Regression Discontinuity Design

An index provider adds any stock whose market capitalization crosses $10 billion, and drops any stock that falls below it. You want to know whether index inclusion itself causes a valuation bump, separate from whatever made the stock big enough to qualify in the first place. Comparing the average valuation of all index members against all non-members is hopeless — index members are systematically bigger, more liquid, more mature companies, and any of that could explain a valuation difference on its own, with nothing to do with the index. What you actually want is a comparison between the stock that just barely crossed $10 billion and the stock that just barely missed it — two companies that, moments before the cutoff, were nearly identical, and where crossing the line was close to arbitrary.

An analogy: the driving test cutoff

Think of a driving test scored out of 100, where 60 is a pass. Someone who scores 59 and someone who scores 61 are, in every practical sense, equally skilled drivers — the one-point difference is closer to noise (a slightly harder question, a moment's hesitation) than to a real gap in ability. Yet one gets a license and the other doesn't. If you want to know whether having a license changes someone's driving behavior — do they drive more, take more risks — comparing the 61-scorers to the 59-scorers isolates that effect almost as cleanly as a randomized experiment would, because nothing else about them differs systematically at that exact boundary.

That's the entire logic of regression discontinuity design: find a rule with a sharp cutoff, trust that units on either side of the line right at the boundary are comparable, and measure the jump in outcome exactly at the threshold.

The design, one symbol at a time

Let $R$ (the running variable, also called the forcing variable) be the quantity that determines treatment — here, market capitalization. Let $c$ be the cutoff — $10 billion. Treatment $D$ is assigned deterministically by the rule

$$D_i = \mathbb{1}(R_i \geq c) .$$

In plain English: a unit is treated exactly when its running variable clears the cutoff, and not otherwise — there's no randomness in who gets treated, only in where each unit happens to land relative to the line.

The causal effect at the cutoff is defined as the gap between two limits — the outcome $Y$ as $R$ approaches $c$ from above, minus $Y$ as $R$ approaches $c$ from below:

$$\tau_{RDD} = \lim_{r \downarrow c} E[Y \mid R = r] \;-\; \lim_{r \uparrow c} E[Y \mid R = r] .$$

In plain English: fit a smooth trend to the treated units and a separate smooth trend to the untreated units, and read off the size of the jump between the two trends exactly at the cutoff — not the overall difference in averages across the whole sample, just the gap right at the boundary.

In practice this is implemented by restricting to a bandwidth, $h$, around the cutoff — only using observations with $c - h \leq R \leq c + h$ — and running two local regressions, one on each side:

$$Y_i = \alpha + \beta (R_i - c) + \tau\, D_i + \gamma\, D_i (R_i - c) + \varepsilon_i, \quad \text{for } |R_i - c| \leq h .$$

In plain English: fit a line (or a mild curve) to the data on each side of the cutoff, let the two sides have different slopes, and the coefficient $\tau$ on the treatment indicator is the estimated jump — the RDD effect.

Worked example 1: the jump near a threshold

Ten stocks with market cap near the $10 billion cutoff (in $ billions) and their subsequent valuation multiple (P/E):

Stock	Market cap ($B)	Index member?	P/E
1	8.5	No	14.0
2	9.0	No	14.4
3	9.4	No	14.7
4	9.7	No	15.0
5	9.9	No	15.2
6	10.1	Yes	17.3
7	10.3	Yes	17.5
8	10.6	Yes	17.8
9	11.0	Yes	18.1
10	11.5	Yes	18.5

The below-cutoff points trend from 14.0 up to 15.2 as cap rises from 8.5 to 9.9 — a slope of about $(15.2-14.0)/(9.9-8.5) = 1.2/1.4 = 0.857$ P/E points per $B. Extrapolating that trend to exactly $10B gives $15.2 + 0.857 \times 0.1 = 15.29$. The above-cutoff points trend from 17.3 at 10.1 up to 18.5 at 11.5, a slope of $(18.5-17.3)/(11.5-10.1) = 1.2/1.4 = 0.857$ P/E points per $B as well — extrapolating backward to $10B gives $17.3 - 0.857 \times 0.1 = 17.21$.

The estimated jump at the cutoff is

$$\tau_{RDD} = 17.21 - 15.29 = 1.92 .$$

So index inclusion is associated with roughly a 1.9-point P/E premium, measured right at the boundary where the two groups of stocks were, moments earlier, essentially identical in size. Notice the underlying trend (rising P/E with market cap) is the same 0.857 slope on both sides — it's the discontinuous jump on top of that shared trend that RDD isolates, not the trend itself.

Worked example 2: how the estimate moves with bandwidth

Bandwidth choice trades off bias and precision. Using only the closest points to the cutoff reduces the risk that you're picking up an unrelated trend, but leaves fewer points to average over. Suppose you widen the sample:

Bandwidth (± $B around $10B)	Stocks included	Estimated jump $\tau$
0.2 (very narrow)	6, 7 vs. 4, 5 only	2.15
0.5 (used above)	6–8 vs. 3–5	1.92
1.5 (wide)	all 10 stocks	1.55

With the narrowest bandwidth, only the two closest points on each side are used, giving a jump of 2.15 — high precision loss (only 4 points) but minimal risk of mixing in the background trend. Widening to 1.5, using all ten stocks, pulls the estimate down to 1.55, because a wider window is more likely to average in points where the underlying (non-causal) relationship between market cap and valuation is doing more of the work, biasing the local jump toward the global trend. This is why RDD estimates are always reported alongside a bandwidth sensitivity check — a jump that only shows up at one specific bandwidth and vanishes at slightly wider or narrower windows is not trustworthy.

What this means in practice

RDD applies wherever an arbitrary, sharp rule assigns treatment: index inclusion at a market-cap or float threshold, credit approval at a credit-score cutoff, exchange tick-size pilots applied by ticker-symbol rule, minimum-price listing requirements. It's attractive because, right at the cutoff, it approximates a randomized experiment without needing one — but it only tells you the effect for units near the cutoff, not for the whole population. A stock at $10.05B and one at $50B may respond to index inclusion completely differently, and RDD is silent about the second one. Quants use this for event-driven strategies around known threshold rules, and for evaluating market-structure changes rolled out by size or score bands.