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# Cumulative LPDSs for the last 50 quarters of the usmacro.update dataset, for eleven different shrinkage priors: (1) the full hierarchical NGG prior, (2) the hierarchical NGG prior with fixed a ξ = a τ = c ξ = c τ = 0.1, (3) the NGG prior with a ξ = a τ = c ξ = c τ = 0.1 and κ 2 B = λ 2 B = 20, (4) the hierarchical horseshoe prior, (5) the horseshoe prior κ 2 B = λ 2 B = 20, (6) the full hierarchical NG prior, (7) the hierarchical NG prior with fixed a ξ = a τ = 0.1, (8) the NG prior with a ξ = a τ = 0.1 and κ 2 B = λ 2 B = 20, (9) the hierarchical Bayesian Lasso, and (10) the Bayesian Lasso with κ 2 B = λ 2 B = 20 and (11) ridge regression with κ 2 B = λ 2 B = 20.

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Time-varying parameter (TVP) models are widely used in time series analysis to flexibly deal with processes which gradually change over time. However, the risk of overfitting in TVP models is well known. This issue can be dealt with using appropriate global-local shrinkage priors, which pull time-varying parameters towards static ones. In this pape...

## Citations

... As shown by [37], such a prior has certain advantages compared to (5) and allows for the introduction of stochastic volatility in model (1), see [33] and Section 5.1. From the viewpoint of variable selection, prior (6) is a ridge prior in a standard regression model, conditional on the hidden path z = (β 0 , . . . ...

... The shrinkTVP package The R package shrinkTVP [33] offers efficient implementations of MCMC algorithms for TVP models with continuous shrinkage priors, specifically the triple gamma prior and its many special and limiting cases. It is designed to provide an easy entry point for fitting TVP models with shrinkage priors, while also giving more experienced users the option to adapt the model to their needs. ...

In this chapter, we review variance selection for time-varying parameter (TVP) models for univariate and multivariate time series within a Bayesian framework. We show how both continuous as well as discrete spike-and-slab shrinkage priors can be transferred from variable selection for regression models to variance selection for TVP models by using a non-centered parametrization. We discuss efficient MCMC estimation and provide an application to US inflation modeling.

... We also consider a modification of the UCSV model-a time-varying parameter model 2 To provide more intuition behind the UCSV example, we present in Appendix A, supplementary materials an additional conceptual exercise based on the basic SV model, which is a building block of the UCSV model. introduced in Section 3.3-with univariate state vector and SV for the observation errors, which can be fitted using off-theshelf R package shrinkTVP (Knaus et al. 2021). The second application involves the dataset on the Northern lapwing (Vanellus vanellus), which has been extensively analyzed in statistical ecology, see Besbeas et al. (2002), Brooks, King, and Morgan (2004), or King et al. (2008). ...

... However, to the best of our knowledge, there are no available packages for Bayesian estimation of the previous UCSV model (17). However, if we simplify the model we can use the shrinkTVP R package (Knaus et al. 2021), which fits the class of time-varying parameter (TVP) models. In particular, we replace the stochastic volatility for the trend process τ t with a homoscedastic noise, model the stochastic volatility of inflation, h t , as an AR (1) process (instead of a random walk) and use different priors. ...

... In our experiments we set both κ 2 B and λ 2 B to 0.02, which corresponds to relatively vague priors. For the parameters of the SV process h t we adopt priors from Kastner and Frühwirth-Schnatter (2014) and Knaus et al. (2021) given as Alternative packages. shrinkTVP implements a DA algorithm using Gibbs sampling with MH steps and applies a number of specialist algorithms to improve mixing. ...

We propose a novel efficient model-fitting algorithm for state space models. State space models are an intuitive and flexible class of models, frequently used due to the combination of their natural separation of the different mechanisms acting on the system of interest: the latent underlying system process; and the observation process. This flexibility, however, often comes at the price of more complicated model-fitting algorithms due to the associated analytically intractable likelihood. For the general case a Bayesian data augmentation approach is often employed, where the true unknown states are treated as auxiliary variables and imputed within the MCMC algorithm. However, standard “vanilla” MCMC algorithms may perform very poorly due to high correlation between the imputed states and/or parameters, often leading to model-specific bespoke algorithms being developed that are nontransferable to alternative models. The proposed method addresses the inefficiencies of traditional approaches by combining data augmentation with numerical integration in a Bayesian hybrid approach. This approach permits the use of standard “vanilla” updating algorithms that perform considerably better than the traditional approach in terms of improved mixing and lower autocorrelation, and has the potential to be incorporated into bespoke model-specific algorithms. To demonstrate the ideas, we apply our semi-complete data augmentation algorithm to different application areas and models, leading to distinct implementation schemes and improved mixing and demonstrating improved mixing of the model parameters. Supplementary materials for this article are available online.

... Hosszejni and Kastner (2021) present an update of R package stochvol (Kastner 2016) to fit univariate stochastic volatility (SV) models (that now can handle linear mean models, conditionally heavy tails, and the leverage effect in combination with SV) and describe the factorstochvol R package for multivariate SV models. Time-varying parameter (TVP) models using global-local shrinkage priors to avoid overfitting with the shrinkTVP R package are described in Knaus, Bitto-Nemling, Cadonna, and Frühwirth-Schnatter (2021). Kuschnig and Vashold (2021) fit vector auto-regression (VAR) models for multivariate time series with hierarchical prior selection using the R package BVAR. ...

... For simulating values from the posterior distributions, they use the BUGS language via JAGS (see e.g., Bonner et al. 2021;Erler et al. 2021;Mayrink et al. 2021;Weber et al. 2021), Stan (see e.g., Bürkner 2021; Merkle et al. 2021;Weber et al. 2021), or nimble Bonner et al. 2021), interfaced with R by means of the corresponding packages rjags (Plummer, Stukalov, and Denwood 2021), rstan (Stan Development Team 2021) and nimble . Alternatively, other papers (e.g., Corradin et al. 2021;Hosszejni and Kastner 2021;Knaus et al. 2021;Venturini and Piccarreta 2021) write sampling functions in C++ which are then integrated into R by using the Rcpp (Eddelbuettel and François 2011) and RcppArmadillo (Eddelbuettel and Sanderson 2014) packages. Finally, some papers do not use MCMC but numerical approximations such as Eggleston et al. (2021), Fasiolo et al. (2021) and Van Niekerk et al. (2021). ...

In this summary we introduce the papers published in the special issue on Bayesian statistics. This special issue comprises 20 papers on Bayesian statistics and Bayesian inference on different topics such as general packages for hierarchical linear model fitting, survival models, clinical trials, missing values, time series, hypothesis testing, priors, approximate Bayesian computation, and others.

... This brief discussion shows that the SVD algorithm scales well and renders estimation of huge dimensional models feasible. Notes: The figure shows the actual and theoretical time necessary to obtain a draw ofβ using our proposed SVD algorithm for Z being block-diagonal and lower triangular, an AWOL sampler (implemented in R through the shrinkTVP package of Knaus et al., 2021) and the FFBS algorithm. The dashed red lines refer to the SVD approach with a lower triangular Z and a ridge-prior, the orange dashed line refers to the SVD algorithm with block-diagonal Z, the dashed green lines refer to the AWOL sampler, while the dashed blue lines indicate the FFBS. ...

In this paper, 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. JEL: C11, C30, E3, E44

... Hyperparameters are chosen in the following way: d 1 = d 2 = e 1 = e 2 = 0.001, ν ξ = ν τ = 5 and b ξ = b τ = 10. For further details of the model specification and MCMC sampling, we refer to Bitto and Frühwirth-Schnatter (2019) and, especially for notation and choice of hyperparameters, Bitto-Nemling et al. (2019). ...

... The parameters are estimated by means of MCMC sampling drawing 100,000 samples and a burn-in period of 50,000 samples using the R package shrinkTVP (Bitto-Nemling et al. 2019). Convergence diagnostics indicate efficient sampling. ...

We study inflation dynamics in emerging, small open economies of Central and Eastern Europe (CEE) and find new empirical evidence of the existence of the New Keynesian Phillips Curve (NKPC). Acknowledging specification uncertainty, a comprehensive set of alternative proxies for the NKPC’s components is assessed. Our results indicate the superiority of labor market measures for economic slack, support the use of survey inflation expectations and confirm the NKPC’s open economy version. Further, we investigate the stability of the NKPC over time, performing Bayesian inference in a time-varying parameter stochastic volatility version of the model. The results do not suggest the NKPC to have flattened in CEE challenging recent evidence in advanced economies. Inflationary dynamics have not decoupled from the state of the domestic economy. Therefore, a balanced approach to monetary policy which neither neglects the domestic nor external drivers of inflation and focuses on anchoring inflation expectations is well justified.

... There is also a growing literature which extends these methods to the TVP case. Examples include Belmonte et al. (2014), Kalli and Griffin (2014), Eisenstat et al. (2016), Hauzenberger et al. (2019), Kowal et al. (2019) and Bitto-Nemling et al. (2019). ...

Time-varying parameter (TVP) regression models can involve a huge number of coefficients. Careful prior elicitation is required to yield sensible posterior and predictive inferences. In addition, the computational demands of Markov Chain Monte Carlo (MCMC) methods mean their use is limited to the case where the number of predictors is not too large. In light of these two concerns, this paper proposes a new dynamic shrinkage prior which reflects the empirical regularity that TVPs are typically sparse (i.e. time variation may occur only episodically and only for some of the coefficients). A scalable MCMC algorithm is developed which is capable of handling very high dimensional TVP regressions or TVP Vector Autoregressions. In an exercise using artificial data we demonstrate the accuracy and computational efficiency of our methods. In an application involving the term structure of interest rates in the eurozone, we find our dynamic shrinkage prior to effectively pick out small amounts of parameter change and our methods to forecast well.

China's monetary policy framework has evolved considerably in the past two decades, increasingly moving from using quantity-based instruments and targets to using price-based instruments and targets. This paper assesses the effectiveness of monetary policy in China by examining the influence of monetary policy on market interest rates using an event-study approach. We find that the effectiveness of price-based instruments in impacting market interest rates increases over time, and that price-based instruments are as effective as quantity instruments during the period since the completion of interest rates liberalization. Furthermore, central bank communications, an increasingly important aspect of monetary policy, affect medium- and long-term market interest rates. Our findings are robust to the use of an alternative measure of monetary policy surprise and an alternative estimation method.