Answer First
Endogeneity biases OLS estimates because OLS requires regressors to be uncorrelated with the error term. When a regressor is correlated with the error term, OLS incorrectly attributes part of the unexplained variation to that regressor, producing biased and inconsistent coefficient estimates.
Problem Setup
Consider the linear model:
\[ y = \beta_0 + \beta_1 x + u \]
OLS requires:
\[ Cov(x, u) = 0 \]
Endogeneity occurs when:
\[ Cov(x, u) \neq 0 \]
The probability limit of the OLS estimator becomes:
\[ plim(\hat{\beta}_1) = \beta_1 + \frac{Cov(x, u)}{Var(x)} \]
Step-by-Step Solution
1. Identify the source of endogeneity
Endogeneity arises from:
- Omitted variables
- Simultaneity (x and y determine each other)
- Measurement error
2. Understand the OLS requirement
OLS assumes regressors are exogenous, meaning they are unrelated to the error term.
3. Show how correlation with the error term biases OLS
When \(x\) is correlated with \(u\), OLS attributes part of the error term’s variation to \(x\), distorting the coefficient.
4. Derive the bias expression
The bias equals:
\[ Bias(\hat{\beta}_1) = \frac{Cov(x, u)}{Var(x)} \]
The sign of the bias depends on the sign of \(Cov(x, u)\).
5. Explain why the estimator becomes inconsistent
Even with infinite data, the correlation between \(x\) and \(u\) does not disappear. The estimator converges to the wrong value.
Intuition
OLS tries to isolate the effect of each regressor. When a regressor is correlated with the error term, OLS cannot distinguish the regressor’s effect from the unobserved factors in the error term. This misattribution produces biased and inconsistent estimates.
Common Exam Mistakes
- Thinking endogeneity only comes from omitted variables.
- Confusing endogeneity with multicollinearity.
- Believing large samples fix endogeneity (they do not).
- Assuming endogeneity affects only standard errors (it biases coefficients).
Why This Matters
Endogeneity is the central challenge in empirical economics, finance, and policy analysis. Without addressing it, regression results cannot be interpreted causally. Instrumental variables, panel methods, and experimental designs all exist to solve this problem.
Final Summary
Endogeneity biases OLS estimates because regressors become correlated with the error term. This violates the core OLS assumption and causes the estimator
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