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Ancillarity-Sufficiency Interweaving Strategy (ASIS) for Boosting MCMC Estimation of Stochastic Volatility Models
Gregor Kastner, Sylvia Frühwirth-Schnatter
TL;DR
The paper addresses the lack of a uniformly effective parameterization for MCMC estimation of stochastic volatility models across different parameter values. It interweaves centered and non-centered parameterizations using ASIS, finding robust sampling efficiency across parameter settings and exchange-rate applications.
Problem
MCMC efficiency varies with stochastic-volatility parameter values: centered sampling performs poorly when volatility of volatility is small, while non-centered sampling has deficiencies for highly persistent latent series.
Method
ASIS samples key parameters in both centered and non-centered parameterizations within each iteration, using global interweaving strategies with either parameterization as baseline.
Results
Interweaving strategies perform well across underlying parameter values, outperforming or matching the ideal parameterization with an added computational cost of around 2%.
Takeaways & Limitations
ASIS overcomes shortcomings of centered and non-centered parameterizations, supporting efficient Bayesian inference without selecting a parameterization beforehand.
Takeaways & Limitations
The centered SV formulation assumes independent standard-normal innovations and a stationary-distribution initial state; the prior on σ² is also not bounded away from zero.
Abstract
from arXiv · showhide
Bayesian inference for stochastic volatility models using MCMC methods highly depends on actual parameter values in terms of sampling efficiency. While draws from the posterior utilizing the standard centered parameterization break down when the volatility of volatility parameter in the latent state equation is small, non-centered versions of the model show deficiencies for highly persistent latent variable series. The novel approach of ancillarity-sufficiency interweaving has recently been shown to aid in overcoming these issues for a broad class of multilevel models. In this paper, we demonstrate how such an interweaving strategy can be applied to stochastic volatility models in order to greatly improve sampling efficiency for all parameters and throughout the entire parameter range. Moreover, this method of "combining best of different worlds" allows for inference for parameter constellations that have previously been infeasible to estimate without the need to select a particular parameterization beforehand.
1 Introduction
The paper examines how centered and non-centered parameterizations affect MCMC efficiency in stochastic volatility models and introduces ASIS to interweave them. It evaluates the approach through simulation and exchange-rate applications.
- Stochastic volatility models represent time-varying volatility with latent autoregressive processes and provide a nondeterministic alternative to GARCH-type designs.
- The fully non-centered parameterization standardizes the latent process and shifts the scale through ω = e^µ, with prior work finding it effective over moderate parameter ranges.
- The centered parameterization can produce highly correlated, slowly converging latent-state draws, motivating alternative state-sampling methods such as mixture approximations and FFBS.
- MCMC efficiency depends heavily on the data-generating parameter values, so no single centered or non-centered parameterization is uniformly best.
- ASIS interweaves centered and non-centered parameterizations to build a robustly efficient sampler without requiring parameterization selection beforehand.
- The paper compares parameterizations using extensive simulations and applies the methods to daily exchange-rate data.
2 Bayesian Inference in the SV Model
Bayesian inference uses an auxiliary-mixture representation to make the non-Gaussian SV model conditionally Gaussian, enabling efficient latent-volatility sampling and parameter updates. The section specifies priors, blocking strategies, and mixture-indicator sampling for centered and non-centered parameterizations.
- Prior distributions: The priors include a beta-transformed persistence parameter whose support guarantees stationarity and a volatility-of-volatility prior that is not bounded away from zero.The latter choice is motivated by reduced prior influence when the true volatility of volatility is small.
- MCMC methodology: The SV likelihood is handled by approximating log(ϵ_t^2) with a ten-component mixture, yielding a linear and conditionally Gaussian state-space model.The mixture indicator r_t identifies one of ten components with tabulated means and variances.
- Centered parameterization: Centered sampling uses one-, two-, or three-block updates, with Metropolis-Hastings required because the selected priors are non-conjugate.The two-block sampler jointly updates µ and φ conditional on σ2, while the three-block sampler updates parameters separately.
- Non-centered parameterization: Non-centered sampling leaves only φ in the state equation, while µ and σ2 can be Gibbs-updated jointly or separately through a regression formulation.The conditional observation equation is rewritten as a homoskedastic regression model.
- MCMC methodology: The latent volatilities are sampled jointly with AWOL by exploiting the tridiagonal precision matrix induced by the AR(1) state process.Cholesky factorization and band back-substitution avoid FFBS and matrix inversion.
- MCMC methodology: The indicator probabilities P(r_t = k|·) combine mixture weights with component likelihood terms and are sampled by inverse transform sampling.Computing these probabilities requires T × 10 exponential-function calls, but the step can be parallelized.
3 Interweaving C and NC by ASIS
ASIS interweaves centered and non-centered parameterizations by sampling parameters in both forms within each MCMC iteration. The resulting global interweaving strategy combines their parameter-specific advantages while adding little computational cost.
- Motivation: In the SV model, h is sufficient for µ and σ in the centered form, whereas ˜h is ancillary for these parameters in the non-centered form.This ancillary-sufficiency pairing motivates interweaving the two latent-variable representations.
- Convergence rationale: The interwoven sampler’s convergence rate is bounded by R√r_Cr_NC, where R is the maximal posterior correlation between h and ˜h.The bound links interwoven convergence to the individual parameterization rates and the posterior correlation.
- GIS-C: The GIS-C algorithm samples h in C, transforms to NC, redraws µ, φ, and σ there, and then returns to C through the deterministic transformation h_t = µ + σ˜h_t.The parameters are sampled once in each parameterization during every iteration.
- Computational cost: The additional transformations and redraws have minimal impact on total sampling time because the interweaving steps are computationally cheap.This cost assessment is stated for the GIS implementation.
- GIS-NC: GIS-NC reverses the baseline: it samples ˜h in NC, moves to C, redraws the parameters in C, and samples the indicators in NC.The latent volatilities and indicators are sampled in the non-centered baseline representation.
- Interpretation: Interweaving differs from alternating complete MCMC cycles across parameterizations and from compromising through partial non-centering.The strategy instead redraws the parameters in both representations within one iteration.
4 Simulation Results
Simulation efficiency depends strongly on the persistence and volatility-of-volatility parameters, with centered and non-centered parameterizations failing in opposite regions. Interweaving remains efficient across the examined parameter range while adding little computational cost.
- Simulation design: 45 parameter settings varied φtrue across 0 to 0.99 and σtrue across 0.1 to 0.5, covering plausible and extreme regimes.The simulations used 500 independently generated datasets for each setting.
- Simulation design: 2.3–2.4 seconds per 1000 draws was the mean runtime for C and GIS-NC, with times roughly constant across parameter values.For T = 500, the corresponding runtime was 0.23–0.24 seconds.
- Raw parameterizations: When σtrue = 0.1, C produces large autocorrelation for draws from p(µ|y) and p(σ|y), whereas NC performs substantially better.This illustrates the small-volatility-of-volatility regime in which the centered parameterization mixes poorly.
- Raw parameterizations: When σtrue = 0.5, C shows little autocorrelation while NC chains do not mix well.The persistence and level parameters are held constant relative to the preceding setup.
- Efficiency overview: For µ, C is efficient across a wide range except when both σtrue and φtrue are small, while NC has IFs of 1000 and above when both are large.The interwoven samplers are at least as efficient as the ideal parameterization, with interweaving costing around 2% in the reported setup.
- Efficiency overview: Interweaving is always at least twice as effective as the suboptimal raw parameterization in effective sample size, and it performs well for σ across all underlying parameter values.For φ, differences between C and NC are less pronounced because φ is not involved in the reparameterization; interwoven two-block samplers have IFs of 30 or below.
5 Application to Exchange Rate Data
The methodology is applied to 3140 daily observations across 23 currencies, with results illustrating highly persistent Euro–US dollar volatility and efficiency patterns across exchange rates.
- Posterior estimation uses a Beta prior for φ, vague priors for µ and σ², a 10,000-draw burn-in, and 1,000,000 posterior draws.
- For EUR/US$, the posterior means are φ = 0.993 and σ = 0.07, producing a much smoother estimated volatility path than the absolute log-returns.
- The application reports inefficiency factors and effective-sample-size gains for the parameterizations and blocking strategies across all 23 exchange rates.
6 Concluding Remarks
The paper concludes that ASIS improves MCMC sampling efficiency over centered and non-centered parameterizations at little extra computational cost. It also identifies extensions to richer stochastic-volatility models as promising but incompletely investigated.
- ASIS overcomes shortcomings of centered and non-centered parameterizations, outperforming them for all parameters at very little extra computational cost.
- The baseline used for interweaving has only minor influence on the resulting sampling efficiency.
- The interweaving concept is presented as general and potentially applicable to models with non-Gaussian innovations, asymmetry, or both.
- Preliminary leverage-model results support this extension possibility, but a thorough investigation remains beyond the article’s scope.