ECMT3120 Chap.1 Statistical Models and OLS Assumptions
Statistical Models and OLS Assumptions
Econometric reasoning begins before estimation. This chapter separates a complete data generating process from a statistical model and an observed sample, then asks what random sampling and conditional mean independence mean in economic terms.
It shows why a regression equation alone is incomplete, how clustered or time ordered observations challenge independence, and why a precise OLS slope can still mix a target effect with omitted determinants.
What this chapter covers
- 01
Data generating processes and parameter indexed statistical models
- 02
Independent and identically distributed sampling as a modelling choice
- 03
Regression equations, structural errors and omitted determinants
- 04
Conditional mean independence and coefficient interpretation
- 05
Descriptive, predictive and causal claims
- 06
Omitted variable bias direction
- 07
Sampling dependence across firms, schools, households and time
Worked example · free
Audit an electricity use regression
- +1Attach units to the slope: one extra square metre is associated with 3.4 additional electricity units per month in this sample.
- +1Treat the intercept as an algebraic anchor because zero floor area lies outside meaningful support.
- +1Place occupancy, insulation, appliances, orientation and heating technology in the omitted component.
- +1Explain why those omitted determinants may vary systematically with apartment size.
- +1Report a conditional association rather than the physical effect of adding floor space.
- +1State what stronger design or control argument would support a causal claim.
Key terms
- Data generating process
- A complete probability mechanism for the variables and dependence that produce observations in a population.
- Statistical model
- A collection of possible data generating processes indexed by unknown parameters or unknown features.
- Random sampling
- A modelling assumption that observations follow the same distribution and are independent across sampled units.
- Structural error
- The collection of unobserved determinants of the outcome left outside the specified regression function.
- Conditional mean independence
- The condition that the structural error averages to zero after conditioning on the included regressors.
- Omitted variable bias
- Distortion in an estimated coefficient when an excluded determinant affects the outcome and correlates with an included regressor.
Statistical Models and OLS Assumptions FAQ
Why is a regression equation not a complete econometric model?
An equation names variables and parameters but does not state how observations are sampled, how the error behaves, whether units are dependent, or how regressors relate to omitted determinants. Those assumptions determine whether an estimator has a sampling distribution and whether its coefficient receives a descriptive or causal interpretation.
What does conditional mean independence mean economically?
It means that after conditioning on the included regressors, observations with different regressor values do not systematically differ in the average omitted component. If larger regressor values select units with higher ability, motivation or latent demand, the condition fails and OLS mixes effects.
Can a small standard error repair omitted variable bias?
No. A small standard error describes sampling precision around the quantity OLS estimates under the fitted model. Omitted variable bias changes the centre of that quantity by linking a regressor to the structural error. More precision can therefore produce a very precise estimate of the wrong target.
Exam move
For each regression, begin with a four line audit: define the observational unit, list plausible omitted determinants, explain which are related to the key regressor, and state the strongest interpretation the design supports. Practise bias direction by writing the sign of the omitted factor's outcome effect and its covariance with the regressor before multiplying the signs.
When data repeat firms, people, schools or dates, mark the likely dependence cluster. Finish by separating descriptive association, prediction and causal effect. This routine creates the foundation for the IV and panel chapters, where the same error correlation is addressed with a new source of variation rather than ignored. Build a small bank of counterexamples.
For random sampling, use a classroom, store chain and time series, and explain the dependence mechanism in each. For conditional mean zero, vary the omitted factor so one example creates upward bias and another creates downward bias. Rehearse interpretations with units and support: state what a one-unit change means, whether the intercept describes an observed case, and which variables are held fixed.
When you check a proposed causal sentence, underline the variation that supports it and circle every assumption required to move from association to intervention. If the assumptions are absent, rewrite the ending as a descriptive or predictive claim. On a timed problem, reserve the final minute for a three-part audit: sampling structure, error-regressor link, and scope of interpretation.
That habit prevents a technically correct coefficient calculation from carrying a stronger conclusion than the model earns.
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