在現實生活中，我們學過股票的報酬隨著每天的股價變化而改變。本文中，我們透過copula之下的馬可夫鍊模型去探討股價的對數報酬的相關性。由於在股票市場上報酬有厚尾的性質，所以我們使用邊際分布為非標準學生T分配。但是自由度的最大概似估計量在此模型假設下無法求得。所以我們決定用貝氏理論方法去估計模型的參數。藉由馬可夫鍊蒙地卡羅法中的Metropolis-Hastigs 演算法可以估計出我們模型中的參數，再來利用條件機率的方式去產生有相關性的模擬資料驗證貝氏理論方法在copula之下的馬可夫鍊模型下是可以去估計的。在實證分析中，我們將選用S&P 500指數作為我們的實證分析資料。

In the real world, we learn that log returns change from the variety of the stock price everyday. In this paper, we propose a copula-based Markov model to perform the log return for the stock price. Owing to the fat tail feature in stock market, we select non-standardized Student's t-distribution being the marginal distribution. However, the maximum likelihood estimator of the degree of freedom can not be found. We decide to use bayes inference to estimate the parameters of copula-based Markov model. Using Metropolis-Hastings algorithm within Markov chain Monte Carlo method can find the bayes estimator of our model and developing the algorithm for generating correlated data by extending the conditional approach to test bayes inference can work for the copula-based Markov model. In the empirical analysis, the S&P 500 Index is analyzed for illustration.

[1] Berger, J. O. (1985) Statistical Decision Theory and Bayesian Analysis, 2nd Edition.
Springer-Verlag: New York.
[2] Carter, C. K., Kohn, R. (1994) Markov chain Monte Carlo in conditionally Gaussian
state space models. Biometrika, 83, 589-601.
[3] Darsow, W. F., Nguten, B., Olsen, E. T. (1992) Copulas and Markov Processes.
Illinois Journal of Mathematics, 36, 600-642.
[4] Emura, T. Long, T. H., Sun, L. H. (2017) R routines performing estimation and
statistical process control under copula-based time series models. Communications
in Statistics - Simulation and Computation, 46(4), 3067-3087.
[5] Frees, E. W., Valadez, E. (1998) Understanding the relationships using copulas. North
American Actuarial Journal, 2, 1-25.
[6] Gelman, A., Rubin, D. (1992) Inference from Iterative simulation using Multiple
sequences. Statistical Science, 7, 457-473.
[7] Geman, S., Geman, D. (1984) Stochastic relaxation, Gibbs distributions and the
Bayesian restoration of images. IEEE Trans. on Pattern Analysis and Machine Intelligence.
6, 721-741.
[8] Hasting, W. K. (1970) Monte Carlo sampling methods using Markov chains and their
applications. Biometrika, 57, 97-109.
32
[9] Huang, Y. L., Fan, T. H. (2013) Bayesian Reliablity Analysis of Constant-stress
Accelerated Degradation Based on Gamma Process with Random Effect and Time-
Scale Transformation. Master thesis.
[10] Joe, H. (1997) Multivariate Models and dependence. Chapman & hall.
[11] Kiefer, N. (1978) Discrete Parameter Variation: Efficient Estimation of a Switching
Regression Model. Econometrica, 2, 427-434.
[12] Lindgren, G. (1978) Markov Regime Models for Mixed Distributions and Switching
Regressions. Scandinavian Journal of Statistics, 5, 81-91.
[13] Long, T. H., Emura, T. (2014) A control chart using copula-based Markov chain
models. Journal of the Chinese Statistical Association, 52(4), 466-496.
[14] Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., Teller, E. (1953)
Equation of state calculations by fast computing machines. Journal of Chemical
Physics, 21, 1087-1092.
[15] Michael, J., Nicholas, P. (2002) MCMC Methods for Financial Econometrics. Handbook
of Financial Econometrics.
[16] Nelsen, R. B. (2006) An Introduction to Copulas, 2nd Edition. Springer Series in
Statistics, Springer-Verlag: New York.
[17] Neslehova, J. (2014) On rank correlation measures for non-continuous random variables.
Multivariate Analysis, 98, 544-567.
[18] R development Core Team (2014) R: a language and environment for statistical
computing. Foundation for Statistical Computing, R version 3.2.1.
[19] Ross, S. M. (2006) Simulation, 4th Edition. Elevier.
[20] Smith, A. F. M., Roberts, G. O. (1993) Bayesian Computation via the Gibbs sampler
and related Markov chain Monte Carlo method (with discussion). J. R. Statist. Soc.,
B, 55, 3-23.