Omitted-variable bias refers to the bias that occurs in statistical analyses, particularly in regression models, when a relevant variable is left out of the model. This can lead to incorrect estimates of the relationships between the included variables. When an important variable that affects both the dependent variable (the outcome) and one or more independent variables (the predictors) is omitted, it can cause the estimated coefficients of the included independent variables to be biased and inconsistent.
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If the true conditional mean is , the population slope from regressing on alone is , assuming a mean-zero error uncorrelated with the predictors. The omitted predictor can steepen, flatten, or reverse the pooled association. The same identity holds for fitted coefficients and empirical covariances when comparing nested ordinary least-squares fits. Adjustment changes the comparison being described; a causal interpretation requires additional assumptions.