Lesson M10.L01: Endogeneity and Causality

Field	Detail
Module	M10 — Instrumental Variables
Level	Graduate (ECMT5001 S1 2027)
Duration	30–35 min
Learning Objective	Identify the three sources of endogeneity; derive the probability limit of OLS under endogeneity; explain why increasing sample size cannot fix the problem
Provenance	Wooldridge, Introductory Econometrics, Ch 15 (introduction)

Explanation

Why Causality Matters

The goal of most applied econometrics is not prediction but causal inference: what is the effect of a policy, treatment, or variable $x$ on an outcome $y$? OLS recovers causal effects only under MLR.4 — zero conditional mean: $E(u \mid \mathbf{x}) = 0$. When this fails, we have endogeneity, and OLS does not estimate causal parameters even in very large samples.

Definition: Endogeneity

A regressor $x_j$ is endogenous if:

\[\text{Cov}(x_j, u) \neq 0\]

This violates MLR.4 and means OLS is biased and inconsistent — the estimates do not converge to the true causal parameter even as $n \to \infty$.

The Three Sources of Endogeneity

Source 1: Omitted Variables

The true model is:

\[y = \beta_0 + \beta_1 x + \beta_2 q + u\]

but $q$ is unobserved and omitted. The estimated model is:

\[y = \alpha_0 + \alpha_1 x + v, \qquad v = \beta_2 q + u\]

If $\text{Cov}(x, q) \neq 0$, then $\text{Cov}(x, v) \neq 0$ → $x$ is endogenous. The omitted variable bias formula (from M03) shows:

\[\text{plim}\,\hat{\alpha}_1 = \beta_1 + \beta_2 \cdot \frac{\text{Cov}(x,q)}{\text{Var}(x)}\]

Classic example: Wage regressed on education alone. Ability ($q$) is omitted. Since ability is positively correlated with education and positively affects wages, the OLS coefficient on education is upward biased.

Source 2: Simultaneity (Reverse Causality)

Both $y$ and $x$ are jointly determined by a system of equations. The causal arrow runs in both directions:

\[y = \beta_0 + \beta_1 x + u \quad \text{(demand equation)}$$ $$x = \gamma_0 + \gamma_1 y + v \quad \text{(supply equation)}\]

Classic example: Price ($x$) and quantity ($y$) in a market. Price affects demand (quantity), but quantity demanded also affects price through market clearing. Since $x$ is partly determined by $y$, $\text{Cov}(x, u) \neq 0$ — OLS on either equation gives biased estimates of structural parameters.

Another example: Returns to schooling. Do smarter people go to school longer, or does school make people smarter? Both directions may be present, creating a simultaneous relationship between education and wages.

Source 3: Measurement Error in $x$

Suppose the true model has the true regressor $x^*$ but we observe a mismeasured $x = x^* + \epsilon$ where $\epsilon$ is classical measurement error ($\text{Cov}(\epsilon, x^*) = 0$, $\text{Cov}(\epsilon, u) = 0$).

The estimated model uses $x$ instead of $x^*$:

\[y = \beta_0 + \beta_1 x + (u - \beta_1 \epsilon)\]

The composite error is $(u - \beta_1 \epsilon)$, which is correlated with $x = x^* + \epsilon$ through $\epsilon$. This creates attenuation bias:

\[\text{plim}\,\hat{\beta}_1 = \beta_1 \cdot \frac{\text{Var}(x^*)}{\text{Var}(x^*) + \text{Var}(\epsilon)} = \beta_1 \cdot \left(1 - \frac{\text{Var}(\epsilon)}{\text{Var}(x)}\right) < \beta_1 \text{ (if } \beta_1 > 0\text{)}\]

OLS is attenuated toward zero — underestimates the true effect. This is the errors-in-variables problem.

Probability Limit of OLS Under Endogeneity

For the simple regression $y = \beta_0 + \beta_1 x + u$, the probability limit of $\hat{\beta}_1^{\text{OLS}}$ is:

\[\text{plim}\,\hat{\beta}_1^{\text{OLS}} = \beta_1 + \frac{\text{Cov}(x, u)}{\text{Var}(x)}\]

The bias is:

\[\text{Inconsistency} = \frac{\text{Cov}(x, u)}{\text{Var}(x)}\]

Key properties:

If $\text{Cov}(x,u) > 0$: OLS overestimates $\beta_1$ (upward inconsistency)
If $\text{Cov}(x,u) < 0$: OLS underestimates $\beta_1$ (downward inconsistency)
If $\text{Cov}(x,u) = 0$: OLS is consistent (even if $u$ is not zero-mean, as long as the intercept absorbs it)

Classic Examples of Endogeneity

Model	Endogenous variable	Source	Direction of OLS bias
Wage = f(education)	education	Ability in $u$ (omitted)	Upward
Test scores = f(class size)	class size	School resources (omitted); richer schools have smaller classes AND better outcomes	Downward (class size coefficient is biased toward zero or negative)
Crime = f(police)	police	More crime → more police (simultaneity)	Upward (more police appear to increase crime)
Earnings = f(health)	health	Income enables better health (simultaneity)	Upward
Sales = f(price)	price	Demand and supply simultaneous	Ambiguous

Worked Example

Problem: Estimate the effect of class size on student test scores.

\[\text{score}_t = \beta_0 + \beta_1 \text{class\_size}_t + u_t\]

Why is class size endogenous?

Wealthy school districts tend to: 1. Have smaller class sizes (more per-pupil spending) 2. Have students from higher-income families → higher scores

The omitted variable "school district wealth" is positively correlated with scores and negatively correlated with class size ($\text{Cov}(\text{class\_size}, u) < 0$). The bias:

\[\text{plim}\,\hat{\beta}_1 = \beta_1 + \frac{\text{Cov}(\text{class\_size}, u)}{\text{Var}(\text{class\_size})} = \beta_1 + \text{negative number}\]

OLS will overstate the negative effect of class size (i.e., make smaller classes look more beneficial than they truly are). This is a case of negative bias when the true $\beta_1 < 0$.

Magnitude illustration:

Suppose the true causal effect of a 1-student increase in class size on scores is $\beta_1 = -0.5$ points. If $\text{Cov}(\text{class\_size}, u)/\text{Var}(\text{class\_size}) = -0.8$:

\[\text{plim}\,\hat{\beta}_1 = -0.5 + (-0.8) = -1.3\]

OLS reports $-1.3$ — class size appears 2.6× more harmful than it truly is. A policy maker who reduces class size by 10 students would predict a 13-point gain, but the true causal gain is only 5 points.

Implication: Any study that doesn't account for school wealth/resources in the error term will give misleading policy guidance.

Common Misconception

"Using a large sample fixes endogeneity — the law of large numbers takes care of it."

This is false. Consistency requires $\text{Cov}(x, u) = 0$, not just a large $n$. If $\text{Cov}(x, u) = c \neq 0$, the OLS estimator converges as $n \to \infty$ to:

\[\text{plim}\,\hat{\beta}_1 = \beta_1 + \frac{c}{\text{Var}(x)} \neq \beta_1\]

A larger sample narrows the confidence interval around a wrong estimate. With $n = 1{,}000{,}000$ and endogeneity, you get a very precise, very wrong answer. The law of large numbers ensures convergence — it just converges to the wrong limit.

Analogy: A scale that reads 2 kg too heavy is biased. Weighing yourself 10,000 times averages to "2 kg too heavy" — you don't get the correct weight, you just confirm the wrong answer with great precision.

Practice Prompts

A researcher estimates returns to schooling using OLS with $n = 50{,}000$ observations, obtaining $\hat{\beta} = 0.12$ (12% wage increase per year of school). They argue that with such a large sample, endogeneity concerns are irrelevant. Critique this argument using the probability limit formula.
Consider the model $\text{crime}_{it} = \beta_0 + \beta_1 \text{police}_{it} + u_{it}$ estimated by OLS. Explain the simultaneity problem. Is the expected OLS bias positive or negative relative to the true effect of police on crime? (Hint: what does city government do when crime rises?)
A firm measures employee productivity $y$ using self-reported hours worked $x$, which suffers from classical measurement error. If the true elasticity is $\beta_1 = 0.8$ and the reliability ratio $\text{Var}(x^*)/\text{Var}(x) = 0.7$, compute the probability limit of the OLS estimator. Is this attenuation bias economically significant?

Further Resources

Wooldridge, Introductory Econometrics (7th ed.), Ch 15 (intro and 15.1)
Angrist, J.D. & Pischke, J.-S. (2009), Mostly Harmless Econometrics, Ch 4 — Instrumental Variables
Card, D. (1999), "The Causal Effect of Education on Earnings," Handbook of Labor Economics
Angrist, J.D. & Krueger, A.B. (1991), "Does Compulsory School Attendance Affect Schooling and Earnings?" QJE

Model	Endogenous variable	Source	Direction of OLS bias
Wage = f(education)	education	Ability in \(u\) (omitted)	Upward
Test scores = f(class size)	class size	School resources (omitted); richer schools have smaller classes AND better outcomes	Downward (class size coefficient is biased toward zero or negative)
Crime = f(police)	police	More crime → more police (simultaneity)	Upward (more police appear to increase crime)
Earnings = f(health)	health	Income enables better health (simultaneity)	Upward
Sales = f(price)	price	Demand and supply simultaneous	Ambiguous