A quasi-Bayesian local likelihood approach to time varying parameter VAR models

Abstract

The paper establishes a quasi-Bayesian local likelihood (QBLL) estimation methodology for a multivariate model with time varying parameters. The asymptotic validity of the resulting quasiposterior distributions of the drifting parameters is proven in general and, in the special case of a Gaussian VAR model, a closed form time varying Normal-Wishart expression for the quasiposterior distribution of the parameters is provided. In addition, this paper develops several Gibbs algorithms, which can sample from a VAR model with a mixture of time varying and time invariant parameters. The proposed estimators di¤er from existing state space approaches to VAR models in that they estimate parameter time variation nonparametrically, without imposing parametric stochastic processes on the parameters. The QBLL estimators are robust to misspeci…cation of the state equation and exhibit good …nite sample performance, even when compared to correctly speci…ed parametric state space models, as illustrated by a Monte Carlo exercise. In addition, we demonstrate that the QBLL approach provides a remedy to the 'curse of dimensionality'by accommodating large dimensional VAR systems and delivers improvements in the out-of-sample forecasts of key macroeconomic variables. Finally, the paper makes an empirical contribution to the literature on changing in ‡ation dynamics in the U.S., presenting evidence of a fall in in ‡ation persistence and volatility during the Great Moderation period.

Quasi‐Bayesian Estimation of Time‐Varying Volatility in DSGE Models

“…We first assume that we observe a realization from the history of the shocks η t for which we denote by Conditional on such a draw, the model simplifies to In this setting, Petrova () proposes a quasi‐Bayesian methodology for estimating Ω t at each point , which, for a wide class of time‐ varying processes (see Remark 1 below for details), is consistent and asymptotically valid for inference. We outline the quasi‐Bayesian methodology.…”

“…The bandwidth parameter H satisfies H → ∞ and . The normalization of the kernel weights in is proposed in Petrova () and is required to ensure that the prior is asymptotically dominated by the data, and the same rate of convergence as in Giraitis et al (2014, 2016) is achieved.…”

“…In this setting, Petrova (2017) proposes a quasi-Bayesian methodology for estimating Ω t at each point t ∈ {1, ..., T}, which, for a wide class of time-varying processes (see Remark 1 below for details), is consistent and asymptotically valid for inference. We outline the quasi-Bayesian methodology.…”

“…This paper proposes a novel quasi-Bayesian Metropolis-within-Gibbs algorithm that can be used to jointly estimate the time variation in the volatilities of a DSGE model's shocks and the time-invariant structural parameters. The proposed methodology shares similarity with Justiniano and Primiceri (2008) in the way the volatility enters the structural model, but instead of assuming a law of motion, it estimates the changing volatility with the help of a non-parametric estimator, building on the previous work of Petrova (2017) and Giraitis et al (2014). The resulting volatility estimates differ from those in Justiniano and Primiceri (2008) and Diebold et al (2017) in two ways.…”

“…It is the Bayesian treatment of this paper that facilitates the construction of such an algorithm and, to our best knowledge, this is the first procedure that can accommodate mixtures of time-varying volatilities and time-invariant parameters in a DSGE framework without imposing parametric assumptions on the volatility processes. The novelty is the facilitation of such mixtures: the frequentist method of Giraitis et al (2014Giraitis et al ( , 2016 cannot handle mixtures, while Petrova (2017) only deals with conjugate posterior mixtures where Metropolis steps are not required.…”