Omitted Variable Bias

ECON 3209 · Week 16, Lecture 1 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

By the end of this lecture, you should be able to:

  1. define omitted variable bias (OVB) and explain when it arises
  2. use the OVB formula to determine the direction of bias
  3. distinguish between short and long regressions in practice
  4. understand the role of proxy variables when the true regressor is unavailable
  5. illustrate OVB with Python examples relevant to credit scoring and banking data

Why Omitted Variables Matter

  • We often want to estimate the effect of one regressor, say \(X_1\), on \(Y\).
  • Trouble begins when another relevant factor \(X_2\) is left out of the model.
  • If \(X_2\) affects \(Y\) and is correlated with \(X_1\), the estimated coefficient on \(X_1\) absorbs part of \(X_2\)’s effect.
  • This is omitted variable bias.

The OVB Formula

If the true model is \(Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + u\) but we omit \(X_2\), then

\[\text{plim}\hat{\beta}_1 = \beta_1 + \beta_2 \frac{\text{Cov}(X_1,X_2)}{\text{Var}(X_1)}\]

  • Bias exists when two conditions hold:
    1. \(\beta_2 \neq 0\) (the omitted variable matters), and
    2. \(\text{Cov}(X_1, X_2) \neq 0\) (the omitted variable is correlated with the included regressor).

Direction of Bias

Effect of omitted variable on \(Y\) Correlation with included regressor Bias direction
Positive Positive Upward
Positive Negative Downward
Negative Positive Downward
Negative Negative Upward

Always think in two signs: the omitted effect and the correlation sign.

Short and Long Regressions in Python

Interpretation of the Comparison

  • The short regression mixes the effect of income with the effect of credit score if credit score is related to income.
  • The long regression separates these channels more clearly.
  • A big change in the income coefficient after adding credit score is a warning sign that OVB was important.

Proxy Variables

  • Sometimes the true omitted variable is not observed.
  • A proxy variable is an imperfect but related measure.
  • Example: if repayment discipline is unobserved, branch inspection score may capture part of it.
  • A proxy rarely removes all bias, but it can reduce it.

Using a Proxy in Python

Kerala Credit Score Example

Suppose we estimate loan default using only borrower income.

But if credit score is omitted and higher-income borrowers also tend to have stronger credit history, then the income coefficient partly captures the effect of credit quality.

That is why econometrics always asks: what important variable may still be missing?

A Simple Sensitivity Check

Practical Checklist

  • Start from economic theory, not only data availability.
  • Ask which omitted factors could affect both the dependent variable and a key regressor.
  • Compare short and long regressions.
  • Use better data, fixed effects, instruments, or proxies when possible.
  • Remember: OVB is about causal credibility, not only statistical significance.

Exercise

Using the starter code, compare the income coefficient across the short, proxy, and full regressions. Which coefficient would you report to a cooperative bank manager, and why?

Summary

  • ✅ OVB appears when a relevant omitted variable both affects Y and is correlated with an included regressor.
  • ✅ The OVB formula helps determine whether the bias is upward or downward.
  • ✅ Comparing short and long regressions is a useful diagnostic step.
  • ✅ Proxy variables may reduce bias, but theory and data quality remain crucial.

Next Lecture

  • We test model specification with the Ramsey RESET test.
  • You will see how fitted values squared and cubed can reveal functional-form problems.
  • Today’s message about misspecification continues in a formal testing framework.