A Gibbs Sampling Algorithm to Estimate the Parameters of a Volatility Model: An Application to Ozone Data


In this work we consider a stochastic volatility model, commonly used in financial time series studies, to analyse ozone data. The model considered depends on some parameters and in order to estimate them a Markov chain Monte Carlo algorithm is proposed. The algorithm considered here is the so-called Gibbs sampling algorithm which is programmed using the language R. Its code is also given. The model and the algorithm are applied to the weekly ozone averaged measurements obtained from the monitoring network of Mexico City.

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V. Romo, E. Rodrigues and G. Tzintzun, "A Gibbs Sampling Algorithm to Estimate the Parameters of a Volatility Model: An Application to Ozone Data," Applied Mathematics, Vol. 3 No. 12A, 2012, pp. 2178-2190. doi: 10.4236/am.2012.312A299.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] W. J. Gauderman, E. Avol, F. Gililand, H. Vora, D. Thomas, K. Berhane, R. McConnel, N. Kuenzli, F. Lurmman, E. Rappaport, H. Margolis, D. Bates and J. Peter, “The Effects of Air Pollution on Lung Development from 10 to 18 Years of Age,” New England Journal of Medicine, Vol. 351, 2004, pp. 1057-1067. doi:10.1056/NEJMoa040610
[2] W. Davis, L. Zaikowski and S. C. Nodvin, “Acid Rain,” In: J. Cutler, Ed., Encyclopedia of Earth, Cleveland, Washington DC, 2011. http://www.eoearth.org/article/Acid_rain
[3] WHO (World Health Organization), “Air Quality Guidelines—2005. Particulate Matter, Ozone, Nitrogen Dioxide and Sulfur Dioxide,” World Health Organization Regional Office for Europe, 2006.
[4] M. L. Bell, R. Peng and F. Dominici, “The Exposure-Response Curve for Ozone and Risk of Mortality and the Adequacy of Current Ozone Regulations,” Environmental Health Perspectives, Vol. 114, No. 4, 2005, pp. 532-536. doi:10.1289/ehp.8816
[5] D. Loomis, V. H. Borja-Arbuto, S. I. Bangdiwala and C. M. Shy, “Ozone Exposure and Daily Mortality in Mexico City: A Time Series Analysis,” Health Effects Institute Research Report, Vol. 75, 1996, pp. 1-46.
[6] M. R. O’Neill, D. Loomis and V. H. Borja-Aburto, “Ozone, Area Social Conditions and Mortality in Mexico City,” Environmental Research, Vol. 94, No. 3, 2004, pp. 234-242. doi:10.1016/j.envres.2003.07.002
[7] N. Gouveia and T. Fletcher, “Time Series Analysis of Air Pollution and Mortality: Effects by Cause, Age and Socio-Economics Status,” Journal of Epidemiology and Community Health, Vol. 54, 2000, pp. 750-755. doi:10.1136/jech.54.10.750
[8] R. L. Smith, “Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone,” Statistical Sciences, Vol. 4, No. 4, 1989, pp. 367-393. doi:10.1214/ss/1177012400
[9] J.-N. Pan and S.-T. Chen, “Monitoring Long-Memory Air Quality Data Using ARFIMA Model,” Environmetrics, Vol. 19, No. 2, 2008, pp. 209-219. doi:10.1002/env.882
[10] L. J. álvarez, A. A. Fernández-Bremauntz, E. R. Rodrigues and G. Tzintzun, “Maximum a Posteriori Estimation of the Daily Ozone Peaks in Mexico City,” Journal of Agricultural, Biological Statistics, Vol. 10, No. 3, 2005, pp. 276-290. doi:10.1198/108571105X59017
[11] L. C. Larsen, R. A. Bradley and G. L. Honcoop, “A New Method of Characterizing the Variability of Air Quality-Related Indicators,” Air and Waste Management Association’s International Specialty Conference of Tropospheric Ozone and the Environment, Los Angeles, EUA, 19-22 March 1990.
[12] G. Huerta and B. Sansó, “Time-Varying Models for Extreme Values,” Environmental and Ecological Statistics, Vol. 14, No. 3, 2007, pp. 285-299. doi:10.1007/s10651-007-0014-3
[13] J. S. Javits, “Statistical Interdependencies in the Ozone National Ambient Air Quality Standard,” Journal of Air Pollution Control Association, Vol. 30, No. 1, 1980, pp. 58-59. doi:10.1080/00022470.1980.10465918
[14] A. E. Raftery, “Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone,” Statistical Sciences, Vol. 4, No. 4, 1989, pp. 378-381. doi:10.1214/ss/1177012401
[15] J. A. Achcar, A. A. Fernández-Bremauntz, E. R. Rodrigues and G. Tzintzun, “Estimating the Number of Ozone Peaks in Mexico City Using a Non-Homogeneous Poisson Model,” Environmetrics, Vol. 19, No. 5, 2008, pp. 469-485. doi:10.1002/env.890
[16] J. A. Achcar, E. R. Rodrigues and G. Tzintzun, “Using Non-Homogeneous Poisson Models with Multiple Change-Points to Estimate the Number of Ozone Exceedances in Mexico City,” Environmetrics, Vol. 22, No. 1, 2011, pp. 1-12. doi:10.1002/env.1029
[17] E. R. Rodrigues and J. A. Achcar, “Applications of Discrete-Time Markov Chains and Poisson Processes to Air Pollution Modeling and Studies,” Springer, New York, 2012.
[18] L. W. Davis, “The Effect of Driving Restrictions on Air Quality in Mexico City,” Journal of Political Economy, Vol. 116, No. 1, 2008, pp. 39-81. doi:10.1086/529398
[19] G. McKinley, M. Zuk, M. Hojer, M. Avalos, I. Gonzaléz, R. Iniestra, L. Laguna, M. A. Martnez, P. Osnaya, L. M. Reynales, R. Valdés and J. Martnez, “Quantification of Local and Global Benefits from Air Pollution Control in Mexico City,” Environmental Science & Technology, Vol. 39, No. 7, 2005, pp. 1954-1961. doi:10.1021/es035183e
[20] M. Zavala, S. C. Herndon, E. C. Wood, T. B. Onasch, W. B. Knighton, L. C. Marr, C. E. Kolb and L. T. Molina, “Evaluation of Mobile Emissions Contributions to Mexico City’s Emissions Inventory Using on Road and Cross-Road Emission Measurements and Ambient Data,” Atmospheric Chemistry and Physics, Vol. 9, 2009, pp. 6305-6317. doi:10.5194/acp-9-6305-2009
[21] J. A. Achcar, H. C. Zozolotto and E. R. Rodrigues, “Bivariate Stochastic Volatility Models Applied to Mexico City Ozone Pollution Data,” In: G. C. Romano and A. G. Conti, Eds., Air Quality in the 21st Century, Nova Publishers, New York, 2010, pp. 241-267.
[22] J. A. Achcar, E. R. Rodrigues and G. Tzintzun, “Using Stochastic Volatility Models to Analyse Weekly Ozone Averages in Mexico City,” Environmental and Ecological Statistics, Vol. 18, No. 2, 2011, pp. 271-290. doi:10.1007/s10651-010-0132-1
[23] H. C. Zozolotto, “Aplicacao de Modelos de Volatilidade Estocástica em Dados de Polui??o do ar de Duas Grandes Cidades: Cidade do México e S?o Paulo,” Master’s Dissertation, Universidade de S?o Paulo, Ribeir?o Preto, 2010.
[24] J. A. Achcar, H. C. Zozolotto, E. R. Rodrigues and P. H. N. Saldiva, “Two Multivariate Stochastic Volatility Models Applied to Air Pollution Data from Sao Paulo, Brazil,” Advances and Applications in Statistics, Vol. 20, 2011, pp. 1-23.
[25] NOM, “Modificación a la Norma Oficial Mexicana NOM-020-SSA1-1993,” Diario Oficial de la Federación, 2002.
[26] D. J. Spiegelhalter, A. Thomas, N. G. Best and W. R. Gilks, “Winbugs: Bayesian Inference Using Gibbs Sampling,” MRC Biostatistics Unit, Cambridge, 1999.
[27] E. Ghysels, A. C. Harvey and E. Renault, “A Stochastic Volatility,” In: C. R. Rao and G. S. Maddala, Eds., Statistical Models in Finance, North-Holland, Amsterdam, 1996. doi:10.1016/S0169-7161(96)14007-4
[28] S. Kim, N. Shepard and S. Chib, “Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models,” Review of Economic Studies, Vol. 65, No. 3, 1998, pp. 361-393. doi:10.1111/1467-937X.00050
[29] R. Meyer and J. Yu, “BUGS for a Bayesian Analysis of Stochastic Volatility Models,” Econometrics Journal, Vol. 3, No. 2, 2000, pp. 198-215. doi:10.1111/1368-423X.00046
[30] J. Yu and R. Meyer, “Multivariate Stochastic Volatility models: Bayesian Estimation and Model Comparison,” Econometric Reviews, Vol. 25, No. 2-3, 2006, pp. 361-384. doi:10.1080/07474930600713465
[31] R. F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, Vol. 50, No. 4, 1982, pp. 987-1007. doi:10.2307/1912773
[32] T. Bollerslev, “Generalized Autoregressive Conditional Heterocedasticity,” Journal of Econometrics, Vol. 31, No. 3, 1986, pp. 307-327. doi:10.1016/0304-4076(86)90063-1
[33] A. F. M. Smith and G. O. Roberts, “Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods,” Journal of the Royal Statistical Society Series B, Vol. 55, 1993, pp. 3-23.
[34] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equation of State Calculations by Fast Computing Machines,” The Journal of Chemical Physics, Vol. 21, No. 6, 1953, pp. 1087-1092. doi:10.1063/1.1699114
[35] W. K. Hastings, “Monte Carlo Sampling-Based Methods Using Markov Chains and Their Applications,” Biometrika, Vol. 57, No. 1, 1970, pp. 97-109. doi:10.1093/biomet/57.1.97
[36] S. Chib and E. Greenberg, “Understanding the Metropolis-Hastings Algorithm,” The American Statistician, Vol. 49, No. 4, 1995, pp. 327-335. doi:10.1080/00031305.1995.10476177
[37] J. Geweke, “Evaluating the Accuracy of Sampling-Based Approaches to the Calculation of Posterior Moments,” In: J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, Eds., Bayesian Statistics, Vol. 4, Clarendon Press, Oxford, 1992, pp. 169-193.
[38] A. Raftery and S. Lewis, “How Many Iterations in the Gibbs Sampler?” In: J. Bemardo, J. Berger, A. Dawid and A. Smith, Eds., Bayesian Statistics, Vol. 4, Claredon Press, Oxford, 1992, pp. 763-774.
[39] P. Heidelberger and P. Welch, “A Spectral Method for Confidence Interval Generation and Run Length Control in Simulations,” Communications of the Association for Computing Machinery, Vol. 24, No. 4, 1983, pp. 233-245. doi:10.1145/358598.358630
[40] A. Gelman and D. B. Rubin, “Inference from Iterative Simulation Using Multiple Sequences,” Statistical Sciences, Vol. 7, No. 4, 1992, pp. 457-511. doi:10.1214/ss/1177011136

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