Why Endogeneity Is a Property of an Equation
A regressor is not inherently endogenous or exogenous. The classification depends on the structural equation being studied and on the error defined relative to that equation. For example, a variable may be determined jointly with an outcome in a market model, while appearing as a regressor in the demand equation. The demand error can then depend on that regressor, even though the equation is a useful description of demand.
This distinction matters because a structural coefficient describes a specified relationship, not merely the slope of an arbitrary association. If the error in that equation varies systematically with a regressor, ordinary least squares generally does not isolate the structural coefficient. To see how this happens, it helps to express endogeneity as a condition on the error and then trace that condition to its economic or measurement source.
The condition \(E[U\mid X]=0\) says that, at every relevant value of \(X\), the average structural error is zero. It is stronger than the unconditional restriction \(E[U]=0\), which only says that positive and negative errors average to zero overall. It is also useful to distinguish conditional-mean endogeneity from every possible form of statistical dependence. A conditional distribution can change with \(X\) even when its conditional mean remains zero.
Conditional Means, Covariance, and the OLS Slope
For a scalar regressor with finite second moments and \(\operatorname{Var}(X)>0\), the population OLS slope from projecting \(Y\) on an intercept and \(X\) is
The second equality follows by substituting \(Y=\beta_0+\beta_1X+U\) into the covariance. Thus, when \(\operatorname{Cov}(X,U)\ne 0\), the population OLS slope differs from the structural coefficient by \(\operatorname{Cov}(X,U)/\operatorname{Var}(X)\). This is a statement about the population projection; an estimate from a finite sample will also vary from sample to sample.
Conditional-mean exogeneity implies zero covariance, provided the relevant expectations exist: \(E[XU]=E[XE(U\mid X)]=0\) and \(E[U]=0\), so \(\operatorname{Cov}(X,U)=0\). The converse does not generally hold. For instance, if \(X\) is standard normal and \(U=X^2-1\), then \(E[U\mid X]=X^2-1\), which is not identically zero, while \(\operatorname{Cov}(X,U)=E[X^3-X]=0\). Therefore, a zero covariance rules out linear covariance bias in the scalar population slope, but by itself does not establish conditional-mean exogeneity.
Omitted Variables: A Structural Error That Absorbs a Cause
Suppose the outcome depends on both a regressor of interest and another variable that is left out of the fitted equation. The omitted variable becomes part of the error in the restricted equation. If it is related to the included regressor, the restricted error will generally depend on that regressor. This is endogeneity relative to the restricted equation, even if the error in the fuller structural equation is conditionally mean zero.
Worked Example: An Omitted Variable Related to the Regressor
Consider the hypothetical structural equation \(Y=3+2X+2Z+\varepsilon\). The researcher wants the coefficient on \(X\) but omits \(Z\), fitting an equation with an intercept and \(X\) only. Suppose the hypothetical data-generating process has \(\varepsilon=0\) and these three observations:
| \(X\) | \(Z\) | \(Y=3+2X+2Z\) |
|---|---|---|
| 0 | 1 | 5 |
| 1 | 1.5 | 8 |
| 2 | 2 | 11 |
Check the structural assumptions. In this example, \(Z=1+0.5X\) exactly and \(\varepsilon=0\), so the full structural equation is satisfied by all three rows. The restricted equation is \(Y=3+2X+U\), where \(U=2Z+\varepsilon=2Z\). Thus \(U\) is 2, 3, and 4 as \(X\) is 0, 1, and 2: the restricted error plainly depends on \(X\).
Compute the restricted slope. The means are \(\bar{x}=1\) and \(\bar{y}=8\). The sample covariance numerator is \(\sum_i(x_i-\bar{x})(y_i-\bar{y})=3+0+3=6\), and the sum of squared deviations in \(X\) is \(1+0+1=2\). Therefore the fitted slope is \(6/2=3\), rather than the structural coefficient 2.
Trace the difference. The restricted error has mean 3, and its centered values are \(-1,0,1\). Its covariance with \(X\), using divisor 3, is \([(-1)(-1)+0+(1)(1)]/3=2/3\). The variance of \(X\) is \([1+0+1]/3=2/3\), so the population-slope difference is \((2/3)/(2/3)=1\). The omitted-variable relationship contributes \(2(0.5)=1\) to the slope, giving \(2+1=3\). This calculation illustrates how omitting a relevant, related variable can make the restricted structural error endogenous.
Simultaneity: The Regressor Responds to the Structural Shock
In a simultaneous system, two or more variables are determined together. A regressor in one structural equation may respond to that equation’s error because the error shifts the equilibrium that determines the regressor. Solving the system for the jointly determined variable gives a reduced form: an equation expressing that variable in terms of the system’s shocks and predetermined quantities. The reduced form makes the dependence visible.
Worked Example: Price Is Endogenous in a Demand Equation
Consider a hypothetical market with demand \(Q=10-2P+U_d\) and supply \(Q=1+P+U_s\). Here \(U_d\) is the demand shock and \(U_s\) is the supply shock. Assume the market clears, both shocks have mean zero, they are independent normal random variables, and their variances are 4 and 1, respectively. These assumptions define a simultaneous equilibrium model; no claim is being made that they describe an actual market.
Derive the reduced form for price. Equating quantity demanded and supplied gives \(10-2P+U_d=1+P+U_s\). Rearranging yields \(3P=9+U_d-U_s\), so \(P=3+(U_d-U_s)/3\). In particular, a demand shock changes the equilibrium price.
Check dependence in the demand equation. From the reduced form, \(\operatorname{Cov}(P,U_d)=\operatorname{Var}(U_d)/3=4/3\), because the shocks are independent. Also, \(\operatorname{Var}(P)=[\operatorname{Var}(U_d)+\operatorname{Var}(U_s)]/9=5/9\). The covariance is nonzero, so price is correlated with the demand error.
Because the shocks are jointly normal, \(P\) and \(U_d\) are jointly normal, and the conditional mean of \(U_d\) given \(P\) is linear:
This conditional mean is not identically zero. Thus price is conditionally mean endogenous in the demand equation. The source is simultaneity: the demand shock shifts demand, and market clearing transmits part of that shock into equilibrium price. A regression of quantity on price in this equation cannot treat the demand shock as unrelated to price.
Measurement Error: The Observed Regressor Changes the Error
Measurement error creates a different route to endogeneity. The structural equation may use a true quantity \(X^*\), while the researcher observes \(W=X^*+V\), where \(V\) is measurement error. Replacing \(X^*\) with \(W\) does not simply preserve the original error: rearranging the equation shows that the observed-regressor error includes a term involving \(V\). Since \(W\) also includes \(V\), dependence can result.
Worked Example: Classical Measurement Error and Attenuation
Suppose the hypothetical structural equation is \(Y=1+2X^*+\varepsilon\), but only \(W=X^*+V\) is observed. Assume \(X^*\), \(V\), and \(\varepsilon\) are mutually independent, have mean zero, and have variances 4, 1, and 1, respectively. For the conditional-mean calculation below, also assume they are jointly normal. These are classical measurement-error assumptions: in particular, the measurement error \(V\) has mean zero and is independent of the true regressor and the structural disturbance.
Find the error in the equation using the observed regressor. Since \(X^*=W-V\), substitution gives \(Y=1+2W+U_{\mathrm{obs}}\), where \(U_{\mathrm{obs}}=\varepsilon-2V\). The observed regressor is \(W=X^*+V\). Therefore, \(\operatorname{Cov}(W,U_{\mathrm{obs}})=\operatorname{Cov}(X^*+V,\varepsilon-2V)=-2\operatorname{Var}(V)=-2\). The observed-regressor error is not uncorrelated with \(W\).
Compute the population OLS slope. The covariance of \(W\) and \(Y\) is \(\operatorname{Cov}(X^*+V,1+2X^*+\varepsilon)=2\operatorname{Var}(X^*)=8\), using the assumed independence. The variance of \(W\) is \(4+1=5\). Thus the population slope from regressing \(Y\) on \(W\) and an intercept is \(8/5=1.6\), below the true coefficient 2.
The same result follows from the error covariance: the slope is \(2+\operatorname{Cov}(W,U_{\mathrm{obs}})/\operatorname{Var}(W)=2-2/5=1.6\). Under the joint-normality assumption, \(E[U_{\mathrm{obs}}\mid W]=(-2/5)W\), which is not identically zero. Measurement error therefore makes the observed regressor endogenous in the equation written with the true structural coefficient. The reduction from 2 to 1.6 is the attenuation effect in this particular classical, single-regressor setting; it should not be generalized to every measurement-error design.
Comparing the Three Sources
| Source | Structural feature | Why the regressor and error depend |
|---|---|---|
| Omitted variable | A relevant variable is excluded from the fitted equation. | The excluded variable enters the restricted error and is related to an included regressor. |
| Simultaneity | Variables are jointly determined by a system of equations. | A structural shock shifts equilibrium, which changes a regressor in that equation. |
| Measurement error | The equation uses a true regressor but the data contain a noisy version. | Rewriting the model with the observed regressor places measurement error in the equation error as well. |
These mechanisms are conceptually distinct, although more than one can occur in the same application. In each case, the diagnostic task is to name the structural equation, define its error, and determine whether the regressor is related to that error. The same variable may be exogenous in one equation and endogenous in another; the label belongs to the variable as it appears in a specified equation.
Common Mistake: Treating Zero Covariance as the Whole Condition
A common error is to argue, “The regressor and error have zero covariance, so the regressor is exogenous.” Zero covariance is weaker than conditional-mean exogeneity: it rules out a particular linear relationship but does not establish that the conditional average error is zero at every regressor value. Conversely, independence without a zero-mean error does not establish conditional-mean exogeneity either. State the condition actually needed for the model, and keep it distinct from the simpler covariance check.
Another frequent mistake is to define the error only after seeing the fitted regression and then treat its relationship with the regressor as a structural fact. The error must be defined relative to the structural equation and its specified coefficient. If a relevant variable is omitted or a true regressor is replaced by a noisy measure, the error in the altered equation changes; that change is precisely what can create endogeneity.
Check Your Understanding
Answer each question by identifying the equation and error condition involved.
- In \(Y=\beta_0+\beta_1X+U\), what does conditional-mean exogeneity require, and how does it differ from the restriction \(E[U]=0\)?
- Why does independence between \(U\) and \(X\) imply conditional-mean exogeneity when \(E[U]=0\), but not necessarily when \(E[U]\ne 0\)?
- In a restricted equation that omits \(Z\) from \(Y=\beta_0+\beta_1X+\gamma Z+\varepsilon\), write the restricted error and state when it may depend on \(X\).
- For the demand-and-supply system in the worked example, derive the reduced form for price and explain why price is related to the demand shock.
- If the true regressor is \(X^*\) but the observed regressor is \(W=X^*+V\), what term involving measurement error appears in the equation error when the model is rewritten using \(W\)?