Econometrics is not a collection of tests to memorise. It is a disciplined way of asking whether the variation we observe in data can answer an economic question.
1. Begin with the question, not the formula
Suppose we want to study how years of education relate to wages. A simple population model is:
The coefficient β1 is the systematic change in expected wage associated with one additional year of education. The error term u collects everything affecting wages that the model has not explicitly included: ability, experience, occupation, location, measurement error and much more.
The difficult question is therefore not “Can software draw a line?” It is “Does the variation in education isolate the relationship we claim it does?” That is why assumptions matter.
2. What ordinary least squares does
For each observation, the residual is the vertical distance between the observed outcome and the fitted line. OLS chooses the intercept and slope that minimise the sum of squared residuals:
Squaring prevents positive and negative residuals from cancelling and gives larger errors more weight. It produces a unique and convenient estimator when the explanatory variables contain sufficient independent variation.
OLS always finds the best-fitting coefficients by this rule when the calculation is possible. It does not automatically make those coefficients causal, unbiased or precisely estimated. Those properties depend on the model, data and assumptions.
3. Read the assumptions as economic statements
Linearity in parameters
The coefficients enter the model linearly. The explanatory variables themselves may be transformed: a model containing income squared can still be linear in its parameters.
If it fails: A wrongly specified functional form can leave a systematic pattern in the residuals and distort interpretation.
Random sampling and variation
The observations must support the comparison the model is trying to make, and the explanatory variable must vary in the sample.
If it fails: Selection problems or a variable with no useful variation prevent the sample from identifying the relationship of interest.
No perfect multicollinearity
No explanatory variable can be an exact linear combination of the others. The model needs independent variation to separate their effects.
If it fails: With perfect multicollinearity, the relevant coefficients cannot be estimated separately. High, but imperfect, correlation mainly reduces precision.
Zero conditional mean
Once the included explanatory variables are held fixed, the remaining error has an expected value of zero: E(u | X) = 0.
If it fails: Omitted causes, simultaneity and measurement error can make X move with the error term, producing biased and inconsistent OLS estimates.
Homoscedastic, uncorrelated errors
For the classical efficiency and standard-error results, the error variance is constant and errors are not correlated across observations.
If it fails: Heteroscedasticity or autocorrelation may leave coefficients unbiased under exogeneity while making conventional standard errors and tests unreliable.
4. A distinction that prevents many mistakes
A coefficient and its standard error answer different questions. The coefficient estimates the size and direction of a relationship. The standard error measures the sampling uncertainty around that estimate.
Heteroscedasticity is the classic examination example. If zero conditional mean still holds, heteroscedasticity does not by itself bias the OLS coefficient. It does, however, invalidate the usual homoscedastic standard-error formula. A candidate who writes only “OLS is biased” has confused estimation with inference.
5. Correlation is not yet causation
Returning to education and wages, ability may raise both education and wages. If ability is omitted, part of its effect enters the error term. Education is then correlated with the error, violating zero conditional mean. The fitted slope may describe an association, but it cannot be interpreted automatically as the causal return to education.
Adding relevant controls, using panel data, finding an instrumental variable or exploiting a natural experiment are possible strategies—but each introduces assumptions of its own. A good econometric answer names those assumptions rather than treating a technique as magic.
6. A five-step method for examination questions
- 1Name the estimator or problem being tested.
- 2Write the relevant assumption in words and, where useful, in notation.
- 3State which property is affected: unbiasedness, consistency, efficiency, or the validity of standard errors.
- 4Distinguish the coefficient estimate from its estimated uncertainty.
- 5Choose a diagnostic or remedy only after identifying the cause.
Final checkpoint
Before accepting any regression result, ask four questions: What is being estimated? Where does the identifying variation come from? Which assumption connects that variation to the economic claim? Are the reported standard errors appropriate for the data?
Once those questions become habitual, econometrics stops looking like a wall of formulas. The formulas become compact descriptions of an argument you already understand.