Characterizing and Communicating the Balance of Risks of Macroeconomic Forecasts: A Predictive Density Approach for Colombia

Since July 2021, Banco de la República strengthened its forecasting process and communication instruments, by involving predictive densities on the projections of its models, PATACON and 4GM. This paper presents the main theoretical and empirical elements of the predictive density approach for macroeconomic forecasting. This model-based methodology allows to characterize the balance of risks of the economy, and quantify their effects through a joint probability distribution of forecasts. We estimate this distribution based on the simulation of DSGE models, preserving the general equilibrium relationships and their macroeconomic consistency. We also illustrate the technical criteria used to represent the prospective factors of risk through the probability distributions of shocks.

Information Geometry of the Exponential Family of Distributions with Progressive Type-II Censoring

In geometry and topology, a family of probability distributions can be analyzed as the points on a manifold, known as statistical manifold, with intrinsic coordinates corresponding to the parameters of the distribution. Consider the exponential family of distributions with progressive Type-II censoring as the manifold of a statistical model, we use the information geometry methods to investigate the geometric quantities such as the tangent space, the Fisher metric tensors, the affine connection and the α-connection of the manifold. As an application of the geometric quantities, the asymptotic expansions of the posterior density function and the posterior Bayesian predictive density function of the manifold are discussed. The results show that the asymptotic expansions are related to the coefficients of the α-connections and metric tensors, and the predictive density function is the estimated density function in an asymptotic sense. The main results are illustrated by considering the Rayleigh distribution.

Predictive Density Aggregation

In this paper we propose a novel approach to obtain the predictive density of global GDP growth. It hinges upon a bottom-up probabilistic model that estimates and combines single countries’ predictive GDP growth densities, taking into account cross-country interdependencies. Speci?cally, we model non-parametrically the contemporaneous interdependencies across the United States, the euro area, and China via a conditional kernel density estimation of a joint distribution. Then, we characterize the potential ampli?cation e?ects stemming from other large economies in each region—also with kernel density estimations—and the reaction of all other economies with para-metric assumptions. Importantly, each economy’s predictive density also depends on a set of observable country-speci?c factors. Finally, the use of sampling techniques allows us to aggregate individual countries’ densities into a world aggregate while preserving the non-i.i.d. nature of the global GDP growth distribution. Out-of-sample metrics con?rm the accuracy of our approach.

Bayesian Estimate of Parameters for ARMA Model Forecasting

This paper presents a Bayesian approach to finding the Bayes estimator of parameters for ARMA model forecasting under normal-gamma prior assumption with a quadratic loss function in mathematical expression. Obtaining the conditional posterior predictive density is based on the normal-gamma prior and the conditional predictive density, whereas its marginal conditional posterior predictive density is obtained using the conditional posterior predictive density. Furthermore, the Bayes estimator of parameters is derived from the marginal conditional posterior predictive density.