Quantile regression generalizes traditional least squares regression by fitting a distinct regression line for each quantile of the distribution of the variable of interest. Least squares regression only produces coefficients that allow us to fit the mean of the dependent variable conditional on some explanatory/predictor variables. In that respect, quantile regression is more appropriate for making inferences about predictive distributions and assessing the forecast uncertainty. At the same time, quantile regression estimates are more robust to outliers in the dependent variable. Therefore, quantile regression can be used to investigate predictive relationships between the dependent and exogenous variables, when typical regression modelling fails to indicate the existence of predictability in these exogenous variables; see Koenker (2005).

This paper examines the forecasting performance of Bayesian quantile regression. The final aim is to produce quantile forecasts for inflation using several potential explanatory variables, and to examine the role of model uncertainty in quantile forecasts. Bayesian model averaging (BMA) and selection (BMS) methods have been used traditionally to deal with model uncertainty in forecasting regressions. Following Alhamzawi and Yu (2012) and Yu, Chen, Reed, and Dunson (2013), it is shown that applying BMA to the quantile regression model allows each quantile of inflation to be forecast using a different set of predictors, whereas estimation using Bayesian methods is quite straightforward. Using model selection and averaging in a quantile regression setting allows us to approximate complex forms of the posterior predictive density of inflation, in spite of the fact that the quantile regression model specified in this paper is inherently linear.1 Although a large body of empirical literature using quantile regression exists, applications of (Bayesian) model averaging are scarce. The only exception is the study by Crespo-Cuaresma, Foster, and Stehrer (2011); however, they approximated Bayesian inference by using least squares and the Bayesian information criterion (BIC). This paper integrates two vastly expanding bodies of literature. On the one hand, there are several studies that have developed estimation, inference and forecasting in (Bayesian) quantile regression models, such as those byGaglianone and Lima (2012), Geraci and Bottai (2007), Gerlach, Chen, and Chan (2011), Lancaster and Jun (2010), Meligkotsidou, Vrontos, and Vrontos (2009), Schüler (2014), Tsionas (2003) and Yu and Moyeed (2001). On the other hand, there is a vast literature in macroeconomic and financial forecasting that shows the superiority of Bayesian model averaging and selection methods to other alternatives; see Koop and Korobilis (2012) and Wright (2008), among others.

Our empirical evaluation of the quantile regression BMA method is based on the real-time forecasting of quarterly US consumer price index inflation, observed over the period 1947Q1–۲۰۱۵Q3, using 16 potential predictors, also measured in real-time. We show which predictors are relevant for each quantile of inflation at various forecast horizons, and compare our results to Bayesian model averaging in the mean regression specification, as well as to various popular nonlinear regression specifications that have been shown to forecast inflation well. Based on predictive likelihoods (Geweke & Amisano, 2010), the quantile regression BMA provides density forecasts that are superior to those from either regular regression BMA or naive quantile regression methods without BMA.