Fast and Flexible Bayesian Inference in Time-varying Parameter Regression Models

Abstract

In this article, we write the time-varying parameter (TVP) regression model involving K explanatory variables and T observations as a constant coefficient regression model with KT explanatory variables. In contrast with much of the existing literature which assumes coefficients to evolve according to a random walk, a hierarchical mixture model on the TVPs is introduced. The resulting model closely mimics a random coefficients specification which groups the TVPs into several regimes. These flexible mixtures allow for TVPs that feature a small, moderate or large number of structural breaks. We develop computationally efficient Bayesian econometric methods based on the singular value decomposition of the KT regressors. In artificial data, we find our methods to be accurate and much faster than standard approaches in terms of computation time. In an empirical exercise involving inflation forecasting using a large number of predictors, we find our models to forecast better than alternative approaches and document different patterns of parameter change than are found with approaches which assume random walk evolution of parameters.

Keywords:

• Clustering
• Hierarchical priors
• Singular value decomposition
• Time-varying parameter regression

Supplemental Materials

The supplementary material consists of three sections. In Section A, we provide additional technical details such as all full conditional posterior distributions and the resulting MCMC sampler. Section B provides a brief overview on the dataset used in the empirical work while Section C includes additional empirical results such as convergence diagnostics and robustness checks.

Acknowledgments

We thank the participants of the 6th NBP Workshop on Forecasting (Warsaw, 2019), the 11th European Seminar on Bayesian Econometrics (Madrid, 2021) and internal seminars at the University of Salzburg, the FAU Erlangen-Nuremberg and the ECB, four anonymous referees as well as Anna Stelzer, Michael Pfarrhofer and Paul Hofmarcher for helpful comments and suggestions.

Notes

1 Other approaches which remain agnostic on the transition distribution of the coefficients are, for example, Kalli and Griffin (2018) and Kapetanios, Marcellino, and Venditti (2019).

2 This setup can be easily extended to VAR models. In particular, recent articles (see, e.g., Carriero, Clark, and Marcellino 2019; Koop, Korobilis, and Pettenuzzo 2019; Tsionas, Izzeldin, and Trapani 2019; Cadonna, Frühwirth-Schnatter, and Knaus 2020; Huber, Koop, and Onorante 2021; Kastner and Huber 2020; Carriero et al. 2021) work with a structural VAR specification which allows for the equations to be estimated separately. Accordingly, the size of the system does not penalize the estimation time. This extension is part of our current research agenda.

3 In the case of lower triangular Z, the ${\tilde{\mathit{\beta }}}_t$ ’s can be interpreted as the shocks to the latent states with the actual value of the TVPs in time t given by ${\sum }_{s=1}^t{\tilde{\mathit{\beta }}}_s$.

4 Notice that if the condition number (i.e., the ratio of the largest and the smallest element in $\mathit{\lambda }$ ) is very large, numerical issues can arise. This is the case if ${\mathit{x}}_t\approx \mathbf{0}$. In our simulations and real data exercises, we never encountered computational issues. If these arise, then a simple solution would be to use a truncated SVD and discard eigenvalues smaller than a threshold very close to zero.

5 Using the Minnesota prior in combination with the clustering specification introduced in this sub-section is less sensible. That is, its form, involving different treatments of coefficients on lagged dependent variables and exogenous variables and smaller prior variances for longer lag length already, in a sense, clusters the coefficients into groups. A similar argument holds for a lower triangular matrix Z since that would translate into a random walk with a (potentially) time-varying drift term.

6 For a detailed discussion on the relationship between sparse finite mixtures and Dirichlet process mixtures, see Frühwirth-Schnatter and Malsiner-Walli (2019).

7 The R package fipp, which is available on CRAN, allows for investigating how influential the prior on a is and whether alternative specifications substantially change the posterior of the number of nonempty groups.

8 The AWOL sampler is implemented in R through the shrinkTVP package (Knaus et al. 2021).

9 A survey of these techniques is given in Hassani and Silva (2015).

10 For the SVD versions of the UCM models, we only present results for the g-prior with clustering as the other priors imply white-noise behavior for inflation which is not sensible.

Additional information

Funding

The first two authors gratefully acknowledge financial support by the Austrian Science Fund (FWF): ZK 35 and by funds of the Oesterreichische Nationalbank (Austrian Central Bank, Anniversary Fund, project number 18127).