In nonparametric estimation, the bandwidth (often written (h)) is a tuning parameter that controls how much nearby data influence an estimate. A small bandwidth uses a narrow neighborhood (less smoothing), while a large bandwidth uses a wide neighborhood (more smoothing).
The core idea is the bias-variance tradeoff: smoothing reduces noise but can blur real structure.
Core Mechanics
Bandwidth is easiest to see in a kernel density estimator:
[ \hat f_h(x) = \frac{1}{nh}\sum_{i=1}^n K!\left(\frac{x - X_i}{h}\right) ]
where (K(\cdot)) is the kernel (a weighting function) and (h) sets the scale of the neighborhood. As (h) increases, the estimate becomes smoother.
The same concept appears in kernel regression and local polynomial regression: (h) determines how local the fit is around a point (x).
Choosing A Bandwidth (Why It Is Hard)
There is no single “correct” bandwidth for every purpose:
- If (h) is too small, estimates can be extremely noisy (high variance).
- If (h) is too large, estimates can miss real features like sharp changes or local curvature (high bias).
Empirical practice often includes:
- a data-driven choice rule (for example, plug-in or cross-validated bandwidths), and
- sensitivity checks showing that key conclusions do not depend on one arbitrary (h).
Bandwidth In Regression Discontinuity (RDD)
In a regression discontinuity design, the bandwidth is the window around the cutoff used for estimation. A narrow window makes treated and control observations more comparable but uses less data; a wide window uses more data but may introduce bias because observations far from the cutoff are less comparable.
This is why RDD studies often report estimates at multiple bandwidths and include robustness checks.