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We develop and evaluate quarterly out-of-sample individual and composite density forecasts for U.S. hog prices. Individual density forecasts are generated using time series models and the implied distributions of USDA and Iowa State University outlook forecasts. Composite density forecasts are constructed using linear and logarithmic combinations of the individual forecasts and several weighting schemes. Density forecasts are evaluated on predictive accuracy (sharpness), goodness of fit (calibration), and their economic value in a hedging simulation. Logarithmic combinations using equal and mean square error weights outperform all individual density forecasts and are modestly better than linear composites. Comparison of the outlook forecasts to the best composite demonstrates the usefulness of the composite procedure, and identifies the economic value that more accurate expected price probability distributions can provide to producers.

The combination of forecast densities, whether they result from a set of models,a group of consulted experts, or other sources, is becoming increasingly importantin the fields of economics, policy, and finance, among others. Requiring methodologythat goes beyond standard Bayesian model uncertainty and model mixing -with its well-known limitations based on a clearly proscribed theoretical basis - multiple`density combination' methods have been proposed. While some proposals havedemonstrated empirical success, most apparently lack a core philosophical and theoreticalfoundation. Interesting recent examples generalize the common `linear opinionpool' with flexible mixing weights that depend on the forecast variable itself - i.e., outcome-dependent mixing. This dissertation takes a foundational subjectiveBayesian perspective in order to show that such a density combination scheme isin fact justified as one example of Bayesian agent opinion analysis, or `predictivesynthesis'. This logically coherent framework clearly delineates the underlying assumptionsas well as the theoretical constraints and limitations of many combination`rules', defining a broad class of Bayesian models for the general problem. A numberof examples, including applications to sets of predictive densities for foreign exchangeand United States inflation time series data, provide illustrations.Chapters 1-2 introduce and describe the ideas involved in Bayesian predictivesynthesis (BPS) as a method of subjective analysis. Chapters 3-4 describe differentpossible formulations of outcome-dependent mixing. Chapter 5 places the analysis into a time series context and covers relevant inference techniques. Chapters 6 and 7apply the time series analysis to euro currency forecasts and United States inflationdata. Chapter 8 concludes.

We introduce a Combined Density Nowcasting (CDN) approach to Dynamic Factor Models (DFM) that in a coherent way accounts for time-varying uncertainty of several model and data features in order to provide more accurate and complete density nowcasts. The combination weights are latent random variables that depend on past nowcasting performance and other learning mechanisms. The combined density scheme is incorporated in a Bayesian Sequential Monte Carlo method which re-balances the set of nowcasted densities in each period using updated information on the time-varying weights. Experiments with simulated data show that CDN works particularly well in a situation of early data releases with relatively large data uncertainty and model incompleteness. Empirical results, based on US real-time data of 120 leading indicators, indicate that CDN gives more accurate density nowcasts of US GDP growth than a model selection strategy and other combination strategies throughout the quarter with relatively large gains for the two first months of the quarter. CDN also provides informative signals on model incompleteness during recent recessions. Focusing on the tails, CDN delivers probabilities of negative growth, that provide good signals for calling recessions and ending economic slumps in real time.

We introduce a Combined Density Nowcasting (CDN) approach to Dynamic Factor Models (DFM) that in a coherent way accounts for time-varying uncertainty of several model and data features in order to provide more accurate and complete density nowcasts. The combination weights are latent random variables that depend on past nowcasting performance and other learning mechanisms. The combined density scheme is incorporated in a Bayesian Sequential Monte Carlo method which re-balances the set of nowcasted densities in each period using updated information on the time-varying weights. Experiments with simulated data show that CDN works particularly well in a situation of early data releases with relatively large data uncertainty and model incompleteness. Empirical results, based on US real-time data of 120 leading indicators, indicate that CDN gives more accurate density nowc asts of US GDP growth than a model selection strategy and other combination strategies throughout the quarter with relatively large gains for the two first months of the quarter. CDN also provides informative signals on model incompleteness during recent recessions. Focusing on the tails, CDN delivers probabilities of negative growth, that provide good signals for calling recessions and ending economic slumps in real time.

This is the authors’ manuscript to the article also published at http://www.ssrn.com/ We propose a Bayesian combination approach for multivariate predictive densities which relies upon a distributional state space representation of the combination weights. Several speci cations of multivariate time-varying weights are introduced with a particular focus on weight dynamics driven by the past performance of the predictive densities and the use of learning mechanisms. In the proposed approach the model set can be incomplete, meaning that all models can be individually misspeci ed. A Sequential Monte Carlo method is proposed to approximate the ltering and predictive densities. The combination approach is assessed using statistical and utility-based performance measures for evaluating density forecasts of simulated data, US macroeconomic time series and surveys of stock market prices. Simulation results indicate that, for a set of linear autoregressive models, the combination strategy is successful in selecting, with probability close to one, the true model when the model set is complete and it is able to detect parameter instability when the model set includes the true model that has generated subsamples of data. Also, substantial uncertainty appears in the weights when predictors are similar; residual uncertainty reduces when the model set is complete; and learning reduces this uncertainty. For the macro series we nd that incompleteness of the models is relatively large in the 70's, the beginning of the 80's and during the recent nancial crisis, and lower during the Great Moderation; the predicted probabilities of recession accurately compare with the NBER business cycle dating; model weights have substantial uncertainty attached. With respect to returns of the S&P 500 series, we nd that an investment strategy using a combination of predictions from professional forecasters and from a white noise model puts more weight on the white noise model in the beginning of the 90's and switches to giving more weight to the professional forecasts over time. Information on the complete predictive distribution and not just on some moments turns out to be very important, above all during turbulent times such as the recent nancial crisis. More generally, the proposed distributional state space representation o ers a great exibility in combining densities.

This paper presents the MATLAB package DeCo (density combination) which is based on the paper by Billio, Casarin, Ravazzolo, and van Dijk (2013) where a constructive Bayesian approach is presented for combining predictive densities originating from different models or other sources of information. The combination weights are time-varying and may depend on past predictive forecasting performances and other learning mechanisms. The core algorithm is the function DeCo which applies banks of parallel Sequential Monte Carlo algorithms to filter the time-varying combination weights. The DeCo procedure has been implemented both for standard CPU computing and for graphical process unit (GPU) parallel computing. For the GPU implementation we use the MATLAB parallel computing toolbox and show how to use general purpose GPU computing almost effortlessly. This GPU implementation provides a speed up of the execution time of up to seventy times on a standard CPU MATLAB implementation on a multicore CPU. We show the use of the package and the computational gain of the GPU version through some simulation experiments and empirical applications.

We propose a Bayesian combination approach for multivariate predictive densities which relies upon a distributional state space representation of the combination weights. Several specifications of multivariate time-varying weights are introduced with a particular focus on weight dynamics driven by the past performance of the predictive densities and the use of learning mechanisms. In the proposed approach the model set can be incomplete, meaning that all models can be individually misspecified. A Sequential Monte Carlo method is proposed to approximate the filtering and predictive densities. The combination approach is assessed using statistical and utility-based performance measures for evaluating density forecasts. Simulation results indicate that, for a set of linear autoregressive models, the combination strategy is successful in selecting, with probability close to one, the true model when the model set is complete and it is able to detect parameter instability when the model set includes the true model that has generated subsamples of data. For the macro series we find that incompleteness of the models is relatively large in the 70's, the beginning of the 80's and during the recent financial crisis, and lower during the Great Moderation. With respect to returns of the S&P 500 series, we find that an investment strategy using a combination of predictions from professional forecasters and from a white noise model puts more weight on the white noise model in the beginning of the 90's and switches to giving more weight to the professional forecasts over time.

We summarize the general combination approach by Billio et al. [2010]. In the combination model the weights follow logistic autoregressive processes, change over time and their dynamics are possible driven by the past forecasting performances of the predictive densities. For illustrative purposes we apply it to combine White Noise and GARCH models to forecast the Amsterdam Exchange index and use the combined predictive forecasts in an investment asset allocation exercise.

We propose a multivariate combination approach to prediction based on a distributional state space representation of the weights belonging to a set of Bayesian predictive densities which have been obtained from alternative models. Several specifications of multivariate time-varying weights are introduced with a particular focus on weight dynamics driven by the past performance of the predictive densities and the use of learning mechanisms. In the proposed approach the model set can be incomplete, meaning that all models are individually misspecified. The approach is assessed using statistical and utility-based performance measures for evaluating density forecasts of US macroeconomic time series and surveys of stock market prices. For the macro series we find that incompleteness of the models is relatively large in the 70's, the beginning of the 80's and during the recent financial crisis; structural changes like the Great Moderation are empirically identified by our model combination and the predicted probabilities of recession accurately compare with the NBER business cycle dating. Model weights have substantial uncertainty attached and neglecting this may seriously affect results. With respect to returns of the S&P 500 series, we find that an investment strategy using a combination of predictions from professional forecasters and from a white noise model puts more weight on the white noise model in the beginning of the 90's and switches to giving more weight to the left tail of the professional forecasts during the start of the financial crisis around 2008.

Using a Bayesian framework this paper provides a multivariate combination approach to prediction based on a distributional state space representation of predictive densities from alternative models. In the proposed approach the model set can be incomplete. Several multivariate time-varying combination strategies are introduced. In particular, a weight dynamics driven by the past performance of the predictive densities is considered and the use of learning mechanisms. The approach is assessed using statistical and utility-based performance measures for evaluating density forecasts of US macroeconomic time series and of surveys of stock market prices.

Recent work has found that, without the benefit of hindsight, it can prove difficult for policy-makers to pin down accurately the current position of the output gap; real-time estimates are unreliable. However, attention primarily has focused on output gap point estimates alone. But point forecasts are better seen as the central points of ranges of uncertainty; therefore some revision to real-time estimates may not be surprising. To capture uncertainty fully density forecasts should be used. This paper introduces, motivates and discusses the idea of evaluating the quality of real-time density estimates of the output gap. It also introduces density forecast combination as a practical means to overcome problems associated with uncertainty over the appropriate output gap estimator. An application to the Euro area illustrates the use of the techniques. Simulated out-of-sample experiments reveal that not only can real-time point estimates of the Euro area output gap be unreliable, but so can measures of uncertainty associated with them. The implications for policy-makers use of Taylor-type rules are discussed and illustrated. We find that Taylor-rules that exploit real-time output gap density estimates can provide reliable forecasts of the ECB's monetary policy stance only when alternative density forecasts are combined