Consequences of Multicollinearity

ECON 3209 · Week 11, Lecture 2 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

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

1. explain why multicollinearity inflates standard errors
2. interpret the symptom of high R² with insignificant t-statistics
3. recognize coefficient instability across samples or specifications
4. distinguish effects on explanation from effects on prediction
5. use simulations to compare low- and high-collinearity cases

OLS Is Still Unbiased — But Less Precise

If the zero conditional mean assumption holds, multicollinearity does not bias OLS coefficients.

Its main effect is to increase sampling variance.

\[Var(\hat\beta_j) = \frac{\sigma^2}{SST_j(1-R_j^2)}\]

When \(R_j^2\) is large, the denominator shrinks and the variance of \(\hat\beta_j\) rises.

Inflated Standard Errors

• Larger standard errors mean wider confidence intervals.
• Wider intervals make it harder to reject null hypotheses.
• Important variables may appear statistically insignificant.

Multicollinearity makes it hard to separate who gets credit for explaining variation in \(Y\).

High R² but Weak Individual t-Tests

This pattern is common under strong collinearity: - the model fits the data well overall - regressors are jointly informative - but each variable looks weak on its own

Why? Because highly related regressors compete to explain the same movement in the dependent variable.

Unstable Coefficients Across Samples

With severe multicollinearity: - coefficient signs may flip across samples - magnitudes may change sharply after adding or dropping a related variable - substantive interpretation becomes fragile

This does not always imply the model is wrong. It may simply mean the data cannot sharply distinguish separate effects.

Python Demo: Low vs High Collinearity

Python Demo: High R² with Weak t-Stats

Python Demo: Coefficient Instability Across Samples

Prediction Can Still Be Good

Even if individual slopes are unstable, the model may still predict well because the regressors move together in a predictable way.

Multicollinearity is usually a problem for explanation and inference more than for pure prediction.

Python Demo: Compare Prediction Error

What Should the Researcher Conclude?

• Do not automatically drop an economically essential variable.
• Focus on theory, data design, and whether separate effects are truly identifiable.
• Report uncertainty honestly.
• Consider combined hypotheses or alternative specifications when variables are conceptually linked.

🏋️ Exercise

1. Simulate one dataset with low regressor correlation and one with high correlation.
2. Estimate the same MLR in both datasets.
3. Compare standard errors, t-statistics, and RMSE.
4. Write a two-sentence conclusion about inference versus prediction.

Summary

✅ Multicollinearity usually increases the variance and standard errors of OLS coefficients.

✅ A model can have high overall fit while individual regressors look insignificant.

✅ Coefficient estimates become unstable when regressors compete to explain the same variation.

✅ Prediction may remain acceptable even when interpretation becomes fragile.

✅ The right response depends on the research goal: explanation, causal inference, or prediction.

Next Lecture

Lecture 3 — VIF & Remedies in Python

We will cover: - the VIF diagnostic - computing VIF in Python - remedies such as dropping or combining variables - a preview of ridge regression