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Estimation of Multivariate Stochastic Volatility Models: A Comparative Monte Carlo Study

Year 2016, Volume: 8 Issue: 2, 19 - 52, 30.09.2016
https://doi.org/10.33818/ier.278044

Abstract

In this paper, we compare the small sample performances of Quasi Maximum Likelihood (QML) and Monte Carlo Likelihood (MCL) methods through Monte Carlo studies for several multivariate stochastic volatility models, among which we consider two new models that account for leverage effects. Our results confirm previous findings within the literature, namely, that the MCL estimator has better finite sample performance compared to the QML estimator. QML estimator's performance is closer to that of MCL estimator when the volatility processes have higher variance or when the correlations are high and/or time varying, but it performs relatively worse when leverage is introduced. Finally, we include an empirical illustration by estimating an MSV model with leverage using a trivariate data from the major European stock markets.

References

  • Aguilar, O. and M. West (2000). Bayesian Dynamic Factor Models and Portfolio Allocation. Journal of Business and Economic Statistics, 18, 338–357.
  • Alizadeh, S., M.W. Brandt and F.X. Diebold (2002). Range based estimation of stochastic volatility models. Journal of Finance, 57, 1047–1091.
  • Andersen, T., H. Chung and B. Sorensen (1999). Efficient method of moments estimation of a stochastic volatility model: a Monte Carlo study. Journal of Econometrics, 91, 61–87.
  • Andersen, T. and T. Bollerslev (1998). Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review, 39 (4), 885–905.
  • Asai, M. and M. McAleer (2005). Dynamic asymmetric leverage in stochastic volatility models. Econometric Reviews, 24, 317–332.
  • Asai, M. and M. McAleer (2006). Asymmetric Multivariate Stochastic Volatility. Econometric Reviews, 25 (2-3), 453–473.
  • Asai, M., M. McAleer and J. Yu (2006). Multivariate Stochastic Volatility: A Review. Econometric Reviews, 25 (2-3), 145–175.
  • Asai, M. and M. McAleer (2009). The structure of dynamic correlations in multivariate stochastic models. Journal of Econometrics, 150, 182–192.
  • Asai, M. and M. McAleer (2011). Dynamic conditional correlations for asymmetric processes. Journal of the Japan Statistical Society, 41 (2), 143–157.
  • Asai, M. and M. McAleer (2015). Leverage and feedback effects on multifactor Wishart stochastic volatility for option pricing. Journal of Econometrics, 187 (2), 436–446.
  • Asai, M. and A. Unite (2008). The relationship between stock return volatility and trading volume: the case of the Philippines. Applied Financial Economics, 18 (16), 1333–1341.
  • Bauwens, L., L. Sebastien and J.V.K. Rombouts (2006). Multivariate GARCH Models: A Survey. Journal of Applied Econometrics, 21, 79–109.
  • Black, F. (1976). Studies in Stock Price Volatility Changes. In Proceedings of the 1976 Business Meeting of the Business and Economic Section, American Economic Association, 177–181.
  • Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31 (3), 307–327
  • Bollerslev, T., R.Y. Chou and K.F. Kroner (1992). ARCH modelling in finance: A review of the theory and empirical evidence. Journal of Econometrics, 52, 5-59
  • Bollerslev, T. and M.J. Wooldridge (1992). Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews, 11 (2), 143-172.
  • Brandt, M.W. and Q. Kang (2004). On the relationship between the conditional mean and volatility of stocks returns: a latent VAR approach. Journal of Financial Economics, 72, 217–258.
  • Breidt, F.J. and A. Carriquiry (1996). Improved quasi-maximum likelihood estimation for stochastic volatility models. In: (Eds.) Modelling and Prediction: Honouring Seymour Geisel, ed. A. Zellner and J.S. Lee. New York: Springer.
  • Broto, C. and E. Ruiz (2004). Estimation methods for stochastic volatility models: a survey. Journal of Economic Surveys, 18 (5), 613–649.
  • Carnero, A., D. Pena and E. Ruiz (2004). Persistence and Kurtosis in GARCH and Stochastic Volatility Models. Journal of Financial Econometrics, 2 (2), 319–342.
  • Chib, S., F. Nardari and N. Shephard (2006). Analysis of High Dimensional Multivariate Stochastic Volatility Models. Journal of Econometrics, 134 (2), 341–371
  • Chib, S., Y. Omori and M. Asai (2009). Multivariate Stochastic Volatility. Handbook of Financial Time Series, Part 2, 365-400.
  • Christie, A. (1982). The Stochastic Behavior of Common Stock Variances- Value, Leverage, and Interest Rate Effects. Journal of Financial Economic Theory, 10, 407–432.
  • Danielsson, J. (1998). Multivariate Stochastic Volatility Models: Estimation and A Comparison With VGARCH models. Journal of Empirical Finance, 5, 155–173.
  • Danielsson, J. and J.F. Richard (1993). Accelerated Gaussian Importance Sampler with Application to Dynamic Latent Variable Models. Journal of Applied Econometrics, 8, 153–154
  • Danielsson, J. (1994). Stochastic volatility in asset prices estimation with simulated maximum likelihood, Journal of Econometrics, 64, Issues 1-2, 375-400
  • De Jong, P. and N. Shephard (1995). The simulation smoother for time series models. Biometrika, 82 (2), 339–350.
  • Dunsmuir, W. (1979). A Central Limit Theorem for Parameter Estimation in Stationary Time Series and Its Applications to Models for a Signal Observed White Noise. Annals of Statistics, 7, 490–506.
  • Durbin, J. and S.J. Koopman (1997). Monte Carlo maximum likelihood estimation for non- Gaussian state space models. Biometrika, 84 (3), 669–684.
  • Durbin, J. and S.J. Koopman (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89, (3), 603-615
  • Engle, R.F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50 (4), 987–1008.
  • Fuller, W.A. (1996). Introduction to Statistical Time Series. New York: Wiley.
  • Ghysels, E., A.C. Harvey and E. Renault (1996). Stochastic volatility. In Statistical Models in Finance, ed. C.R. Rao and G.S. Maddala. Amsterdam: North-Holland, 119–191.
  • Hansen, P.R. and A. Lunde (2005). A forecast comparison of volatility models: does anything beat a GARCH(1,1)? Journal of Applied Econometrics, 20, 873–889.
  • Harvey, A. (1989). Forecasting Structural Models and the Kalman Filter. Cambridge University Press: Cambridge
  • Harvey, A., E. Ruiz and N. Shephard (1994). Multivariate stochastic variance models. The Review of Economic Studies, 61, 247–264.
  • Harvey, A.C. and N.G. Shephard (1996). Estimation of an asymmetric stochastic volatility model for asset returns. Journal of Business and Economic Statistics, 14, 429–434.
  • Hwang, S. and S.E. Satchell (2000). Market risk and the concept of fundamental volatility: measuring volatility across asset and derivative markets and testing for the impact of derivatives markets on financial markets. Journal of Banking and Finance, 24, 759–785.
  • Hull, J. and A. White (1987). Hedging the Risks from Writing Foreign Currency Options. Journal of International Money and Finance, 6, 131–152.
  • Jacquier, E., N. Polson and P. Rossi (1994). Bayesian Analisis of Stochastic Volatility Models, Journal of Business and Economic Statistics, 12, 371-389
  • Jungbacker, B. and S.J. Koopman (2005). On Importance Sampling for State Space Models, Tinbergen http://ssrn.com/abstract=873472 (accessed June 17, 2016). Discussion Paper No. 05-117. Available at SSRN:
  • Jungbacker, B. and S.J. Koopman (2006). Monte Carlo likelihood estimation for three multivariate stochastic volatility models. Econometric Reviews, 25 (2-3), 385–408.
  • Kim, S., N. Shephard and S. Chib (1998). Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models. Review of Economic Studies, 65 (3), 361–393.
  • Liesenfeld, R. (1998). Dynamic bivariate mixture models: modeling the behavior of prices and trading volume. Journal of Business and Economic Statistics, 16, 101–109.
  • Liesenfeld, R. (2001). A generalized bivariate mixture model for stock price volatility and trading volume. Journal of Econometrics, 104, 141–178.
  • Liesenfeld, R. and R.C. Jung (2000). Stochastic volatility models: conditional normality versus heavy tailed distributions. Journal of Applied Econometrics, 15, 137–160.
  • Liesenfeld, R. and J.F. Richard (2003). Univariate and Multivariate Stochastic Volatility Models: Estimation and Diagnostics. Journal of Empirical Finance, 10, 505-531.
  • Liesenfeld, R. and J.F. Richard (2006). Classical and Bayesian Analysis of Univariate and Multivariate Stochastic Volatility Models. Econometric Reviews, 25 (2–3), 335–360
  • Lutkepohl, H. (1996). Handbook of Matrices. New York: Wiley.
  • Maasoumi, E. and M. McAleer (2006). Multivariate Stochastic Volatility: An Overview. Econometric Reviews, 25 (2–3), 139–144.
  • Meyer, R. and J. Yu (2000). BUGS for a bayesian analysis of stochastic volatility models. Econometrics Journal, 3, 198–215.
  • Nelson, D.B. (1988). Time Series Behavior of Stock Market Volatility and Returns. Unpublished PhD dissertation. Economics Dept.: MIT:
  • Omori, Y. and T. Ishihara (2012) Multivariate Stochastic Volatility Models, in Handbook of Volatility Models and Their Applications. In Handbook of Volatility Models and Their Applications, ed. L. Bauwens, C. Hafner and S. Laurent. Hoboken, NJ: John Wiley & Sons Inc.
  • Ruiz, E. (1994). Quasi-maximum likelihood estimation of stochastic volatility models. Journal of Econometrics, 63 (1), 289–306
  • Sandmann, G. and S.J. Koopman (1998). Estimation of stochastic volatility models via Monte Carlo maximum likelihood. Journal of Econometrics, 87 (2), 271–301
  • Shephard, N. (1996). Statistical Aspects of ARCH and Stochastic Volatility. In Time Series Models in Econometrics, Finance and Other Fields, ed. D.R. Cox, D.V. Hinkley and O.E. Barndorff-Nielsen. London: Chapman and Hall, 1–67.
  • Shephard, N. and T.G. Andersen (2009). Stochastic Volatility: Origins and Overview. Handbook of Financial Econometrics, Part 2, 233–254.
  • Shephard, N. and M.K. Pitt (1997). Likelihood Analysis of Non-Gaussian Measurement Time Series. Biometrika, 84, 653–667.
  • Silvennoinen, A. and T. Teräsvirta, (2009). Multivariate GARCH Models. In Handbook of Financial Time Series, ed. T.G. Andersen, R.A. Davis, J.-P. Kreiss and T. Mikosch. New York: Springer.
  • Taylor, S.J. (1982). Financial returns modelled by the product of two stochastic processes - a study of daily sugar prices 1961-79. In Time Series Analysis: Theory and Practice, ed. O.D. Anderson, 1, 203–226. Amsterdam: North-Holland.
  • Taylor, S.J. (1986). Modelling Financial Time Series. Chichester: John Wiley.
  • Taylor, S.J. (1994). Modelling Stochastic Volatility: A Review and Comparati ve Study. Mathematical Finance, 4 (2), 183–204.
  • Tsay, R.S. (2005). Analysis of Financial Time Series: Financial Econometrics. (2ed). New York: Wiley.
  • Tsui, A.K and Q. Yu (1999). Constant conditional correlation in a bivariate garch model: Evidence from the stock market in China. Mathematics and Computers in Simulation, 48, 503–509.
  • Watanabe, T. (1999). A Nonlinear Filtering Approach to Stochastic Volatility Models with an Application to Daily Stock Returns. Journal of Applied Econometrics, 14, 101–121
  • Yu, J. (2002). Forecasting volatility in the New Zealand stock market. Applied Financial Economics, 12, 193–202.
  • Yu, J. and R. Meyer (2006). Multivariate Stochastic Volatility Models: Bayesian Estimation and Model Comparison. Econometric Reviews, 25, 2–3.
Year 2016, Volume: 8 Issue: 2, 19 - 52, 30.09.2016
https://doi.org/10.33818/ier.278044

Abstract

References

  • Aguilar, O. and M. West (2000). Bayesian Dynamic Factor Models and Portfolio Allocation. Journal of Business and Economic Statistics, 18, 338–357.
  • Alizadeh, S., M.W. Brandt and F.X. Diebold (2002). Range based estimation of stochastic volatility models. Journal of Finance, 57, 1047–1091.
  • Andersen, T., H. Chung and B. Sorensen (1999). Efficient method of moments estimation of a stochastic volatility model: a Monte Carlo study. Journal of Econometrics, 91, 61–87.
  • Andersen, T. and T. Bollerslev (1998). Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review, 39 (4), 885–905.
  • Asai, M. and M. McAleer (2005). Dynamic asymmetric leverage in stochastic volatility models. Econometric Reviews, 24, 317–332.
  • Asai, M. and M. McAleer (2006). Asymmetric Multivariate Stochastic Volatility. Econometric Reviews, 25 (2-3), 453–473.
  • Asai, M., M. McAleer and J. Yu (2006). Multivariate Stochastic Volatility: A Review. Econometric Reviews, 25 (2-3), 145–175.
  • Asai, M. and M. McAleer (2009). The structure of dynamic correlations in multivariate stochastic models. Journal of Econometrics, 150, 182–192.
  • Asai, M. and M. McAleer (2011). Dynamic conditional correlations for asymmetric processes. Journal of the Japan Statistical Society, 41 (2), 143–157.
  • Asai, M. and M. McAleer (2015). Leverage and feedback effects on multifactor Wishart stochastic volatility for option pricing. Journal of Econometrics, 187 (2), 436–446.
  • Asai, M. and A. Unite (2008). The relationship between stock return volatility and trading volume: the case of the Philippines. Applied Financial Economics, 18 (16), 1333–1341.
  • Bauwens, L., L. Sebastien and J.V.K. Rombouts (2006). Multivariate GARCH Models: A Survey. Journal of Applied Econometrics, 21, 79–109.
  • Black, F. (1976). Studies in Stock Price Volatility Changes. In Proceedings of the 1976 Business Meeting of the Business and Economic Section, American Economic Association, 177–181.
  • Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31 (3), 307–327
  • Bollerslev, T., R.Y. Chou and K.F. Kroner (1992). ARCH modelling in finance: A review of the theory and empirical evidence. Journal of Econometrics, 52, 5-59
  • Bollerslev, T. and M.J. Wooldridge (1992). Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews, 11 (2), 143-172.
  • Brandt, M.W. and Q. Kang (2004). On the relationship between the conditional mean and volatility of stocks returns: a latent VAR approach. Journal of Financial Economics, 72, 217–258.
  • Breidt, F.J. and A. Carriquiry (1996). Improved quasi-maximum likelihood estimation for stochastic volatility models. In: (Eds.) Modelling and Prediction: Honouring Seymour Geisel, ed. A. Zellner and J.S. Lee. New York: Springer.
  • Broto, C. and E. Ruiz (2004). Estimation methods for stochastic volatility models: a survey. Journal of Economic Surveys, 18 (5), 613–649.
  • Carnero, A., D. Pena and E. Ruiz (2004). Persistence and Kurtosis in GARCH and Stochastic Volatility Models. Journal of Financial Econometrics, 2 (2), 319–342.
  • Chib, S., F. Nardari and N. Shephard (2006). Analysis of High Dimensional Multivariate Stochastic Volatility Models. Journal of Econometrics, 134 (2), 341–371
  • Chib, S., Y. Omori and M. Asai (2009). Multivariate Stochastic Volatility. Handbook of Financial Time Series, Part 2, 365-400.
  • Christie, A. (1982). The Stochastic Behavior of Common Stock Variances- Value, Leverage, and Interest Rate Effects. Journal of Financial Economic Theory, 10, 407–432.
  • Danielsson, J. (1998). Multivariate Stochastic Volatility Models: Estimation and A Comparison With VGARCH models. Journal of Empirical Finance, 5, 155–173.
  • Danielsson, J. and J.F. Richard (1993). Accelerated Gaussian Importance Sampler with Application to Dynamic Latent Variable Models. Journal of Applied Econometrics, 8, 153–154
  • Danielsson, J. (1994). Stochastic volatility in asset prices estimation with simulated maximum likelihood, Journal of Econometrics, 64, Issues 1-2, 375-400
  • De Jong, P. and N. Shephard (1995). The simulation smoother for time series models. Biometrika, 82 (2), 339–350.
  • Dunsmuir, W. (1979). A Central Limit Theorem for Parameter Estimation in Stationary Time Series and Its Applications to Models for a Signal Observed White Noise. Annals of Statistics, 7, 490–506.
  • Durbin, J. and S.J. Koopman (1997). Monte Carlo maximum likelihood estimation for non- Gaussian state space models. Biometrika, 84 (3), 669–684.
  • Durbin, J. and S.J. Koopman (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89, (3), 603-615
  • Engle, R.F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50 (4), 987–1008.
  • Fuller, W.A. (1996). Introduction to Statistical Time Series. New York: Wiley.
  • Ghysels, E., A.C. Harvey and E. Renault (1996). Stochastic volatility. In Statistical Models in Finance, ed. C.R. Rao and G.S. Maddala. Amsterdam: North-Holland, 119–191.
  • Hansen, P.R. and A. Lunde (2005). A forecast comparison of volatility models: does anything beat a GARCH(1,1)? Journal of Applied Econometrics, 20, 873–889.
  • Harvey, A. (1989). Forecasting Structural Models and the Kalman Filter. Cambridge University Press: Cambridge
  • Harvey, A., E. Ruiz and N. Shephard (1994). Multivariate stochastic variance models. The Review of Economic Studies, 61, 247–264.
  • Harvey, A.C. and N.G. Shephard (1996). Estimation of an asymmetric stochastic volatility model for asset returns. Journal of Business and Economic Statistics, 14, 429–434.
  • Hwang, S. and S.E. Satchell (2000). Market risk and the concept of fundamental volatility: measuring volatility across asset and derivative markets and testing for the impact of derivatives markets on financial markets. Journal of Banking and Finance, 24, 759–785.
  • Hull, J. and A. White (1987). Hedging the Risks from Writing Foreign Currency Options. Journal of International Money and Finance, 6, 131–152.
  • Jacquier, E., N. Polson and P. Rossi (1994). Bayesian Analisis of Stochastic Volatility Models, Journal of Business and Economic Statistics, 12, 371-389
  • Jungbacker, B. and S.J. Koopman (2005). On Importance Sampling for State Space Models, Tinbergen http://ssrn.com/abstract=873472 (accessed June 17, 2016). Discussion Paper No. 05-117. Available at SSRN:
  • Jungbacker, B. and S.J. Koopman (2006). Monte Carlo likelihood estimation for three multivariate stochastic volatility models. Econometric Reviews, 25 (2-3), 385–408.
  • Kim, S., N. Shephard and S. Chib (1998). Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models. Review of Economic Studies, 65 (3), 361–393.
  • Liesenfeld, R. (1998). Dynamic bivariate mixture models: modeling the behavior of prices and trading volume. Journal of Business and Economic Statistics, 16, 101–109.
  • Liesenfeld, R. (2001). A generalized bivariate mixture model for stock price volatility and trading volume. Journal of Econometrics, 104, 141–178.
  • Liesenfeld, R. and R.C. Jung (2000). Stochastic volatility models: conditional normality versus heavy tailed distributions. Journal of Applied Econometrics, 15, 137–160.
  • Liesenfeld, R. and J.F. Richard (2003). Univariate and Multivariate Stochastic Volatility Models: Estimation and Diagnostics. Journal of Empirical Finance, 10, 505-531.
  • Liesenfeld, R. and J.F. Richard (2006). Classical and Bayesian Analysis of Univariate and Multivariate Stochastic Volatility Models. Econometric Reviews, 25 (2–3), 335–360
  • Lutkepohl, H. (1996). Handbook of Matrices. New York: Wiley.
  • Maasoumi, E. and M. McAleer (2006). Multivariate Stochastic Volatility: An Overview. Econometric Reviews, 25 (2–3), 139–144.
  • Meyer, R. and J. Yu (2000). BUGS for a bayesian analysis of stochastic volatility models. Econometrics Journal, 3, 198–215.
  • Nelson, D.B. (1988). Time Series Behavior of Stock Market Volatility and Returns. Unpublished PhD dissertation. Economics Dept.: MIT:
  • Omori, Y. and T. Ishihara (2012) Multivariate Stochastic Volatility Models, in Handbook of Volatility Models and Their Applications. In Handbook of Volatility Models and Their Applications, ed. L. Bauwens, C. Hafner and S. Laurent. Hoboken, NJ: John Wiley & Sons Inc.
  • Ruiz, E. (1994). Quasi-maximum likelihood estimation of stochastic volatility models. Journal of Econometrics, 63 (1), 289–306
  • Sandmann, G. and S.J. Koopman (1998). Estimation of stochastic volatility models via Monte Carlo maximum likelihood. Journal of Econometrics, 87 (2), 271–301
  • Shephard, N. (1996). Statistical Aspects of ARCH and Stochastic Volatility. In Time Series Models in Econometrics, Finance and Other Fields, ed. D.R. Cox, D.V. Hinkley and O.E. Barndorff-Nielsen. London: Chapman and Hall, 1–67.
  • Shephard, N. and T.G. Andersen (2009). Stochastic Volatility: Origins and Overview. Handbook of Financial Econometrics, Part 2, 233–254.
  • Shephard, N. and M.K. Pitt (1997). Likelihood Analysis of Non-Gaussian Measurement Time Series. Biometrika, 84, 653–667.
  • Silvennoinen, A. and T. Teräsvirta, (2009). Multivariate GARCH Models. In Handbook of Financial Time Series, ed. T.G. Andersen, R.A. Davis, J.-P. Kreiss and T. Mikosch. New York: Springer.
  • Taylor, S.J. (1982). Financial returns modelled by the product of two stochastic processes - a study of daily sugar prices 1961-79. In Time Series Analysis: Theory and Practice, ed. O.D. Anderson, 1, 203–226. Amsterdam: North-Holland.
  • Taylor, S.J. (1986). Modelling Financial Time Series. Chichester: John Wiley.
  • Taylor, S.J. (1994). Modelling Stochastic Volatility: A Review and Comparati ve Study. Mathematical Finance, 4 (2), 183–204.
  • Tsay, R.S. (2005). Analysis of Financial Time Series: Financial Econometrics. (2ed). New York: Wiley.
  • Tsui, A.K and Q. Yu (1999). Constant conditional correlation in a bivariate garch model: Evidence from the stock market in China. Mathematics and Computers in Simulation, 48, 503–509.
  • Watanabe, T. (1999). A Nonlinear Filtering Approach to Stochastic Volatility Models with an Application to Daily Stock Returns. Journal of Applied Econometrics, 14, 101–121
  • Yu, J. (2002). Forecasting volatility in the New Zealand stock market. Applied Financial Economics, 12, 193–202.
  • Yu, J. and R. Meyer (2006). Multivariate Stochastic Volatility Models: Bayesian Estimation and Model Comparison. Econometric Reviews, 25, 2–3.
There are 67 citations in total.

Details

Subjects Business Administration
Other ID JA46TF48DP
Journal Section Articles
Authors

M. Hakan Eratalay This is me

Publication Date September 30, 2016
Submission Date September 30, 2016
Published in Issue Year 2016 Volume: 8 Issue: 2

Cite

APA Eratalay, M. H. (2016). Estimation of Multivariate Stochastic Volatility Models: A Comparative Monte Carlo Study. International Econometric Review, 8(2), 19-52. https://doi.org/10.33818/ier.278044
AMA Eratalay MH. Estimation of Multivariate Stochastic Volatility Models: A Comparative Monte Carlo Study. IER. December 2016;8(2):19-52. doi:10.33818/ier.278044
Chicago Eratalay, M. Hakan. “Estimation of Multivariate Stochastic Volatility Models: A Comparative Monte Carlo Study”. International Econometric Review 8, no. 2 (December 2016): 19-52. https://doi.org/10.33818/ier.278044.
EndNote Eratalay MH (December 1, 2016) Estimation of Multivariate Stochastic Volatility Models: A Comparative Monte Carlo Study. International Econometric Review 8 2 19–52.
IEEE M. H. Eratalay, “Estimation of Multivariate Stochastic Volatility Models: A Comparative Monte Carlo Study”, IER, vol. 8, no. 2, pp. 19–52, 2016, doi: 10.33818/ier.278044.
ISNAD Eratalay, M. Hakan. “Estimation of Multivariate Stochastic Volatility Models: A Comparative Monte Carlo Study”. International Econometric Review 8/2 (December 2016), 19-52. https://doi.org/10.33818/ier.278044.
JAMA Eratalay MH. Estimation of Multivariate Stochastic Volatility Models: A Comparative Monte Carlo Study. IER. 2016;8:19–52.
MLA Eratalay, M. Hakan. “Estimation of Multivariate Stochastic Volatility Models: A Comparative Monte Carlo Study”. International Econometric Review, vol. 8, no. 2, 2016, pp. 19-52, doi:10.33818/ier.278044.
Vancouver Eratalay MH. Estimation of Multivariate Stochastic Volatility Models: A Comparative Monte Carlo Study. IER. 2016;8(2):19-52.