Periodic autoregressive stochastic volatility



This paper proposes a stochastic volatility model (PAR-SV) in which the log-volatility follows a first-order periodic autoregression. This model aims at representing time series with volatility displaying a stochastic periodic dynamic structure, and may then be seen as an alternative to the familiar periodic GARCH process. The probabilistic structure of the proposed PAR-SV model such as periodic stationarity and autocovariance structure are first studied. Then, parameter estimation is examined through the quasi-maximum likelihood (QML) method where the likelihood is evaluated using the prediction error decomposition approach and Kalman filtering. In addition, a Bayesian MCMC method is also considered, where the posteriors are given from conjugate priors using the Gibbs sampler in which the augmented volatilities are sampled from the Griddy Gibbs technique in a single-move way. As a-by-product, period selection for the PAR-SV is carried out using the (conditional) deviance information criterion (DIC). A simulation study is undertaken to assess the performances of the QML and Bayesian Griddy Gibbs estimates in finite samples while applications of Bayesian PAR-SV modeling to daily, quarterly and monthly S&P 500 returns are considered.


Periodic stochastic volatility Periodic autoregression QML via prediction error decomposition and Kalman filtering Bayesian Griddy Gibbs sampler Single-move approach DIC 

Mathematics Subject Classification

Primary 62M10 Secondary 60F99 



I am deeply grateful to the Editor-In-Chief Prof. Marc Hallin, an Associate Editor and two Referees for their careful reading, helpful comments and useful suggestions that have greatly helped me in substantially improving the earlier version of the manuscript.


  1. Aknouche A, Al-Eid E (2012) Asymptotic inference of unstable periodic \(ARCH\) processes. Stat. Inference Stoch. Process. 15:61–79MathSciNetCrossRefMATHGoogle Scholar
  2. Aknouche A, Bibi A (2009) Quasi-maximum likelihood estimation of periodic \(GARCH\) and periodic \(ARMA\)-\(GARCH\) processes. J. Time Ser. Anal. 30:19–46MathSciNetCrossRefMATHGoogle Scholar
  3. Berg A, Meyer R, Yu J (2004) Deviance information criterion for comparing stochastic volatility models. J. Bus. Econ. Stat. 22:107–120MathSciNetCrossRefGoogle Scholar
  4. Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New YorkMATHGoogle Scholar
  5. Bollerslev T, Ghysels E (1996) Periodic autoregressive conditional heteroskedasticity. J. Bus. Econ. Stat. 14:139–151Google Scholar
  6. Box GEP, Tiao GC (1973) Bayesian inference in statistical analysis. Addison-Wesley, ReadingMATHGoogle Scholar
  7. Breidt FJ (1997) A threshold autoregressive stochastic volatility.
  8. Breidt FJ, Crato N, De Lima P (1998) The detection and estimation of long memory in stochastic volatility. J. Econ. 83:325–348MathSciNetCrossRefMATHGoogle Scholar
  9. Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2nd edn. Springer, New YorkCrossRefMATHGoogle Scholar
  10. Carvalho CM, Lopes HF (2007) Simulation-based sequential analysis of Markov switching stochastic volatility models. Comput Stat Data Anal 51:4526–4542MathSciNetCrossRefMATHGoogle Scholar
  11. Celeux G, Forbes F, Robert CP, Titterington DM (2006) Deviance information criteria for missing data models. Bayesian Anal 1:651–674MathSciNetCrossRefMATHGoogle Scholar
  12. Chan CC, Grant AL (2014) Issues in comparing stochastic volatility models using the deviance information criterion. CAMA Working Paper 51/2014. Australian National UniversityGoogle Scholar
  13. Chib S, Nardari F, Shephard N (2002) Markov chain Monte Carlo methods for stochastic volatility models. J. Econ. 108:281–316MathSciNetCrossRefMATHGoogle Scholar
  14. Davis RA, Mikosch T (2009) Probabilistic properties of stochastic volatility models. In: Mikosch Thomas (ed) Handbook of financial time series, vol Part 2. Springer, Berlin, pp 255–267CrossRefGoogle Scholar
  15. Dunsmuir W (1979) A central limit theorem for parameter estimation in stationary vector time series and its applications to models for a signal observed with noise. Ann Stat 7:490–506MathSciNetCrossRefMATHGoogle Scholar
  16. Francq C, Zakoïan JM (2010) GARCH models: structure, statistical inference and financial applications. Wiley, New YorkCrossRefMATHGoogle Scholar
  17. Francq C, Zakoïan JM (2006) Linear-representation bases estimation of stochastic volatility models. Scand J Stat 33:785–806CrossRefMATHGoogle Scholar
  18. Geweke J (1989) Bayesian inference in econometric models using Monte Carlo integration. Econometrica 57:1317–1339MathSciNetCrossRefMATHGoogle Scholar
  19. Geyer CJ (1992) Practical Markov Chain Monte Carlo. Stat Sci 7:473–511CrossRefGoogle Scholar
  20. Ghysels E, Harvey AC, Renault E (1996) Stochastic volatility. In: Rao CR, Maddala GS (eds) Statistical Methods in Finance. North-Holland, AmsterdamGoogle Scholar
  21. Ghysels E, Osborn D (2001) The econometric analysis of seasonal time series. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  22. Gladyshev EG (1961) Periodically correlated random sequences. Sov Math 2:385–388MATHGoogle Scholar
  23. Harvey AC, Ruiz E, Shephard N (1994) Multivariate stochastic variance models. Rev Econ Stud 61:247–264CrossRefMATHGoogle Scholar
  24. Hylleberg S, Engle RF, Granger CWJ, Yoo BS (1990) Seasonal integration and cointegration. J Econ 44:215–238MathSciNetCrossRefMATHGoogle Scholar
  25. Jacquier E, Polson NG, Rossi PE (1994) Bayesian analysis of stochastic volatility models. J Bus Econ Stat 12:413–417MATHGoogle Scholar
  26. Kim S, Shephard N, Chib S (1998) Stochastic volatility: likelihood inference and comparison with \(ARCH\) models. Rev Econ Stud 65:361–393CrossRefMATHGoogle Scholar
  27. Koopman SJ, Ooms M, Carnero MA (2007) Periodic seasonal \(Reg\)-\(ARFIMA\)-\(GARCH\) models for daily electricity spot prices. J Am Stat Assoc 102:16–27MathSciNetCrossRefMATHGoogle Scholar
  28. Ljung L, Söderström T (1983) Theory and practice of recursive identification. MIT Press, CambridgeMATHGoogle Scholar
  29. Meyn S, Tweedie R (2009) Markov chains and stochastic stability, 2nd edn. Springer, New YorkCrossRefMATHGoogle Scholar
  30. Nakajima J, Omori Y (2009) Leverage, heavy-tails and correlated jumps in stochastic volatility models. Comput Stat Data Anal 53:2335–2353MathSciNetCrossRefMATHGoogle Scholar
  31. Omori Y, Chib S, Shephard N, Nakajima J (2007) Stochastic volatility with leverage: fast and efficient likelihood inference. J Econ 140:425–449MathSciNetCrossRefMATHGoogle Scholar
  32. Osborn DR, Savva CS, Gill L (2008) Periodic dynamic conditional correlations between stock markets in Europe and the US. J Financ Econ 6:307–325Google Scholar
  33. Regnard N, Zakoïan JM (2011) A conditionally heteroskedastic model with time-varying coefficients for daily gas spot prices. Energy Econ 33:1240–1251CrossRefGoogle Scholar
  34. Ritter C, Tanner MA (1992) Facilitating the Gibbs sampler: the Gibbs stopper and the Griddy-Gibbs sampler. J Am Stat Assoc 87:861–870CrossRefGoogle Scholar
  35. Ruiz E (1994) Quasi-maximum likelihood estimation of stochastic variance models. J Econ 63:284–306CrossRefGoogle Scholar
  36. Sakai H, Ohno S (1997) On backward periodic autoregressive processes. J Time Ser Anal 18:415–427MathSciNetCrossRefMATHGoogle Scholar
  37. Shiller RJ (2015) Irrational exuberance, 3rd edn. Princeton University Press, PrincetonCrossRefGoogle Scholar
  38. Sigauke C, Chikobvu D (2011) Prediction of daily peak electricity demand in South Africa using volatility forecasting models. Energy Econ 33:882–888CrossRefGoogle Scholar
  39. So MKP, Lam K, Li WK (1998) A stochastic volatility model with Markov switching. J Bus Econ Stat 16:244–253MathSciNetGoogle Scholar
  40. Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc B64:583–639MathSciNetCrossRefMATHGoogle Scholar
  41. Taylor JW (2006) Density forecasting for the efficient balancing of the generation and consumption of electricity. Int J Forecast 22:707–724CrossRefGoogle Scholar
  42. Taylor SJ (1982) Financial returns modelled by the product of two stochastic processes—a study of daily sugar prices, 1961–1979. In: Anderson OD (ed) Time series analysis: theory and practice. North-Holland, Amsterdam, pp 203–226Google Scholar
  43. Tiao GC, Grupe MR (1980) Hidden periodic autoregressive-moving average models in time series data. Biometrika 67:365–373MathSciNetMATHGoogle Scholar
  44. Tsay RS (2010) Analysis of financial time series: financial econometrics, 3rd edn. Wiley, New YorkCrossRefMATHGoogle Scholar
  45. Tsiakas I (2006) Periodic stochastic volatility and fat tails. J Financ Econ 4:90–135Google Scholar
  46. Zhu L, Carlin BP (2000) Comparing hierarchical models for spatio-temporally misaligned data using the deviance information criterion. Stat Med 19:2265–2278CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of Science and Technology Houari BoumedieneAlgiersAlgeria
  2. 2.Mathematics Department, College of ScienceQassim UniversityQassimSaudi Arabia

Personalised recommendations