Bayesian Quantile-on-Quantile Regression
Also known as: Bayesian QQR, Bayesian QQ regression, Bayes quantile-on-quantile, BQQ regression
Bayesian Quantile-on-Quantile (BQQ) Regression extends the Sim-Zhou quantile-on-quantile framework by replacing frequentist local linear estimation with Bayesian posterior inference. For each pair of quantiles (theta of the outcome, tau of the predictor), the method yields a full posterior distribution over the slope, enabling uncertainty quantification across the entire bivariate quantile surface — a key advantage when sample sizes are moderate and tail quantiles are sparse.
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When to use it
Use Bayesian QQ regression when you suspect that the relationship between two variables differs fundamentally across quantiles of both variables — for example, energy-equity linkages, crypto volatility spillovers, or inequality-growth nexus — and you need uncertainty bands, especially in the tails where data are sparse. It is particularly valuable when sample sizes are moderate (roughly 100–500 observations) and frequentist local linear estimates at extreme quantile pairs are noisy. Prefer the standard frequentist QQ approach when the sample is large (n > 1000) and computational speed matters. Do not use BQQ when the conditional relationship is well-approximated as linear and homogeneous across quantiles — standard Bayesian OLS or quantile regression is simpler and better calibrated.
Strengths & limitations
- Provides full posterior uncertainty at every (theta, tau) cell, including credible intervals for tail dependence where point estimates are unreliable.
- Captures heterogeneous dependence: the slope can vary freely across both the predictor quantile and the outcome quantile, revealing asymmetric and nonlinear co-movement patterns.
- Incorporates prior information — useful when theory constrains the direction of tail dependence (e.g., negative co-movement in bear markets).
- The ALD likelihood links naturally to existing Bayesian quantile regression theory (Yu and Moyeed, 2001), providing a coherent inferential framework.
- Produces interpretable surfaces: researchers can visually identify which quantile combinations drive the overall association.
- Computationally expensive: MCMC must be run for each point on the (theta, tau) grid, and a 10×10 grid already requires hundreds of thousands of posterior draws.
- The ALD working likelihood yields valid quantile estimates but is a pseudo-likelihood; formal Bayesian model comparison via marginal likelihoods requires caution.
- Extreme quantile pairs (theta or tau near 0 or 1) have very few effective observations, so posteriors remain wide regardless of the prior, and results are sensitive to bandwidth choice.
- No widely adopted software package standardises the method; implementations vary, making replication harder than for frequentist QQ regression.
Frequently asked
How does Bayesian QQ regression differ from standard Bayesian quantile regression?
Standard Bayesian quantile regression estimates the theta-th quantile of the outcome as a single linear function of the predictor, holding the predictor fixed. BQQ additionally lets the slope vary across quantiles of the predictor (indexed by tau), producing a two-dimensional surface rather than a single coefficient vector.
How many grid points should I use for (theta, tau)?
Common practice is a 9×9 or 10×10 equally-spaced grid from 0.1 to 0.9 in each dimension. Finer grids provide smoother surfaces but multiply computation; avoid grid points below 0.05 or above 0.95 unless the sample is large.
What bandwidth should I choose for the local approximation?
Bandwidth governs how many observations contribute to each local regression. A common rule of thumb is h = n^(-1/5); alternatively, cross-validation within the Bayesian framework can select h by predictive log-likelihood. Sensitivity analysis over two or three bandwidth choices is advisable.
Can I use BQQ with panel data?
Yes, but panel extensions require additional modelling choices — fixed or random effects for the intercept at each (theta, tau) cell. As of 2024, dedicated software for panel BQQ is limited; researchers typically pool the data or run the model separately per cross-section.
How do I assess convergence of the MCMC chains?
Apply standard diagnostics: R-hat below 1.01 for all parameters, effective sample size above 400 per cell, and trace-plot inspection. Because the model is run independently at each (theta, tau) point, convergence problems at extreme cells do not invalidate interior estimates.
Sources
- Sim, N., & Zhou, H. (2015). Oil prices, US stock return, and the dependence between their quantiles. Journal of Banking and Finance, 55, 1–8. DOI: 10.1016/j.jbankfin.2015.01.013 ↗
- Yu, K., & Moyeed, R. A. (2001). Bayesian quantile regression. Statistics and Probability Letters, 54(4), 437–447. DOI: 10.1016/S0167-7152(01)00124-9 ↗
How to cite this page
ScholarGate. (2026, June 3). Bayesian Quantile-on-Quantile Regression. ScholarGate. https://scholargate.app/en/econometrics/bayesian-quantile-on-quantile-regression
Which method?
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