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Omitted variable bias

Omitted variable bias distorts a regression coefficient when the model excludes a variable that both affects the outcome and correlates with an included regressor.

Also known asOVB

ByHoang TruongUpdated

FrameworkOrdinary least squares (OLS)

See it move

Loading infographic...

The infographic is a Venn diagram with two overlapping circles labelled 'Included regressor' and 'Omitted variable'. The intersection is marked 'Bias leaks here', illustrating that shared variation between the two variables causes the included regressor's estimated coefficient to absorb part of the omitted variable's effect on the outcome Y. The note gives the rule for the direction of the bias: it equals the sign of the omitted variable's effect on Y multiplied by the sign of its correlation with the included regressor.

Where it fits
SubjectData Analysis & StatisticsAdvancedTopicRegression Diagnostics & ProblemsAdvanced

The formula

LaTeX
Bias(β^1)=β2⋅Cov(X1,X2)Var(X1)\text{Bias}(\hat{\beta}_1) = \beta_2 \cdot \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_1)}

Variables

OLS estimate of the slope on the included regressor X₁
True effect of the omitted variable X₂ on the outcome
Covariance between the included regressor and the omitted variable
Variance of the included regressor X₁

If β₂ and Cov(X₁, X₂) share the same sign the bias is upward; opposite signs produce downward bias.