How do you choose the bandwidth and functional form for a regression discontinuity estimate?

## The estimation problem The quantity you want is the gap between two limits at the cutoff: the average outcome approached from above minus the average approached from below. Neither limit is observed directly, so both are approximated by regression, and every approximation choice — how far from the cutoff to look, what shape to fit, how to weight points — moves the answer. ## Local linear as the default The standard estimator restricts attention to observations with `|X - c| <= h` and fits a separate straight line on each side, then reads the two fitted values at `c`. Equivalently, fit `Y = a + b*(X - c) + t*D + g*D*(X - c) + error` on `|X - c| <= h` where `D` marks the treated side, and `t` is the estimated jump. Separate slopes matter: forcing a common slope makes any difference in curvature between the sides leak into `t`. A kernel weight is usually applied so that points nearer the cutoff count more — a triangular weight declining linearly to zero at the bandwidth edge is the common choice. The intuition is that the observations closest to the threshold carry the identifying information, and the far ones are there mainly to pin down the slope. ## The bandwidth tradeoff Bandwidth trades bias against variance, and the direction is worth stating exactly: - **Wider `h`**: more observations, so lower variance and a tighter interval. But the straight line must now approximate the true curve over a longer stretch, and if the outcome-score relationship bends, that misfit is absorbed into the estimated jump. Bias rises. - **Narrower `h`**: the straight-line approximation is better over a shorter stretch, so bias falls. But fewer observations mean a larger standard error, and past some point the estimate is too noisy to be useful. At the mean-squared-error-optimal bandwidth, the estimator is deliberately left with some bias — that is what minimising MSE means. Consequently a conventional confidence interval centred on the estimate under-covers, and the modern practice is to report a bias-corrected estimate with an interval widened to account for the correction. Say this if asked why the interval from an automatic selector looks wider than the naive one. ## Why global high-order polynomials are discouraged An appealing but poor alternative is to fit one high-order polynomial — say fourth order — in the running variable across the entire data range, with a treatment indicator for the jump. Three problems: 1. **Distant points get weight at the boundary.** The estimate at the cutoff depends on observations far away, which the design gives no reason to trust for the local comparison. 2. **Boundary behaviour is unstable.** Polynomial fits are noisiest at the edges of their range, and the cutoff is exactly an edge for each side's fit. 3. **The order is arbitrary and the answer moves with it.** A jump that appears at fourth order and vanishes at third order is not evidence about treatment; it is evidence about the polynomial. The practical consequence is that a global fourth-order fit can manufacture a visible jump at the cutoff from ordinary curvature. If you use one at all, use it as a secondary check, not as the headline specification, and show the local linear result beside it. ## The sensitivity plot The single most convincing artefact is a plot of the estimated jump, with its confidence band, against a range of bandwidths — say from a quarter of the selected value to twice it. What you want to see is a flat region: the estimate roughly constant, with the interval widening as `h` shrinks. What condemns an analysis is an estimate that changes sign or size systematically as the window moves, or one that is significant at a single bandwidth and nowhere else. Report the plot, not just the number. Complementary checks: repeat with and without kernel weighting, with a local quadratic instead of a local linear fit, and with a placebo cutoff where no rule exists. ## Practical constraints - **Discrete running variables.** If the score takes few distinct values, there may be no meaningful choice of a small bandwidth, and the standard errors need to reflect the coarseness of the score rather than the number of units. - **Too few observations near the cutoff.** Sometimes the honest conclusion is that the design cannot be run at a defensible bandwidth. Widening `h` until something is significant is specification searching. - **Pre-registration.** Choosing the bandwidth after seeing which one gives a significant answer invalidates the inference. Fix the rule — a named data-driven selector, plus the sensitivity range you will show — before looking at the estimates. ## How this is asked Usually as a follow-up after the basic design: "and how would you actually fit it?" A strong answer names local linear as the default, describes the bias-variance direction correctly, mentions data-driven bandwidth selection with bias-corrected inference, rejects the global high-order polynomial with a reason, and finishes on the sensitivity plot as the thing that makes the estimate believable.