Combining Predictive Densities using a Bayesian Nonlinear Filtering - PowerPoint PPT Presentation

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Combining Predictive Densities using a Bayesian Nonlinear Filtering Approach Monica Billio Roberto Casarin University of Venice University of Brescia Francesco Ravazzolo Herman K. van Dijk Norges Bank Erasmus University Rotterdam Luxembourg, September 28, 2010 Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Motivations: • Combining densities in a multivariate setting. Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Motivations: • Combining densities in a multivariate setting. • Random and time-varying weights. Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Motivations: • Combining densities in a multivariate setting. • Random and time-varying weights. • Parameters and model uncertainty are taken into account. Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Motivations: • Combining densities in a multivariate setting. • Random and time-varying weights. • Parameters and model uncertainty are taken into account. • Applications to Macro and Finance. Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Previous Works: BMA (General) • Barnes (1963): the first mention of model combination. Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Previous Works: BMA (General) • Barnes (1963): the first mention of model combination. • Roberts (1965): obtained a distribution which includes the predictions from two experts (or models). This distribution is essentially a weighted average of the posterior distributions of two models. This is similar to a Bayesian Model Averaging (BMA) procedure. Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Previous Works: BMA (General) • Barnes (1963): the first mention of model combination. • Roberts (1965): obtained a distribution which includes the predictions from two experts (or models). This distribution is essentially a weighted average of the posterior distributions of two models. This is similar to a Bayesian Model Averaging (BMA) procedure. • Bates and Granger (1969): seminal paper about combining predictions from different forecasting models. Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Previous Works: BMA (General) • Barnes (1963): the first mention of model combination. • Roberts (1965): obtained a distribution which includes the predictions from two experts (or models). This distribution is essentially a weighted average of the posterior distributions of two models. This is similar to a Bayesian Model Averaging (BMA) procedure. • Bates and Granger (1969): seminal paper about combining predictions from different forecasting models. • Useful reviews: Hoeting et al. (1999) (on BMA with historical perspective), Granger (2006) and Timmermann (2006) (forecasts combination). Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Previous Works: BMA (State-space models) • Granger and Ramanathan (1984): combine the forecasts with unrestricted regression coefficients as weights; Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Previous Works: BMA (State-space models) • Granger and Ramanathan (1984): combine the forecasts with unrestricted regression coefficients as weights; • Terui and Van Dijk (2002): generalize the least squares model weights by representing the dynamic forecast combination as a state space. In their work the weights are assumed to follow a random walk process; Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Previous Works: BMA (State-space models) • Granger and Ramanathan (1984): combine the forecasts with unrestricted regression coefficients as weights; • Terui and Van Dijk (2002): generalize the least squares model weights by representing the dynamic forecast combination as a state space. In their work the weights are assumed to follow a random walk process; • Guidolin and Timmermann (2009): introduced Markov-switching weights; Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Previous Works: BMA (State-space models) • Granger and Ramanathan (1984): combine the forecasts with unrestricted regression coefficients as weights; • Terui and Van Dijk (2002): generalize the least squares model weights by representing the dynamic forecast combination as a state space. In their work the weights are assumed to follow a random walk process; • Guidolin and Timmermann (2009): introduced Markov-switching weights; • Hoogerheide et al. (2010) and Groen et al. (2009): robust time-varying weights and accounting for both model and parameter uncertainty in model averaging. Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Our contributions: non-linear combination of densities We extend the state-space representation of Terui and Van Dijk (2002) and Hoogerheide et al. (2010): Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Our contributions: non-linear combination of densities We extend the state-space representation of Terui and Van Dijk (2002) and Hoogerheide et al. (2010): • We assume time-varying (and logistic-transformed) weights and propose a distributional state-space representation of the predictive densities and of the combination scheme. Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo

Outline Motivation and Previous Works Contributions of our work The Methodology Empirical Results Our contributions: non-linear combination of densities We extend the state-space representation of Terui and Van Dijk (2002) and Hoogerheide et al. (2010): • We assume time-varying (and logistic-transformed) weights and propose a distributional state-space representation of the predictive densities and of the combination scheme. • This representation (see Harrison and West (1987) for a review). It is general enough to include: linear and Gaussian models, dynamic mixtures and Markov-switching models, as special cases. Monica Billio Roberto Casarin University of Venice Combining Predictive Densities using a Bayesian Nonlinear Filte University of Brescia Francesco Ravazzolo