High-Dimensional Posterior Consistency in Bayesian Vector Autoregressive Models

Abstract

Vector autoregressive (VAR) models aim to capture linear temporal interdependencies amongst multiple time series. They have been widely used in macroeconomics and financial econometrics and more recently have found novel applications in functional genomics and neuroscience. These applications have also accentuated the need to investigate the behavior of the VAR model in a high-dimensional regime, which provides novel insights into the role of temporal dependence for regularized estimates of the model’s parameters. However, hardly anything is known regarding properties of the posterior distribution for Bayesian VAR models in such regimes. In this work, we consider a VAR model with two prior choices for the autoregressive coefficient matrix: a non-hierarchical matrix-normal prior and a hierarchical prior, which corresponds to an arbitrary scale mixture of normals. We establish posterior consistency for both these priors under standard regularity assumptions, when the dimension p of the VAR model grows with the sample size n (but still remains smaller than n). A special case corresponds to a shrinkage prior that introduces (group) sparsity in the columns of the model coefficient matrices. The performance of the model estimates are illustrated on synthetic and real macroeconomic data sets.

Convergence analysis of a collapsed Gibbs sampler for Bayesian vector autoregressions

“…Proof of Lemma 3.1. The full conditional distribution of B is immediate from dropping terms not depending on B in (11). Consider next the integrand in (12).…”

“…The priors on α and Σ are common in macroeconomics [23] and the prior on Σ includes the inverse Wishart (D ∈ S r ++ , a > 2r) and Jeffreys prior (D = 0, a = r + 1) as special cases. In other work on similar models it is often assumed that [Z, X] has full rank or that the prior for [A T , B T ] T is proper [1,2,11,13,25,44]. Treating A and B differently is appealing in the current setting: it adheres to the common practice of using flat priors for regression coefficients such as B, while m and C can be chosen to reflect the fact that many commonly studied time series are known to be near non-stationary in the unit root sense.…”

“…It implies that if the prior on Σ is a proper inverse Wishart density, so that a > 2r and D ∈ S r ++ , then the posterior is proper if f (α) is proper and X has full column rank. In particular, r or q can be arbitrarily large in comparison to n. Thus, this setting is compatible with large VARs [3,11,24]. The second set of conditions allows for the use of improper priors also on α and Σ when n is large in comparison to all of p, q, and r. The full column rank of [Y, Z, X] is natural in large-n settings.…”

Sectoral Slowdowns in the UK: Evidence from Transmission Probabilities and Economic Linkages

“…Researchers have used Bayesian techniques (known as Bayesian VAR) to improve the inference and the forecasting performance of their VAR models, as well as to avoid overfitting or other ill-behaved estimation problems. This is often done using Bayesian priors that have an effect comparable to the frequentist notion of regularization (Bańbura et al, 2010;Koop, 2013;Ghosh et al, 2019).…”

Finite sample theory for high-dimensional functional/scalar time series with applications

“…In the context of functional time series, many standard univariate or low-dimensional time series methods have been recently adapted to the functional domain with theoretical properties explored from a standard asymptotic perspective, see, e.g., Bosq (2000); Bathia, Yao and Ziegelmann (2010); Hörmann and Kokoszka (2010); Panaretos and Tavakoli (2013); Aue, Norinho and Hörmann (2015); Hörmann, Kidziński and Kokoszka (2015); Pham and Panaretos (2018); Li, Robinson and Shang (2020) and reference therein. In the context of high-dimensional time series, some lower-dimensional structural assumptions are often incorporated on the model parameter space and different regularized estimation procedures have been developed for the respective learning tasks including, e.g., high-dimensional sparse linear regression (Basu and Michailidis, 2015;Wu and Wu, 2016;Han and Tsay, 2020) and high-dimensional sparse vector autoregression (Guo, Wang and Yao, 2016;Lin and Michailidis, 2017;Gao et al, 2019;Ghosh, Khare and Michailidis, 2019;Zhou and Raskutti, 2019;Wong, Li and Tewari, 2020;Lin and Michailidis, 2020).…”