Robust Quantile-on-Quantile (RQQR) Regression
Robust Quantile-on-Quantile Regression · Also known as: RQQR, robust QQ regression, robust quantile-on-quantile, outlier-robust QQR
Robust Quantile-on-Quantile Regression extends the QQ framework of Sim and Zhou (2015) by adding resistance to outliers and heavy-tailed distributions. It estimates how each quantile of one variable responds to each quantile of another, producing a full dependence surface while guarding against leverage points that can distort standard QQ estimates.
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When to use it
Use RQQR when you need to characterise the full quantile-to-quantile dependence between two continuous variables and the data contain outliers, heavy-tailed distributions, or crisis-driven extreme observations common in financial and macroeconomic time series. It is especially appropriate for energy economics, financial contagion, and commodity market studies where tail dependence is the main research question. Avoid it when the sample is small (fewer than 100–200 observations), since the local nonparametric estimation requires adequate observations near each quantile of X; also avoid it when interpretability of a single linear coefficient is more important than the full dependence surface.
Strengths & limitations
- Reveals heterogeneous dependence across the full joint distribution, not just at the mean or a single quantile.
- Robust variant guards against outliers and heavy tails that frequently contaminate financial and macroeconomic time series.
- Nonparametric kernel weighting imposes no functional form assumption on the X–Y relationship.
- Produces an interpretable two-dimensional slope surface that can be visualised as a heat map.
- Applicable to both time series and cross-sectional data without distributional assumptions on the error term.
- Computationally intensive: the local quantile regression must be solved on a grid of (τ, θ) pairs, which grows quadratically with grid resolution.
- Requires moderately large samples (at least 100–200 time periods) for stable kernel estimates near the distribution tails.
- The choice of bandwidth h and kernel function materially affects the shape of the estimated surface; no universally optimal selector exists.
- Inference via bootstrap is time-consuming and results may be sensitive to the bootstrap scheme chosen for dependent data.
Frequently asked
How does RQQR differ from standard quantile regression?
Standard quantile regression estimates how the conditional quantile of Y shifts with a one-unit change in X, assuming a single linear slope across all values of X. QQ regression further allows that slope to vary depending on which quantile of X the observation falls in, producing a surface of slopes. The robust variant additionally downweights outliers in each local fit.
What does a 'quantile-on-quantile slope surface' look like?
It is a heat map with the quantile of X on one axis and the quantile of Y on the other. Each cell shows the estimated slope β(τ, θ): a positive and significant value means that being near the τ-th quantile of X is associated with higher Y at the θ-th quantile, and significance is typically overlaid as dots or hatching.
How should I choose the bandwidth h?
Common practice is to use a rule-of-thumb such as Silverman's bandwidth adapted to the sample size, or to select h via cross-validation on the quantile loss. Sensitivity analysis — reporting results for two or three bandwidths — is strongly recommended to show that conclusions are not driven by this tuning choice.
Can RQQR handle panel data?
Yes; the method can be applied to pooled panel data by using the pooled empirical CDF of X, though fixed-effect adjustments or within-group transformations may be required before estimation to remove unobserved heterogeneity.
What software implements RQQR?
No off-the-shelf package is universally adopted. Researchers typically implement it in R or MATLAB using quantile regression routines (e.g., the quantreg package in R) combined with custom kernel-weighting loops. StatWise provides a guided implementation.
Sources
- Sim, N., & Zhou, H. (2015). Oil prices, US stock return, and the dependence between their quantiles. Journal of Banking & Finance, 55, 1–8. DOI: 10.1016/j.jbankfin.2015.01.013 ↗
- Quantile regression. Wikipedia. link ↗
How to cite this page
ScholarGate. (2026, June 3). Robust Quantile-on-Quantile Regression. ScholarGate. https://scholargate.app/en/econometrics/robust-quantile-on-quantile-regression
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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