OmittedVariableBias

Omitted variable bias is a bias that arises in statistical estimation when a model leaves out one or more relevant variables that affect the dependent variable and are correlated with the included regressors. This can lead to biased and inconsistent estimates of the coefficients for the included variables, making it difficult to draw valid conclusions about causal relationships.

In the context of linear regression, suppose the true model is Y = β0 + β1 X1 + β2 X2 + ε. If X2 is omitted and is correlated with X1 (Cov(X1, X2) ≠ 0) and X2 also affects Y (β2 ≠ 0), the OLS estimate of β1 is biased. In the simple case, the asymptotic bias in β1_hat can be expressed as β2 times the influence of X2 on X1, often written as β2 × Cov(X1, X2) / Var(X1). More generally, omitting relevant variables that influence Y induces bias in the estimated effects of the included variables.

OVB commonly arises in observational studies where random assignment is not present. Examples include estimating the effect of education on earnings without accounting for ability, or assessing class size effects without controlling for teacher quality or family background. The bias can lead to overstated or understated conclusions about causal effects.

Remedies focus on improving model specification and identification strategies. These include adding relevant covariates, using fixed effects or panel data to control for unobserved heterogeneity, applying instrumental variables, employing natural experiments, randomized experiments, or propensity score methods. It is important to recognize that multicollinearity and model misspecification can complicate interpretation, but they are distinct from omitted variable bias.