数据范儿_范老师-AI时代科研统计新范式-计量科研AI导航

1. Introduction: Two Languages, One Word

If you read both types of papers, you will find that the term "causal relationship" is used in two seemingly completely different contexts—

Context A (common in labor economics, economics of education, and development economics):

"We use propensity score matching to estimate the treatment effect of a training program on wages. Under the conditional independence assumption, the ATT estimate is 0.12, implying that participating in training raises wages by approximately 12%."

Context B (more traditional empirical research):

"After controlling for individual characteristics, family background, and region fixed effects, the coefficient on years of education is 0.08 and significant at the 1% level. This suggests that, holding other factors constant, each additional year of education raises wages by approximately 8%."

Both statements claim to estimate a "causal effect." But they use two completely different language systems. Context A uses the potential outcomes framework (Rubin Causal Model, RCM)—with counterfactuals, treatment effects, and assignment mechanisms as core concepts. Context B uses multiple regression—with partial derivatives, control variables, and the FWL theorem as core concepts.

Are the "causal effects" claimed by these two frameworks the same thing? Is one more credible than the other? Under what conditions do they give the same answer?

This article aims to bridge these two languages.

Core message: The potential outcomes framework and multiple regression are not two "competing causal theories"—they are two expressions of the same identification logic. The potential outcomes framework is more transparent (you are forced to state assumptions explicitly), while multiple regression is more convenient (but assumptions tend to remain hidden). Understanding the conditions under which the two are equivalent is a key skill for judging the credibility of an empirical paper.

2. The Potential Outcomes Framework: Defining Causality with "Counterfactuals"

2.1 Core Idea—Each Individual Has Two Potential Selves

The potential outcomes framework was systematized by Rubin (1974). Its starting point is a "counterfactual" definition:

For each individual i, define two potential outcomes:

Yᵢ(1): the potential outcome for individual i if receiving treatment (e.g., participating in training, attending college, taking a drug).
Yᵢ(0): the potential outcome for individual i if not receiving treatment.

The individual treatment effect is the difference between these two potential outcomes:

$\tau_i = Y_i(1) - Y_i(0)$

How clean is this definition? It relies on no model, no assumption, and no regression equation whatsoever. Causality is simply the difference between two potential outcomes—flawless by definition.

2.2 The Fundamental Problem—You Can Never Observe Both Potential Outcomes

But a clean definition is one thing; practical implementation is another. For any given individual, you can only observe one potential outcome. For someone who participated in training, you cannot see what would have happened had they "not participated"; for someone who did not participate, you cannot see what would have happened had they "participated."

This is what Holland (1986) calls the fundamental problem of causal inference—causal effects are defined at the individual level, but individual-level causal effects are never directly observable.

2.3 The Solution—Moving from Individuals to Groups, Introducing Observable Assumptions

Since individual causal effects are unobservable, the goal of empirical research shifts to estimating the average treatment effect (ATE):

$\text{ATE} = \mathbb{E}[Y(1) - Y(0)]$

To identify the ATE, a key condition must be satisfied—unconfoundedness, also known as the conditional independence assumption (CIA):

$Y(0), Y(1) \perp T \mid \mathbf{X}$

This means that after controlling for covariates X, who receives treatment (T=1) and who does not (T=0) is no longer correlated with the potential outcomes. Conditional on X, treatment assignment is "as if random."

If the CIA holds, the ATE can be identified as follows:

$\text{ATE} = \mathbb{E}_{\mathbf{X}} \left[ \mathbb{E}[Y|T=1, \mathbf{X}] - \mathbb{E}[Y|T=0, \mathbf{X}] \right]$

That is, within each stratum of X, compare the mean difference between the treatment group and the control group, then take the weighted average according to the distribution of X.

2.4 What Exactly Is the "Causal Language" of the Potential Outcomes Framework?

To summarize in one sentence: In the potential outcomes framework, the "causal effect" is defined on counterfactuals—it does not depend on any functional form assumption, but only on a substantive assumption (whether the assignment mechanism is equivalent to "approximately random after controlling for X").

3. Multiple Regression: Approaching Causality with "Partial Derivatives"

Recall the FWL theorem (fully developed in the previous article): the $\hat{\beta}_1$ in a multiple regression equals the relationship between Y and the variation in X₁ that remains after removing the part of X₁ that is linearly related to all control variables.

But to interpret this $\hat{\beta}_1$ as a causal relationship, an equally substantive condition is required—zero omitted variable bias. That is, after controlling for the observed variables, there are no unobserved variables that affect both X and Y.

Notice how similar this statement is to the CIA in the potential outcomes framework—in fact, they are mathematically the same condition, just stated differently.

Therefore, the "causal effect" claimed by multiple regression also holds only under the same CIA. The difference is that the potential outcomes framework puts the CIA front and center—you must write in your paper "we assume that treatment assignment is approximately random after controlling for X"—whereas regression users sometimes do not even mention this condition, jumping directly to "the coefficient is β, representing the causal effect."

4. Equivalence and Non-Equivalence: Under What Conditions Do They Give the Same Answer?

4.1 An Underappreciated Result: OLS = Variance-Weighted Treatment Effect

Angrist & Pischke (2009, Mostly Harmless Econometrics) provide an influential result:

Under the conditional independence assumption (CIA), the multiple regression coefficient $\hat{\beta}_1$ is not the ATE, nor the ATT—it is a conditional-variance-weighted average of treatment effects.

Specifically, suppose the treatment variable T (0/1) and control variables X. Under the CIA and correct linear model specification:

$\beta_1 = \mathbb{E}\left[ \omega(X) \cdot \tau(X) \right]$

where $\tau(X) = \mathbb{E}[Y(1)-Y(0)|X]$ is the treatment effect conditional on X, and the weight $\omega(X) \propto \text{Var}(T|X)$ —that is, the conditional variance of the treatment variable.

What does this mean?

The larger the variance of the treatment variable at a given value of X, the greater the weight that the treatment effect at that X value receives in the OLS estimate.

Intuitively: if within a stratum of X, almost everyone is treated (or almost no one is treated), then the treatment effect in that stratum contributes little to the OLS coefficient—because the data in that stratum "cannot distinguish" between treated and untreated populations. Conversely, within a stratum of X where roughly half are treated and half are not (maximum variance), the treatment effect in that stratum receives the greatest weight.

This weighting scheme is chosen automatically by OLS—it is not the natural weight of any treatment effect definition (ATE, ATT), but rather a mathematical property of the regression itself.

4.2 When Treatment Effects Are Homogeneous: Regression Coefficient = ATE

If treatment effects are homogeneous—that is, every individual experiences the same treatment effect, with $\tau(X) \equiv \tau$ constant across all X—then regardless of how weights are allocated, the weighted average equals τ:

$\beta_1 = \tau = \text{ATE} = \text{ATT}$

In this special case, the ATE under the potential outcomes framework and the multiple regression coefficient are exactly equivalent.

But homogeneous treatment effects is an extremely strong assumption. In the returns-to-education example, it implies that "college has the same effect on everyone"—regardless of ability, field of study, or labor market. This is clearly counterintuitive.

4.3 When Treatment Effects Are Heterogeneous: Regression Coefficient ≠ ATE, ≠ ATT

If treatment effects are heterogeneous—that is, different individuals experience different treatment effects—the OLS coefficient is neither the ATE nor the ATT. It is a weighted average of treatment effects that gives greater weight to strata with higher variance.

A comparison:

Estimator	Definition	Weighting Scheme
ATE	$\mathbb{E}[Y(1)-Y(0)]$	Equal weight according to population distribution
ATT	$\mathbb{E}[Y(1)-Y(0)\\|T=1]$	Average only over the treatment group
OLS coefficient (under CIA)	$\mathbb{E}[\omega(X)\tau(X)]$	Weighted by the conditional variance of the treatment variable

This means that even if the CIA holds, the number you read from a multiple regression does not equal the ATE as defined in the potential outcomes framework—unless you are willing to additionally assume homogeneous treatment effects, or you use more flexible estimation methods (such as interaction term models, matching methods, etc.) that allow for heterogeneous treatment effects.

5. Which Is More Credible?—Not a Question of Superiority, but of Transparency

5.1 The Advantage of the Potential Outcomes Framework: Transparent Assumptions

The potential outcomes framework almost forces you to do three things:

Clearly define the treatment: What is T? Is it binary or continuous? How is treatment assigned?
Clearly define the counterfactual: Who is compared with whom? Are you estimating the ATE, ATT, or LATE?
Clearly state the identification assumption: Is the CIA reasonable? After controlling for which variables are you willing to believe the CIA holds?

These steps cannot be skipped. If you skip them, reviewers will fill them in for you.

The "credibility" of the potential outcomes framework lies not in its more advanced mathematics—but in the fact that you put all the "ifs" on the table, allowing readers to judge for themselves whether these "ifs" hold.

5.2 The Risk of Multiple Regression: Hidden Assumptions

The danger of multiple regression is that—you can run a regression and obtain significant coefficients without ever interrogating any identification assumption. The stars in the regression table (***) give you a false sense of certainty: p < 0.01, three stars, it must be true.

But returning to the discussion above—if the CIA does not hold, this p-value measures "the probability of rejecting the null hypothesis under a misspecified model," not "the probability that the effect is real." It guarantees nothing.

This is why contemporary empirical economics increasingly treats "regression" merely as an estimation tool, placing the identification argument in a completely separate section. In the identification argument, you need to explain: why, after controlling for these variables, can the residual variation in the treatment variable be "considered approximately random"? This is not a question that a regression table can answer.

5.3 A Table Comparing the Two Frameworks

Dimension	Potential Outcomes Framework (RCM)	Multiple Regression
Definition of causality	Counterfactual: Y(1)-Y(0)	Partial derivative: ∂Y/∂X (holding others constant)
Identification condition	CIA (unconfoundedness)	Omitted variable bias = 0
Relationship to the CIA	Directly is the CIA	Equivalent, just not stated explicitly
Target quantity estimated	ATE/ATT/LATE (clearly defined)	Under CIA = variance-weighted treatment effect
Heterogeneous treatment effects	Can be flexibly estimated (matching, stratification, ML, etc.)	Standard OLS only gives variance-weighted average
Visibility of assumptions	High—forced to be written out by the framework	Low—easily obscured by the rhetoric of "control variables"
Most prone to abuse	"Black-box" use of propensity score matching	Blind operation of "the more control variables, the better"
In one sentence	"Assumptions are open cards; judge for yourself"	"Assumptions are hole cards; you need to know how to reveal them"

6. When Do the Two Approaches Converge or Coincide?—Practical Judgments

Based on the foregoing, here are some practical judgment criteria:

CIA holds + homogeneous treatment effects → The two are exactly equivalent. The regression coefficient = ATE. However, the homogeneous effects assumption is too restrictive for most applications.
CIA holds + heterogeneous but modest treatment effects → The two are close. If treatment effects differ little across populations, the gap between the variance-weighted average and the simple average is also small. In practice, this "closeness" may be sufficient to support your qualitative conclusions.
CIA holds + substantial heterogeneity → The two may differ considerably. In this case, it is better to use more flexible methods such as matching, inverse probability weighting (IPW), or interaction term regressions—allowing the model to let treatment effects vary with X.
CIA does not hold → Neither approach yields causal effects. In this case, you need to turn to more credible identification strategies (IV, DID, RDD, etc.), rather than debating "which is better" between regression and RCM.

Key conclusion: The potential outcomes framework and multiple regression belong to the same class of methods in terms of identification logic—both rely on "treatment assignment being approximately random after controlling for observable variables." Their difference lies in the fact that the potential outcomes framework makes your assumptions more visible, and without additional assumptions (such as homogeneous treatment effects), the estimation target under this framework is clearer.

7. Summary

Returning to the four questions in the title:

How do the "causal effects" in the two frameworks differ? —The potential outcomes framework defines causality on counterfactuals; multiple regression approximates causality through partial derivatives. But both rely on the same substantive condition (CIA / no omitted variable bias).
Which is more credible? —It is not about credibility but transparency. The potential outcomes framework does not automatically make your causal inference stronger, but it makes you see more clearly "where your inference is strong and where it is weak." Credibility comes from the reasonableness of assumptions, not the choice of framework.
Under what conditions are the two close? —When the CIA holds and treatment effects are homogeneous, the two are equivalent. When the CIA holds and heterogeneity is modest, the two are close.
Under what conditions are they equivalent? —When the CIA holds and treatment effects are strictly homogeneous, the regression coefficient = ATE.

One-sentence conclusion:

"The potential outcomes framework and multiple regression are not two competing causal philosophies—they are different expressions of the same logic. The significance of the framework lies not in which is more 'advanced,' but in whether you are willing to put your assumptions on the table for all to scrutinize, or hide them in the footnotes of a regression table."

8. Presentation Suggestions for Bilibili/WeChat Official Account

Bilibili video: For the core logical chain, consider a "side-by-side comparison" visual—the left panel displays the counterfactual language of the Rubin causal framework, the right panel displays the partial derivative language of regression, with a recurring line of text at the bottom: "They rely on the same assumption: CIA." When discussing OLS ≠ ATE, use a balance scale animation to show "different weights on each side," visually conveying the meaning of conditional variance weighting.
WeChat official account: The comparison table of the two frameworks is well suited for a left-right split infographic. Angrist & Pischke's "OLS = variance-weighted average" result can be presented as a highlighted quote card—it is the core informational increment of this article.
Recommended titles:
- Main title: 《Potential Outcomes vs. Multiple Regression: How Do the Two "Causal" Languages Differ?》
- Alternative title: 《Does the Coefficient You Get from Regression Equal the ATE?—In Most Cases, No》
Key quote:

"The potential outcomes framework will not automatically make your causal inference stronger—but it will make you see more clearly where your inference is strong and where it is weak."