Deriving the Omitted-Variable Bias Formula

Why Derive the Bias Formula?

When a regression includes two predictors but a fitted model includes only one, the coefficient of the included predictor can differ from its coefficient in the two-predictor model. The difference is not mysterious: it depends on how the omitted predictor relates to the outcome and how it is associated with the included predictor. The omitted-variable bias formula makes those two ingredients explicit.

The result below is an identity between population least-squares slopes. It describes a difference between two linear summaries of the same population. It does not, by itself, establish that either slope is causal. Causal interpretation requires additional subject-matter and design assumptions.

Set Up the Two Population Projections

Let \(Y\) be an outcome, \(X\) the predictor whose coefficient is of interest, and \(Z\) a second predictor that may be omitted. Assume the variables have finite second moments and that \(\operatorname{Var}(X)>0\). Consider the population linear projection of \(Y\) on an intercept, \(X\), and \(Z\):

Here \(\beta_0,\beta_1,\beta_2\) are the population least-squares coefficients, and \(\varepsilon\) is the projection residual. The least-squares normal equations imply that the residual has mean zero and is uncorrelated with each included predictor:

These orthogonality statements are properties of the population linear projection. They do not require that the conditional mean of \(Y\) be exactly linear in \(X\) and \(Z\).

Now consider the population linear projection of \(Y\) on an intercept and \(X\) alone:

The reduced-model slope is \(\gamma_1\). The full-model slope on \(X\) is \(\beta_1\), meaning the slope in the two-predictor linear projection with \(Z\) included. We define the omitted-variable bias for this comparison as the difference \(\gamma_1-\beta_1\). This is a difference between population coefficients, not a sampling error.

Deriving the Formula

For a population linear projection with an intercept and one predictor, the slope equals the covariance of the predictor and outcome divided by the predictor’s variance. Thus \(\gamma_1=\operatorname{Cov}(X,Y)/\operatorname{Var}(X)\). We can evaluate this covariance using the full two-predictor projection.

In equations, the covariance calculation is:

The last equality follows from \(\operatorname{Cov}(X,\varepsilon)=0\), a normal-equation property of the full projection. Dividing both sides by \(\operatorname{Var}(X)\), which is positive by assumption, gives the reduced-model slope. Subtracting the full-model slope \(\beta_1\) yields the result.

The ratio \(\operatorname{Cov}(X,Z)/\operatorname{Var}(X)\) is the population slope from linearly projecting \(Z\) on \(X\). It describes how much the linear prediction of \(Z\) changes per one-unit increase in \(X\). The bias therefore combines two quantities: the change in the outcome associated with a unit of \(Z\), holding \(X\) fixed in the full projection, and the change in \(Z\) associated with a unit of \(X\).

The units provide a useful check. If \(Y\) is measured in points, \(Z\) in years, and \(X\) in hours, then \(\beta_2\) has units of points per year and \(\operatorname{Cov}(X,Z)/\operatorname{Var}(X)\) has units of years per hour. Their product has units of points per hour, the same as both slopes \(\gamma_1\) and \(\beta_1\).

Worked Applications

Interpreting the Product and Its Limits

The formula separates the sources of bias. If \(\beta_2=0\), then \(Z\) has no slope in the full linear projection after accounting for \(X\), and this formula gives zero bias. If \(\operatorname{Cov}(X,Z)=0\), the formula also gives zero bias, even when \(\beta_2\) is nonzero. Otherwise, the sign of the bias is the sign of the product of \(\beta_2\) and \(\operatorname{Cov}(X,Z)\), because \(\operatorname{Var}(X)\) is positive.

When the standard deviations are positive, the association ratio can also be written using the population correlation \(r_{XZ}\):

This form emphasizes that the association’s direction alone does not determine the numerical size of bias: the scale of \(Z\) relative to \(X\) and the size of \(\beta_2\) also matter. Changing measurement units changes the numerical coefficients and association ratio in compensating ways; the resulting bias remains on the scale of \(Y\) per unit of \(X\).

The identity applies to population linear projections with the stated comparison. In a sample, estimated coefficients vary from sample to sample, and sample estimates need not equal the population coefficients. Also, the identity does not identify a causal effect: a regression coefficient can be interpreted causally only with appropriate assumptions beyond this algebraic result.

Common Mistake

A common mistake is to say, “The omitted predictor affects the outcome, so omitting it must bias the slope on \(X\).” That reasoning overlooks the association factor. A nonzero \(\beta_2\) is not sufficient: if \(X\) and \(Z\) have zero covariance in the population, then \(\beta_2\operatorname{Cov}(X,Z)/\operatorname{Var}(X)=0\). The reverse shortcut is also wrong: an omitted predictor associated with \(X\) does not create bias under this formula if \(\beta_2=0\). Check both factors and their units before interpreting the product.