Why Do We Need Causal Inference? Three Sources of Endogeneity

Structure of This Article

This is the first article in the causal inference series, designed to help you build the core intuition of "why causal inference is needed." The article is divided into three parts:

Principles: The three sources of endogeneity and their mathematical expressions
Intuition: Building intuition using the minimum wage and employment example
Code: Stata simulation demonstrating OLS bias

Layer 1: Principles

The Core Problem of Causal Inference

The central goal of empirical research in economics is to answer causal questions: "What is the causal effect of $X$ on $Y$ ?"

Using the Potential Outcomes Framework, the treatment effect for individual $i$ is:

\tau_i = Y_i(1) - Y_i(0)

where $Y_i(1)$ and $Y_i(0)$ are the potential outcomes for the individual under treatment and no treatment, respectively. The fundamental problem of causal inference is that we can never observe both potential outcomes for the same individual simultaneously.

Three Sources of Endogeneity

When we use OLS to estimate $Y = \alpha + \beta X + \varepsilon$ , $\hat{\beta}$ being a consistent estimator of $\beta$ requires $\text{Cov}(X, \varepsilon) = 0$ . The endogeneity problem arises when $\text{Cov}(X, \varepsilon) \neq 0$ , and it has three sources:

1. Omitted Variable Bias (OVB)

If a variable $W$ that affects both $X$ and $Y$ is omitted:

\text{plim } \hat{\beta}_{OLS} = \beta + \underbrace{\frac{\text{Cov}(X, W)}{\text{Var}(X)}}_{\text{OVB}} \cdot \gamma

2. Reverse Causality

$Y$ may in turn affect $X$ . For example: higher GDP leads to more education investment, rather than education alone raising GDP.

3. Measurement Error

If $X$ suffers from classical measurement error $X^* = X + u$ , then:

\text{plim } \hat{\beta}_{OLS} = \beta \cdot \frac{\sigma_X^2}{\sigma_X^2 + \sigma_u^2} < \beta

This produces attenuation bias.

Layer 2: Intuition

Example: Does Raising the Minimum Wage Reduce Employment?

Suppose you observe that states that raised their minimum wage actually saw employment rates increase. Can you directly conclude that "raising the minimum wage promotes employment"?

No! Because:

OVB: Economically booming states are more likely to raise the minimum wage (economic conditions are omitted)
Reverse causality: Strong employment conditions → increased political pressure → raise the minimum wage
Selection bias: The labor market structures of different states are inherently different

This is precisely the problem that Card and Krueger (1994, AER) addressed using difference-in-differences (DID).

Layer 3: Stata Code

Simulation Demonstration: OLS Bias Under Endogeneity

// ═══════════════════════════════════════════════
// Demonstration: OLS bias in the presence of endogeneity
// ═══════════════════════════════════════════════
 
clear all
set seed 12345
set obs 1000
 
// True DGP
gen ability = rnormal(0, 1)           // Unobservable ability (omitted variable)
gen education = 12 + 2*ability + rnormal(0, 1)  // Education is affected by ability
gen wage = 10 + 3*education + 5*ability + rnormal(0, 2)
// True return to education = 3
 
// OLS regression (omitting ability)
reg wage education, robust
// Coefficient > 3, indicating upward bias!
 
// After adding the control variable
reg wage education ability, robust
// Coefficient ≈ 3, close to the true value

Key Takeaway: When there are omitted variables, the OLS estimator is biased and inconsistent. This is the fundamental reason we need causal inference methods such as DID, IV, and RDD.

Types of Treatment Effects

Abbreviation	Full Name	Meaning
ATE	Average Treatment Effect	Average treatment effect across all individuals
ATT	Average Treatment Effect on the Treated	Average treatment effect for the treatment group
ATU	Average Treatment Effect on the Untreated	Average treatment effect for the untreated group
LATE	Local Average Treatment Effect	Treatment effect for marginal individuals (Compliers)

\text{ATE} = E[Y(1) - Y(0)]

\text{ATT} = E[Y(1) - Y(0) \mid D = 1]

When treatment effects are heterogeneous, $\text{ATE} \neq \text{ATT} \neq \text{LATE}$ . Understanding which parameter your estimator identifies is a prerequisite for correctly interpreting empirical results.

References

Angrist, J. D., & Pischke, J. S. (2009). Mostly Harmless Econometrics. Princeton University Press.
Card, D., & Krueger, A. B. (1994). Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania. American Economic Review, 84(4), 772-793. DOI
Rubin, D. B. (1974). Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies. Journal of Educational Psychology, 66(5), 688-701.