Semiparametric stochastic volatility modelling using penalized splines

Abstract

Stochastic volatility (SV) models mimic many of the stylized facts attributed to time series of asset returns, while maintaining conceptual simplicity. The commonly made assumption of conditionally normally distributed or Student-t-distributed returns, given the volatility, has however been questioned. In this manuscript, we introduce a novel maximum penalized likelihood approach for estimating the conditional distribution in an SV model in a nonparametric way, thus avoiding any potentially critical assumptions on the shape. The considered framework exploits the strengths both of the hidden Markov model machinery and of penalized B-splines, and constitutes a powerful alternative to recently developed Bayesian approaches to semiparametric SV modelling. We demonstrate the feasibility of the approach in a simulation study before outlining its potential in applications to three series of returns on stocks and one series of stock index returns.

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References

  1. Abanto-Valle CA, Bandyopadhyay D, Lachos V, Enriquez I (2010) Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions. Comput Stat Data Anal 54:2883–2898

    MATH  MathSciNet  Article  Google Scholar 

  2. Abraham B, Balakrishna N, Sivakumar R (2006) Gamma stochastic volatility models. J Forecast 25:153–171

    MathSciNet  Article  Google Scholar 

  3. Arlot S, Celisse A (2010) Model selection. Stat Surv 4:40–79

    MATH  MathSciNet  Article  Google Scholar 

  4. Bartolucci F, De Luca G (2001) Maximum likelihood estimation of a latent variable time-series model. Appl Stoch Models Bus Ind 17:5–17

    MATH  MathSciNet  Article  Google Scholar 

  5. Bartolucci F, De Luca G (2003) Likelihood-based inference for asymmetric stochastic volatility models. Comput Stat Data Anal 42:445–449

    MATH  Article  Google Scholar 

  6. Bergmeier C, Benítez JM (2012) On the use of cross-validation for time series predictor evaluation. Inf Sci 191:192–213

    Article  Google Scholar 

  7. Chib S, Nardari F, Shephard N (2002) Markov chain Monte Carlo methods for stochastic volatility models. J Econom 108:281–316

    MATH  MathSciNet  Article  Google Scholar 

  8. Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quant Financ 1:223–236

    Article  Google Scholar 

  9. de Boor C (1978) A practical guide to splines. Springer, Berlin

    MATH  Book  Google Scholar 

  10. Delatola E-I, Griffin JE (2011) Bayesian nonparametric modelling of the return distribution with stochastic volatility. Bayesian Anal 6:901–926

    MathSciNet  Article  Google Scholar 

  11. Delatola E-I, Griffin JE (2013) A Bayesian semiparametric model for volatility with a leverage effect. Comput Stat Data Anal 60:97–110

    MathSciNet  Article  Google Scholar 

  12. Durham GB (2006) Monte Carlo methods for estimating, smoothing, and filtering one- and two-factor stochastic volatility models. J Econom 133:273–305

    MathSciNet  Article  Google Scholar 

  13. Eilers PHC, Marx BD (1996) Flexible smoothing with \(B\)-splines and penalties. Stati Sci 11:89–121

    MATH  MathSciNet  Article  Google Scholar 

  14. Fridman M, Harris L (1998) A maximum likelihood approach for non-Gaussian stochastic volatility models. J Bus Econ Stat 16:284–291

    Google Scholar 

  15. Fernandez C, Steel MFJ (1998) On Bayesian modeling of fat tails and skewness. J Am Stat Assoc 93:359–371

  16. Gallant AR, Hsieh D, Tauchen GE (1997) Estimation of stochastic volatility models with diagnostics. J Econom 81:159–192

  17. Gneiting T, Raftery AE (2007) Strictly proper scoring rules, prediction, and estimation. J Am Stat Assoc 102:359–378

    MATH  MathSciNet  Article  Google Scholar 

  18. Harvey AC, Shephard N (1996) Estimation of an asymmetric stochastic volatility model for asset returns. J Bus Econ Stat 14:429–434

    Google Scholar 

  19. Harvey CR, Siddique A (2000) Conditional skewness in asset pricing tests. J Financ 55:1263–1295

    Article  Google Scholar 

  20. Jacquier E, Polson NG, Rossi PE (2004) Bayesian analysis of stochastic volatility models with fat-tails and correlated errors. J Econom 122:185–212

    MathSciNet  Article  Google Scholar 

  21. Jondeau E, Rockinger M (2003) Conditional volatility, skewness and kurtosis: existence, persistence and co-movements. J Econ Dyn Control 27:1699–1737

    MATH  MathSciNet  Article  Google Scholar 

  22. Jensen MJ, Maheu JM (2010) Bayesian semiparametric stochastic volatility modeling. J Econom 157:306–316

  23. Kim S, Shephard N, Chib S (1998) Stochastic volatility: likelihood inference and comparison with ARCH models. Rev Econ Stud 65:361–393

    MATH  Article  Google Scholar 

  24. Krivobokova T, Crainiceanu CM, Kauermann G (2008) Fast adaptive penalized splines. J Comput Graph Stat 17:1–20

    MathSciNet  Article  Google Scholar 

  25. Langrock R (2011) Some applications of nonlinear and non-Gaussian state-space modelling by means of hidden Markov models. J Appl Stat 38:2955–2970

    MathSciNet  Article  Google Scholar 

  26. Langrock R, King R (2013) Maximum likelihood estimation of mark-recapture-recovery models in the presence of continuous covariates. Ann Appl Stat 7:1709–1732

    MATH  MathSciNet  Article  Google Scholar 

  27. Langrock R, Kneib T, Sohn A, DeRuiter SL (2014) Nonparametric inference in hidden Markov models using P-splines. Biometrics (to appear). arXiv:1309.0423v2

  28. Langrock R, MacDonald IL, Zucchini W (2012) Some nonstandard stochastic volatility models and their estimation using structured hidden Markov models. J Empir Financ 19:147–161

    Article  Google Scholar 

  29. Nakajima J, Omori Y (2012) Stochastic volatility model with leverage and asymmetrically heavy-tailed error using GH skew Student’s -distribution. Comput Stat Data Anal 56:3690–3704

    MATH  MathSciNet  Article  Google Scholar 

  30. Racine J (2000) Consistent cross-validatory model-selection for dependent data: \(h\nu \)-block cross-validation. J Econom 99:39–61

    MATH  Article  Google Scholar 

  31. Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472

    MATH  MathSciNet  Article  Google Scholar 

  32. Ruppert D (2002) Selecting the number of knots for penalized splines. J Comput Graph Stat 11:735–757

    MathSciNet  Article  Google Scholar 

  33. Ruppert D, Wand MP, Carroll RJ (2003) Semiparametric regression. Cambridge University Press, Cambridge

  34. Schellhase C, Kauermann G (2012) Density estimation and comparison with a penalized mixture approach. Comput Stat 27:757–777

    MATH  MathSciNet  Article  Google Scholar 

  35. Shephard N (1996) Statistical aspects of ARCH and stochastic volatility. In: Cox DR, Hinkley DV, Barndorff-Nielsen OE (eds) Time series models: in econometrics, finance and other fields. Chapman & Hall, London, pp 1–67

    Chapter  Google Scholar 

  36. Zucchini W, MacDonald IL (2009) Hidden Markov Models for time series: an introduction using R. Chapman & Hall, London

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Acknowledgments

The authors would like to thank the two anonymous referees who provided useful comments on an earlier version of this manuscript.

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Correspondence to Roland Langrock.

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Appendix: HMM essentials

Appendix: HMM essentials

This appendix reviews some HMM basics. A standard \(m\)-state HMM has the same two process structure as SV models and SSMs, only that the unobserved process is a Markov chain and hence discrete-valued rather than continuous-valued. Consider an HMM with observable process \(\{ X_t \}_{t=1}^T\) and underlying Markov chain \(\{ S_t \}_{t=1}^T\). Given the current state of \(S_t\), the variable \(X_t\) is usually assumed to be conditionally independent from previous and future observations and states. The Markov chain is typically considered to be of first order, and the probabilities of transitions between the different states are summarized in the \(m \times m\) transition probability matrix \(\varvec{\Gamma }=\left( \gamma _{ij} \right) \), where \(\gamma _{ij}=\Pr \bigl (S_{t+1}=j\vert S_t=i \bigr )\), \(i,j=1,\ldots ,m\). The initial state probabilities are summarized in the vector \(\varvec{\pi }\), where \(\pi _{i} = \Pr (S_1=i)\), \(i=1,\ldots ,m\). It is usually convenient and appropriate to assume \(\varvec{\pi }\) to be the stationary distribution. For the described HMM, with observations given by \(x_1,\ldots ,x_T\) and underlying states denoted by \(s_1,\ldots ,s_T\), the likelihood is given by

$$\begin{aligned} {\mathcal {L}}^{\text {HMM}} = f(x_1, \ldots , x_T)&= \sum _{s_1=1}^m \ldots \sum _{s_T=1}^m f(x_1, \ldots , x_T | s_1, \ldots , s_T) f(s_1, \ldots , s_T) \\&= \sum _{s_1=1}^m \ldots \sum _{s_T=1}^m \pi _{s_1} \prod _{t=1}^T f (x_t | s_t) \prod _{t=2}^T \gamma _{s_{t-1},s_t}. \end{aligned}$$

In this form the likelihood involves \(m^T\) summands, which would make a numerical maximization infeasible in most cases. However, there is a much more efficient way of calculating the likelihood \({\mathcal {L}}^{\text {HMM}}\), given by a recursive scheme called the forward algorithm. To see this, we consider the vectors of forward variables, defined as \(\varvec{\alpha }_t = \bigl ( {\alpha }_t (1), \ldots , {\alpha }_t (m) \bigr )\), \(t=1,\ldots ,T\), where \({\alpha }_t (j) = f (x_1, \ldots , x_t, S_t=j)\), \(j=1,\ldots ,m\). We then have the recursion:

$$\begin{aligned} {\varvec{\alpha }}_1 = {\varvec{\pi }} {\mathbf {Q}}(x_1), \qquad {\varvec{\alpha }}_{t+1} = {\varvec{\alpha }}_{t} {\varvec{\Gamma }} {\mathbf {Q}}(x_{t+1}), \end{aligned}$$
(10)

where \({\mathbf {Q}}(x_t)= \text {diag} \bigl ( f_1 (x_{t}), \ldots , f_m (x_{t}) \big )\), with \(f_i(x_t) =f (x_{t} | S_{t}=i)\). The recursion (10) can be derived in a straightforward manner using the HMM dependence structure. The likelihood can then be written as a matrix product:

$$\begin{aligned} {\mathcal {L}}^{\text {HMM}} = \sum _{i=1}^m {\alpha }_T(i) = {\varvec{\pi }} {\mathbf {Q}}(x_1) {\varvec{\Gamma }} {\mathbf {Q}}(x_{2}) \ldots {\varvec{\Gamma }} {\mathbf {Q}}(x_{T}) {\mathbf {1}}, \end{aligned}$$

where \({\mathbf {1}}\in {\mathbb {R}}^m\) is a column vector of ones. For a missing observation \(x_t\), the associated matrix \({\mathbf {Q}}(x_{t})\) is simply replaced by the \(m\times m\) identity matrix. For more details on HMMs, see for example Zucchini and MacDonald (2009).

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Cite this article

Langrock, R., Michelot, T., Sohn, A. et al. Semiparametric stochastic volatility modelling using penalized splines. Comput Stat 30, 517–537 (2015). https://doi.org/10.1007/s00180-014-0547-5

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Keywords

  • B-splines
  • Cross-validation
  • Forward algorithm
  • Hidden Markov model
  • Numerical integration
  • Penalized likelihood