Introduction
A regression slope can summarize how an outcome varies with a regressor without representing the parameter of interest in an economic or causal model. The difference matters when the regressor is related to unobserved determinants of the outcome. In an instrumental-variables (IV) model, the starting point is therefore not a particular estimator but a structural equation: an equation that states how the outcome is generated and names the parameter the analysis seeks to recover.
This tutorial focuses on a single regressor. It defines endogeneity as a relationship between that regressor and the structural error, shows algebraically why this relationship invalidates the usual OLS interpretation, and clarifies the target of IV estimation. Whether a proposed instrument can identify that target depends on additional conditions considered separately.
The Linear Structural Equation
The word structural signals that the equation specifies a relationship in the model, not merely a convenient summary of the observed data. In an applied setting, the error \(u_i\) may include omitted variables, unobserved circumstances, or other influences on the outcome. It is not the same object as an OLS residual. The error is part of the population model; a residual is calculated from a fitted sample regression.
A coefficient’s interpretation depends on the variables and their units. If \(y_i\) is measured in dollars and \(x_i\) in hours, then \(\beta_1\) is measured in dollars per hour. The model alone does not make \(\beta_1\) a causal effect. That interpretation requires an appropriate structural interpretation and conditions that support recovering the parameter from the observed data.
For the OLS comparison below, assume observations are independently and identically distributed, the relevant second moments are finite, and \(\operatorname{Var}(x_i)>0\). These conditions allow sample averages to converge to their population counterparts and ensure that the regressor varies. They do not, by themselves, guarantee that OLS estimates \(\beta_1\).
Worked Example: Interpreting a Structural Slope in Context
Suppose a hypothetical model relates a patient’s systolic blood pressure \(y_i\), measured in mmHg, to a medication dose \(x_i\), measured in mg: $$ y_i = \beta_0 + \beta_1 x_i + u_i. $$ Assume the structural parameter is \(\beta_1=-0.8\) mmHg per mg. The model’s structural comparison for a 5 mg difference in dose, holding the other determinants represented by \(u_i\) fixed, is $$ (-0.8\ \text{mmHg per mg})(5\ \text{mg})=-4\ \text{mmHg}. $$ Thus the model associates that dose difference with a 4 mmHg lower outcome, with the units cancelling correctly. This describes the structural slope, not necessarily the slope OLS would estimate from observed patients. For example, if patients with more severe underlying illness are more likely to receive higher doses and illness also affects blood pressure, dose can be related to \(u_i\). The calculation does not remove that concern or establish that the structural parameter is identified.
Endogeneity and the OLS Problem
Endogeneity can arise in several ways. An omitted determinant of \(y_i\) may also be related to \(x_i\); the outcome and regressor may be jointly determined; or \(x_i\) may be measured with error. In each case, the structural equation can remain a useful description, but the regressor may carry information about the structural error. The error then does not behave like unrelated noise from the perspective of OLS.
With an intercept in the regression, the OLS slope has the exact sample representation $$ \hat{\beta}_1^{\mathrm{OLS}} = \beta_1+ \frac{\sum_{i=1}^{n}(x_i-\bar{x})(u_i-\bar{u})} {\sum_{i=1}^{n}(x_i-\bar{x})^2}, $$ where \(\bar{x}\) and \(\bar{u}\) are the sample means of \(x_i\) and \(u_i\). The numerator records how the regressor and structural error move together in the sample. If their population covariance is nonzero, this term does not generally vanish as the sample grows.
Why Endogeneity Shifts the OLS Slope
Under the sampling and moment conditions stated above, the law of large numbers gives convergence of the sample covariance and variance terms to their population counterparts. Therefore the OLS slope converges to the following probability limit:
This result follows directly from substituting the structural equation into the OLS slope. After centering \(y_i\) and \(x_i\), the intercept drops out, so the sample slope is \(\sum_i(x_i-\bar{x})(y_i-\bar{y})/\sum_i(x_i-\bar{x})^2\). Since \(y_i=\beta_0+\beta_1x_i+u_i\), the centered outcome is \(\beta_1(x_i-\bar{x})+(u_i-\bar{u})\). The numerator consequently separates into \(\beta_1\sum_i(x_i-\bar{x})^2\) plus \(\sum_i(x_i-\bar{x})(u_i-\bar{u})\). Dividing by the denominator gives the exact sample representation above. Under the stated conditions, the ratio of the covariance term to the variance term converges to \(\operatorname{Cov}(x_i,u_i)/\operatorname{Var}(x_i)\), yielding the probability limit.
If \(\operatorname{Cov}(x_i,u_i)=0\), the probability limit equals \(\beta_1\), so the OLS slope is consistent for the structural slope under these conditions. If the covariance is nonzero, the probability limit differs from \(\beta_1\), and OLS is inconsistent for that parameter. The sign and size of the discrepancy depend on the covariance relative to the variation in \(x_i\). This is a statement about the large-sample target; it does not say that every finite-sample OLS estimate must lie on the same side of \(\beta_1\).
The bias term is not generally a fixed number that can be inferred from the OLS output alone. It depends on the relationship between \(x_i\) and the unobserved structural error. A precisely estimated OLS association can therefore be precise about the wrong target if the intended target is \(\beta_1\) and endogeneity is present.
Worked Example: OLS on a Hypothetical Endogenous Sample
Consider four hypothetical observations generated by \(y_i=10+3x_i+u_i\). The values are:
| \(i\) | \(x_i\) | \(u_i\) | \(y_i=10+3x_i+u_i\) |
|---|---|---|---|
| 1 | 1 | 2 | 15 |
| 2 | 2 | 3 | 19 |
| 3 | 3 | 4 | 23 |
| 4 | 4 | 5 | 27 |
The structural slope is \(\beta_1=3\) outcome units per unit of \(x\). The sample means are \(\bar{x}=2.5\) and \(\bar{y}=21\). The OLS slope is $$ \hat{\beta}_1^{\mathrm{OLS}} = \frac{\sum_i(x_i-\bar{x})(y_i-\bar{y})} {\sum_i(x_i-\bar{x})^2} = \frac{(-1.5)(-6)+(-0.5)(-2)+(0.5)(2)+(1.5)(6)} {(-1.5)^2+(-0.5)^2+(0.5)^2+(1.5)^2} = \frac{20}{5}=4. $$
The sample covariance numerator involving the structural error is $$ \sum_i(x_i-\bar{x})(u_i-\bar{u}) = (-1.5)(-1.5)+(-0.5)(-0.5)+(0.5)(0.5)+(1.5)(1.5) =5. $$ Thus the exact OLS decomposition gives \(\hat{\beta}_1^{\mathrm{OLS}}=3+5/5=4\). In this hypothetical sample the OLS slope exceeds the structural slope by 1. The example illustrates the algebra; four observations do not establish a population probability limit.
Worked Example: A Population Endogeneity Calculation
Suppose a hypothetical population has centered variables \(z_i\), \(e_i\), and \(\epsilon_i\), mutually independent, with variances \(1\), \(3\), and \(2\), respectively. Let $$ x_i=z_i+e_i,\qquad u_i=2z_i+\epsilon_i,\qquad y_i=5-1.5x_i+u_i. $$ Here \(z_i\) could represent an unobserved factor affecting both the regressor and outcome. By independence, \(\operatorname{Cov}(e_i,z_i)=\operatorname{Cov}(e_i,\epsilon_i)=\operatorname{Cov}(z_i,\epsilon_i)=0\). Therefore $$ \operatorname{Var}(x_i)=\operatorname{Var}(z_i)+\operatorname{Var}(e_i)=1+3=4 $$ and $$ \operatorname{Cov}(x_i,u_i) = \operatorname{Cov}(z_i+e_i,2z_i+\epsilon_i) = 2\operatorname{Var}(z_i)=2. $$ The OLS probability limit is consequently $$ \operatorname{plim}\hat{\beta}_1^{\mathrm{OLS}} = -1.5+\frac{2}{4} = -1.0. $$
In this example, the structural slope is \(-1.5\) outcome units per unit of \(x\), but the large-sample OLS target is \(-1.0\). The discrepancy arises because the shared component \(z_i\) makes \(x_i\) positively correlated with \(u_i\). A larger sample does not make that systematic component disappear. Under suitable identifying assumptions, IV estimation seeks the structural slope \(-1.5\), rather than the OLS probability limit \(-1.0\).
What IV Estimation Targets
The purpose of IV estimation is to recover structural parameters such as \(\beta_1\) when the regressor is endogenous, provided the chosen instrument and model satisfy the conditions needed for identification and estimation. An instrument is a variable used to isolate informative variation in an endogenous regressor. Merely having a variable called an instrument does not guarantee that it is suitable: its relevance and its relationship to the structural error must be assessed before an IV estimator can be justified.
The target is not automatically the OLS slope, nor is it simply any alternative association between \(x_i\) and \(y_i\). It is the parameter in the stated structural equation. Whether the data and a proposed instrument can recover that parameter is a separate identification question. The present model defines the target; it does not, by itself, establish that the target is identified.
This distinction is useful in applied work. Before comparing estimates, state which structural parameter is the target and why OLS may fail to recover it. Then assess the additional assumptions required by the estimator being considered. An IV estimate is not a correction by definition: its interpretation depends on whether those assumptions are credible in the application.
Common Mistakes
- Calling any OLS residual the structural error. The residual is calculated after fitting a sample regression; \(u_i\) is the unobserved disturbance in the structural model. They are related in an analysis but are not the same object.
- Assuming a large sample fixes endogeneity. More observations can reduce sampling uncertainty, but when \(\operatorname{Cov}(x_i,u_i)\ne0\), OLS converges to \(\beta_1+\operatorname{Cov}(x_i,u_i)/\operatorname{Var}(x_i)\), not to \(\beta_1\).
- Equating correlation with a particular cause. A nonzero covariance signals endogeneity in this model, but it does not identify whether the source is omitted variables, joint determination, measurement error, or something else.
- Assuming IV always recovers the structural parameter. The wrong reasoning is that using a different estimator is enough to solve endogeneity. The correction is that IV requires suitable identifying assumptions; the structural equation alone does not establish them.
- Interpreting the structural slope as automatically causal. A structural equation names a model parameter, but causal interpretation requires a defensible model and assumptions. The equation’s notation alone cannot supply that justification.
Check Your Understanding
Use the structural equation and OLS probability limit to answer these questions.
- In \(y_i=\beta_0+\beta_1x_i+u_i\), identify the structural slope, the structural error, and the OLS residual.
- In words, what does \(\operatorname{Cov}(x_i,u_i)\ne0\) mean in this model?
- If \(\beta_1=2\), \(\operatorname{Cov}(x_i,u_i)=-3\), and \(\operatorname{Var}(x_i)=6\), what is the probability limit of the OLS slope?
- Does increasing the sample size remove the asymptotic discrepancy when the regressor and structural error have nonzero covariance? Explain.
- What parameter does IV seek to recover in this model, and why does that goal not by itself establish that a proposed instrument is valid?