Bayesian Analysis of Extended Auto Regressive Model with Stochastic Volatility

  • Praveen Kumar TripathiEmail author
  • Satyanshu Kumar Upadhyay
Research article


This paper proposes an extension of the autoregressive model with stochastic volatility error. The Bayes analysis of the proposed model using vague priors for the parameters of the conditional mean equation and informative prior for the parameters of conditional volatility equation is done. The Gibbs sampler with intermediate Metropolis steps is used to find out posterior inferences for the parameters of autoregressive model and independent Metropolis–Hastings algorithm is used to simulate the volatility of the mean equation. The two data sets in the form of gross domestic product growth rate of India at constant prices and exchange rate of Indian rupees relative to US dollar are considered for numerical illustration. These data are used after assuring the stationarity by differencing the data once. The retrospective as well as prospective short term predictions of the data are provided based on the two simple components of the general autoregressive process. The findings based on the real data are expected to assist the policy makers and managers to make economic and business strategies more precisely.


Autoregressive model Stochastic volatility GDP growth rate Exchange rate Gibbs sampler Metropolis algorithm Retrospective and prospective predictions 

Mathematics Subject Classification

37M10 60J22 62C10 62M20 62F15 65C05 



The authors express their thankfulness to the Editor and the Referees for their valuable comments and suggestions that improved the earlier version of the manuscript.


  1. Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econom 31(3):307–327MathSciNetzbMATHGoogle Scholar
  2. Box GE, Pierce DA (1970) Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. J Am Stat Assoc 65(332):1509–1526MathSciNetzbMATHGoogle Scholar
  3. Box GE, Jenkins GM, Reinsel GC, Ljung GM (2015) Time series analysis: forecasting and control. Wiley, New YorkzbMATHGoogle Scholar
  4. Broto C, Ruiz E (2004) Estimation methods for stochastic volatility models: a survey. J Econ Surv 18(5):613–649Google Scholar
  5. Chib S, Greenberg E (1994) Bayes inference in regression models with ARMA(\(p, q\)) errors. J Econom 64:183–206MathSciNetzbMATHGoogle Scholar
  6. Chib S, Nardari F, Shephard N (2002) Markov chain Monte Carlo methods for stochastic volatility models. J Econom 108(2):281–316MathSciNetzbMATHGoogle Scholar
  7. Cogley T, Sargent TJ (2005) Drifts and volatilities: monetary policies and outcomes in the post WWII US. Rev Econ Dyn 8(2):262–302Google Scholar
  8. Cogley T, Primiceri G, Sargent T (2010) Inflation-gap persistence in the US. Am Econ J Macroecon 2:43–69Google Scholar
  9. Engle RF (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econom J Econom Soc 50(4):987–1007MathSciNetzbMATHGoogle Scholar
  10. Geweke J (1989) Bayesian inference in econometric models using Monte Carlo integration. Econom J Econom Soc 57(6):1317–1339MathSciNetzbMATHGoogle Scholar
  11. Harvey A, Ruiz E, Shephard N (1994) Multivariate stochastic variance models. Rev Econ Stud 61(2):247–264zbMATHGoogle Scholar
  12. Huber F, Krisztin T, Piribauer P (2017) Forecasting global equity indices using large Bayesian VARs. Bull Econ Res 69(3):288–308MathSciNetzbMATHGoogle Scholar
  13. Hyndman RJ, Athanasopoulos G (2013) Forecasting: principles and practice.
  14. Jacquier E, Polson N, Rossi P (1994) Bayesian analysis of stochastic volatility models. J Bus Econ Stat 12(4):371–389. zbMATHGoogle Scholar
  15. Jacquier E, Polson NG, Rossi PE (2004) Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. J Econom 122(1):185–212MathSciNetzbMATHGoogle Scholar
  16. Kim S, Shephard N, Chib S (1998) Stochastic volatility: likelihood inference and comparison with ARCH models. Rev Econ Stud 65(3):361–393zbMATHGoogle Scholar
  17. Kitagawa G (1987) Non-gaussian state-space modeling of nonstationary time series. J Am Stat Assoc 82(400):1032–1041MathSciNetzbMATHGoogle Scholar
  18. Ljung GM, Box GEP (1978) On a measure of lack of fit in time series models. Biometrika 65:297–303zbMATHGoogle Scholar
  19. Lopes HF (2013) Stochastic volatility models.
  20. Mandelbrot B (1963) The variation of certain speculative prices. J Bus 36:394–419Google Scholar
  21. Monahan JF (1984) A note on enforcing stationarity in autoregressive-moving average models. Biometrika 71(2):403–404MathSciNetGoogle Scholar
  22. Nelson DB (1991) Conditional heteroskedasticity in asset returns: a new approach. Econom J Econom Soc 59:347–370MathSciNetzbMATHGoogle Scholar
  23. Platanioti K, McCoy EJ, Stephens DA (2005) A review of stochastic volatility: univariate and multivariate models. Working paperGoogle Scholar
  24. Sandmann G, Koopman SJ (1998) Estimation of stochastic volatility models via Monte Carlo maximum likelihood. J Econom 87(2):271–301MathSciNetzbMATHGoogle Scholar
  25. Smith AFM, Roberts GO (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J R Stat Soc Ser B 55:3–25MathSciNetzbMATHGoogle Scholar
  26. Taylor SJ (1982) Financial returns modelled by the product of two stochastic processes—a study of the daily sugar prices 1961–75. Time Ser Anal Theory and Pract North-Holland 1:203–226Google Scholar
  27. Upadhyay SK, Vasishta N, Smith AFM (2001) Bayes inference in life testing and reliability via Markov chain Monte Carlo simulation. Sankhya Ser A 63:15–40zbMATHGoogle Scholar
  28. Upadhyay SK, Gupta A, Dey DK (2012) Bayesian modeling of bathtub shaped hazard rate using various Weibull extensions and related issues of model selection. Sankhya Ser B 74:15–43MathSciNetzbMATHGoogle Scholar

Copyright information

© The Indian Society for Probability and Statistics (ISPS) 2019

Authors and Affiliations

  • Praveen Kumar Tripathi
    • 1
    Email author
  • Satyanshu Kumar Upadhyay
    • 2
  1. 1.Department of MathematicsDIT UniversityDehradunIndia
  2. 2.Department of Statistics and DST Center for Interdisciplinary Sciences, Institute of ScienceBanaras Hindu UniversityVaranasiIndia

Personalised recommendations