Bayesian model comparison for time‐varying parameter VARs with stochastic volatility

We develop importance sampling methods for computing two popular Bayesian model comparison criteria, namely, the marginal likelihood and the deviance information criterion (DIC) for time‐varying parameter vector autoregressions (TVP‐VARs), where both the regression coefficients and volatilities are drifting over time. The proposed estimators are based on the integrated likelihood, which are substantially more reliable than alternatives. Using US data, we find overwhelming support for the TVP‐VAR with stochastic volatility compared to a conventional constant coefficients VAR with homoskedastic innovations. Most of the gains, however, appear to have come from allowing for stochastic volatility rather than time variation in the VAR coefficients or contemporaneous relationships. Indeed, according to both criteria, a constant coefficients VAR with stochastic volatility outperforms the more general model with time‐varying parameters.     

Forecasting using variational Bayesian inference in large vector autoregressions with hierarchical shrinkage

Many recent papers in macroeconomics have used large vector autoregressions (VARs) involving 100 or more dependent variables. With so many parameters to estimate, Bayesian prior shrinkage is vital to achieve reasonable results. Computational concerns currently limit the range of priors used and render difficult the addition of empirically important features such as stochastic volatility to the large VAR. In this paper, we develop variational Bayesian methods for large VARs that overcome the computational hurdle and allow for Bayesian inference in large VARs with a range of hierarchical shrinkage priors and with time-varying volatilities. We demonstrate the computational feasibility and good forecast performance of our methods in an empirical application involving a large quarterly US macroeconomic data set.     

Forecasting with Medium and Large Bayesian VARS

This paper is motivated by the recent interest in the use of Bayesian VARs for forecasting, even in cases where the number of dependent variables is large. In such cases factor methods have been traditionally used, but recent work using a particular prior suggests that Bayesian VAR methods can forecast better. In this paper, we consider a range of alternative priors which have been used with small VARs, discuss the issues which arise when they are used with medium and large VARs and examine their forecast performance using a US macroeconomic dataset containing 168 variables. We find that Bayesian VARs do tend to forecast better than factor methods and provide an extensive comparison of the strengths and weaknesses of various approaches. Typically, we find that the simple Minnesota prior forecasts well in medium and large VARs, which makes this prior attractive relative to computationally more demanding alternatives. Our empirical results show the importance of using forecast metrics based on the entire predictive density, instead of relying solely on those based on point forecasts. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient estimation of Bayesian VARMAs with time‐varying coefficients

Empirical work in macroeconometrics has been mostly restricted to using vector autoregressions (VARs), even though there are strong theoretical reasons to consider general vector autoregressive moving averages (VARMAs). A number of articles in the last two decades have conjectured that this is because estimation of VARMAs is perceived to be challenging and proposed various ways to simplify it. Nevertheless, VARMAs continue to be largely dominated by VARs, particularly in terms of developing useful extensions. We address these computational challenges with a Bayesian approach. Specifically, we develop a Gibbs sampler for the basic VARMA, and demonstrate how it can be extended to models with time‐varying vector moving average (VMA) coefficients and stochastic volatility. We illustrate the methodology through a macroeconomic forecasting exercise. We show that in a class of models with stochastic volatility, VARMAs produce better density forecasts than VARs, particularly for short forecast horizons.     

Forecasting with High‐Dimensional Panel VARs

This paper develops methods for estimating and forecasting in Bayesian panel vector autoregressions of large dimensions with time‐varying parameters and stochastic volatility. We exploit a hierarchical prior that takes into account possible pooling restrictions involving both VAR coefficients and the error covariance matrix, and propose a Bayesian dynamic learning procedure that controls for various sources of model uncertainty. We tackle computational concerns by means of a simulation‐free algorithm that relies on analytical approximations to the posterior. We use our methods to forecast inflation rates in the eurozone and show that these forecasts are superior to alternative methods for large vector autoregressions.     

Bayesian doubly adaptive elastic-net Lasso for VAR shrinkage

We develop a novel Bayesian doubly adaptive elastic-net Lasso (DAELasso) approach for VAR shrinkage. DAELasso achieves variable selection and coefficient shrinkage in a data-based manner. It deals constructively with explanatory variables which tend to be highly collinear by encouraging the grouping effect. In addition, it also allows for different degrees of shrinkage for different coefficients. Rewriting the multivariate Laplace distribution as a scale mixture, we establish closed-form conditional posteriors that can be drawn from a Gibbs sampler. An empirical analysis shows that the forecast results produced by DAELasso and its variants are comparable to those from other popular Bayesian methods, which provides further evidence that the forecast performances of large and medium sized Bayesian VARs are relatively robust to prior choices, and, in practice, simple Minnesota types of priors can be more attractive than their complex and well-designed alternatives.     

Combining VAR and DSGE forecast densities

A popular macroeconomic forecasting strategy utilizes many models to hedge against instabilities of unknown timing; see (among others) Stock and Watson (2004), Clark and McCracken (2010), and Jore et al. (2010). Existing studies of this forecasting strategy exclude dynamic stochastic general equilibrium (DSGE) models, despite the widespread use of these models by monetary policymakers. In this paper, we use the linear opinion pool to combine inflation forecast densities from many vector autoregressions (VARs) and a policymaking DSGE model. The DSGE receives a substantial weight in the pool (at short horizons) provided the VAR components exclude structural breaks. In this case, the inflation forecast densities exhibit calibration failure. Allowing for structural breaks in the VARs reduces the weight on the DSGE considerably, but produces well-calibrated forecast densities for inflation.     

Combining survey long-run forecasts and nowcasts with BVAR forecasts using relative entropy

This paper constructs hybrid forecasts that combine forecasts from vector autoregressive (VAR) model(s) with both short- and long-term expectations from surveys. Specifically, we use the relative entropy to tilt one-step-ahead and long-horizon VAR forecasts to match the nowcasts and long-horizon forecasts from the Survey of Professional Forecasters. We consider a variety of VAR models, ranging from simple fixed-parameter to time-varying parameters. The results across models indicate meaningful gains in multi-horizon forecast accuracy relative to model forecasts that do not incorporate long-term survey conditions. Accuracy improvements are achieved for a range of variables, including those that are not tilted directly but are affected through spillover effects from tilted variables. The accuracy gains for hybrid inflation forecasts from simple VARs are substantial, statistically significant, and competitive to time-varying VARs, univariate benchmarks, and survey forecasts. We view our proposal as an indirect approach to accommodating structural change and moving end points.     

Forecasting large datasets with Bayesian reduced rank multivariate models

The paper addresses the issue of forecasting a large set of variables using multivariate models. In particular, we propose three alternative reduced rank forecasting models and compare their predictive performance for US time series with the most promising existing alternatives, namely, factor models, large‐scale Bayesian VARs, and multivariate boosting. Specifically, we focus on classical reduced rank regression, a two‐step procedure that applies, in turn, shrinkage and reduced rank restrictions, and the reduced rank Bayesian VAR of Geweke ( 1996 ). We find that using shrinkage and rank reduction in combination rather than separately improves substantially the accuracy of forecasts, both when the whole set of variables is to be forecast and for key variables such as industrial production growth, inflation, and the federal funds rate. The robustness of this finding is confirmed by a Monte Carlo experiment based on bootstrapped data. We also provide a consistency result for the reduced rank regression valid when the dimension of the system tends to infinity, which opens the way to using large‐scale reduced rank models for empirical analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

VAR FORECASTING USING BAYESIAN VARIABLE SELECTION

This paper develops methods for automatic selection of variables in Bayesian vector autoregressions (VARs) using the Gibbs sampler. In particular, I provide computationally efficient algorithms for stochastic variable selection in generic linear and nonlinear models, as well as models of large dimensions. The performance of the proposed variable selection method is assessed in forecasting three major macroeconomic time series of the UK economy. Data‐based restrictions of VAR coefficients can help improve upon their unrestricted counterparts in forecasting, and in many cases they compare favorably to shrinkage estimators. Copyright © 2011 John Wiley & Sons, Ltd.     

Forecasting with Bayesian Vector Autoregressions Estimated Using Professional Forecasts

We propose a Bayesian shrinkage approach for vector autoregressions (VARs) that uses short‐term survey forecasts as an additional source of information about model parameters. In particular, we augment the vector of dependent variables by their survey nowcasts, and claim that each variable modelled in the VAR and its nowcast are likely to depend in a similar way on the lagged dependent variables. In an application to macroeconomic data, we find that the forecasts obtained from a VAR fitted by our new shrinkage approach typically yield smaller mean squared forecast errors than the forecasts obtained from a range of benchmark methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Extracting a common stochastic trend: Theory with some applications

This paper investigates the statistical properties of estimators of the parameters and unobserved series for state space models with integrated time series. In particular, we derive the full asymptotic results for maximum likelihood estimation using the Kalman filter for a prototypical class of such models—those with a single latent common stochastic trend. Indeed, we establish the consistency and asymptotic mixed normality of the maximum likelihood estimator and show that the conventional method of inference is valid for this class of models. The models we explicitly consider comprise a special–yet useful–class of models that may be employed to extract the common stochastic trend from multiple integrated time series. Such models can be very useful to obtain indices that represent fluctuations of various markets or common latent factors that affect a set of economic and financial variables simultaneously. Moreover, our derivation of the asymptotics of this class makes it clear that the asymptotic Gaussianity and the validity of the conventional inference for the maximum likelihood procedure extends to a larger class of more general state space models involving integrated time series. Finally, we demonstrate the utility of this class of models extracting a common stochastic trend from three sets of time series involving short- and long-term interest rates, stock return volatility and trading volume, and Dow Jones stock prices.     

Moving average stochastic volatility models with application to inflation forecast

We introduce a new class of models that has both stochastic volatility and moving average errors, where the conditional mean has a state space representation. Having a moving average component, however, means that the errors in the measurement equation are no longer serially independent, and estimation becomes more difficult. We develop a posterior simulator that builds upon recent advances in precision-based algorithms for estimating these new models. In an empirical application involving US inflation we find that these moving average stochastic volatility models provide better in-sample fitness and out-of-sample forecast performance than the standard variants with only stochastic volatility.     

Exchange rates and oil prices: A multivariate stochastic volatility analysis

This paper uses the multivariate stochastic volatility (MSV) and the multivariate GARCH (MGARCH) models to investigate the volatility interactions between the oil market and the foreign exchange (FX) market, in an attempt to extract information intertwined in the two for better volatility forecast. Our analysis takes into account structural breaks in the data. We find that when the markets are relatively calm (before the 2008 crisis), both oil and FX markets respond to shocks simultaneously and therefore no interaction is detected in daily data. However, during turbulent time, there is bi-directional volatility interaction between the two. In other words, innovations that hit one market also have some impact on the other at a later date and thus using such a dependence significantly improves the forecasting power of volatility models. The MSV models outperform others in fitting the data and forecasting exchange rate volatility. However, the MGARCH models do better job in forecasting oil volatility.     

Dynamic functional time-series forecasts of foreign exchange implied volatility surfaces

This paper presents static and dynamic versions of univariate, multivariate, and multilevel functional time-series methods to forecast implied volatility surfaces in foreign exchange markets. We find that dynamic functional principal component analysis generally improves out-of-sample forecast accuracy. Specifically, the dynamic univariate functional time-series method shows the greatest improvement. Our models lead to multiple instances of statistically significant improvements in forecast accuracy for daily EUR–USD, EUR–GBP, and EUR–JPY implied volatility surfaces across various maturities, when benchmarked against established methods. A stylised trading strategy is also employed to demonstrate the potential economic benefits of our proposed approach.     

Macroeconomic forecasting and structural change

The aim of this paper is to assess whether modeling structural change can help improving the accuracy of macroeconomic forecasts. We conduct a simulated real‐time out‐of‐sample exercise using a time‐varying coefficients vector autoregression (VAR) with stochastic volatility to predict the inflation rate, unemployment rate and interest rate in the USA. The model generates accurate predictions for the three variables. In particular, the forecasts of inflation are much more accurate than those obtained with any other competing model, including fixed coefficients VARs, time‐varying autoregressions and the naïve random walk model. The results hold true also after the mid 1980s, a period in which forecasting inflation was particularly hard. Copyright © 2011 John Wiley & Sons, Ltd.     

Large time‐varying parameter VARs: A nonparametric approach

In this paper we introduce a nonparametric estimation method for a large Vector Autoregression (VAR) with time‐varying parameters. The estimators and their asymptotic distributions are available in closed form. This makes the method computationally efficient and capable of handling information sets as large as those typically handled by factor models and Factor Augmented VARs. When applied to the problem of forecasting key macroeconomic variables, the method outperforms constant parameter benchmarks and compares well with large (parametric) Bayesian VARs with time‐varying parameters. The tool can also be used for structural analysis. As an example, we study the time‐varying effects of oil price shocks on sectoral U.S. industrial output. According to our results, the increased role of global demand in shaping oil price fluctuations largely explains the diminished recessionary effects of global energy price increases.     

Marginalized Predictive Likelihood Comparisons of Linear Gaussian State‐Space Models with Applications to DSGE,DSGE‐VAR,and VAR Models

The predictive likelihood is useful for ranking models in forecast comparison exercises using Bayesian inference. We discuss how it can be estimated, by means of marzginalization, for any subset of the observables in linear Gaussian state‐space models. We compare macroeconomic density forecasts for the euro area of a DSGE model to those of a DSGE‐VAR, a BVAR and a multivariate random walk over 1999:Q1–2011:Q4. While the BVAR generally provides superior forecasts, its performance deteriorates substantially with the onset of the Great Recession. This is particularly notable for longer‐horizon real GDP forecasts, where the DSGE and DSGE‐VAR models perform better. Copyright © 2016 John Wiley & Sons, Ltd.     

Realized volatility forecast with the Bayesian random compressed multivariate HAR model

We develop a Bayesian random compressed multivariate heterogeneous autoregressive (BRC-MHAR) model to forecast the realized covariance matrices of stock returns. The proposed model randomly compresses the predictors and reduces the number of parameters. We also construct several competing multivariate volatility models with the alternative shrinkage methods to compress the parameter’s dimensions. We compare the forecast performances of the proposed models with the competing models based on both statistical and economic evaluations. The results of statistical evaluation suggest that the BRC-MHAR models have the better forecast precision than the competing models for the short-term horizon. The results of economic evaluation suggest that the BRC-MHAR models are superior to the competing models in terms of the average return, the Shape ratio and the economic value.     

Variational Bayes approximation of factor stochastic volatility models

Estimation and prediction in high dimensional multivariate factor stochastic volatility models is an important and active research area, because such models allow a parsimonious representation of multivariate stochastic volatility. Bayesian inference for factor stochastic volatility models is usually done by Markov chain Monte Carlo methods (often by particle Markov chain Monte Carlo methods), which are usually slow for high dimensional or long time series because of the large number of parameters and latent states involved. Our article makes two contributions. The first is to propose a fast and accurate variational Bayes methods to approximate the posterior distribution of the states and parameters in factor stochastic volatility models. The second is to extend this batch methodology to develop fast sequential variational updates for prediction as new observations arrive. The methods are applied to simulated and real datasets, and shown to produce good approximate inference and prediction compared to the latest particle Markov chain Monte Carlo approaches, but are much faster.