Endogeneity

Definition

Endogeneity is the problem that arises when an explanatory variable is correlated with the error term.

$Y = \beta_0 + \beta_1 X + u \quad \text{where} \quad Cov(X, u) \neq 0$

In this case the OLS estimator $\hat{\beta}_1$ is biased and inconsistent.

Intuitive Understanding

In estimating price elasticity, endogeneity is the most fundamental challenge.

Firms do not set prices at random. They raise prices when they expect demand to be high and lower them when they expect it to be low. This optimizing behavior creates a spurious correlation between price and unobserved demand factors.

Key Properties

Sources of Endogeneity

Source	Description	Example
Simultaneity	Price and quantity are determined simultaneously	Market equilibrium
Omitted variables	Omission of a variable that affects both price and demand	Quality, brand
Reverse causality	Demand affects price	Demand-forecast-based pricing
Measurement error	Measurement error in the explanatory variable	Missing records of price discounts

Direction of the Bias

Price endogeneity almost always leads to underestimation of elasticity (a positive bias).

$\hat{\beta}_{OLS} = \beta + \frac{Cov(X, u)}{Var(X)}$

Since higher quality → higher price and higher demand:

• $Cov(\text{price}, \text{quality}) > 0$
• $Cov(\text{quality}, \text{demand}) > 0$

Result: although the true elasticity is negative, the estimate can approach 0 or even become positive.

Example

Simulation

import numpy as np
from sklearn.linear_model import LinearRegression

np.random.seed(42)
n = 5000

# Unobserved quality (confounder)
quality = np.random.randn(n)

# Price: depends on quality (endogenous!)
true_price_effect = -2.0
price = 20 + 3 * quality + np.random.randn(n) * 2

# Demand: depends on both price and quality
demand = 100 + true_price_effect * price + 10 * quality + np.random.randn(n) * 5

# Naive OLS (quality uncontrolled) - biased!
naive_model = LinearRegression()
naive_model.fit(price.reshape(-1, 1), demand)
print(f"True price effect: {true_price_effect}")
print(f"Naive OLS estimate: {naive_model.coef_[0]:.3f}")  # biased toward ~ -0.5

# After controlling for quality - consistent
X_controlled = np.column_stack([price, quality])
controlled_model = LinearRegression()
controlled_model.fit(X_controlled, demand)
print(f"After controlling for quality: {controlled_model.coef_[0]:.3f}")  # ~ -2.0

Solution Approaches

Approach	Core idea	Assumption
Experiments	Randomly assign prices	Ethical/cost constraints
Instrumental variables	Exploit exogenous price variation	Exclusion restriction
Control strategy	Control for sufficient covariates	Unconfoundedness
Structural models	Explicitly model the economic structure	Functional-form assumptions

Related Concepts

• Instrumental Variables - the primary remedy for endogeneity
• Confounder - a variable that induces endogeneity
• A-B Testing - causal estimation free of endogeneity
• Price Elasticity - the target for which endogeneity is a problem

References

• Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data.
• Comprehensive Personalized Pricing Guide, Part I, §3