Decoupling Shrinkage and Selection for the Bayesian Quantile Regression

Abstract

This paper extends the idea of decoupling shrinkage and sparsity for continuous priors to Bayesian Quantile Regression (BQR). The procedure follows two steps: In the first step, we shrink the quantile regression posterior through state of the art continuous priors and in the second step, we sparsify the posterior through an efficient variant of the adaptive lasso, the signal adaptive variable selection (SAVS) algorithm. We propose a new variant of the SAVS which automates the choice of penalisation through quantile specific lossfunctions that are valid in high dimensions. We show in large scale simulations that our selection procedure decreases bias irrespective of the true underlying degree of sparsity in the data, compared to the un-sparsified regression posterior. We apply our two-step approach to a high dimensional growth-at-risk (GaR) exercise. The prediction accuracy of the un-sparsified posterior is retained while yielding interpretable quantile specific variable selection results. Our procedure can be used to communicate to policymakers which variables drive downside risk to the macro economy

Nonlinearities in Macroeconomic Tail Risk through the Lens of Big Data Quantile Regressions

“…For the large datasets we consider in this paper, M + K T and thus suitable shrinkage priors are necessary to obtain precise inference. Kohns and Szendrei (2021) and Mitchell et al (ming) use flexible shrinkage priors in large-scale QRs and show that these work well for tail forecasting.…”

“…A recent exception isKohns and Szendrei (2021) who estimate large-scale quantile regressions and then apply ex-post sparsification to sharpen predictive inference.…”

Nonlinearities in macroeconomic tail risk through the lens of big data quantile regressions

“…Conditional on the quantile, this model resembles a generalized additive model (GAM); see Hastie and Tibshirani (1987). This specification differs from much of the literature (e.g., Adrian et al, 2019;Carriero et al, 2022;Kohns & Szendrei, 2021;Mitchell et al, 2022) which sets g 𝔮 (x t ) = 0 for all t. We approximate g 𝔮 (x t ) using nonlinear transformations of x t :…”

“…For the large datasets we consider in this paper, M + K ≫ T and thus suitable shrinkage priors are necessary to obtain precise inference. Kohns and Szendrei (2021) and Mitchell et al (2022) use flexible shrinkage priors in large-scale QRs and show that these work well for tail forecasting. We build on their findings by considering a range of different priors on 𝜷 𝔮 and 𝜸 𝔮 .…”

“…Our approach is fast and allows for computing all results of our forecasting exercise without the use of high performance computing environments. As compared with, for example, Kohns and Szendrei (2021) and Mitchell et al (2022), who estimate large QRs under flexible Bayesian shrinkage priors, our techniques are highly scalable and can be applied to models even larger than the ones we consider in this paper.…”