Endogeneity

Occurs when a model excludes a variable that influences both the dependent variable and one or more included independent variables. The excluded variable's effect is improperly absorbed by the included regressor and the error term, creating a correlation between them.

• Mechanism: The omitted variable acts as a common cause, inducing a spurious association.
• Direction of Bias: Determined by the product of the covariance between the included and omitted variable and the effect of the omitted variable on the outcome.
• Example: Regressing wage on education level without controlling for innate ability. Ability affects both education choice and wage, biasing the estimated return to schooling upward.

Arises when the dependent variable and at least one independent variable are jointly determined in equilibrium. The regressor is not just causing the outcome; the outcome is also causing the regressor, creating a feedback loop.

• Mechanism: The structural equation is one part of a system of simultaneous equations, making the regressor endogenous by construction.
• Classic Example: Estimating a demand curve by regressing quantity on price. Observed price-quantity pairs are equilibrium intersections of supply and demand curves; price is correlated with supply-side shocks in the error term.
• Solution: Requires Instrumental Variables (IV) or structural modeling to isolate exogenous variation.

Occurs when an independent variable is measured with noise. The observed proxy equals the true value plus a random error. This classical measurement error attenuates the coefficient estimate toward zero and introduces correlation between the regressor and the composite error term.

• Mechanism: The mismeasured regressor is correlated with the measurement error component embedded in the regression disturbance.
• Consequence: Even if the true model is perfectly specified, OLS becomes inconsistent. This is often called attenuation bias.
• Example: Using self-reported income in a consumption function. Reporting errors in income correlate with the regression residual, biasing the marginal propensity to consume downward.

A form of endogeneity caused by analyzing a non-randomly selected sample of the population. If selection into the sample is correlated with the outcome variable through unobserved factors, the error term in the outcome equation becomes correlated with the regressors.

• Mechanism: The selection process truncates the error distribution conditionally on the covariates, violating the zero-conditional-mean assumption.
• Heckman Correction: A two-step procedure that models the selection probability using a Probit model and includes the resulting Inverse Mills Ratio as a control in the outcome equation.
• Example: Studying wage determinants using only data on employed individuals. The decision to work is endogenous and correlated with unobserved wage potential.

Specific to panel data models where a lagged dependent variable is included as a regressor. The lagged outcome is mechanically correlated with the individual-specific fixed effects in the error term, rendering standard estimators biased.

• Mechanism: The fixed effect is a time-invariant component of the composite error. Since the lagged dependent variable contains this fixed effect, strict exogeneity fails.
• Arellano-Bond Estimator: A Generalized Method of Moments (GMM) approach that uses deeper lags of the dependent variable and differences of exogenous variables as instruments.
• Example: Modeling firm investment with last year's investment as a predictor. The firm's unobserved, persistent management quality affects both past and current investment.