Bayesian vector heterogeneous autoregressive modelling

Abstract

The Bayesian vector autoregressive (BVAR) model with the Minnesota prior proposed by Litterman [Litterman RB. Forecasting with bayesian vector autoregressions-five years of experience. J Business Economic Statist.1986;4(1):25–38.] has been very successful in multivariate time series modelling, providing better forecasting performance. However, the conventional Minnesota prior for BVAR depends only on the latest lag; in turn, it is not suitable for multivariate long memory time series forecasting. This study extends BVAR to (possibly) high-dimensional long memory time series. To this end, we incorporate a vector heterogeneous autoregressive (VHAR) structure to accommodate long-term persistence and impose priors on distant lags as well. Our proposed Bayesian VHAR (BVHAR) models, the so-called BVHAR-S and BVHAR-L, are easy to implement by using the Normal-inverse-Wishart prior and added dummy variable approach of [Bańbura M, Giannone D, Reichlin L. Large bayesian vector auto regressions. J Appl Econom. 2010;25(1):71–92.]. A supporting simulation study also proves posterior consistency. We further apply our models to nine CBOE Volatility Indices (VIXs) and show that our BVHAR models perform best in forecasting with the narrowest forecasting region.

Keywords:

• Bayesian vector heterogeneous autoregressive model (BVHAR)
• Bayesian vector autoregressive (BVAR) model
• posterior consistency
• long memory

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 A multivariate t-distribution (MVT) $(\mathbf{X}|\mathit{\mu },\mathrm{\Sigma },\nu )\sim t_{\nu }(\mathit{\mu },\mathrm{\Sigma })$ is specified by probability density function $f\left(\mathbf{x}\right)=\frac{\mathrm{\Gamma }\left(\right(\nu +k)/2}{\mathrm{\Gamma }\left(\nu /2\right){\nu }^{k/2}{\pi }^{k/2}}det\left(\mathrm{\Sigma }{)}^{-1/2}\right\{1+\frac{1}{\nu }(\mathbf{x}-\mathit{\mu }{)}^{\mathrm{\prime }}{\mathrm{\Sigma }}^{-1}(\mathbf{x}-\mathit{\mu }){\}}^{-(\nu +k)/2}$.

Additional information

Funding

This work was supported by National Research Foundation of Korea [NRF-2022R1F1A1066209].