Purpose
Regression analysis is often used to estimate the effect of one variable on another. However, obtaining a statistically significant coefficient does not necessarily mean that the estimated relationship is causal.
One of the most serious challenges in applied econometrics is endogeneity . Endogeneity occurs when an explanatory variable is correlated with the error term. When this happens, the estimated coefficients may be biased and misleading.
In this chapter, we learn what endogeneity is, why it occurs, and how economists use instrumental variables to address the problem.
Applied question
Does education increase earnings?
Suppose we estimate the following relationship:
\[
Income_i =
\beta_0 +
\beta_1 Education_i +
u_i
\]
where income represents annual earnings and education represents years of schooling.
Most people expect education to increase earnings. However, individuals differ in many ways that are difficult to observe.
Some individuals may have:
greater motivation
better problem-solving skills
higher natural ability
stronger family support
These factors may influence both education and earnings. As a result, the estimated relationship between education and income may not reflect the true causal effect of education.
Economic background
Economists are often interested in causal questions.
Examples include:
Does education increase income?
Does fertilizer increase crop yield?
Does advertising increase sales?
Does foreign aid promote economic growth?
Does trade liberalization increase exports?
Simple correlations rarely provide convincing answers. The challenge is that many economic variables influence one another simultaneously.
As a result, causal interpretation requires caution.
Key idea
The classical regression model assumes:
\[
\operatorname{Cov}(X,u)=0
\]
This means that the explanatory variable is unrelated to the error term.
Endogeneity occurs when:
\[
\operatorname{Cov}(X,u)\neq0
\]
When this assumption fails, OLS estimates become biased.
Unlike heteroskedasticity or multicollinearity, endogeneity threatens the validity of the coefficient estimate itself.
A simple example
Suppose we estimate:
\[
Income_i =
\beta_0 +
\beta_1 Education_i +
u_i
\]
The error term contains many omitted factors:
ability
motivation
family background
social networks
Suppose more able individuals obtain more education.
Ability therefore affects education and income. Ability enters the error term because it is unobserved.
Consequently:
\[
\operatorname{Cov}(Education,u)\neq0
\]
The OLS estimate is biased.
Understanding omitted variable bias
Omitted variable bias occurs when three conditions hold:
A relevant variable is omitted.
The omitted variable affects the dependent variable.
The omitted variable is correlated with an explanatory variable.
In our example:
Because ability satisfies both conditions, it creates bias.
Visualizing the problem
A simple causal diagram helps clarify the issue.
Ability
↘
↘
Education → Income
↗
↗
Family Background
Interpretation
Education influences income. However, ability and family background influence both education and income.
If these variables are omitted, the estimated effect of education captures more than education alone.
Simulating endogeneity
We create a dataset where ability affects both education and income.
import numpy as npimport pandas as pd4107 )= 500 = np.random.normal(0 , 1 , n)= (12 + 2 * ability+ np.random.normal(0 , 1 , n)= (20000 + 3000 * education+ 5000 * ability+ np.random.normal(0 , 3000 , n)= pd.DataFrame({"Income" : income,"Education" : education,"Ability" : ability
0
60731.721132
13.185934
0.841017
1
34712.026965
8.004777
-1.312692
2
50578.249472
10.439781
-0.706391
3
47284.142849
11.268915
-0.708966
4
48901.418652
11.621520
-0.031475
Estimating the naive model
Suppose ability is unobserved. We estimate:
\[
Income_i=
\beta_0+
\beta_1 Education_i+
u_i
\]
import statsmodels.api as sm= sm.add_constant(data["Education" ])= data["Income" ]= sm.OLS(y, X).fit()print (ols_model.summary())
OLS Regression Results
==============================================================================
Dep. Variable: Income R-squared: 0.890
Model: OLS Adj. R-squared: 0.890
Method: Least Squares F-statistic: 4026.
Date: Sat, 13 Jun 2026 Prob (F-statistic): 8.94e-241
Time: 08:26:47 Log-Likelihood: -4846.2
No. Observations: 500 AIC: 9696.
Df Residuals: 498 BIC: 9705.
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const -3088.8505 948.775 -3.256 0.001 -4952.946 -1224.755
Education 4925.2845 77.619 63.455 0.000 4772.784 5077.786
==============================================================================
Omnibus: 3.071 Durbin-Watson: 1.931
Prob(Omnibus): 0.215 Jarque-Bera (JB): 2.613
Skew: 0.077 Prob(JB): 0.271
Kurtosis: 2.681 Cond. No. 66.5
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
Interpretation
The coefficient on education may appear large and highly significant.
However, the estimate includes both the effect of education and the effect of ability. The coefficient is biased upward.
Why endogeneity is serious
Consider the problems studied previously.
Heteroskedasticity
Standard errors become unreliable
Autocorrelation
Standard errors become unreliable
Multicollinearity
Precision declines
Endogeneity
The coefficient itself may be wrong
This is why endogeneity is often considered the most serious problem in applied econometrics.
Simultaneity
Endogeneity can also arise through reverse causality.
Consider:
\[
Sales=f(Advertising)
\]
Firms often increase advertising when sales rise.
Thus:
advertising affects sales
sales affect advertising
Both variables influence each other simultaneously. OLS struggles to separate cause from effect.
Measurement error
Another source of endogeneity is measurement error.
Suppose farmers report fertilizer use incorrectly. If explanatory variables are measured with error, coefficient estimates may become biased.
Measurement error is common in surveys and self-reported data.
Instrumental variables
Economists often use instrumental variables to address endogeneity.
An instrument is a variable that:
affects the endogenous explanatory variable
does not directly affect the dependent variable
Example instrument
Suppose we want to estimate the effect of education on income.
A possible instrument might be distance to the nearest university.
Distance affects educational attainment. However, distance should not directly determine future earnings once education is accounted for.
The logic is:
Distance to University
↓
Education
↓
Income
Requirements for a good instrument
A valid instrument must satisfy two conditions.
Relevance
The instrument must be correlated with the endogenous variable.
\[
\operatorname{Cov}(Z,X)\neq0
\]
Exogeneity
The instrument must not affect the error term.
\[
\operatorname{Cov}(Z,u)=0
\]
Finding valid instruments is often the most difficult part of empirical research.
Two-Stage Least Squares
Instrumental variable estimation is commonly implemented through Two-Stage Least Squares.
Stage 1
Predict education using the instrument.
\[
Education_i=
\gamma_0+
\gamma_1 Instrument_i+
v_i
\]
Stage 2
Use the predicted education values to estimate income.
\[
Income_i=
\beta_0+
\beta_1 \widehat{Education}_i+
u_i
\]
The resulting estimate is less vulnerable to endogeneity bias.
Basic Python implementation
The following code shows the structure of an IV regression. It requires the linearmodels package and an instrument variable in the dataset.
# Uncomment after installing linearmodels and adding an instrument variable. # from linearmodels.iv import IV2SLS # # iv_model = IV2SLS( # dependent=y, # exog=np.ones(len(data)), # endog=data["Education"], # instruments=data["Distance"] # ).fit() # # print(iv_model.summary)
The details of IV estimation are beyond the scope of this course. The important point is understanding why economists use instruments.
Why instrumental variables are difficult
Finding a valid instrument is challenging.
Researchers must convince readers that:
the instrument affects the endogenous variable
the instrument does not directly affect the outcome
Many empirical debates focus on whether instruments are truly valid. Weak or invalid instruments can produce misleading results.
Do not assume that a statistically significant coefficient is causal. Statistical significance and causal identification are different issues.
Common mistakes
Mistake 1: Equating correlation with causation
A significant coefficient does not automatically imply causality.
Mistake 2: Ignoring omitted variables
Unobserved factors frequently influence both explanatory and dependent variables.
Mistake 3: Assuming reverse causality never exists
Many economic relationships operate in both directions.
Mistake 4: Using weak instruments
An instrument that barely affects the endogenous variable provides little information.
Mistake 5: Believing IV solves everything
Instrumental variables address some sources of endogeneity, but they do not automatically guarantee credible results.
Key takeaways
Endogeneity occurs when an explanatory variable is correlated with the error term.
Omitted variables, simultaneity, and measurement error are common sources of endogeneity.
Endogeneity can bias coefficient estimates.
Biased coefficients threaten causal interpretation.
Correlation does not imply causation.
Instrumental variables provide one strategy for addressing endogeneity.
Valid instruments must be relevant and exogenous.
Establishing causality is often the most challenging task in applied economics.
Looking ahead
Throughout this chapter, we examined individual econometric problems that can weaken empirical conclusions. In the next chapter, we bring everything together and learn how economists evaluate model credibility.