Combining, evaluating and constraining predictive distributions

This thesis consists in three essays on predictive distributions, in particular their combination, calibration and constraint. Chapter (2), entitled “Combination of Probabilistic Forecasts: a comparison between inference approaches”, aims to compare two inference approaches for combinations found in the literature. One is arguably more common in macroeconomics and finance literature, the other is more used in statistics. Both inference approaches have pros and cons, but no analysis has been found in literature about which approach is the most accurate. The paper’s results find clear evidence favouring one approach or the other based on the problem and data at hand. Chapter (3) is entitled “Are Central Banks’ Fan charts Reliable? On Calibration of Density Path Forecasts”. Central Banks regularly publish fan charts of macroeconomic variables, communicating forecasts for several horizons. Although fan charts contains three types of information: point forecasts (the path), the probability around each point forecast (bands around it), and variable’s dynamics across horizons (i.e. the path), existing absolute evaluation approaches neglect the latter. Practitioners evaluate the calibration of fan charts testing the forecast accuracy horizon by horizon, not considering any joint calibration of the path. This paper describes fan charts as density path forecasts, discusses the impact of horizon-dependence in the evaluation and proposes calibration tests to assess whether Central Banks publish reliable forecasts. We proposed several calibration tests, analysing their size and power, demonstrating that, according to our test, the Bank published on average non-calibrated path density forecasts. Chapter (4), entitled “Generalised Constraints for Predictive Distributions: a Bayesian Approach” investigates the concept of constraining density forecasts. Often policymakers wish to impose a feature to predictive distributions (such as moments constraint, tails behaviour, shifts in support). Although moment constraining is well discussed in the literature (i.e. by exponential tilting), little study has been done on constraining specific parts of the density’s support. This forecast constraining shifts individual predictive densities using Bayesian Importance Sampling. This approach is applied to forecast US GDP under the Covid-19 pandemic: density forecasts from statical models are constrained to the survey of professional forecasters (SPF) in the left tail. xii