12. Endogeneity and Threats to Identification
Confounding, measurement error, simultaneity and sample selection
Endogenous regressors
For the linear orthogonality condition, endogeneity means E[Xu]≠0 for a regressor and the structural disturbance. A failure of E[u|X]=0 is a broader failure of conditional mean exogeneity: it need not imply E[Xu]≠0, because a nonlinear conditional relationship can still have zero covariance. The central problem is interpretive: OLS combines the structural effect with sources of variation in the explanatory variable caused by unobserved factors. Endogeneity can arise from omitted variables, reverse causality, simultaneous determination, measurement errors, sample selection or inappropriate conditioning choices.
Consider a labour-market equation describing the causal return to a year of schooling. Ability, parental resources and networks may influence wages and the decision to acquire schooling. Omission of these factors can make schooling endogenous. Similarly, a higher market price might correlate with demand shocks because demand and supply jointly determine equilibrium price and quantity; a regression of quantity on price cannot directly isolate a demand curve.
A general omitted-variable bias interpretation
The simple omitted-variable formula provides an initial diagnostic. Suppose Y=β0+β1X+β2Z+u, Cov(X,u)=0 and Var(X)>0. A population regression of Y on an intercept and X that omits Z has slope β1+β2 Cov(X,Z)/Var(X). The issue does not disappear when analysts add arbitrarily many variables; the chosen controls must include the right confounders and should generally be measured before treatment when the target is a total causal effect.
Bad controls deserve special attention. A post-treatment mediator can absorb part of the causal effect one hopes to estimate. Conditioning on a variable influenced by both the treatment and unobserved determinants of the outcome can open a spurious statistical path. Sound regression specification requires a theory of the timing and causal relationships among variables—not only an algorithm that maximises R2.
Measurement error
If a dependent variable is measured with classical independent noise, the outcome becomes noisier and OLS coefficients often remain consistent under standard assumptions, although inference becomes less precise. If the observed regressor X equals the true value X* plus classical additive measurement error, X=X*+w, where w has mean zero and is independent of X* and the structural disturbance u, OLS of Y on X in a simple model has probability limit
The term is the reliability ratio, between zero and one under these assumptions, producing attenuation toward zero. More complex measurement error—correlated with true values, other regressors or the outcome—can bias coefficients in any direction. A generic assertion that measurement error always attenuates estimates is incorrect, especially in multivariate regression.
Simultaneous equations and equilibrium
Imagine linear market demand and supply:
Observed transactions satisfy Qᵈ=Qˢ, so equilibrium price responds to both demand and supply shocks. A regression of Q on P across equilibria need not recover either a1 or b1. To identify the demand slope, the researcher seeks a source of price variation that shifts supply without directly shifting demand (for example a plausibly exogenous supply cost shifter), subject to economic scrutiny of its exclusion restriction.
Missing data, selection and external validity
A sample may exclude people who do not respond, firms that exit the market, workers who do not participate in employment, or households that decline to disclose wealth. If sample inclusion is related to unobserved determinants of Y after conditioning on X, the observed regression may be unrepresentative. A cleanly estimated population effect for the sampled group may also fail to transport to an economically distinct target population.
Sample selection models can sometimes recover a target under distributional or exclusion assumptions, but there is no universal correction. Diagnose who was included, why and at which stage, and whether variables determining inclusion are recorded.
Internal and external validity
Internal validity concerns whether the empirical comparison identifies the target effect in the studied population. Omitted confounding, simultaneity and measurement error threaten it. External validity concerns whether an internally valid estimate generalises across populations, institutions, times or treatment intensities. An effect of smaller school classes in one region may differ in another region with different teacher supply or baseline class sizes.
A credible study explicitly distinguishes internal and external validity, and explains the source of identification, rather than assuming a statistically significant coefficient settles both.
Sensitivity analysis
Useful robustness checks include changes in functional form, economically motivated control sets, subsample analyses, influential-observation diagnostics, alternative measures, placebo outcomes and examination of likely violations of the identifying assumption. Robustness to several arbitrary specifications is not a substitute for a credible source of exogenous variation. It is evidence about specification sensitivity, not a formal guarantee of causal identification.
Endogeneity, bad controls and collider bias
A collider is a variable caused by two or more variables that may otherwise be unrelated. Conditioning on a collider can create statistical dependence between its causes. For example, among applicants admitted to a highly selective scholarship programme, motivation and family connections could be negatively associated even if they were independent in the applicant population, because either quality may increase admission probability. A regression restricted to selected recipients can therefore generate relationships not present in the full population.
The distinction between confounders, mediators and colliders is conceptually important. A confounder is a common cause of treatment and outcome; adjusting for it can reduce bias. A mediator is partly caused by treatment and then affects outcome; adjusting for it changes the causal target. A collider is a common consequence of other variables; conditioning on it can induce bias. Temporal ordering and economic institutions should guide the identification of these categories rather than post-hoc correlations.
How to construct a sensitivity argument
Suppose an OLS education coefficient decreases from 0.10 to 0.07 as parental background and test scores are included. This change suggests measured factors help explain the original association, but it does not directly quantify the residual confounding due to unmeasured ability. A sensitivity argument can ask how strongly an unobserved variable would need to relate to both education and wages to explain the remaining 0.07, compared with the strengths of observed controls. Such comparisons must disclose assumptions and should never be treated as proof of exogeneity.