Deriving Omitted-Variable Bias in Linear Regression

Why an Omitted Confounder Shifts a Treatment Coefficient

A treatment coefficient from a linear regression is often used to summarize the relationship between treatment and outcome after adjustment for recorded covariates. But if a variable related to both treatment and outcome is left out, the fitted coefficient can absorb some of that variable’s outcome association. The resulting shift is not mysterious: under a suitable linear model, it is the product of two quantities—how strongly the omitted variable predicts the outcome, and how strongly it remains associated with treatment after adjustment for included covariates.

This tutorial derives that relationship. The formula describes a particular linear-regression setting; it does not imply that every omitted variable causes bias, or that a regression coefficient is automatically a causal effect. The assumptions connecting the model to a causal treatment effect must be stated separately.

Definition: Let $A$ be treatment, $Y$ the outcome, $Z$ a vector of included covariates, and $U$ an omitted variable. In the additive linear model $$ Y=\tau A+\beta U+\gamma^\top Z+\varepsilon, $$ the omitted-variable bias in the coefficient of $A$ is the difference between the coefficient from a regression of $Y$ on $A$ and $Z$, which omits $U$, and the coefficient $\tau$ in the full model. Under the conditions developed below, that difference is $\beta$ times the residual association of $U$ with $A$ after adjusting for $Z$.

The Model and the Residual Association

The model is additive: the conditional mean of $Y$ changes by $\tau$ per unit of $A$, by $\beta$ per unit of $U$, and according to the linear term $\gamma^\top Z$. The error has conditional mean zero, $E(\varepsilon\mid A,U,Z)=0$. This condition ensures that, once the variables in the full model are included, the error has no remaining systematic linear association with them. For the regression coefficient to represent a constant treatment effect, the model also excludes treatment-by-$U$ and treatment-by-$Z$ interactions.

To define residual association, linearly regress $A$ on an intercept and $Z$, and call the residual $A^\perp$. Likewise, linearly regress $U$ on an intercept and $Z$, and call the residual $U^\perp$. These residuals are the parts of treatment and the omitted variable not linearly explained by the included covariates. Define their residual association as the slope from regressing $U^\perp$ on $A^\perp$:

\delta = \frac{\operatorname{Cov}(A^\perp,U^\perp)} {\operatorname{Var}(A^\perp)}.

Thus $\delta$ has units of $U$ per unit of treatment. It is conditional on the adjustment set $Z$: a strong marginal association between $A$ and $U$ may disappear after adjusting for $Z$, while a small marginal association may remain or even change direction. The bias depends on this residual association, not simply on the unadjusted relationship.

The population version of the omitted regression is the linear projection of $Y$ on $A$ and $Z$. Its treatment coefficient is denoted $\tilde{\tau}$. The key result is:

Theorem: If the full additive model holds, $E(\varepsilon\mid A,U,Z)=0$, and $\operatorname{Var}(A^\perp)>0$, then the population treatment coefficient from the regression of $Y$ on $A$ and $Z$, omitting $U$, satisfies $$ \tilde{\tau} = \tau+\beta \frac{\operatorname{Cov}(A^\perp,U^\perp)} {\operatorname{Var}(A^\perp)} = \tau+\beta\delta. $$ The omitted-variable bias is $\tilde{\tau}-\tau=\beta\delta$.

Deriving the Formula

The derivation uses the partial-regression result: in a regression containing $A$ and $Z$, the coefficient of $A$ can be calculated by first removing the linear association of $Z$ with $A$ and $Y$, then regressing the residualized outcome on the residualized treatment. Equivalently, in the population, the coefficient is $\operatorname{Cov}(A^\perp,Y^\perp)/\operatorname{Var}(A^\perp)$, where $Y^\perp$ is the residual from linearly projecting $Y$ on an intercept and $Z$.

Residualizing the full model on $Z$ removes its intercept and linear $Z$ component. The remaining expression is $Y^\perp=\tau A^\perp+\beta U^\perp+\varepsilon^\perp$, where $\varepsilon^\perp$ is the residualized error. Substitution into the coefficient formula gives the result:

\begin{aligned} \tilde{\tau} &= \frac{\operatorname{Cov}(A^\perp,Y^\perp)} {\operatorname{Var}(A^\perp)}\\ &= \frac{ \tau\operatorname{Var}(A^\perp) +\beta\operatorname{Cov}(A^\perp,U^\perp) +\operatorname{Cov}(A^\perp,\varepsilon^\perp)} {\operatorname{Var}(A^\perp)}\\ &= \tau+ \beta\frac{\operatorname{Cov}(A^\perp,U^\perp)} {\operatorname{Var}(A^\perp)}. \end{aligned}

For the last equality, the conditional mean-zero assumption implies that the error is uncorrelated with $A$, $U$, and $Z$. Since $A^\perp$ is a linear combination of $A$ and $Z$, it is also uncorrelated with the error residualized on $Z$, so $\operatorname{Cov}(A^\perp,\varepsilon^\perp)=0$. The denominator must be positive: if $A$ is completely determined by $Z$ in the population, there is no residual treatment variation and the adjusted treatment coefficient is not identified by this regression.

The same argument applies to a finite sample under the corresponding linear model. The exact OLS decomposition includes a term for the sample association between residualized treatment and residualized error:

\hat{\tau}_{\text{omit}} = \tau+ \beta\frac{\sum_i A_i^\perp U_i^\perp}{\sum_i (A_i^\perp)^2} + \frac{\sum_i A_i^\perp\varepsilon_i^\perp}{\sum_i (A_i^\perp)^2}.

The middle term is the sample omitted-variable component. The last term is sampling error from the outcome disturbance; it need not be exactly zero in a particular sample. This distinction matters: the population bias formula describes the systematic shift under the stated model, not the full difference between a sample estimate and $\tau$.

Worked Calculations

Worked Example: Bias Without Additional Covariates

Suppose a hypothetical population follows $Y=\tau A+\beta U+\varepsilon$, with no $Z$. Treatment $A$ is measured in treatment units, $U$ is a motivation score, and $Y$ is a symptom score in points. Assume the full model is correct, the error has conditional mean zero, $\tau=2.0$ symptom points per treatment unit, and $\beta=3.0$ symptom points per motivation-score unit. Suppose the population slope from regressing $U$ on $A$ is $0.40$ motivation-score units per treatment unit.

With no covariates to residualize, this slope is $\delta=\operatorname{Cov}(A,U)/\operatorname{Var}(A)=0.40$. The omitted-variable bias is therefore $$ \beta\delta = (3.0\ \text{points per motivation unit}) (0.40\ \text{motivation units per treatment unit}) = 1.20\ \text{points per treatment unit}. $$

The regression omitting motivation has population coefficient $$ \tilde{\tau}=\tau+\beta\delta=2.0+1.20=3.20 \quad\text{symptom points per treatment unit}. $$

The shift is positive because motivation both predicts a higher symptom score in this stipulated model and is more prevalent as treatment increases. This example checks the formula’s assumptions by stipulating the additive model and mean-zero error; it illustrates the algebra, not evidence that treatment causes the outcome.

Worked Example: Bias After Adjusting for a Recorded Covariate

In a second hypothetical population, a regression adjusts for $Z$, a baseline severity measure, but omits $U$, a health-literacy score. The full model is additive with conditional mean-zero error. Suppose $\beta=5.0$ outcome points per health-literacy unit. After linearly removing $Z$, the residualized treatment has variance $1.50$ treatment-units squared, and the residualized treatment and literacy have covariance $0.60$ treatment-units times literacy units. The true full-model treatment coefficient is $\tau=-1.0$ outcome point per treatment unit.

The residual association is $$ \delta = \frac{0.60\ \text{treatment units}\times\text{literacy units}} {1.50\ \text{treatment-units squared}} = 0.40\ \text{literacy units per treatment unit}. $$

The omitted-variable component is $$ \beta\delta = (5.0\ \text{points per literacy unit}) (0.40\ \text{literacy units per treatment unit}) = 2.0\ \text{points per treatment unit}. $$

Consequently, the population coefficient from the regression that adjusts for $Z$ but omits $U$ is $\tilde{\tau}=-1.0+2.0=1.0$ outcome point per treatment unit. The sign reverses in this hypothetical calculation. Adjustment for $Z$ is already reflected in the residual covariance and variance; using the marginal association between treatment and literacy instead would not calculate the stated adjusted coefficient.

Worked Example: Binary Treatment and an Omitted Binary Confounder

Consider a hypothetical population with binary treatment $A$ and binary confounder $U$. Half the population is treated; $P(U=1\mid A=1)=0.60$ and $P(U=1\mid A=0)=0.30$. The full model is additive, $Y=\tau A+\beta U+\varepsilon$, with conditional mean-zero error, $\beta=4.0$ outcome points per unit of $U$, and $\tau=-0.50$ outcome points. There are no other covariates in this calculation.

For a binary treatment, the slope from regressing $U$ on $A$ equals the difference in mean $U$ between the treated and untreated groups. The difference here is $0.60-0.30=0.30$ units of $U$ per treatment unit. Equivalently, $\operatorname{Var}(A)=0.5(1-0.5)=0.25$ and $\operatorname{Cov}(A,U)=0.5(1-0.5)(0.60-0.30)=0.075$, so $$ \delta=\frac{0.075}{0.25}=0.30. $$

The bias is $4.0(0.30)=1.20$ outcome points. Thus the population regression coefficient that omits $U$ is $$ \tilde{\tau}=-0.50+1.20=0.70\ \text{outcome points}. $$

The model’s true coefficient is negative, while the omitted-variable regression coefficient is positive. This reversal follows from the assumed data-generating model and group proportions; it is not a general conclusion about binary treatment or confounding.

What Determines the Direction and Size of Bias?

The product $\beta\delta$ separates two roles. The coefficient $\beta$ measures how the omitted variable relates to the outcome in the full model, holding treatment and included covariates fixed. The quantity $\delta$ measures how much the omitted variable changes, on average in a linear sense, with treatment after accounting for those included covariates. If either factor is zero, the formula gives zero omitted-variable bias. A variable can predict the outcome but cause no bias in this model if it has no residual association with treatment; likewise, residual association with treatment does not shift the coefficient if the variable has no outcome coefficient.

If $\beta$ and $\delta$ have the same sign, the bias is positive; if they have opposite signs, it is negative. “Positive bias” means the omitted-variable coefficient is greater than $\tau$, not necessarily that it is positive. Several omitted variables contribute additively when the same linear-model conditions hold: each variable’s outcome coefficient is multiplied by its residual association with treatment, and the contributions sum.

The formula also makes clear why the adjustment set matters. Adding or removing a recorded covariate changes the residualized treatment and omitted variable, so it can change $\delta$. It does not generally change $\beta$ in the simple additive model, but in applications with misspecification or interactions, coefficients may depend on the model specification. The formula is therefore a description under a stated model, not a guarantee that a particular adjustment set removes confounding.

Common Mistake: Using a Marginal Association

A common mistake is to calculate the relationship between $A$ and $U$ without regard to the covariates already in the regression, then multiply that marginal association by $\beta$. That is only appropriate when there are no included covariates (or when the relevant marginal and residual associations happen to agree). In an adjusted regression, the required quantity is the association between the parts of $A$ and $U$ left after each has been linearly adjusted for the same $Z$.

Another mistake is to call the entire difference between a sample estimate and a causal effect “omitted-variable bias.” The decomposition separates the systematic omitted-variable component from the sample error term. And even if the omitted-variable component is zero, causal interpretation still requires that the full model and the conditions connecting its coefficient to the causal effect be credible. In a constant-effect structural interpretation, that includes a correctly specified outcome model and no unaccounted source of treatment-outcome confounding after conditioning on the variables in the full model.

Key takeaway: Under the additive linear model with mean-zero error, omitting $U$ shifts the adjusted treatment coefficient by $\beta\delta$. Here $\beta$ is $U$'s outcome coefficient, and $\delta$ is the slope relating residualized $U$ to residualized treatment after adjustment for included covariates. The formula quantifies a model-based shift; it does not establish that the model or its causal interpretation is correct.

Check Your Understanding

Use the model and units carefully when answering.

If an omitted variable has outcome coefficient $-2$ points per unit and residual association $0.5$ units per treatment unit, what is the omitted-variable bias and its direction?
Why does an adjusted regression require the residual association of $U$ with $A$, rather than their marginal association?
If $U$ is associated with treatment after adjustment but has zero outcome coefficient in the full model, what does the formula imply about the coefficient’s bias under the stated assumptions?
What does $\operatorname{Var}(A^\perp)=0$ imply for estimating an adjusted treatment coefficient?
In the binary-treatment example, which quantities determine the residual association when there are no other covariates?