Time-varying sparsity in dynamic regression models

A novel Bayesian method for inference in dynamic regression models is proposed where both the values of the regression coefficients and the importance of the variables are allowed to change over time. We focus on forecasting and so the parsimony of the model is important for good performance. A prior is developed which allows the shrinkage of the regression coefficients to suitably change over time and an efficient Markov chain Monte Carlo method for posterior inference is described. The new method is applied to two forecasting problems in econometrics: equity premium prediction and inflation forecasting. The results show that this method outperforms current competing Bayesian methods.     

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.     

Bayesian VARs: Specification Choices and Forecast Accuracy

In this paper we discuss how the point and density forecasting performance of Bayesian vector autoregressions (BVARs) is affected by a number of specification choices. We adopt as a benchmark a common specification in the literature, a BVAR with variables entering in levels and a prior modeled along the lines of Sims and Zha (International Economic Review 1998; 39 : 949–968). We then consider optimal choice of the tightness, of the lag length and of both; evaluate the relative merits of modeling in levels or growth rates; compare alternative approaches to h‐step‐ahead forecasting (direct, iterated and pseudo‐iterated); discuss the treatment of the error variance and of cross‐variable shrinkage; and assess rolling versus recursive estimation. Finally, we analyze the robustness of the results to the VAR size and composition (using also data for France, Canada and the UK, while the main analysis is for the USA). We obtain a large set of empirical results, but the overall message is that we find very small losses (and sometimes even gains) from the adoption of specification choices that make BVAR modeling quick and easy, in particular for point forecasting. This finding could therefore further enhance the diffusion of the BVAR as an econometric tool for a vast range of applications. Copyright © 2013 John Wiley & Sons, Ltd.     

Modelling and Forecasting Fiscal Variables for the Euro Area*

In this paper, we assess the possibility of producing unbiased forecasts for fiscal variables in the Euro area by comparing a set of procedures that rely on different information sets and econometric techniques. In particular, we consider autoregressive moving average models, Vector autoregressions, small‐scale semistructural models at the national and Euro area level, institutional forecasts (Organization for Economic Co‐operation and Development), and pooling. Our small‐scale models are characterized by the joint modelling of fiscal and monetary policy using simple rules, combined with equations for the evolution of all the relevant fundamentals for the Maastricht Treaty and the Stability and Growth Pact. We rank models on the basis of their forecasting performance using the mean square and mean absolute error criteria at different horizons. Overall, simple time‐series methods and pooling work well and are able to deliver unbiased forecasts, or slightly upward‐biased forecast for the debt–GDP dynamics. This result is mostly due to the short sample available, the robustness of simple methods to structural breaks, and to the difficulty of modelling the joint behaviour of several variables in a period of substantial institutional and economic changes. A bootstrap experiment highlights that, even when the data are generated using the estimated small‐scale multi‐country model, simple time‐series models can produce more accurate forecasts, because of their parsimonious specification.     

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.     

Forecasting with Global Vector Autoregressive Models: a Bayesian Approach

This paper develops a Bayesian variant of global vector autoregressive (B‐GVAR) models to forecast an international set of macroeconomic and financial variables. We propose a set of hierarchical priors and compare the predictive performance of B‐GVAR models in terms of point and density forecasts for one‐quarter‐ahead and four‐quarter‐ahead forecast horizons. We find that forecasts can be improved by employing a global framework and hierarchical priors which induce country‐specific degrees of shrinkage on the coefficients of the GVAR model. Forecasts from various B‐GVAR specifications tend to outperform forecasts from a naive univariate model, a global model without shrinkage on the parameters and country‐specific vector autoregressions. Copyright © 2016 John Wiley & Sons, Ltd.     

Data-based priors for vector error correction models

We propose two data-based priors for vector error correction models. Both priors lead to highly automatic approaches which require only minimal user input. For the first one, we propose a reduced rank prior which encourages shrinkage towards a low-rank, row-sparse, and column-sparse long-run matrix. For the second one, we propose the use of the horseshoe prior, which shrinks all elements of the long-run matrix towards zero. Two empirical investigations reveal that Bayesian vector error correction (BVEC) models equipped with our proposed priors scale well to higher dimensions and forecast well. In comparison to VARs in first differences, they are able to exploit the information in the level variables. This turns out to be relevant to improve the forecasts for some macroeconomic variables. A simulation study shows that the BVEC with data-based priors possesses good frequentist estimation properties.     

Forecasting using a large number of predictors: Is Bayesian shrinkage a valid alternative to principal components?

This paper considers Bayesian regression with normal and double-exponential priors as forecasting methods based on large panels of time series. We show that, empirically, these forecasts are highly correlated with principal component forecasts and that they perform equally well for a wide range of prior choices. Moreover, we study conditions for consistency of the forecast based on Bayesian regression as the cross-section and the sample size become large. This analysis serves as a guide to establish a criterion for setting the amount of shrinkage in a large cross-section.     

Forecasting macroeconomic variables using collapsed dynamic factor analysis

We explore a new approach to the forecasting of macroeconomic variables based on a dynamic factor state space analysis. Key economic variables are modeled jointly with principal components from a large time series panel of macroeconomic indicators using a multivariate unobserved components time series model. When the key economic variables are observed at a low frequency and the panel of macroeconomic variables is at a high frequency, we can use our approach for both nowcasting and forecasting purposes. Given a dynamic factor model as the data generation process, we provide Monte Carlo evidence of the finite-sample justification of our parsimonious and feasible approach. We also provide empirical evidence for a US macroeconomic dataset. The unbalanced panel contains quarterly and monthly variables. The forecasting accuracy is measured against a set of benchmark models. We conclude that our dynamic factor state space analysis can lead to higher levels of forecasting precision when the panel size and time series dimensions are moderate.     

Forecasting performance of multivariate time series models with full and reduced rank: an empirical examination

Since Quenouille's influential work on multiple time series, much progress has been made towards the goal of parameter reduction and model fit. Relatively less attention has been paid to the systematic evaluation of out-of-sample forecast performance of multivariate time series models. In this paper, we update the hog data set studied by Quenouille (and other researchers who followed him). We re-estimate his model with extended observations (1867–1966), and generate recursive one- to four-steps-ahead forecasts for the period of 1967 through 2000. These forecasts are compared to forecasts from an unrestricted vector autoregression, a reduced rank regression model, an index model and a cointegration-based error correction model. The error correction model that takes into account both nonstationarity of the data and rank reduction performs best at all four forecasting horizons. However, differences among competing models are statistically insignificant in most cases. No model consistently encompasses the others at all four horizons.     

Leading Indicators for Euro‐area Inflation and GDP Growth*

In this paper, we evaluate the role of a set of variables as leading indicators for Euro‐area inflation and GDP growth. Our leading indicators are taken from the variables in the European Central Bank's (ECB) Euro‐area‐wide model database, plus a set of similar variables for the US. We compare the forecasting performance of each indicator ex post with that of purely autoregressive models. We also analyse three different approaches to combining the information from several indicators. First, ex post, we discuss the use as indicators of the estimated factors from a dynamic factor model for all the indicators. Secondly, within an ex ante framework, an automated model selection procedure is applied to models with a large set of indicators. No future information is used, future values of the regressors are forecast, and the choice of the indicators is based on their past forecasting records. Finally, we consider the forecasting performance of groups of indicators and factors and methods of pooling the ex ante single‐indicator or factor‐based forecasts. Some sensitivity analyses are also undertaken for different forecasting horizons and weighting schemes of forecasts to assess the robustness of the results.     

Forecasting inflation using time‐varying Bayesian model averaging

This paper presents a Bayesian model averaging regression framework for forecasting US inflation, in which the set of predictors included in the model is automatically selected from a large pool of potential predictors and the set of regressors is allowed to change over time. Using real‐time data on the 1960–2011 period, this model is applied to forecast personal consumption expenditures and gross domestic product deflator inflation. The results of this forecasting exercise show that, although it is not able to beat a simple random‐walk model in terms of point forecasts, it does produce superior density forecasts compared with a range of alternative forecasting models. Moreover, a sensitivity analysis shows that the forecasting results are relatively insensitive to prior choices and the forecasting performance is not affected by the inclusion of a very large set of potential predictors.     

Bayesian assessment of dimensionality in reduced rank regression

We consider Bayesian inference about the dimensionality in the multivariate reduced rank regression framework, which encompasses several models such as MANOVA, factor analysis and cointegration models for multiple time series. The fractional Bayes approach is used to derive a closed form approximation to the posterior distribution of the dimensionality and some asymptotic properties of the approximation are proved. Finite sample properties are studied by simulation and the method is applied to growth curve data and cointegrated multivariate time series.     

Factor MIDAS for Nowcasting and Forecasting with Ragged‐Edge Data: A Model Comparison for German GDP*

In this article, we merge two strands from the recent econometric literature. First, factor models based on large sets of macroeconomic variables for forecasting, which have generally proven useful for forecasting. However, there is some disagreement in the literature as to the appropriate method. Second, forecast methods based on mixed‐frequency data sampling (MIDAS). This regression technique can take into account unbalanced datasets that emerge from publication lags of high‐ and low‐frequency indicators, a problem practitioner have to cope with in real time. In this article, we introduce Factor MIDAS, an approach for nowcasting and forecasting low‐frequency variables like gross domestic product (GDP) exploiting information in a large set of higher‐frequency indicators. We consider three alternative MIDAS approaches (basic, smoothed and unrestricted) that provide harmonized projection methods that allow for a comparison of the alternative factor estimation methods with respect to nowcasting and forecasting. Common to all the factor estimation methods employed here is that they can handle unbalanced datasets, as typically faced in real‐time forecast applications owing to publication lags. In particular, we focus on variants of static and dynamic principal components as well as Kalman filter estimates in state‐space factor models. As an empirical illustration of the technique, we use a large monthly dataset of the German economy to nowcast and forecast quarterly GDP growth. We find that the factor estimation methods do not differ substantially, whereas the most parsimonious MIDAS projection performs best overall. Finally, quarterly models are in general outperformed by the Factor MIDAS models, which confirms the usefulness of the mixed‐frequency techniques that can exploit timely information from business cycle indicators.     

Forecasting cryptocurrencies under model and parameter instability

This paper studies the predictability of cryptocurrency time series. We compare several alternative univariate and multivariate models for point and density forecasting of four of the most capitalized series: Bitcoin, Litecoin, Ripple and Ethereum. We apply a set of crypto-predictors and rely on dynamic model averaging to combine a large set of univariate dynamic linear models and several multivariate vector autoregressive models with different forms of time variation. We find statistically significant improvements in point forecasting when using combinations of univariate models, and in density forecasting when relying on the selection of multivariate models. Both schemes deliver sizable directional predictability.     

Bayesian identification,selection and estimation of semiparametric functions in high-dimensional additive models

In this paper we propose an approach to both estimate and select unknown smooth functions in an additive model with potentially many functions. Each function is written as a linear combination of basis terms, with coefficients regularized by a proper linearly constrained Gaussian prior. Given any potentially rank deficient prior precision matrix, we show how to derive linear constraints so that the corresponding effect is identified in the additive model. This allows for the use of a wide range of bases and precision matrices in priors for regularization. By introducing indicator variables, each constrained Gaussian prior is augmented with a point mass at zero, thus allowing for function selection. Posterior inference is calculated using Markov chain Monte Carlo and the smoothness in the functions is both the result of shrinkage through the constrained Gaussian prior and model averaging. We show how using non-degenerate priors on the shrinkage parameters enables the application of substantially more computationally efficient sampling schemes than would otherwise be the case. We show the favourable performance of our approach when compared to two contemporary alternative Bayesian methods. To highlight the potential of our approach in high-dimensional settings we apply it to estimate two large seemingly unrelated regression models for intra-day electricity load. Both models feature a variety of different univariate and bivariate functions which require different levels of smoothing, and where component selection is meaningful. Priors for the error disturbance covariances are selected carefully and the empirical results provide a substantive contribution to the electricity load modelling literature in their own right.     

Bayesian modeling of joint and conditional distributions

In this paper, we study a Bayesian approach to flexible modeling of conditional distributions. The approach uses a flexible model for the joint distribution of the dependent and independent variables and then extracts the conditional distributions of interest from the estimated joint distribution. We use a finite mixture of multivariate normals (FMMN) to estimate the joint distribution. The conditional distributions can then be assessed analytically or through simulations. The discrete variables are handled through the use of latent variables. The estimation procedure employs an MCMC algorithm. We provide a characterization of the Kullback–Leibler closure of FMMN and show that the joint and conditional predictive densities implied by the FMMN model are consistent estimators for a large class of data generating processes with continuous and discrete observables. The method can be used as a robust regression model with discrete and continuous dependent and independent variables and as a Bayesian alternative to semi- and non-parametric models such as quantile and kernel regression. In experiments, the method compares favorably with classical nonparametric and alternative Bayesian methods.     

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.     

Comparing the forecasting performances of linear models for electricity prices with high RES penetration

We compare alternative univariate versus multivariate models and frequentist versus Bayesian autoregressive and vector autoregressive specifications for hourly day-ahead electricity prices, both with and without renewable energy sources. The accuracy of point and density forecasts is inspected in four main European markets (Germany, Denmark, Italy, and Spain) characterized by different levels of renewable energy power generation. Our results show that the Bayesian vector autoregressive specifications with exogenous variables dominate other multivariate and univariate specifications in terms of both point forecasting and density forecasting.     

Estimation and forecasting in vector autoregressive moving average models for rich datasets

We address the issue of modelling and forecasting macroeconomic variables using rich datasets by adopting the class of Vector Autoregressive Moving Average (VARMA) models. We overcome the estimation issue that arises with this class of models by implementing an iterative ordinary least squares (IOLS) estimator. We establish the consistency and asymptotic distribution of the estimator for weak and strong VARMA(p,q) models. Monte Carlo results show that IOLS is consistent and feasible for large systems, outperforming the MLE and other linear regression based efficient estimators under alternative scenarios. Our empirical application shows that VARMA models are feasible alternatives when forecasting with many predictors. We show that VARMA models outperform the AR(1), ARMA(1,1), Bayesian VAR, and factor models, considering different model dimensions.