Publications

Refereed Papers:

1. "High-dimensional Sparse Multivariate Stochastic Volatility Models" (with Benjamin Poignard), Journal of Time Series Analysis, 2023, 44, 4-22. Supplementary Material. Github Link.

2. "Estimation of High Dimensional Vector Autoregression via Sparse Precision Matrix" (with Benjamin Poignard), Econometrics Journal, 2023, 26, 307–326.

3. "Dynamic Network Poisson Autoregression with Application to COVID-19 Count Data" (with Amanda M. Y. Chu and Mike K. P. So), to appear in Journal of Data Science.

4. "Linkage Vector Autoregressive Model" (with Mike K. P. So), to appear in Applied Stochastic Models in Business and Industry.

5. "Feasible Panel GARCH Models: Variance-Targeting Estimation and Empirical Application", Econometrics and Statistics, 2023, 25, 23-38.

6. "Realized BEKK-CAW Models"  (with Mike K. P. So), Journal of Time Series Econometrics, 2023, 15, 49-77.

7. "Estimation of Realized Asymmetric Stochastic Volatility Models using Kalman Filter", Econometrics, 2023, 11(3), 18.

8. "Bayesian Nonlinear Quantile Effects on Modelling Realized Kernels" (with Manh Cuong Dong and Cathy W. S. Chen), International Journal of Finance and Economics, 2023, 28, 981-995.

9. "Realized Matrix-Exponential Stochastic Volatility with Asymmetry, Long Memory and Higher-Moment Spillovers" (with Chialin Chang and Michael McAleer), Journal of Econometrics, 2022, 227, 285-304.

10. "Multivariate Hyper-Rotated GARCH-BEKK" (with Michael McAleer), Journal of Time Series Econometrics, 2022, 14, 175-198. Supplementary Material.

11. "A New Structural Multivariate GARCH-BEKK Model: Causality of Green, Sustainable and Fossil Energy ETFs" (with Chialin Chang, Michael McAleer, and Laurent Pauwels), Communications in Statistics – Case Studies and Data Analysis, 2022, 8, 215-233.

12. "Bayesian Analysis of Realized Matrix-Exponential GARCH Models" (with Michael McAleer), Computational Economics, 2022, 59, 103-123. 

13. "Quasi-Maximum Likelihood Estimation of Conditional Autoregressive Wishart Models" (with Mike K.P. So), Journal of Time Series Analysis, 2021, 42, 271-294.

14. "On a Bivariate Hysteretic Autoregressive Model with Conditional Asymmetry in Correlations" (with Cathy W.S. Chen and Hong Than-Thi), Computational Economics, 2021, 58, 413-433. 

15. "Asymptotic and Finite Sample Properties for Multivariate Rotated GARCH Models" (with Chialin Chang, Michael McAleer, and Laurent Pauwels), Econometrics, 2021, 9(2), 21.

16. "A Simulation Smoother for Long Memory Time Series with Correlated and Heteroskedastic Additive Noise" (with Mike K. P. So), Communications in Statistics - Simulation and Computation, 2021, 50, 388-399.

17. "Cointegrated Dynamics for A Generalized Long Memory Process: Application to Interest Rates" (with Shelton Peiris, Michael McAleer, and David Allen), Journal of Time Series Econometrics, 2020, 20(1), 1-18. 

18. "Forecasting Volatility and Co-volatility of Crude Oil and Gold Futures: Effects of Leverage, Jumps, Spillovers, and Geopolitical Risks" (with Rangan Gupta and Michael McAleer), International Journal of Forecasting, 2020, 36(3), 933-948. 

19. "Realized Stochastic Volatility Models with Generalized Gegenbauer Long Memory" (with Michael McAleer and Shelton Peiris), Econometrics and Statistics, 2020, 16, 42-54.

20. "The Impact of Jumps and Leverage in Forecasting the Co-Volatility of Oil and Gold Futures" (with Rangan Gupta and Michael McAleer), Energies, 2019, 12(17), 3379.

21. "On Quasi-Maximum Likelihood Estimation of Realized Stochastic Volatility Model via Kalman Filter" (in Japanese), Journal of the Japan Statistical Society, Japanese Issue, 2019, 48(2), 215-238.  

22. “Realized Stochastic Volatility with General Asymmetry and Long Memory” (with Chia-Lin Chang and Michael McAleer), Journal of Econometrics, 2017, 199(2), 202-212.

23. “The Impact of Jumps and Leverage in Forecasting Co-volatility” (with Michael McAleer), Econometric Reviews, 2017, 36(6-9), 638-650. 

24. “Estimating and Forecasting Generalized Fractional Long Memory Stochastic Volatility Models” (with Michael McAleer and Shelton Peiris), Journal of Risk and Financial Management, 2017, 10(4), 23.

25. “Forecasting the Volatility of Nikkei 225 Futures” (with Michael McAleer), Journal of Futures Markets, 2017, 37(11), 1141-1152.

26. “Stochastic Multivariate Mixture Covariance Model” (with Mike K.P. So, Raymond W.M. Li, and Yue Jiang), Journal of Forecasting, 2017, 36(2), 139-155.

27. “A Fractionally Integrated Wishart Stochastic Volatility Model” (with Michael McAleer), Econometric Reviews, 2016, 36(1-3), 42-59.

28. “Matrix Exponential Stochastic Volatility with Cross Leverage” (with Tsunehiro Ishihara and Yasuhiro Omori), Computational Statistics & Data Analysis, 2016, 100, 331-350.

29. “Generalized Fractional Processes with Long Memory and Time Dependent Volatility Revisited” (with Shelton Peiris), Econometrics, 2016, 4(3), 37.

30. “Forecasting Co-Volatilities via Factor Models with Asymmetry and Long Memory in Realized Covariance” (with Michael McAleer), Journal of Econometrics, 2015, 189(2), 251-262.

31. “Leverage and Feedback Effects on Multifactor Wishart Stochastic Volatility for Option Pricing” (with Michael McAleer), Journal of Econometrics, 2015, 187(2), 436-446.

32. “Bayesian Analysis of General Asymmetric Multivariate GARCH Models and News Impact Curves”, Journal of the Japan Statistical Society, 2015, 45(2), 129-144.

33. “Forecasting Value-at-Risk Using Block Structure Multivariate Stochastic Volatility ” (with Massimiliano Caporin and Michael McAleer), International Review of Economics & Finance, 2015, 40, 40-50.

34. “Long Memory and Asymmetry for Matrix-Exponential Dynamic Correlation Processes” (with Mike K.P. So), Journal of Time Series Econometrics, 2015, 7(1), 69-94.

35. “Heterogeneous Asymmetric Dynamic Conditional Correlation Model with Stock Return and Range”, Journal of Forecasting, 2013, 32(5), 469-480.

36. “Stress Testing Correlation Matrices for Risk Management” (with Mike K.P. So and Jerry Wong), North American Journal of Economics and Finance, 2013, 26, 310-322.

37. “Stochastic Covariance Models” (with Mike K.P. So), Journal of the Japan Statistical Society, 2013, 43(2), 127-162.

38. “Forecasting Volatility via Stock Return, Range, Trading Volume and Spillover Effects: The Case of Brazil” (with Ivan Brugal), North American Journal of Economics and Finance, 2013, 25(2), 202-213.

39. “Asymmetry and Long Memory in Volatility Modeling” (with Michael McAleer and Marcelo C. Medeiros), Journal of Financial Econometrics, 2012, 10(3), 495-512.

40. “Modelling and Forecasting with Noisy Realized Volatility” (with Michael McAleer and Marcelo C. Medeiros), Computational Statistics & Data Analysis, 2012, 56(1), 217-230.

41. “Forecasting Volatility Using Range Data: Analysis for Emerging Equity Markets in Latin America” (with Ivan Brugal), Applied Financial Economics, 2012, 22(6), 461-470.

42. “Dynamic Conditional Correlations for Asymmetric Processes” (with Michael McAleer), Journal of the Japan Statistical Society, 2011, 41(2), 143-157.

43. “Alternative Asymmetric Stochastic Volatility Models” (with Michael McAleer), Econometric Reviews, 2011, 30(5), 548-564.

44. “General Asymmetric Stochastic Volatility Models Using Range Data: Estimation and Empirical Evidence from Emerging Equity Markets” (with Angelo Unite), Applied Financial Economics, 2010, 20(13), 1041-1049.

45. “Multivariate Stochastic Volatility, Leverage and News Impact Surfaces” (with Michael McAleer), Econometrics Journal, 2009, 12(2), 292-309.

46. “The Structure of Dynamic Correlations in Multivariate Stochastic Volatility Models” (with Michael McAleer), Journal of Econometrics, 2009, 150(2), 182-192.

47. “Bayesian Analysis of Stochastic Volatility Models with Mixture-of-Normal Distributions”, Mathematics and Computers in Simulation, 2009, 79(8), 2579-2596.

48. “A Portfolio Index GARCH Model” (with Michael McAleer), International Journal of Forecasting, 2008, 24(3), 449-461.

49. “The Relationship between Stock Return Volatility and Trading Volume: The Case of The Philippines” (with Angelo Unite), Applied Financial Economics, 2008, 18(16), 1333-1348.

50. “Autoregressive Stochastic Volatility Models with Heavy-Tailed Distributions: A Comparison with Multifactor Volatility Models”, Journal of Empirical Finance, 2008, 15(2), 332-341.

51. “A Distribution-Free Test for Symmetry with an Application to S&P Index Returns” (with Ulziijargal Dashzeveg), Applied Economics Letters, 2008, 15(6), 461-464.

52. “Portfolio Single Index (PSI) Multivariate Conditional and Stochastic Volatility Models” (with Michael McAleer and Bernardo da Veiga), Mathematics and Computers in Simulation, 2008, 78(2-3), 209-214.

53. “Non-Trading Day Effects in Asymmetric Conditional and Stochastic Volatility Models” (with Michael McAleer), Econometrics Journal, 2007, 10(1), 113-123.

54. “Multivariate Stochastic Volatility: A Review”, (with Jun Yu and Michael McAleer), Econometric Reviews, 2006, 25(2-3), 145-175.

55. “Asymmetric Multivariate Stochastic Volatility”, (with Michael McAleer), Econometric Reviews, 2006, 25(2-3), 453-473.

56. “Comparison of MCMC Methods for Estimating GARCH Models”, Journal of the Japan Statistical Society, 2006, 36(2), 199-212.

57. “Dynamic Asymmetric Leverage in Stochastic Volatility Models” (with Michael McAleer), Econometric Reviews, 2005, 24(3), 317-332.

58. “Comparison of MCMC Methods for Estimating Stochastic Volatility Models”, Computational Economics, 2005, 25(3), 281-301.

59. “Testing for Serial Correlation in the Presence of Stochastic Volatility”, Asia-Pacific Financial Markets, 2000, 7(4), 321-337. See also Asai (2019).

60. “Time Series Evidence on a New Keynesian Theory of the Output-Inflation Trade-Off", Applied Economics Letters, 1999, 6(9), 539-541.

61. “Essays in Nonstationary Financial Time Series”, Ph.D. dissertation, University of Tsukuba, 1998.

62. “A New Method to Estimate Stochastic Volatility Models: A Log-GARCH Approach”, Journal of the Japan Statistical Society, 1998, 28(1), 101-114.

63. “The Japanese Stock Market and the Macroeconomy: An Empirical Investigation” (with T. Shiba), Financial Engineering and the Japanese Markets, 1995, 2(3), 259-267.

Book Chapter:

1. “Multivariate Stochastic Volatility” (with Siddhartha Chib and Yasuhiro Omori), in: T.G. Andersen, R.A. Davis, J.-P. Kreiss and T. Mikosch(eds.), Handbook of Financial Time Series, Springer-Verlag, New York, 2009, 365-400.

2. “Option Pricing and MCMC (in Japanese)”, in: H. Wago (ed.), Bayesian Econometrics; Markov chain Monte Carlo methods and their applications, Toyo Keizai, Tokyo, 2005, 295-327.

3. “The Forecasting Performance of Models of Interests Futures: HJM Models and Others (in Japanese)” (with T. Takahashi and T. Shiba), in K. Morimune and T. Kariya, eds., Risk Management and Investment Strategy for Financial Derivatives, Toyo Keizai, Tokyo, 1998, 151-171.

Book:

1. Translation of: W. Greene, Econometric Analysis , 4th ed., Prentice Hall, 1999 (with T. Shiba, T. Takahashi, and T. Nakatsuma, published by Economist-sha, 2000)(『計量経済分析』エコノミスト社).

Reviews:

1. "(Review) Pascual, Hector; Yee, Xin C. Least squares regression principal component analysis: a supervised dimensionality reduction method. Numer. Linear Algebra Appl. 29 (2022), no. 1, Paper No. e2411, 16", Mathematical Review, MR4372447.

2. "(Review) Tang, Fan, and Kong, Precision Matrix Estimation by Inverse Principal Orthogonal Decomposition,  Commun. Math. Res., 36 (2020), pp. 68-92", Mathematical Reviews, MR4199946.  

3. "(Review) Chen, Mykland, and Zhang,  The Five Trolls Under the Bridge: Principal Component Analysis With Asynchronous and Noisy High Frequency Data, J. Amer. Statist. Assoc. (2020), no.115, 1960–1977", Mathematical Reviews, MR4189770.  

4. “(Review) Liu and Ngo, Approximation of eigenvalues of spot cross volatility matrix with a view toward principal component analysis, Jpn. J. Ind. Appl. Math. 34 (2017), no. 3, 747-761”, Mathematical Reviews, MR3719692.

5. “(Review) Gonçalves, McCracken, and Perron, Tests of equal accuracy for nested models with estimated factors, J. Economet. (2017), no.198, 231-252”, Mathematical Reviews, MR3638179.

6. “(Review) Chen and Leng, Dynamic covariance models, J. Amer. Statist. Assoc. 111 (2016), no. 515, 1196-1208”, Mathematical Reviews, MR3561942.

7. “(Review) Tao et al., Large volatility matrix inference via combining low-frequency and high-frequency approaches, J. Amer. Statist. Assoc. 106 (2011), no. 495, 1025-1040”, Mathematical Reviews, MR2894761.

Other Papers:

1. "Analysis for Increasing Vaccination Rate at Soka University (in Japanese)", Journal of Leaner-Centered Higher Education, Soka University, 2023, 12, 101-107.

2. "Asymptotic Theory for Robust Autocorrelation Test under Stochastic Volatility", Soka Economic Studies Quarterly, 2019, 49(1), 55-76.

3. "Analysis on UK Interest Rates via Long Memory Models (in Japanese)", Soka Economic Studies Quarterly, 2018, 48(1), 133-146.

4. "Covariance Matrix of Quasi-Maximum Likelihood Estimator of ARFIMA Models", Soka Economic Studies Quarterly, 2017, 47(1), 55-66.

5. “Asymptotic Theory for Extended Asymmetric Multivariate GARCH Processes” (with Michael McAleer), International Journal of Statistics and Probability, 2017, 6(6), 13-23.

6. “On Forecasting Hedge Ratio of Nikkei 225 Index Futures (in Japanese)”, Options and Futures Report, 2017, 29(7), 1-5.

7. “Options Pricing Using the Fast Fourier Transform”, Soka Economic Studies Quarterly, 2016, 45(1), 49-59.

8. “Volatility (in Japanese)”, Ritsumeikan Economic Review, 2016, 64(5), 3-18.

9. “Forecasting Volatility of Nikkei 225 Index Futures (in Japanese)”, Options and Futures Report, 2015, 27(4), 1-5.

10. “The Relationships between Nikkei 225 Index and Its Futures for Returns and Volatilities (in Japanese)”, Options and Futures Report, 2015, 27(1), 1-5.

11. “Initial Values on Quasi-Maximum Likelihood Estimation for BEKK Multivariate GARCH Models”, Soka Economic Studies Quarterly, 2015, 44(1), 45-52.

12. “Careful Consideration Index from Smokers to NonSmokers”, Soka Economic Studies Quarterly, 2013, 42(1), 55-59.

13. “Testing Granger Causality under Dynamic Covariance”, Soka Economic Studies Quarterly, 2012, 41(1), 37-46.

14. “Overreaction Index for Stock Markets”, Soka Economic Studies Quarterly, 2011, 40(1), 45-50.

15. “Lagrange Multiplier Tests (in Japanese)”, Soka Economic Studies Quarterly, 2010, 39(1), 67-78.

16. “Brownian Motion, Stochastic Differential Equations, and Term Structure Models for Interest Rates (in Japanese)”, Ritsumeikan Economic Review, 2000, 49(4), 111-129.

17. “Markov Chain Monte Carlo Simulation Methods (in Japanese)”, Ritsumeikan Economic Review, 2000, 49(1), 13-42.

18. “Error-Correction, and Granger Causality on VAR (in Japanese)”, Ritsumeikan Economic Review, 1999, 48(6), 41-59.

19. “The Econometric Analysis of Univariate Times Series (in Japanese)”, Ritsumeikan Economic Review, 1999, 48(2), 70-103.

20. “Bayesian Analysis of Stochastic Volatility Models with Heavy-Tailed Distributions”, MTEC Journal, 1999, 12, 19-40.

21. “A Method to Estimate Random Walk Stochastic Volatility Models”, Far East Journal of Theoretical Statistics, 1999, 3(1), 187-212.

Oral Presentations at International Conferences:

1. "Dynamic Network Poisson Autoregression with An Application to Covid-19 Count Data", The 25th International Conference on Computational Statistics (Hybrid Conference), Birkbeck, University of London, UK, August 22-25, 2023.  

2. "Linkage Vector Autoregressive Model", The 6th EAC-ISBA conference (Hybrid Conference), Feng Chia University, Taichung, Taiwan, July 8-9, 2022. 

3. "Realized BEKK-CAW Models", Asian Meeting of the Econometric Society in China 2022, Shenzen (Virtual), China, June 22-24, 2022.

4. "Estimation of high dimensional vector autoregression via sparse precision matrix", The 4th International Conference on Econometrics and Statistics, Virtual Conference, June 24-26, 2021

5. "Accelerated Continuous Space Topic Model for Textual Data", The 14th International Conference on Computational and Financial Econometrics, Virtual Conference, December 19-21, 2020. 

6. “Realized Stochastic Volatility Models with Generalized Gegenbauer Long Memory”,  The  2nd International Conference on Econometrics and Statistics, Hong Kong, June 19-21, 2018

7. “Realized Matrix-Exponential Stochastic Volatility with General Asymmetry, Long Memory and Spillovers”, The 14th International Symposium on Econometric Theory and Applications, Sydney, May 31-June 1, 2018. 

8. “Realized Matrix-Exponential Stochastic Volatility with General Asymmetry, Long Memory and Spillovers”, Time Series Analysis of Higher Moments and Distributions of Financial Data, Institute of Advanced Studies, Hong Kong University of Science and Technology, May 28-30, 2018. 

9. “Realized Matrix-Exponential Stochastic Volatility with Asymmetry, Long Memory and Spillovers”, The 70th European Meeting of Econometric Society, Lisbon, August 21-25, 2017.

10. “Realized Asymmetric Long Memory Stochastic Volatility Models”, The 1st International Conference on Econometrics and Statistics, Hong Kong, June 15-17, 2017.

11. “Bayesian Analysis of Alternative Long Memory Stochastic Volatility Models Using Realized Volatility Measure”, The 10th International Conference on Computational and Financial Econometrics, Sevilla, Spain, December 9-11, 2016.

12. “Bayesian Analysis of Alternative Long Memory Stochastic Volatility Models Using Realized Volatility Measure”, International Society for Bayesian Analysis 2016 World Meeting, Sardinia, Italy, June 13-17, 2016.

13. “The Impact of Jumps and Leverage in Forecasting Co-volatility”, The 11th World Congress of Econometric Society, Montreal, Canada, August 17-21, 2015.

14. “Forecasting Co-Volatilities via Factor Models with Asymmetry and Long Memory in Realized Covariance”, The 8th International Conference on Computational and Financial Econometrics, Pisa, Italy, December 6-8, 2014.

15. “Forecasting Co-Volatilities via Factor Models with Asymmetry and Long Memory in Realized Covariance”, The 5th CEQURA Conference on Advances in Financial and Insurance Risk Management, Munich, Germany, October 1-2, 2014.

16. “Forecasting Co-Volatilities via Factor Models with Asymmetry and Long Memory in Realized Covariance”, The 3rd Institute of Mathematical Statistics Asia Pacific Rim Meetings, Taiwan, June 30 - July 3, 2014.

17. “Asymmetry and Long Memory in Realized Covariance”, 59th ISI World Statistical Congress, Hong Kong, August 25-30, 2013.

18. “A Fractionally Integrated Wishart Stochastic Volatility Model”, Asian Meeting of the Econometric Society, Singapore, August 2-4, 2013.

19. “Leverage and Feedback Effects on Multifactor Wishart Stochastic Volatility”, The 9th International Symposium on Econometric Theory and Applications, Seoul, Korea, July 20-21, 2013.

20. “Leverage and Feedback Effects on Multifactor Wishart Stochastic Volatility”, China Meeting of the Econometric Society, Beijing, China, June 14-16, 2013.

21. “Long Memory and Asymmetry for Matrix-Exponential Dynamic Correlation Processes”, The 6th International Conference on Computational and Financial Econometrics, Oviedo, Spain, December 1-3, 2012.

22. “Extracting Dynamic Correlations from Stock Return and Realized Volatility”, The 32nd Annual International Symposium on Forecasting, Boston, USA, June 24-27, 2012.

23. “Heterogeneous Markets Effects for Asymmetric Dynamic Conditional Correlation Model with Stock Return and Range”, The 4th CSDA International Conference on Computational and Financial Econometrics, London, U.K., December 17-19, 2011.

24. “Continuous Time Dynamic Correlation Model”, European Meeting of the Econometric Society, Oslo, Norway, August 25-29, 2011.

25. “Continuous Time Dynamic Correlation Model”, Asian Meeting of the Econometric Society, Seoul, Korea, August 11-13, 2011.

26. “Continuous Time Dynamic Correlation Model”, Singapore Economic Review Conference 2011, Singapore, August 4-6, 2011.

27. “Continuous Time Dynamic Correlation Model”, The 31st Annual International Symposium on Forecasting, Prague, Czech Republic, June 27-29, 2011.

28. “Measuring Market Risk”, The VII International Interdisciplinary Scientific Research Congress, Santo Domingo, Dominican Republic, June 9-10, 2011.

29. “Continuous Time Dynamic Correlation Model”, The 2011 Tsinghua International Conference in Econometrics, Beijing, China, May 21-22, 2011.

30. “Stochastic Covariance Models”, The 3rd CSDA International Conference on Computational and Financial Econometrics, London, U.K., December 10-12, 2010.

31. “Stochastic Covariance Models”, Econometric Society, World Congress, Shanghai, China, August 17-21, 2010.

32. “Stochastic Covariance Models”, International Symposium on Financial Engineering and Risk Management 2010, Taipei, Taiwan, June 10-12, 2010.

33. “Causality Analysis for Common Latent Variables in Financial Time Series”, Singapore Economic Review Conference 2009, Singapore, August 6-8, 2009.

34. “Dynamic Conditional Correlations for Realized Covariance”, The 29th Annual International Symposium on Forecasting, Hong Kong, China, June 21-24, 2009.

35. “A New Method for Estimation and Forecasting with Noisy Realized Volatility”, The 28th Annual International Symposium on Forecasting, Nice, France, June 22-25, 2008.

36. “Modelling and Forecasting Daily Volatility with Noisy Realized Volatility”, Far Eastern and South Asian Meeting of the Econometric Society, Singapore, July 16-18, 2008.

37. “Multivariate Stochastic Volatility, Leverage and News Impact Surfaces”, European Meeting of the Econometric Society, Budapest, Hungary, August 27-31, 2007.

38. “Multivariate Stochastic Volatility, Leverage and News Impact Surfaces”, Far Eastern Meeting of the Econometric Society, Taipei, Taiwan, July 11-13, 2007.

39. “The Structure of Dynamic Correlations in Multivariate Stochastic Volatility Models”, Symposium on Integration of Regional Economy and Finance, and Economic Cooperation between China and Japan, Jinan University, China, March 25-26, 2007.

Invited Lectures:

1. "On Modeling, Estimation, and Forecasting Realized  Stochastic Volatility Models",  Japan Statistical Society Research Prize Lecture, Japanese Joint Statistical Meeting,  Tokyo, September 11, 2018.

2. “Realized Stochastic Volatility Models with Generalized Gegenbauer Long Memory”, “Research Ideas and Tips for Journal Publication”, Workshops on Financial/Economic Analytics, Hong Kong, March 8-9, 2018.

3. “Economic Development of Japan”, Universidad Tecnologica de Santiago, Santo Domingo, Dominican Republic, June 10, 2011.

4. “Measuring Market Risks”, Superintendence of Banks, Santo Domingo, Dominican Republic, June 7, 2011.

5. “Bayesian Markov chain Monte Carlo method”, The 6th Young Economists' Convention, Junior Philippine Economics Society and De La Salle University Economics Organization, February 3rd, 2007.