Regression Discontinuity With Multiple Running Variables

Why Multiple Running Variables Change the Comparison

In a standard regression-discontinuity design, assignment changes at a cutoff in one running variable. With multiple continuous running variables, assignment may instead depend on a rule involving several scores. The key challenge is not merely fitting a regression with more predictors. It is identifying where assignment actually changes and comparing observations that approach the same location on that boundary from opposite sides.

A comparison of all assigned and unassigned units is generally not a regression-discontinuity comparison: those groups may differ substantially in their scores. Even a comparison of units near one score's cutoff can be invalid if another score means that crossing the first cutoff does not change assignment. The assignment rule and its geometry determine which local comparisons can identify an effect.

Definition: Let $R=(R_1,R_2)$ be two continuous running variables, and let $Z$ indicate assignment to the program or policy. The assignment boundary is the set of score values at which the assignment rule changes between $Z=0$ and $Z=1$. An active boundary point is a point on that set where approaching from opposite sides changes assignment. A local regression-discontinuity comparison holds the location along the boundary fixed and compares limits from opposite sides.

Define the Active Boundary Before Estimating an Effect

For illustration, suppose assignment is determined by a rule $Z=1\{R_1\ge c_1,\ R_2\ge c_2\}$: both scores must meet their cutoffs. The assignment boundary is not the entirety of the two cutoff lines. The line $R_1=c_1$ is active only where $R_2\ge c_2$; the line $R_2=c_2$ is active only where $R_1\ge c_1$. Elsewhere, crossing one cutoff does not change $Z$. The two active segments meet at a corner, where the local geometry requires special care.

More generally, an assignment rule may be written $Z=1\{g(R)\ge 0\}$. Its boundary is $g(R)=0$, but a point on that set is useful for RD only if the rule changes across it. Near a smooth boundary, describe score values using a signed distance $s$, positive on the assigned side and negative on the unassigned side, and a coordinate $q$ locating the position along the boundary. At a fixed boundary location $q_0$, the relevant comparison is between limits as $s$ approaches zero from above and below.

\Delta_Y(q_0)=\lim_{s\to 0^+}E[Y\mid s,q_0]-\lim_{s\to 0^-}E[Y\mid s,q_0].

This is a pointwise, boundary-local contrast. If the boundary has several active segments, effects may differ across them. Combining estimates from different segments therefore requires a stated target and weighting scheme; it is not automatic that one segment's effect applies to another.

Worked Example: Finding the Active Segments

A hypothetical scholarship is assigned when both an examination score is at least 70 and a financial-need score is at least 60. Let $R_1$ be the examination score, $R_2$ the need score, and $Z$ scholarship assignment. Thus $Z=1\{R_1\ge70,\ R_2\ge60\}$.

Consider the point $(70,65)$. The need score is above its cutoff. Comparing applicants with $R_1=69.8$ and $R_1=70.2$, while keeping $R_2$ near 65, changes assignment from 0 to 1. This is an active boundary comparison on the $R_1=70$ segment.

At $(70,55)$, crossing $R_1=70$ does not change assignment: both applicants fail the need-score requirement. That part of the line $R_1=70$ is inactive and cannot support an RD estimate based on that crossing.

The corner $(70,60)$ is different again. Approaching it along either active segment may yield a comparison, but the two approach directions need not represent the same local contrast. A sound analysis should not treat the corner as an ordinary point on a single smooth boundary without additional justification.

Identification at a Boundary Point

Let $Y(1)$ and $Y(0)$ denote the potential outcomes under treatment and no treatment. In a sharp design, treatment receipt $D$ equals assignment $Z$. Identification at a regular active boundary point requires that the conditional mean potential outcomes vary continuously through that point when approached from either side. In practical terms, absent the assignment change, expected outcomes must not themselves jump at the boundary. The assignment rule must be correctly specified, observations must be able to occur on both sides arbitrarily close to the same boundary location, and units must not precisely sort to one side in a way that changes their potential outcomes discontinuously.

Under these conditions, the outcome jump identifies the boundary-local average treatment effect. The reason is that the assigned-side limit reveals the mean treated potential outcome at the boundary, while the unassigned-side limit reveals the mean untreated potential outcome there. Continuity makes the limits comparable at the same boundary location.

\begin{aligned} \Delta_Y(q_0) &=\lim_{s\to0^+}E[Y(1)\mid s,q_0] -\lim_{s\to0^-}E[Y(0)\mid s,q_0]\\ &=E[Y(1)-Y(0)\mid s=0,q_0], \end{aligned}

The final equality uses continuity of the relevant conditional potential-outcome means at the boundary. It does not follow from the observed outcome jump alone. The identifying assumption is substantive: other determinants of outcomes should not change discontinuously at the same active boundary.

In a fuzzy design, assignment changes treatment probability but does not determine treatment receipt perfectly. Define $\Delta_D(q_0)$ as the analogous jump in $E[D\mid s,q_0]$. If the first-stage jump is nonzero, the local Wald ratio is $\Delta_Y(q_0)/\Delta_D(q_0)$. Its causal interpretation as a local average treatment effect for compliers additionally requires the usual fuzzy-RD conditions: assignment affects the outcome only through treatment, assignment does not reduce treatment receipt for any unit (monotonicity), and the relevant potential-outcome and treatment-response quantities are continuous at the boundary. These assumptions must be credible at the particular boundary location, not merely somewhere in the score space.

Worked Example: A Sharp Local Comparison

Suppose a hypothetical scholarship uses the rule in the earlier geometry example. An analyst studies the active segment near $(70,65)$. Within a narrow band around that boundary location, local fits on the two sides estimate limiting mean annual earnings of \$28,400 for applicants just below the examination cutoff and \$30,100 for applicants just above it. Assume treatment is exactly scholarship assignment.

The estimated outcome jump is

\widehat{\Delta}_Y =\$30{,}100-\$28{,}400 =\$1{,}700\text{ per year}.

If the assignment rule is enforced as stated, the first-stage jump in treatment receipt is one, and the sharp estimate is \$1,700 per year at this boundary location. This interpretation depends on continuity of the potential-outcome means, no consequential sorting at the cutoff, and local support on both sides for applicants near $R_2=65$. The two fitted limits are hypothetical estimates, not evidence that those assumptions hold; they describe an effect local to this part of the scholarship boundary, not an effect for all applicants.

Worked Example: A Fuzzy Local Wald Ratio

In a second hypothetical program, crossing a two-score eligibility boundary raises, but does not determine, treatment receipt. At a fixed boundary location, local fits estimate a mean outcome of 72 units just on the assigned side and 68 units just on the unassigned side. They estimate treatment receipt rates of 0.78 and 0.28, respectively.

The outcome jump is $72-68=4$ units, and the treatment-probability jump is $0.78-0.28=0.50$. The local Wald estimate is

\widehat{\tau}_{\mathrm{Wald}} =\frac{72-68}{0.78-0.28} =\frac{4}{0.50} =8.00\text{ outcome units}.

Under the fuzzy-design assumptions stated above, this estimates a treatment effect for compliers local to the chosen boundary point. It is not the four-unit reduced-form outcome jump, which is the effect of assignment, and it is not an average treatment effect for the entire population. If the first-stage jump were near zero, the ratio would be unstable and the design would provide little information about a treatment effect, even if the outcome jump appeared nonzero.

Estimation, Diagnostics, and Common Failure Modes

A practical analysis first identifies active boundary segments and selects a target location or a justified set of locations. It then uses observations close to the boundary in the normal direction and close to the target location along the boundary. Local linear regressions can be fit on each side using both the normal distance and the tangential coordinate; the fitted limits at the same boundary point are contrasted. A bandwidth in the normal direction alone may be insufficient if observations far away along the boundary are included. Both dimensions of locality matter.

The following checks help reveal when the design is not supporting the intended comparison:

Verify the rule. Confirm from the actual assignment process where eligibility changes. A nominal score cutoff is not necessarily an active boundary everywhere.
Check local support. Confirm that observations occur on both sides near the same boundary locations. Sparse or absent support on one side prevents a credible local comparison.
Inspect sorting and score quality. Look for bunching, manipulation, or measurement problems near the boundary. With several scores, examine their joint distribution, not only each score separately.
Check predetermined characteristics. Large discontinuities in covariates that should not be affected by assignment can signal sorting, another coincident change, or a misspecified boundary.
Vary local specifications. Compare reasonable bandwidths and local regression specifications. Large changes may indicate that the estimate relies on observations too far from the boundary or on unstable extrapolation.
Treat corners and intersections cautiously. At a corner, multiple directions can cross the boundary. Separate directional limits may differ, so a single smooth-boundary contrast may not be defined without further assumptions.

A particularly common error is to compare all applicants above both cutoffs with all applicants who miss at least one cutoff, then call the difference an RD effect. That is wrong because those groups can differ in both running variables, and the comparison is not restricted to observations approaching the same active boundary point. The correction is to identify a specific active segment, hold location along it fixed (or model it locally), and compare limits from opposite sides.

Alternatives and Limits

Approach	What it compares	Main caution
Multidimensional RD	Limits from opposite sides of an active boundary at the same local boundary position	Requires local support, a correctly identified boundary, and continuity at that location
Scalar-index RD	Units on opposite sides of a cutoff in a prespecified composite score	Appropriate when assignment truly changes at that index cutoff; the boundary contrast may average over locations on the index level set
Separate one-dimensional RD analyses	Local comparisons on individual cutoff segments or strata	May be useful when segments are distinct, but cannot use inactive segments as if assignment changed there
Regression adjustment without an RD contrast	Outcome differences adjusted for observed scores and covariates	Adjustment alone does not supply the discontinuity-based identification argument

A composite score is often a sensible representation if the real assignment rule is a threshold on that score. But creating an index merely to simplify analysis can hide the true boundary geometry. Conversely, a multidimensional RD may be unnecessarily complex when assignment depends only on one valid scalar index. The method should follow the actual assignment mechanism.

Even a well-identified multidimensional RD effect is local to the boundary location or locations studied. It may not describe people far from the threshold or locations on another segment. The number of running variables also makes local support more demanding: as dimension increases, fewer observations may lie close in all relevant directions. No regression specification can replace missing comparisons or establish continuity by itself.

Key takeaway: In a multiple-running-variable design, the causal comparison is defined by the assignment boundary, not by the number of scores in a regression. Identify where crossing the boundary changes assignment, compare limits at the same boundary location, and interpret the estimate only under the corresponding local continuity and support assumptions.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A program assigns treatment when $R_1\ge 10$ and $R_2\ge 5$. Is the point $(10,3)$ an active point on the $R_1=10$ line? Explain.
At a fixed active boundary location, outcome limits are 40 and 37, while treatment-receipt limits are 0.70 and 0.20. Calculate the local Wald ratio and state its interpretation under the fuzzy-RD assumptions.
Why is a comparison of everyone eligible with everyone ineligible not, by itself, a multidimensional RD estimate?
What is a particular concern when estimating an effect at the intersection of two active boundary segments?

Solutions. (1) No. Since $R_2=3<5$, crossing $R_1=10$ does not satisfy the second requirement, so assignment remains zero on both sides; the point is inactive for that crossing. (2) The outcome jump is $40-37=3$, and the treatment jump is $0.70-0.20=0.50$. The ratio is $3/0.50=6$ outcome units, interpreted as a boundary-local effect for compliers if the fuzzy-RD assumptions hold. (3) The groups may differ in both scores and need not approach the same active boundary location, so their difference is not the required local contrast. (4) Different approach directions may yield different limits; a single smooth-boundary comparison may be unjustified, so the corner needs separate analysis or additional assumptions.