[Paper Review] Bayesian Inference on QGARCH Model Using the Adaptive Construction Scheme

Tetsuya Takaishi|RePEc: Research Papers in Economics|Jul 30, 2009

Complex Systems and Time Series Analysis4 citations

TL;DR

This paper proposes an adaptive construction scheme for Bayesian inference in the Quadratic GARCH (QGARCH) model using Markov Chain Monte Carlo (MCMC) methods. By adaptively updating the proposal density in the Metropolis-Hastings algorithm using sampled data, the method significantly reduces autocorrelation in MCMC samples, achieving an autocorrelation time (τ) for the α parameter that is 90 times smaller than with the standard Metropolis algorithm, demonstrating high efficiency in sampling de-correlated parameters.

ABSTRACT

We study the performance of the adaptive construction scheme for a Bayesian inference on the Quadratic GARCH model which introduces the asymmetry in time series dynamics. In the adaptive construction scheme a proposal density in the Metropolis-Hastings algorithm is constructed adaptively by changing the parameters of the density to fit the posterior density. Using artificial QGARCH data we infer the QGARCH parameters by applying the adaptive construction scheme to the Bayesian inference of QGARCH model. We find that the adaptive construction scheme samples QGARCH parameters effectively, i.e. correlations between the sampled data are very small. We conclude that the adaptive construction scheme is an efficient method to the Bayesian estimation of the QGARCH model.

Motivation & Objective

• To evaluate the performance of the adaptive construction scheme for Bayesian estimation of the QGARCH model, which captures asymmetry in financial volatility.
• To address the challenge of high autocorrelation in MCMC sampling for GARCH-type models with multiple parameters.
• To test whether the adaptive construction scheme, which avoids reliance on maximum likelihood for proposal density tuning, remains effective for the more complex QGARCH model with four parameters.
• To compare the efficiency of the adaptive scheme against the standard Metropolis algorithm in terms of autocorrelation time and convergence.

Proposed method

• The adaptive construction scheme dynamically updates the parameters of a multivariate Student’s t-distribution proposal density during MCMC simulation using real-time sampled data.
• The Metropolis-Hastings algorithm uses this adaptive proposal density to generate Markov chains targeting the posterior distribution of QGARCH parameters.
• The proposal density is updated iteratively using the mean (M) and covariance matrix (Σ) of the sampled parameter values to better match the posterior distribution.
• Bayesian inference is performed using a non-informative (uniform) prior, and posterior expectations are estimated via ergodic averages over MCMC samples.
• Autocorrelation functions (ACF) and autocorrelation times (τ) are computed to quantify the correlation between successive MCMC samples.
• Acceptance rates of the MH step are monitored during simulation to assess convergence and stability of the adaptive scheme.

Experimental results

Research questions

• RQ1 Does the adaptive construction scheme significantly reduce autocorrelation in MCMC samples for the QGARCH model compared to the standard Metropolis algorithm?
• RQ2 Can the adaptive construction scheme maintain high sampling efficiency for the QGARCH model, which has four parameters and exhibits asymmetric volatility dynamics?
• RQ3 How does the adaptive scheme's performance in terms of acceptance rate and convergence compare to traditional MCMC methods in the absence of maximum likelihood tuning?
• RQ4 At what point during simulation does the adaptive parameter update become stable, and can it be safely terminated without degrading performance?

Key findings

• The adaptive construction scheme reduces the autocorrelation time (τ) for the α parameter by a factor of 90 compared to the standard Metropolis algorithm, with τ = 340 ± 100 for Metropolis and τ = 4 ± 1 for the adaptive scheme.
• The acceptance rate in the MH step stabilizes at approximately 70% after an initial transient phase, indicating effective proposal density adaptation.
• The autocorrelation function (ACF) of the adaptive scheme decays rapidly, while the ACF of the Metropolis algorithm remains high for many lags, confirming lower correlation in sampled data.
• The covariance matrix V of the sampled parameters converges to stable values as simulation length increases, indicating reliable estimation of posterior uncertainty.
• The adaptive scheme produces MCMC samples with minimal correlation, leading to significantly smaller statistical errors compared to the standard Metropolis method.
• The adaptive construction scheme is effective even for models with higher dimensionality, such as the QGARCH model with four parameters, confirming its robustness and scalability.