Robust Bayesian Forecast Combination: Application and Extension

Due to the unknown form of model misspecification and the occurrence of structural changes, consensus has been reached that combining forecasts can improve the performance over individual forecasting models in terms of mean square error. However, over the past decades, empirical findings always suggest that simple averaging forecasts often outperform theoretically optimal methods. Therefore, enormous literature has proposed various estimators to explain and solve this long standing puzzle. In this project, a Bayesian shrinkage estimator for forecast combination is proposed. It draws together earlier work of mine and others to form a robust Bayesian estimator that is superior to simple average and yet easy to calculate. In Bayesian econometrics, one of the controversial and difficult aspects is to choose a prior for the parameters before estimation. The empirical puzzle suggests that simple average can be chosen to be the prior location of the regression coefficients. However, one has to figure out whether the traditional natural conjugate prior is appropriate or not since the prior assumes a particular dependence structure among the regression coefficients and the error variance which is impossible to justify. In spite of its convenient form with constant shrinkage weight, it is not adaptive to model performance over time. Besides, it may not be robust to various kinds of error distribution, for example, Cauchy or flat-tailed error, which is one of the characteristics of the financial time series. Therefore, we relax the dependence structure among the coefficients and variance in this project. Unlike existing estimators, the analytical form of our estimator under independent prior now depends on the prediction error, meaning that it adjusts automatically by penalizing the least square estimators when the underlying models do not respond to the shocks quick enough. It is my belief that such an extension of estimating combination weights can usefully serve in several ways to illuminate the potential difference of choosing a realistic prior over traditional ones in the context of forecast combination and the importance of penalizing the least square estimator when the underlying models perform poorly over time. It also paves a way to extend the current results to density forecast combination that can be applied to risk management. A range of Monte Carlo studies that mimics the properties of financial time series and two applications are carried out to compare the robustness of our estimator.