Gauss-Markov Theorem

ECON 3209 · Week 8, Lecture 1 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

State the classical linear regression assumptions clearly.
Explain what BLUE means in the Gauss-Markov theorem.
Understand how violations affect OLS precision or validity.
Simulate assumption violations in Python.
Connect theory to practical regression work in applied economics.

The linear regression model revisited

\[Y_i = \beta_0 + \beta_1 X_i + u_i\]

OLS provides estimates of \(\beta_0\) and \(\beta_1\).
But its good properties depend on assumptions.
The Gauss-Markov theorem tells us when OLS is especially desirable.
The key result is that OLS becomes BLUE under the classical assumptions.

Assumption 1: linear in parameters

The model must be linear in the unknown coefficients.
The variable itself may be transformed, such as \(\log X\), but the coefficients must enter linearly.
Example: \[Y_i = \beta_0 + \beta_1 \log(X_i) + u_i\] is still linear in parameters.
This assumption is about the form of the model, not about straight lines only.

Assumption 2: random sampling

The sample should be drawn randomly from the population.
Each observation should represent the population fairly.
Non-random samples can create systematic bias.
Example: surveying only high-performing societies would distort inference.
Random sampling supports generalisation beyond the sample.

Assumption 3: sample variation and no perfect collinearity

The regressor must vary in the sample.
If every farmer uses exactly the same fertilizer level, slope estimation is impossible.
In multiple regression, one regressor must not be an exact linear combination of others.
OLS needs information in the data to identify coefficients.
No variation means no slope estimate.

Assumption 4: zero conditional mean

\[E(u_i \mid X_i) = 0\]

This is the most important assumption for unbiasedness.
It says the error term is unrelated to the explanatory variable.
Omitted variables that are correlated with \(X_i\) violate this assumption.
Example: soil quality omitted from a fertilizer-yield regression may bias the slope.

Assumption 5: homoskedasticity

\[Var(u_i \mid X_i) = \sigma^2\]

The variance of the error term is constant for all observations.
If dispersion changes with income, farm size, or district size, errors are heteroskedastic.
Heteroskedasticity does not usually bias OLS coefficients.
But it makes the usual standard errors unreliable.

Assumption 6: correct specification

The model should include the relevant variables in the right functional form.
Omitting important regressors or adding badly chosen irrelevant terms can distort interpretation.
Example: a fertilizer-yield model that ignores irrigation quality may be misspecified.
Correct specification supports both unbiased estimation and meaningful policy interpretation.

Assumption 7: normality

\[u_i \mid X_i \sim N(0, \sigma^2)\]

Normality is especially useful for exact t and F tests in small samples.
In large samples, the central limit theorem often gives approximate normal inference even when errors are not perfectly normal.
So normality matters most when the sample is limited and we want precise finite-sample inference.

What does BLUE mean?

Best: minimum variance among linear unbiased estimators.
Linear: estimator is linear in the observed \(Y_i\) values.
Unbiased: average estimate equals the true parameter.
Estimator: a rule that maps sample data into a numerical estimate.
So under the assumptions, OLS is the most precise linear unbiased estimator.

Homoskedastic example in Python

When assumptions fail

Violation of zero conditional mean causes biased and inconsistent estimates.
Heteroskedasticity mainly distorts standard errors and tests.
Non-random sampling reduces external validity.
Perfect collinearity makes estimation impossible.
The remedy depends on the assumption being violated.

Heteroskedastic data simulation

Residual plot for changing variance

Practical checklist for applied work

Inspect the sample design.
Check whether explanatory variables vary enough.
Think carefully about omitted variables and simultaneity.
Plot residuals to detect changing variance.
Use robust standard errors when heteroskedasticity is a concern.

Robust standard errors in Python

Exercise

Write one sentence for each CLR assumption explaining what it means in a fertilizer-yield regression. Then fit a simple OLS model and compare the usual standard error with the robust standard error.

Summary

✅ The Gauss-Markov theorem relies on the CLR assumptions.
✅ The core assumptions now include linearity, random sampling, variation/no perfect collinearity, zero conditional mean, homoskedasticity, correct specification, and normality for small-sample inference.
✅ Under these assumptions, OLS is BLUE.
✅ Zero conditional mean is central for unbiasedness.
✅ Heteroskedasticity affects standard errors more than coefficients.
✅ Diagnostics and robust standard errors are part of practical econometrics.

Next Lecture

We will study the statistical properties of OLS.
The focus will be unbiasedness, consistency, efficiency, and the sampling distribution of \(\hat{\beta}\).