Chapter 33 of 57

Exogeneity vs Endogeneity (and What "Strict" Adds)

Created Jul 5, 2026 Updated Oct 25, 2026

Strip the jargon and there's a single yes/no question underneath both words. In the model y = β·X + ε, does the regressor X move together with the error ε — everything that shapes y but isn't in the equation?

No → X is exogenous. $\mathrm{Cov}(X,\varepsilon)=0$ . The variation in X is clean, and OLS consistently estimates the β the equation claims — which is the causal effect when that equation is a correctly specified structural model.
Yes → X is endogenous. X carries some of ε's variation, OLS blends the two, and the slope is biased.

That's the whole distinction. Exogeneity is the good case, the one every "control for confounders" regression silently hopes it's in.

Where "strict" comes in. In time-series and panel data you can ask the question at different distances in time. Here the sharper conditional-mean form is worth using, because it is what the estimators actually assume — zero covariance is its weaker cousin:

Flavour	What it requires	Reading
Contemporaneous	$E[\varepsilon_t \mid X_t] = 0$	the regressor is clean right now
Predetermined	$E[\varepsilon_t \mid X_t, X_{t-1}, \ldots] = 0$	past shocks may have moved today's `X`, but today's shock did not move today's or any earlier `X`
Strict	$E[\varepsilon_t \mid X_1, \ldots, X_T] = 0$	the regressor is uncorrelated with the shocks of every period, past and future

Strict is the strong one, and the useful way to see what it costs is the lagged outcome. Put y from last period on the right-hand side: $X_t = y_{t-1}$ contains $\varepsilon_{t-1}$ , so the regressor is correlated with a past shock even though it is perfectly uncorrelated with today's. It is predetermined, and it is not strictly exogenous — which is exactly why fixed-effects estimators, which lean on strict exogeneity, break on dynamic panels.

The trap: exogeneity is an assumption, not a fact you can read off the data. There's no residual plot that confirms it — because OLS makes its own residuals uncorrelated with X by construction, no matter how endogenous X truly is.

Hidden-confounder DGP: y = β·X + ε with the structural error ε = ω·u + η, where u is the unobserved confounder that also drives X and η is independent noise. Slide ω up to crank endogeneity. The green view ("OLS residual vs X") is what diagnostics actually plot — and its linear slope is zero for every value of ω, because OLS makes Cov(X, ε̂) = 0 by construction. (Flat in that one sense only: a residual plot can still reveal non-linearity or changing spread. What it cannot reveal is this.) Flip to peek at the structural error and the orange view shows ε itself, which you could never compute: it tilts visibly with X.

So you cannot establish exogeneity from OLS diagnostics: you argue it, from where the variation in X comes from. Random assignment, a physical cutoff, a cost shifter that can't touch y directly — these are claims about the world, defended with domain knowledge. Tests do exist in the other direction: given a valid instrument, a control-function regression gives evidence against exogeneity. Nothing gives evidence for it.

The failure side of this coin — every route by which X and ε get entangled — is endogeneity. The full treatment lives in the Endogeneity note, and the standard repair when exogeneity fails is instrumental variables.