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1. Introduction: A Phrase Repeated Countless Times, Yet Rarely Questioned

Open any empirical paper that uses multiple regression, and you will most likely see this sentence in the section interpreting the coefficients:

"Holding other conditions constant, a one-unit increase in X is associated with an average change of β units in Y."

This sentence is so familiar that you no longer pause to think about its meaning. But if you stop and ask—how exactly is "holding other conditions constant" achieved?

When you run a regression, you are not physically "controlling" anything. You have not locked a group of people in a laboratory with identical diets, identical daily schedules, and identical wage levels. You simply threw a few more variables into the regression equation and clicked a button. So on what basis can you claim that you have "held" them constant?

This article does three things:

Clarify what "holding other conditions constant" means mathematically—and how it differs fundamentally from physical control.
Answer a key question: Can multiple regression automatically establish causal relationships?
Provide an actionable criterion: Under what conditions can multiple regression coefficients approximate causal relationships?

Core message: "Control" in multiple regression is not physical control, but statistical "partialling out"—removing the variation in Y that has already been explained by other variables, and examining the relationship between the remaining variation and Y. This operation does not by itself create causal relationships, but under specific conditions it can approximate them.

2. The Mathematical Truth Behind "Ceteris Paribus": The Frisch-Waugh-Lovell Theorem

2.1 More Than Just a "Partial Derivative"—The Intuition and Limitations of Partial Derivatives

In introductory econometrics textbooks, "holding other conditions constant" is usually explained as a partial derivative. Consider the model:

$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$

Taking the partial derivative with respect to X₁: $\frac{\partial Y}{\partial X_1} = \beta_1$ . The textbook says: β₁ measures "the change in Y when X₁ changes by one unit while X₂ is held constant."

This explanation is entirely correct in mathematical form, but it gives beginners a dangerous intuition—as if "holding X₂ constant" means you physically fix X₂, then vary X₁ alone and observe how Y changes.

You do not. You have observational data in which X₁ and X₂ vary freely across different individuals. You have never "controlled" anything—at least not in the physical sense. The mathematical operation behind "holding other conditions constant" is entirely different from what you might imagine.

2.2 What Actually Happens: Understanding Through the Frisch-Waugh-Lovell Theorem

How is the estimate of β₁ in multiple regression actually computed?

The Frisch-Waugh-Lovell (FWL) theorem provides an equivalent procedure that is more intuitive than any abstract explanation:

Step 1: Regress X₁ on X₂ (treating X₁ as the dependent variable and X₂ as the independent variable). Take the residual from this regression.

$\hat{u} = X_1 - (\hat{\gamma}_0 + \hat{\gamma}_1 X_2)$

What is this residual $\hat{u}$ ? It is the part of X₁ that cannot be explained by a linear function of X₂. In other words, you have "partialled out" the influence of X₂ from X₁ (strictly speaking, you have removed the linear correlation between X₁ and X₂), leaving the part of X₁ that is "orthogonal" to X₂.

Step 2: Regress Y on this residual $\hat{u}$ . The coefficient you obtain is β₁.

$\hat{\beta}_1 = \frac{\text{Cov}(\hat{u}, Y)}{\text{Var}(\hat{u})}$

2.3 A Concrete Example

Suppose you want to study "the effect of education on wages", and you control for one variable: "parental education."

The regression you actually run:

$\ln(\text{wage}) = \beta_0 + \beta_1 \cdot \text{edu} + \beta_2 \cdot \text{parent\_edu} + \varepsilon$

The FWL theorem tells you that $\hat{\beta}_1$ is computed as follows:

First, use parental education to "predict" a person's education, and obtain the residual. This residual represents—the deviation between this person's education and the "typical" education level for people with the same parental education background. For example, if both parents are primary school graduates, the typical education level for their children might be junior high school. If you went to college—your residual is "college minus junior high."
Then use this residual to "predict" wages.

So what β₁ truly measures is:

"Within the group of people whose parents have the same education level, how much higher are the wages of those whose education exceeds the typical level for that group, compared to the typical wage of that group?"

This is the mathematical truth of "holding other conditions constant"—it is not "locking everyone in the same laboratory," but rather statistically grouping together people with "identical other conditions," and then examining the relationship between within-group variation in X₁ and variation in Y.

Dimension	The "Control" You Imagine	The "Control" That Actually Happens
Method	Physically fixing other variables, varying only X₁	Statistically partialling out the linear influence of other variables from X₁
Data requirement	Requires an experimental environment	Only requires sufficient common variation in observational data
Mathematical implementation	Experimental design	Taking residuals
Number of variables controllable	Limited by experimental conditions	Limited by degrees of freedom and multicollinearity

3. Can Multiple Regression Automatically Establish Causal Relationships?

No. This is a fact every econometrics student must accept. And understanding "why not" is more important than knowing "it cannot."

3.1 Regression Coefficients ≠ Causal Effects—A Sobering Example

Let us return to the returns-to-education example and add a variable: individual ability.

The true wage determination process (data-generating process) is:

$\ln(\text{wage}) = \beta_0 + \beta_1 \cdot \text{edu} + \beta_2 \cdot \text{ability} + \varepsilon$

In the true data-generating process, ability affects wages (β₂ > 0), and ability is positively correlated with education (more able people are more likely to pursue more schooling).

Now, if you do not observe ability—that is, the regression you actually run does not include the ability variable, and you only run:

$\ln(\text{wage}) = \beta_0 + \beta_1 \cdot \text{edu} + \varepsilon$

Your $\hat{\beta}_1$ will capture two things simultaneously: (1) the direct effect of education on wages, and (2) the "ability → wages" effect that comes along through the correlation between education and ability. Your estimated return to education is contaminated by ability bias.

This is omitted variable bias—when there is a variable that affects Y and is correlated with X but is not controlled for, the regression coefficient is biased. Mathematically:

$\mathbb{E}[\hat{\beta}_1] = \beta_1 + \beta_2 \cdot \frac{\text{Cov}(\text{edu}, \text{ability})}{\text{Var}(\text{edu})}$

The second term is the bias. As long as Cov(edu, ability) ≠ 0 and ability indeed affects wages (β₂ ≠ 0), the bias exists.

3.2 Conditions Under Which Regression Coefficients Become Causal Effects

Under what conditions can multiple regression coefficients be interpreted as causal effects?

The answer is: the Conditional Independence Assumption (CIA), also known as selection on observables.

After controlling for all variables you include (denoted as Z), the treatment variable X is no longer correlated with potential outcomes. Formally:

$\mathbb{E}[Y(0) | X, \mathbf{Z}] = \mathbb{E}[Y(0) | \mathbf{Z}]$

In plain language: within groups where "other conditions are the same," there should be no systematic differences between those who receive the "treatment" (e.g., one more year of schooling) and those who do not. If, within a group, those with higher education and those with lower education still differ systematically on unobservable dimensions (such as ability), the condition above is violated, and regression coefficients ≠ causal effects.

The core difficulty with this condition is that it is fundamentally untestable with data. You can never prove that "I have controlled for all relevant variables"—because there are always variables you have not measured, or even variables whose existence you are unaware of.

4. Under What Conditions Can Multiple Regression Coefficients Approximate Causal Relationships?

Although we have said that "regression cannot automatically establish causality," in practice a large body of empirical research is still based on multiple regression—because under certain conditions, regression coefficients can come quite close to causal relationships. What conditions? Here are three dimensions to evaluate.

Condition 1: The Richness and Theoretical Persuasiveness of Control Variables

The more control variables and the more critical they are, the smaller the omitted variable bias—but there is an important subtlety here: not all control variables are equally meaningful.

Control variables should primarily target confounders that affect both X and Y. If you control for a variable that affects Y but not X, it will not reduce bias (it may only reduce standard errors); if you control for a variable that affects X but not Y, it likewise does not solve the omitted variable bias problem.

Evaluation criterion: When reading a paper that uses multiple regression, you should ask—what variables does the author control for? Why these particular variables? What variables that could affect both X and Y have been omitted? Does the author discuss whether "after controlling for these variables, unobservable confounders might still exist"?

A good empirical study using multiple regression typically devotes substantial space beyond the regression tables to arguing—why, after controlling for these specific variables, the residual variation in the treatment variable can be "close to random."

Condition 2: The Source of Variation in the Treatment Variable Must Be Scrutinized

Multiple regression uses all variation in X to estimate β₁—including both between-group and within-group variation. But different sources of variation in X imply different degrees of "closeness to randomness."

Consider a classic comparison:

Cross-country cross-sectional regression: "The effect of democracy on economic growth." The variation in X comes from "why some countries are democratic and others are not"—which is entangled with colonial history, cultural traditions, geographic location, resource endowments, and so on. These factors simultaneously affect both democracy and economic growth. After controlling for dozens of variables, do you truly believe all confounders have been exhausted? This is very difficult.
Within-family sibling comparisons: "The effect of education on income." If you use family fixed effects (which essentially means controlling for a set of "family" dummy variables), you are comparing how differences in education among siblings within the same family correspond to differences in their incomes. The variation in X comes from "why children in the same family have different education levels"—which could be birth order, fluctuations in family economic conditions in a particular year, the influence of a particular teacher, etc. These factors, while not perfect, are closer to random than "why this country is democratic and that one is not."

Evaluation criterion: At what level does the variation in X mainly come from? To what extent can this variation be considered "approximately randomly assigned after controlling for observable factors"?

Condition 3: Research Design Precedes Regression—Identification Strategy Provides Justification for Regression

In contemporary empirical economics, regression itself is usually not treated as the identification strategy—it is merely an estimation tool. The identification strategy is separate from the regression: instrumental variables, difference-in-differences, regression discontinuity, fixed effects—these design logics are the main body of causal argumentation. Regression is only responsible for estimating parameters at the final step.

But when fixed effects models are used as the identification strategy, the coefficients from the regression are closer to causal. This is because fixed effects, by using "each individual as their own control," eliminate all time-invariant confounders at the source.

Evaluation criterion: Does the paper's "causal argument" primarily occur in the section discussing the identification strategy, or at the moment the regression table is presented? If the main argument is in the identification strategy section, then the regression coefficients are closer to causal.

5. What Pitfalls Should You Watch Out for Regarding "Control"?

Pitfall 1: Over-controlling

More controls are not always better. Some variables should not be controlled for—controlling for them can introduce new bias.

Classic scenario: controlling for a mediator.

Suppose the true causal chain is: education → occupational prestige → wages. If you want to estimate the total effect of education on wages, you should not control for "occupational prestige"—because part of education's effect on wages operates precisely through "entering more prestigious occupations." After controlling for occupational prestige, you can only estimate the direct effect of education (the part remaining after removing the influence that operates through the occupational pathway), not the total effect you are interested in.

Criterion for bad controls: Is this variable on the causal pathway between X and Y? If so, controlling for it will "block" part of the causal pathway.

Pitfall 2: Control Variables That Are Themselves Endogenous

If your control variable is itself endogenous (i.e., in another equation, it is also affected by some omitted variable), controlling for it may not only fail to reduce bias but may also "infect" other coefficients with bias. A classic example: controlling for "occupation" in a returns-to-education regression—but occupational choice is itself an endogenous process.

Pitfall 3: False Reassurance Under Multicollinearity

When X₁ is highly correlated with the control variables (i.e., multicollinearity is severe), the variance of the regression coefficient inflates (standard errors become very large). In this case, "holding other conditions constant" remains mathematically valid, but statistically it becomes nearly impossible—because the data contain almost no observations where "X₁ changes but X₂ does not." Your estimate relies heavily on linear extrapolation from the model specification rather than on actual evidence in the data.

6. Summary

Three levels of understanding "holding other conditions constant":

Mathematical level (FWL theorem): β₁ measures "the relationship between the residual variation in X₁—after partialling out the linear influence of other control variables—and Y." This has nothing to do with whether you physically "controlled" anything.
Empirical level: Whether multiple regression coefficients can be interpreted as causal effects depends on whether the residual variation in X is independent of potential outcomes after controlling for observable variables (the CIA assumption). This assumption is fundamentally untestable, and its credibility depends on the researcher's argumentation and the reader's judgment.
Operational level: To make regression coefficients closer to causal, you need to combine identification strategies (fixed effects, IV, DID, RDD, etc.), rather than merely piling variables into the regression equation. At the same time, you need to be aware of the pitfalls of over-controlling, endogenous control variables, and multicollinearity.

One-sentence conclusion:

"Holding other conditions constant" is not a magic trick you can achieve by clicking a button—it is an ideal state that can only be approximated under specific assumptions and through specific research designs. Understanding its mathematical essence is the first step, identifying the causal assumptions behind it is the second step, and the most important third step is knowing how far you are from this ideal state in your own research context.

7. Presentation Suggestions for Bilibili/WeChat Official Account

Bilibili video: The two-step procedure of the FWL theorem should be demonstrated with animation—use two side-by-side panels showing "original X₁" and "the residual after removing the influence of X₂," so viewers can intuitively see that "holding other conditions constant" means using this residual in the regression.
WeChat article: For the formula sections (FWL theorem, omitted variable bias formula), it is recommended to present them as handwritten annotated figures to reduce reading pressure. Highlight the CIA assumption with a boxed blockquote.
Recommended titles:
- Main title: 《How Exactly Is "Holding Other Conditions Constant" Achieved in Multiple Regression?》
- Alternative title: 《You Are Not Physically Controlling Anything—The Mathematical Truth Behind "Control Variables" in Multiple Regression》
Key quote:

"Holding other conditions constant is not magic—it is running a regression on residuals. Understanding its mathematical essence is the first step; knowing how far it is from a causal relationship is the second."