The Heckman Selection Correction

Suppose you want to know what drives hedge fund returns, and you regress return on fund characteristics using historical data. There's a hidden problem: funds that performed badly tend to shut down and stop reporting, so your dataset only contains the funds that survived. Regressing on this survivor-only sample doesn't estimate "what drives returns" — it estimates "what drives returns, conditional on having survived," which is a systematically different and more flattering question. The Heckman correction is the standard tool for fixing a regression when the sample you observe was filtered by a process related to the very outcome you're studying.

An analogy: judging restaurants only from happy reviewers

Imagine trying to learn what makes a restaurant good using only its posted reviews. People who had a mediocre-but-not-terrible meal often don't bother reviewing at all; the ones who show up are disproportionately either delighted or furious. If you regress "review score" on "food quality" using only these self-selected reviewers, you'll get a distorted picture, because who chose to leave a review is itself correlated with how the meal actually went. The Heckman correction works by first modeling who chooses to review (or here, which funds survive) as its own equation, and then explicitly accounting for that selection when estimating the main relationship you actually care about.

The method, one piece at a time

The Heckman correction runs in two stages. Stage one is a probit model of whether an observation is selected into the sample at all — for funds, whether they survive to be observed:

$$P(\text{selected}_i = 1) = \Phi(\gamma_0 + \gamma_1 z_i),$$

where $z_i$ is a variable that predicts selection (survival) and $\Phi$ is the normal CDF, as in probit regression (see Probit Versus Logit Models). From this fitted probit, you compute for each selected observation the inverse Mills ratio, $\lambda_i = \phi(\hat{z}_i)/\Phi(\hat{z}_i)$, where $\phi$ is the normal density — a number that measures how "surprising" it is that this particular observation was selected, given its characteristics.

Stage two adds $\lambda_i$ as an extra regressor in the main equation:

$$y_i = \beta_0 + \beta_1 x_i + \rho\sigma \lambda_i + \varepsilon_i .$$

In plain English: rather than just regressing return on fund characteristics using only the survivors, you also control for how unusual each survivor's survival was — funds that "shouldn't" have survived (had a high predicted probability of shutting down but didn't) get a different correction than funds that were always likely to survive. If the coefficient on $\lambda_i$ is significantly different from zero, selection bias was present and the correction meaningfully changes $\beta_1$; if it's near zero, selection wasn't distorting the naive estimate much after all.

Worked example 1: fund survivorship, by the numbers

A naive regression of fund returns on manager tenure, using only surviving funds, gives $\hat\beta_1 = 0.15$ (each extra year of tenure predicts 0.15 percentage points more return). The first-stage probit shows that funds with low past volatility and high past returns were far more likely to survive; among the surviving sample, the average inverse Mills ratio is 0.6, and the second-stage regression's coefficient on $\lambda_i$ comes out at $\rho\sigma = -2.1$ and is highly significant. The corrected estimate of $\beta_1$ drops to 0.04 — most of the naive 0.15 was actually the shutdown filter favoring longer-tenured managers who happened to have gotten lucky, not tenure itself driving returns.

Worked example 2: when correction changes little

A separate study looks at what predicts an IPO's first-day return, using only IPOs that actually priced and listed (some deals are withdrawn beforehand). The first-stage probit for "deal completes" finds that completion is driven almost entirely by market-wide conditions (VIX level at filing), not by the deal's own expected first-day return. The resulting inverse Mills ratio has a coefficient of $\rho\sigma = -0.3$ in the second stage, with a standard error of 0.4 — not statistically distinguishable from zero. Here the naive OLS estimate and the Heckman-corrected estimate come out nearly identical, correctly signaling that selection into the completed-IPO sample wasn't meaningfully related to the outcome being studied.

What this means in practice

The Heckman correction belongs in any study built on data that survived a filter correlated with the outcome: hedge fund and mutual fund databases (survivorship), analyst coverage (analysts only cover firms worth covering), loan performance (only approved loans are observed), or executive compensation (only completed deals report terms). Skipping it doesn't just add noise — it produces a systematically biased coefficient that looks statistically solid.