9934(200008)20:7<603::AID-FUT1>3.0.CO;2-X
Shyy, G., Vijayraghavan, V., & Scott-Quinn, B. (1996). A further investigation of the lead-lag relationship between the cash market and stock index futures market with the use of bid/ask quotes: the case of France. Journal of Futures Markets, 16(4), 405-420. doi:10.1002/(SICI)1096-9934(199606)16:4<405::AID-FUT3>3.0.CO;2-M
Silvapulle, P., & Choi, J. (1999). Testing for linear and nonlinear granger causality in the stock price-volume relation: Korean evidence. Quarterly Review of Economics and Finance, 39(1), 59-76. doi: 10.1016/S1062-9769(99)80004-0. 
Smirlock, M., & Starks, L. (1985). A further examination of stock price changes and transaction volume. Journal of Financial Research, 8(3), 217-225. 
Smirlock, M., & Starks, L. (1988). An empirical analysis of the stock price-volume relationship. Journal of Banking and Finance, 12(1), 31-42. 
Stoll, H. R., & Whaley, R. E. (1990). The dynamics of stock index and stock index futures returns. Journal of Financial and Quantitative Analysis, 25(4), 441-468.
Suominen, M. (2001). Trading volume and information revelation in stock markets. Journal of Financial and Quantitative Analysis, 36(4), 545-565.
Wang, J. (1994). A model of competitive stock trading volume. Journal of Political Economy, 102(1), 127-168.
Warne, A. (2000). Causality and regime inference in a Markov switching VAR. Sveriges riksbank. 
Xu, X. E., & Fung, H. G. (2005). Cross-market linkages between U.S. and Japanese precious metals futures trading. Journal of International Financial Markets, Institutions and Money, 15(2), 107–124. doi:10.1016/j.intfin.2004.03.002
Yang, S. R., & Brorsen, B. W. (1993). Nonlinear dynamics of daily futures prices: conditional heteroskedasticity or chaos? Journal of Futures Markets, 13(2), 175–191. doi:10.1002/fut.3990130205
Ying, C. C. (1966). Stock market prices and volumes of sales. Econometrica, 34(3), 676-685.
Yoruk, N., Erdem, C., & Erdem, M. S. (2006). Testing for linear and nonlinear Granger causality in the stock price-volume relation: Turkish banking firms' evidence. Applied Financial Economics Letters, 2(3), 165-171.
Zivot, E., & Andrews, D. W. K. (1992). Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of Business and Economic Statistics, 10(3), 251-270.