Quantile-on-Quantile (QQ) Regression
Quantile-on-Quantile Regression · Also known as: QQ regression, QQ approach, quantile-on-quantile approach, nonparametric quantile regression
Quantile-on-quantile regression is a nonparametric technique that estimates how the quantiles of one variable depend on the quantiles of another. By combining standard quantile regression with local linear smoothing, it produces a full two-dimensional surface of slope coefficients indexed by both the quantile of the outcome and the quantile of the predictor, revealing heterogeneous and asymmetric dependency structures invisible to standard regression.
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When to use it
QQ regression is appropriate when you suspect that the relationship between two variables is heterogeneous across both the distribution of the outcome and the distribution of the predictor — a situation common in finance (oil prices and equity returns), energy economics, and macro linkages during crisis periods. It is especially valuable when tail dependence, asymmetry, or regime-specific effects are theoretically expected. Use it when standard quantile regression already reveals non-constant slopes and you want to understand whether that heterogeneity is also driven by the predictor's own level. Do not use QQ regression when the sample is small (fewer than roughly 200 observations), when the relationship is believed to be globally linear and symmetric, or when a parsimonious model is required for policy reporting — in those cases, standard OLS or quantile regression is more appropriate and easier to communicate.
Strengths & limitations
- Reveals asymmetric and tail-specific dependencies between two variables that standard regression, GARCH, or even plain quantile regression would mask.
- Nonparametric: no distributional assumptions about the error term are required.
- Provides a rich graphical summary (the slope surface) that communicates heterogeneity intuitively.
- Robust to outliers and heavy-tailed distributions, as the quantile check function is less sensitive to extremes than squared-error loss.
- Nests standard quantile regression as a special case when the slope is constant across predictor quantiles.
- Requires a moderately large sample (ideally 200+ observations) because each local regression uses only a fraction of the data.
- Results are sensitive to bandwidth choice; poor bandwidth selection can produce over-smoothed or noisy surfaces.
- Inference (confidence bands for the slope surface) requires bootstrap resampling and can be computationally intensive.
- Interpretation of the two-dimensional slope surface can be complex and may be difficult to convey to non-specialist audiences.
Frequently asked
How does QQ regression differ from standard quantile regression?
Standard quantile regression estimates a single slope for each outcome quantile (theta), averaged over all values of X. QQ regression additionally conditions on the quantile of X (tau), so the slope is estimated at every (theta, tau) combination. The result is a surface rather than a single curve of quantile slopes.
What bandwidth should I use?
The bandwidth h controls the width of the kernel window in the predictor quantile space. Silverman's rule-of-thumb or leave-one-out cross-validation are common starting points. Sensitivity analysis across several bandwidth values is strongly recommended before interpreting the surface.
How do I test whether a specific region of the slope surface is statistically significant?
Bootstrap confidence intervals are the standard approach. Draw many bootstrap samples, refit the QQ surface each time, and use the empirical distribution of the bootstrap slopes at each (theta, tau) cell to construct pointwise or simultaneous confidence bands.
Can QQ regression be applied to panel data?
The original method is designed for cross-sectional or time-series pairs. Extensions to panel settings exist in the literature (pooled QQ panel regression), but they require additional assumptions about within-unit heterogeneity and are less standardised than the baseline approach.
How many observations do I need?
As a practical rule, at least 150-200 observations are advisable so that the local estimates at extreme quantile combinations (e.g., theta = 0.1, tau = 0.9) are based on a reasonable number of data points. With smaller samples, estimates at the tails are unreliable.
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 ↗
- Koenker, R., & Bassett, G. (1978). Regression quantiles. Econometrica, 46(1), 33-50. DOI: 10.2307/1913643 ↗
How to cite this page
ScholarGate. (2026, June 3). Quantile-on-Quantile Regression. ScholarGate. https://scholargate.app/en/econometrics/quantile-on-quantile-regression
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