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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 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 rebalances 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 U.S. real-time data of 120 monthly variables, indicate that CDN gives more accurate density nowcasts of U.S. 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 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.

This paper presents the Matlab package DeCo (Density Combination) which is based on the paper by Billio et al. (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 Purposes GPU computing almost effortless. This GPU implementation comes with a speed up of the execution time up to seventy times compared to 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.

__Abstract__ Interactions between the eurozone and US booms and busts and among major eurozone economies are analyzed by introducing a panel Markov-switching VAR model well suitable for a multi-country cyclical analysis. The model accommodates changes in low and high data frequencies and endogenous time-varying transition matrices of the country-specific Markov chains. The transition matrix of each Markov chain depends on its own past history and on the history of the other chains, thus allowing for modeling of the interactions between cycles. An endogenous common eurozone cycle is derived by aggregating country-specific cycles. The model is estimated using a simulation based Bayesian approach in which an efficient multi-move strategy algorithm is defined to draw common time-varying Markov-switching chains. Our results show that the US and eurozone cycles are not fully synchronized over the 1991-2013 sample period, with evidence of more recessions in the Eurozone. Shocks affect the US 1-quarter in advance of the eurozone, but these spread very rapidly among economies. An increase in the number of eurozone countries in recession increases the probability of the US to stay within recession, while the US recession indicator has a negative impact on the probability to stay in recession for eurozone countries. Turning point analysis shows that the cycles of Germany, France and Italy are closer to the US cycle than other countries. Belgium, Spain, and Germany, provide more timely information on the aggregate recession than Netherlands and France.

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.

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.

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.

Patton and Timmermann (2011, 'Forecast Rationality Tests Based on Multi-Horizon Bounds', Journal of Business & Economic Statistics, forthcoming) propose a set of useful tests for forecast rationality or optimality under squared error loss, including an easily implemented test based on a regression that only involves (long-horizon and short-horizon) forecasts and no observations on the target variable. We propose an extension, a simulation-based procedure that takes into account the presence of errors in parameter estimates. This procedure can also be applied in the field of 'backtesting' models for Value-at-Risk. Applications to simple AR and ARCH time series models show that its power in detecting certain misspecifications is larger than the power of well-known tests for correct Unconditional Coverage and Conditional Coverage.

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.

Several lessons learnt from a Bayesian analysis of basic macroeconomic time series models are presented for the situation where some model parameters have substantial posterior probability near the boundary of the parameter region. This feature refers to near-instability within dynamic models, to forecasting with near-random walk models and to clustering of several economic series in a small number of groups within a data panel. Two canonical models are used: a linear regression model with autocorrelation and a simple variance components model. Several well-known time series models like unit root and error correction models and further state space and panel data models are shown to be simple generalizations of these two canonical models for the purpose of posterior inference. A Bayesian model averaging procedure is presented in order to deal with models with substantial probability both near and at the boundary of the parameter region. Analytical, graphical and empirical results using U.S. macroeconomic data, in particular on GDP growth, are presented.

We forecast the term structure of U.S. Treasury zero-coupon bond yields by analyzing a range of models that have been used in the literature. We assess the relevance of parameter uncertainty by examining the added value of using Bayesian inference compared to frequentist estimation techniques, and model uncertainty by combining forecasts from individual models. Following current literature we also investigate the benefits of incorporating macroeconomic information in yield curve models. Our results show that adding macroeconomic factors is very beneficial for improving the out-of-sample forecasting performance of individual models. Despite this, the predictive accuracy of models varies over time considerably, irrespective of using the Bayesian or frequentist approach. We show that mitigating model uncertainty by combining forecasts leads to substantial gains in forecasting performance, especially when applying Bayesian model averaging.

In the literature the effects of weather on electricity sales are well-documented. However, studies that have investigated the impact of weather on electricity prices are still scarce (e.g. Knittel and Roberts, 2005), partly because the wholesale power markets have only recently been deregulated. We introduce the weather factor into well-known forecasting models to study its impact. We find that weather has explanatory power for the day-ahead power spot price. Using weather forecasts improves the forecast accuracy, and in particular new models with power transformations of weather forecast variables are significantly better in term of out-of-sample statistics than popular mean reverting models. For different power markets, such as Norway, Eastern Denmark and the Netherlands, we build specific models. The dissimilarity among these models indicates that weather forecasts influence not only the demand of electricity but also the supply side according to different electricity producing methods.

Several frequentist and Bayesian model averaging schemes, including a new one that simultaneously allows for parameter uncertainty, model uncertainty and time varying model weights, are compared in terms of forecast accuracy over a set of simulation experiments. Artificial data are generated, characterized by low predictability, structural instability, and fat tails, which is typical for many financial-economic time series. Sensitivity of results with respect to misspecification of the number of included predictors and the number of included models is explored. Given the set up of our experiments, time varying model weight schemes outperform other averaging schemes in terms of predictive gains both when the correlation among individual forecasts is low and the underlying data generating process is subject to structural locations shifts. In an empirical application using returns on the S&P 500 index, time varying model weights provide improved forecasts with substantial economic gains in an investment strategy including transaction costs.