Omitted-Variable Bias in Linear Regression

Why omitting a regressor changes a slope

A regression coefficient depends on which variables are included in the projection. If an outcome is related to both an included regressor and an omitted one, the included regressor can partly reflect the omitted variable’s contribution. The resulting slope need not equal the slope on that regressor in the full model.

The central question is not simply whether the omitted variable predicts the outcome. It is whether it predicts the outcome and is linearly associated with the included regressor. The omitted-variable bias formula makes these two ingredients explicit. Here, “bias” first refers to the difference between two population linear-projection coefficients, not automatically to a causal effect or to the finite-sample expectation of an estimator.

Definition: Suppose the population linear projection of \(Y\) on an intercept, \(X\), and \(Z\) is \(Y=\beta_0+\beta_1X+\beta_2Z+U\), where \(X\) and \(Z\) are scalar regressors, \(\operatorname{Var}(X)>0\), and the full-model projection is identified. Its residual satisfies \(E[U]=0\), \(\operatorname{Cov}(X,U)=0\), and \(\operatorname{Cov}(Z,U)=0\). The omitted-variable bias for the slope on \(X\) is the difference between the slope in the projection of \(Y\) on an intercept and \(X\) alone and \(\beta_1\).

These are projection assumptions: the full-model residual is orthogonal to the regressors. A correctly specified conditional mean \(E[Y\mid X,Z]=\beta_0+\beta_1X+\beta_2Z\) is one sufficient condition, but the projection result does not require that stronger condition. The coefficients describe linear-projection relationships; without additional assumptions, they should not be read as causal effects.

The scalar omitted-variable bias formula

Let \(\widetilde{\beta}_1\) denote the population slope from the restricted projection of \(Y\) on an intercept and \(X\) alone. With an intercept, this slope is the covariance of \(X\) and \(Y\) divided by the variance of \(X\). Substituting the full projection into that expression gives the formula.

\begin{aligned} \widetilde{\beta}_1 &= \frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(X)}\\ &= \frac{\operatorname{Cov}\!\left(X,\beta_0+\beta_1X+\beta_2Z+U\right)} {\operatorname{Var}(X)}\\ &= \beta_1+\beta_2\frac{\operatorname{Cov}(X,Z)}{\operatorname{Var}(X)} +\frac{\operatorname{Cov}(X,U)}{\operatorname{Var}(X)}\\ &= \beta_1+\beta_2\frac{\operatorname{Cov}(X,Z)}{\operatorname{Var}(X)}. \end{aligned}

The constant contributes no covariance, and the last equality uses the full-model projection condition \(\operatorname{Cov}(X,U)=0\). Therefore the difference between the restricted and full slopes is

\widetilde{\beta}_1-\beta_1 = \beta_2\frac{\operatorname{Cov}(X,Z)}{\operatorname{Var}(X)}.

Because \(\operatorname{Var}(X)>0\), the sign of the difference is the sign of \(\beta_2\operatorname{Cov}(X,Z)\). Equivalently, if \(\operatorname{Corr}(X,Z)=\rho_{XZ}\) and both variables have positive standard deviations, then

\widetilde{\beta}_1-\beta_1 = \beta_2\rho_{XZ}\frac{\sigma_Z}{\sigma_X}.

Thus, the restricted slope is shifted upward when the omitted variable’s full-model coefficient and its covariance with \(X\) have the same sign. It is shifted downward when their signs differ. If either \(\beta_2=0\) or \(\operatorname{Cov}(X,Z)=0\), the two population slopes are equal. The size of the change depends on the magnitude of both factors, not merely on whether \(Z\) is an important predictor of \(Y\).

Interpretation: The formula describes how the population linear-projection slope on \(X\) changes when \(Z\) is omitted. It is not, by itself, evidence that either coefficient is a causal effect. Causal interpretation requires assumptions beyond the projection conditions stated here.

Worked calculations

Worked Example: Verifying the Formula With Hypothetical Data

Consider the following small, deliberately constructed hypothetical data set. Suppose its outcome values satisfy \(Y=10+3X+4Z+U\), with \(U\) shown in the table.

Observation	\(X\)	\(Z\)	\(U\)	\(Y\)
1	-1	-2	2	1
2	-1	0	-2	5
3	1	2	2	23
4	1	0	-2	11

The means of \(X\), \(Z\), and \(U\) are all zero. The sums of cross-products are \(\sum X_iU_i=0\) and \(\sum Z_iU_i=0\), so \(U\) is orthogonal to both regressors. Also, \(X\) and \(Z\) are not perfectly collinear. The full-model slopes are therefore \(\beta_1=3\) and \(\beta_2=4\), and the projection conditions hold for this data set.

Using divisor \(n=4\) to calculate the centered sample moments, \(\operatorname{Var}(X)=4/4=1\). The cross-product sum for \(X\) and \(Z\) is \(2+0+2+0=4\), so \(\operatorname{Cov}(X,Z)=4/4=1\). The formula predicts the restricted slope:

\widetilde{\beta}_1 =3+4\left(\frac{1}{1}\right) =7.

Directly, the mean of \(Y\) is 10, and the cross-product sum of \(X\) with \(Y-\bar{Y}\) is \(28\). Hence the restricted slope is \(\sum_i(X_i-\bar{X})(Y_i-\bar{Y})/\sum_i(X_i-\bar{X})^2=28/4=7\), agreeing with the formula. Omitting \(Z\) raises the slope from 3 to 7 outcome units per unit of \(X\) in this constructed data set.

Worked Example: Positive Outcome Coefficient, Negative Bias

Suppose a hypothetical population has full-projection coefficients \(\beta_1=2\) and \(\beta_2=6\). Let \(\operatorname{Var}(X)=9\), \(\operatorname{Cov}(X,Z)=-3\), and suppose the full-model residual is orthogonal to both regressors. These moments are feasible, for example with \(\operatorname{Var}(Z)=4\), since \(9(4)-(-3)^2=27>0\).

The omitted-variable bias and restricted slope are

\widetilde{\beta}_1-\beta_1 =6\left(\frac{-3}{9}\right)=-2, \qquad \widetilde{\beta}_1=2-2=0.

Although \(Z\) has a positive full-projection coefficient, it is negatively associated with \(X\). Observations with higher \(X\) tend, in the linear association measured by covariance, to have lower \(Z\); omitting \(Z\) consequently shifts the \(X\) slope downward. In this hypothetical population, the restricted projection has a slope of 0 outcome units per unit of \(X\), compared with 2 in the full projection. This is a statement about the specified projections, not a claim that \(X\) has no causal effect.

Worked Example: An Omitted Predictor With Zero Slope Bias

Suppose a hypothetical population has \(\beta_1=1\), \(\beta_2=5\), \(\operatorname{Var}(X)=4\), and \(\operatorname{Cov}(X,Z)=0\), with the full-model projection conditions satisfied. The omitted variable has a nonzero coefficient in the full projection, but its covariance with \(X\) is zero.

\widetilde{\beta}_1-\beta_1 =5\left(\frac{0}{4}\right)=0, \qquad \widetilde{\beta}_1=1.

Thus, in this population, omitting \(Z\) does not change the slope on \(X\). The conclusion concerns the slope formula: it does not establish that \(Z\) is irrelevant to predicting \(Y\), nor does zero covariance imply that \(X\) and \(Z\) are independent. It says that their linear covariance contributes no shift to this particular restricted slope.

What the formula does—and does not—establish

The result is useful because it separates two questions that are often blurred together. The coefficient \(\beta_2\) describes the linear-projection association between \(Z\) and \(Y\) conditional on \(X\) in the full model. The covariance \(\operatorname{Cov}(X,Z)\) describes the linear association between the included and omitted regressors. Both must contribute for the omitted-variable shift to be nonzero.

The units also provide a check. If \(Y\) is measured in outcome units, \(X\) in units of \(X\), and \(Z\) in units of \(Z\), then \(\beta_2\) has units of outcome per unit of \(Z\). The ratio \(\operatorname{Cov}(X,Z)/\operatorname{Var}(X)\) has units of \(Z\) per unit of \(X\). Their product therefore has units of outcome per unit of \(X\), matching both slopes in the formula.

There is also an exact finite-sample counterpart when both regressions use the same observations and include an intercept. Let \(b_1\) and \(b_2\) be the full OLS slopes on \(X\) and \(Z\), and let \(\widetilde{b}_1\) be the slope from regressing \(Y\) on an intercept and \(X\) alone. The full OLS residual is orthogonal to \(X\), so the same covariance calculation gives

\widetilde{b}_1 = b_1+b_2\frac{\widehat{\operatorname{Cov}}(X,Z)} {\widehat{\operatorname{Var}}(X)}.

Here the sample covariance and variance use the same divisor, which cancels in their ratio. This identity compares two fitted slopes in a particular sample; it does not say that the expected value of the restricted estimator differs from the full coefficient by a fixed amount in every random-sample design. The population formula concerns projection coefficients, while finite-sample estimator bias is an expectation over repeated samples and requires its own analysis.

Common mistake

A frequent error is to reason, “\(Z\) strongly predicts \(Y\), so leaving it out must bias the slope on \(X\).” The formula shows why that is incomplete: a large \(\beta_2\) is not enough if \(\operatorname{Cov}(X,Z)=0\). Another error is to say that a positive \(\beta_2\) always pushes the \(X\) slope upward. If \(X\) and \(Z\) have negative covariance, the product in the formula is negative, so the shift is downward. Always determine both signs and multiply them.

Finally, do not equate the word “bias” here with causal bias without qualification. The derived result is an algebraic comparison of population linear projections under the stated orthogonality conditions. A causal interpretation would require additional assumptions about how the variables are generated and how the regressors relate to unobserved causes.

Key takeaway: For a scalar omitted regressor, the restricted population slope equals the full-model slope plus the omitted variable’s full-model coefficient multiplied by its covariance with the included regressor, divided by the included regressor’s variance. The sign depends on the product of the omitted variable’s outcome coefficient and its covariance with \(X\).

Check Your Understanding

Use the formula and state your reasoning for each answer.

If \(\beta_2<0\) and \(\operatorname{Cov}(X,Z)<0\), is the restricted slope above or below \(\beta_1\)?
Suppose \(\beta_2=3\), \(\operatorname{Cov}(X,Z)=2\), and \(\operatorname{Var}(X)=4\). What is \(\widetilde{\beta}_1-\beta_1\)?
Can an omitted variable have a nonzero full-model coefficient and yet cause no change in the population slope on \(X\)? State a condition that makes this possible.
Why does the omitted-variable formula alone not establish that \(\beta_1\) is a causal effect?
What is the distinction between the population coefficient difference in the formula and finite-sample estimator bias?