Common mistake 1 with Regression discontinuity

High-order polynomials. They produce spurious jumps at the boundary; use local linear (Gelman & Imbens 2019).

Common mistake 2 with Regression discontinuity

Generalising a local estimate to everyone. It says nothing about units far from the cutoff.

Common mistake 3 with Regression discontinuity

Not checking whether other policies use the same threshold - a very common confound.

Common mistake 4 with Regression discontinuity

Reporting only the optimal-bandwidth estimate. The sensitivity table is the evidence that the discontinuity is a feature of the data rather than of the smoother, and an RD without it is not assessable.

EconStatsCraft › Regression discontinuity

Regression discontinuity

Estimates the jump in an outcome at a cutoff. Includes bandwidth and polynomial sensitivity, a density-manipulation check, and the RD plot with binned means.

What it needs

Outcome
Running variable - The score that determines treatment.
Actual treatment (fuzzy only) - Leave empty for a sharp design.

The intuition

A rule that assigns a benefit at exactly 50 marks creates two groups either side of the line who are, in every respect other than the benefit, indistinguishable - nobody scoring 49.8 is meaningfully different from someone scoring 50.2. The threshold manufactures something very close to a randomised experiment, for free, out of bureaucracy.

When to use it

Treatment is determined by whether a continuous score crosses a known cutoff.

Where the estimator comes from

\tau=\lim_{x\downarrow c}E[Y\mid X=x]-\lim_{x\uparrow c}E[Y\mid X=x]. Identification needs only that E[Y^0\mid X] and E[Y^1\mid X] are continuous at c - so any jump must be caused by the treatment. Estimation is local linear regression on each side within a bandwidth h, usually with triangular kernel weights that give observations nearest the cutoff the most influence. Fuzzy designs, where crossing changes the PROBABILITY of treatment, use the Wald ratio \tau_{\text{fuzzy}}=\tau_Y/\tau_D: the jump in outcome divided by the jump in treatment.

The model

\tau=\lim_{x\downarrow c}E[y\mid X=x]\;-\;\lim_{x\uparrow c}E[y\mid X=x]

What it assumes

Units cannot precisely manipulate their score around the cutoff - this is what the density test checks.
Everything else is continuous at the cutoff: no other rule kicks in at the same threshold.
For fuzzy RD: crossing the cutoff genuinely changes the probability of treatment.
The bandwidth was chosen by a rule rather than by trying several and keeping the one that worked. The estimate varies with the bandwidth by construction, so an unreported search over bandwidths is a search over answers.

Statistical properties

The estimate is LOCAL: it is the effect at the cutoff, and says nothing about units far from it.
Local linear dominates high-order polynomials, which produce spurious jumps from boundary behaviour (Gelman & Imbens 2019).
Optimal bandwidth trades bias against variance; CCT provide bias-corrected estimators with robust intervals. The rule of thumb used here is simpler, which is why the sensitivity table matters.
Manipulation of the running variable is the fatal threat, and McCrary's density test is the standard check.
Fuzzy RD identifies a LATE for compliers at the cutoff.
RD has lower power than a randomised experiment of the same size, because it uses only observations near the threshold.

How to read the output

The jump at the cutoff is the estimate - it is LOCAL, valid for units near the threshold only.
The sensitivity table matters more than the headline number. A credible result barely moves across bandwidths.
A significant density discontinuity means people sorted across the threshold and the design is invalid.
Fuzzy RD: the estimate is the jump in outcome divided by the jump in treatment probability.

A worked reading

The jump is 2.5831 (SE 0.5343, p < 0.0001) at a cutoff of 50, fitted on the 335 observations inside the optimal bandwidth of 12.63. Across bandwidths from h/2 to 1.5h and polynomial orders 1 and 2 the estimate ranges from 2.2428 to 3.5187 - the same sign and the same order of magnitude everywhere, but a spread of more than one unit, which is worth reporting rather than smoothing over. The narrowest bandwidth at order 2 is the least precise (SE 1.1889, p = 0.059), as it should be with 182 observations and four parameters. The density check gives a ratio of 0.958 just either side of the cutoff, z = -0.19, p = 0.8494: no sign that units sorted across the threshold. A stable sign, a smooth density and an honest account of the bandwidth sensitivity is what a credible RD looks like.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Density jumps at the cutoff	Manipulation - the design is invalid	Abandon, or use a donut RD excluding the immediate neighbourhood
Estimate flips sign across bandwidths	Not robust	Do not report it as causal
Only significant with a high-order polynomial	Almost certainly an artefact	Report the local linear result
Covariates also jump at the cutoff	Something else changes there too	Look for another rule using the same threshold
Very few observations near the cutoff	Underpowered	Widen the bandwidth and accept more bias, or say the design cannot answer it

Common mistakes

High-order polynomials. They produce spurious jumps at the boundary; use local linear (Gelman & Imbens 2019).
Generalising a local estimate to everyone. It says nothing about units far from the cutoff.
Not checking whether other policies use the same threshold - a very common confound.
Reporting only the optimal-bandwidth estimate. The sensitivity table is the evidence that the discontinuity is a feature of the data rather than of the smoother, and an RD without it is not assessable.

Where to read this properly

Imbens & Lemieux (2008) - the practical guide
Lee & Lemieux (2010) - the survey
McCrary (2008) - the density test
Calonico, Cattaneo & Titiunik (2014) - robust bias-corrected inference
Gelman & Imbens (2019) - why not to use high-order polynomials

All analyses

Data & variables

Each example is simulated from a known model, so you can check whether the estimator recovers the truth - and nothing here is anyone's private data.

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Regression discontinuity

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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