Quantile Aggregation of Density Forecasts
| DOI | http://doi.org/10.1111/obes.12163 |
| Date | 01 August 2017 |
| Author | Fabio Busetti |
| Published date | 01 August 2017 |
495
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
Quantile Aggregation of Density Forecasts*
Fabio Busetti†
†Bank of Italy, Directorate General for Economics, Statistics and Research, Via Nazionale
91, 00184, Rome, Italy (e-mail: fabio.busetti@bancaditalia.it)
Abstract
Quantile aggregation (or ‘Vincentization’) is a simple and intuitive way of combining
probability distributions, originally proposed by S.B. Vincent in 1912. In certain cases,
such as under Gaussianity, the Vincentized distribution belongs to the same family as
that of the individual distributions and it can be obtained by averaging the individual
parameters. This article compares the properties of quantile aggregation with those of the
forecast combination schemes normally adopted in the econometric forecasting literature,
based on linear or logarithmic averages of the individual densities.Analytical results and
Monte Carlo experiments indicate that the properties of quantile aggregation are between
those of the linear and the logarithmic pool. Larger differences among the combination
schemes occur when there are biases in the individual forecasts: in that case quantile
aggregation seems preferable on the whole. The practical usefulness of Vincentization
is illustrated empirically in the context of linear forecasting models for Italian GDP and
quantile predictions of euro area inflation.
I. Introduction
Economic forecasts are increasingly reported as point estimates supplemented by confi-
dence bands or selected quantiles of the predictivedistributions in order to provide measures
of uncertainty and risks around the central outcome. Indeed a common practice for cen-
tral banks is to present forecasts of inflation and output in the form of ‘fan charts’ that
describe in probabilistic terms the evolution of these variables along the forecast horizon.
The (subjective) assessment of the likelihood of alternative macroeconomic scenarios is
obtained, using skewed density forecasts that reflect higher probability for events in either
tail of the distribution. For example, the Bank of England publishes fan charts for inflation
since 1996; see Britton, Fisher and Whitley (1998).
The econometric literature on forecast evaluation has been progressively extended to
density forecasts in order to gauge the relative performance of different prediction models
in terms of their distributions (see, inter alia, Diebold, Gunther and Tay, 1998; Corradi
JEL Classification numbers: C53, E17.
*I thank Malte Knuppel, Juri Marcucci,Andrea Silvestrini, Luca Spriano, Ken Wallis and two anonymousreferees
for useful comments on a previous version of this paper. Michele Caivanoand Lisa Rodano have contributed to the
empirical application on quantile forecasts of euro area inflation. All errors are mine. The views expressed here are
mine and do not necessarily reflect those of the Bank of Italy.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 79, 4 (2017) 0305–9049
doi: 10.1111/obes.12163
496 Bulletin
and Swanson, 2003, 2006; Mitchell and Hall, 2005; Amisano and Giacomini, 2007). In
parallel, the idea of forecast combination, introduced by Bates and Granger (1969), has
been applied to predictive distributions. The basic tools have been borrowed from the
statistics literature on aggregation of subjective distribution functions, where the task is
to form an ‘opinion pool’ (cf. Genest and Zidek, 1986 for a review). Econometric stud-
ies have focused on linear or logarithmic weighting of the individual densities, where the
weights may be data-driven reflecting the past performance of different models. Some
examples are Wallis (2005, 2011), Hall and Mitchell (2007), Mitchell and Wallis (2010)
and Geweke and Amisano (2011). A comprehensive review of the main issues in density
forecasting is Hall and Mitchell (2009). Kascha and Ravazzolo (2010) provide an empir-
ical comparison of the linear vs. logarithmic opinion pool of several forecasting models
of inflation. More recently, time-varying density combination weights have been consid-
ered, showing that they can lead to forecast improvements (see e.g. Waggoner and Zha,
2012; Billio et al., 2013; Kapetanios et al., 2015; Del Negro, Hasegawa and Schorfheide,
2016).
This paper considers combining forecast distributions by quantile aggregation (or ‘Vin-
centization’). This simple and intuitive approach that consists of averaging the quantiles
of the individual distributions was originally proposed in Vincent (1912). Ratcliff (1979)
and Thomas and Ross (1980) show that in certain cases, such as under Gaussianity, the
Vincentized distribution belongs to the same family as that of the individual distributions
and can be obtained by averaging the individualparameters. In a recent work, Lichtendahl,
Grushka-Cockayne and Winkler (2013) provide analytical properties of quantile aggrega-
tion and the linear opinion pool in terms of calibration, sharpness and shape. In econo-
metrics, a related approach has been proposed in Granger, White and Kamstra (1989),
where individual quantiles are modelled separately and, for each of them, a certain lin-
ear combination of the forecasts is taken. In a somewhat comparable vein, i.e. in order
to capture different performances of the models at different quantiles, Kapetanios et al.
(2015) have considered density combination weights that depend on the region of the
distribution.
In this paper, the properties of quantile aggregation are compared with those of the linear
and logarithmic opinion pool from an econometric forecasting perspective. Analytical
results and Monte Carlo experiments indicate that the properties of quantile aggregation are
between those of the linear and logarithmicpool. Larger differences among the combination
schemes occur when there are biases in the (conditional mean of the) individualforecasts. In
that case, quantile aggregation seems preferable overall, in terms of both logarithmic score
and density calibration. The practical usefulness ofVincentization is illustrated empirically
in the context of linear forecasting models for Italian GDP and quantile predictions of euro
area inflation.
The paper proceeds as follows. Section II defines quantile aggregation vis-`a-vis the
linear and logarithmic opinion pools and it presents some general features of the various
aggregation methods, focusing on the Gaussian case. Section III sets out several Monte
Carlo experiments to evaluatethe relative properties of the different forecast density combi-
nations in the context of standard econometric models.The empirical illustrations regarding
linear forecasting models for Italian GDP and quantile models for euro area inflation are
given in section IV. Section V provides concluding remarks.
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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