Panel Quantile & Non-Linear Panel Models Services
A mean effect can hide as much as it reveals: a driver may matter far more for high performers than low, or only once a threshold is crossed. Quantile regression estimates effects across the whole distribution; threshold and smooth-transition models let relationships change with regime—both extended to panel data.
Panel quantile regression estimates how a predictor’s effect varies across the distribution of the outcome—not just at the mean—in panel data, with the method of moments quantile regression (MM-QR) allowing fixed effects. Non-linear panel models such as threshold regression and the panel smooth transition regression (PSTR) let relationships differ across regimes defined by a threshold variable.
What these models do
Ordinary regression estimates the effect of a predictor on the average outcome. But the average is often not where the interesting variation lives. A factor might strongly affect firms at the top of the performance distribution while barely touching those at the bottom; a policy might matter only for low-income households; a relationship might hold in one regime and reverse in another. Two families of methods capture what a mean model misses.
Quantile regression estimates the effect of a predictor at different quantiles of the outcome distribution—the median, the lower tail, the upper tail—rather than only at the mean. This reveals whether an effect is uniform across the distribution or concentrated among high or low values of the outcome, and it is naturally robust to outliers and does not assume the errors are normally distributed. Extending it to panel data raised a technical challenge—incorporating fixed effects—which the method of moments quantile regression (MM-QR) addresses, making distributional analysis feasible in panels with unit heterogeneity.
Non-linear and regime-switching panels
The second family lets the relationship itself change depending on the value of some variable. A threshold regression model splits the data into regimes at an estimated threshold—below the threshold the effect is one thing, above it another (for example, debt affecting growth differently past a certain debt level). Its smooth counterpart for panels, the panel smooth transition regression (PSTR), allows a gradual transition between regimes rather than an abrupt switch, and lets the point of transition differ across units—often more realistic than a hard break.
Use quantile methods when the question is about distributional heterogeneity (does the effect differ for high vs low outcomes?); use threshold or PSTR models when the question is about non-linearity or regimes (does the relationship change past a tipping point?). The two answer different questions, and both go beyond the single average effect that a standard linear model provides. We match the method to which kind of departure from the mean-linear picture your theory and data imply.
Beyond the mean-linear model
| Method | Question it answers | Key feature |
|---|---|---|
| Mean regression (OLS/FE) | Average effect | One coefficient, at the mean |
| Quantile regression | Effect at low, median, high outcomes | Effects across the distribution; robust to outliers |
| MM-QR | Distributional effects in panels | Quantile effects with fixed effects |
| Threshold regression | Does the effect change past a tipping point? | Abrupt regime split at an estimated threshold |
| PSTR | Gradual, unit-varying regime change | Smooth transition between regimes |
Doing distributional and non-linear panels well
For quantile methods, the value is in reading the pattern across quantiles, not cherry-picking one. We estimate a range of quantiles and present how the coefficient changes along the distribution—because a story is only credible if the whole quantile process supports it, and reporting only the quantile that happens to be significant is a form of selective inference. In panels, MM-QR is used so that unit fixed effects are handled properly; inference accounts for the panel structure. Quantile results are also interpreted correctly—as effects at quantiles of the outcome, which is a distributional statement, not a claim about particular individuals.
For threshold and PSTR models, two things matter most. First, the existence of a threshold effect (a genuine non-linearity) should be tested for, not assumed—imposing a regime split where the data show a linear relationship manufactures structure that is not there. Second, the threshold value is estimated and its uncertainty acknowledged, and the number of regimes is chosen with the appropriate tests rather than by eye. As with all panel work, cross-sectional dependence and the integration properties of the series still need attention. We report the linearity/threshold tests, the estimated threshold and regimes, and the standard diagnostics.
Read the whole quantile process; test for a threshold before imposing one. Reporting only the significant quantile is selective inference, and splitting data into regimes where the relationship is actually linear invents structure. The pattern across quantiles, and a formal linearity/threshold test, are what make these results credible.
Software
We deliver panel quantile regression (including MM-QR), threshold regression, and PSTR in established, reproducible tools—Stata and R—with estimation across a range of quantiles, formal linearity and threshold tests, estimated thresholds with their uncertainty, and panel-appropriate inference, all with versioned code.
How we deliver a quantile / non-linear panel study
This work sits within our wider econometrics practice—so the method matches the kind of departure from the mean-linear model your question implies, and the non-linearity is tested rather than assumed.
We start from the question—whether it concerns distributional heterogeneity (quantile methods) or regime change (threshold/PSTR)—and the data structure. For quantile work we estimate across a range of quantiles using MM-QR where panel fixed effects are needed; for non-linear work we test formally for a threshold effect before estimating the regimes and the transition.
Reporting sets out the method and why it fits, the quantile process or the linearity/threshold tests, the estimated coefficients (by quantile or by regime), the threshold value and its uncertainty where relevant, and panel diagnostics—so the departure from the average-effect model is demonstrated, not assumed.
You receive the quantile coefficients across the distribution (or the regime-specific estimates and the estimated threshold), with panel-appropriate inference, the linearity/threshold test results, robustness checks, and reproducible analytical code and analysis-ready files (where appropriate and permitted)—interpreted as distributional or regime effects, clearly stated.
Quantile & non-linear panels across Management & Allied Studies
Effects that differ across the distribution or change past a tipping point are common in economics and finance—so these methods are widely used across the quantitative disciplines we serve.
Finance & Financial Markets
Effects that differ for high- vs low-return or high- vs low-risk firms, and regime changes past leverage or volatility thresholds.
Economics & Public Policy
Policies affecting the tails of a distribution differently, and threshold effects (e.g. debt–growth) with tipping points.
Energy & Environmental Economics
Distributional and regime-dependent relationships among energy, emissions, and growth—a common PSTR setting.
Development Economics
Effects concentrated among low- or high-income units, and non-linear relationships that change past a development threshold.
Management & Organizational Research
Drivers that matter more for high- than low-performing firms, and regime-dependent effects of strategy or size.
Banking & Corporate Finance
Threshold effects in leverage, liquidity, or size, and distributional effects across firm performance.
Panel quantile & non-linear models: common questions
Is the average effect hiding the real story?
When a driver matters differently across the distribution, or a relationship changes past a tipping point, quantile and non-linear panel models reveal what a mean model conceals—with the whole quantile process reported and non-linearity tested, not assumed.