MFIN6201 Chap.5 Least Squares Assumptions and Inference
Least Squares Assumptions and Inference
This chapter links least-squares assumptions to the estimator properties they support. Conditional mean zero is the central condition for an unbiased slope, iid sampling supports repeated-sample variance calculations, and restrictions on extreme observations protect large-sample behaviour. Heteroskedasticity is treated as a standard-error problem rather than automatically as coefficient bias.
Conditional mean zero says the average omitted component is zero at every regressor value, a stronger condition than zero covariance. Random assignment is a clear route because treatment is generated independently of potential omitted influences. Observational applications require an institutional argument about what enters the error and how it moves with X.
Extreme outcomes and high-leverage regressor values deserve different diagnostics because either can dominate squared-error fitting. Homoskedasticity concerns conditional error variance, not conditional error mean. When only variance changes, robust standard errors can repair inference while leaving the point estimate unchanged. When the mean changes with X, robust inference does not remove bias.
Re-centering a constant conditional error mean into the intercept clarifies precisely why a varying conditional mean is the real threat to the slope.
What this chapter covers
- 01
Conditional mean zero
- 02
Iid regression sampling
- 03
Outlier conditions
- 04
Unbiased OLS
- 05
Heteroskedasticity
- 06
Robust standard errors
Worked example · free
Re-centering a constant error mean
- +1Define v=u-c.
- +1Rewrite Y=(beta0+c)+beta1X+v.
- +1Show E[v given X]=c-c=0.
Key terms
- Conditional mean zero
- The condition that the regression error averages to zero at every regressor value.
- Heteroskedasticity
- Variation in the conditional error variance across observations or regressor values.
- Robust standard error
- An uncertainty estimate designed to remain valid under heteroskedasticity.
Least Squares Assumptions and Inference FAQ
Why is conditional mean zero important?
It prevents the regressor from moving systematically with omitted determinants inside the error. Under the remaining sampling conditions, this protects the OLS slope from repeated-sample bias and gives the association a defensible conditional interpretation.
Does heteroskedasticity bias OLS?
Not by itself. If conditional mean zero holds, the OLS coefficient can remain unbiased, but the usual homoskedastic standard-error formula is wrong. Robust standard errors target that uncertainty problem without repairing omitted-variable bias.
Should every outlier be removed?
No. First verify data quality, units and leverage, then report sensitivity. A genuine extreme observation may be important information. Deleting it only because it changes the preferred conclusion turns a diagnostic into result selection.
Can robust standard errors fix endogeneity?
No. They repair uncertainty estimates under unequal conditional variance but do not remove a relationship between X and omitted determinants. Endogeneity requires a better design, defensible controls or another identification approach.
Why does random assignment help?
Assignment created independently of potential outcomes makes treatment groups comparable on average before treatment. That can support conditional mean zero, although attrition, noncompliance or interference may introduce new complications afterward.
Is zero covariance enough for unbiased OLS?
Not generally. Zero covariance is an unconditional linear restriction, while conditional mean zero requires the average error to vanish at every X value. Nonlinear dependence can satisfy the former and violate the latter, leaving the slope interpretation unsupported.
What should a sensitivity check report?
Report which observations, variance method or specification changed, then show the effect on the coefficient and its uncertainty. Explain the substantive reason for the alternative rather than presenting a preferred result as proof that the original assumption held. A transparent comparison is more informative than selective reassurance.
Exam move
Make a table matching each assumption to the result it supports and the symptom of failure. Practise distinguishing coefficient bias from incorrect standard errors. For every proposed remedy, state what it fixes and what it leaves untouched. Rework the conditional-mean re-centering proof until the role of the intercept is intuitive.
Build one counterexample for each common confusion: heteroskedastic errors with an unbiased slope, zero covariance without mean independence, and an extreme observation that is genuine rather than erroneous. For a proposed random experiment, explain why assignment changes the conditional error story. For observational data, list plausible omitted causes before looking at the regression output.
Separate data verification, robust variance estimation and control-variable design into different remedy columns. When interpreting a sensitivity result, say whether the coefficient, uncertainty or sample support changed. Review the algebra E[u-c given X]=0 and connect it to the intercept.
Finish every assumption question by naming the exact estimator property at stake instead of declaring the whole regression simply valid or invalid.
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