Multiple Regression & Inferenceintermediate

Assumptions of Multiple Regression

OLS unbiasedness rests on four assumptions. MLR.1 the model is linear in parameters, MLR.2 the data are a random sample, MLR.3 there is no perfect collinearity among regressors, and MLR.4 the error has zero conditional mean, $E(u\mid x_1,\dots,x_k)=0$ . MLR.4 is the decisive one, since it is what omitted variables, simultaneity, and measurement error violate, breaking the ceteris paribus reading of every slope.

Why it matters

The first three assumptions keep the math well defined and let OLS even run. The fourth is the economics. It says nothing left in the error is systematically tied to your regressors. Lose MLR.4 and your coefficients no longer answer the causal question.

Formulas

MLR.4 zero conditional mean

E(u\mid x_1,x_2,\dots,x_k)=0

Regressors carry no information about the expected error.

MLR.3 no perfect collinearity

\text{no } x_j \text{ is an exact linear function of the others}

Otherwise OLS cannot separate the collinear effects.

Worked examples

Scenario

You include a full set of region dummies plus a constant and Stata drops one.

Solution

Stata omits a category to avoid the dummy-variable trap, an exact MLR.3 violation. Keeping every dummy alongside the intercept would make the regressors perfectly collinear and `regress` could not estimate the coefficients.

NoteMLR.3 forbids exact collinearity, but it allows strong (imperfect) correlation.

Common mistakes

✗Reading MLR.1 as requiring straight-line relationships. It only requires linearity in the parameters, so $x^2$ and $\log x$ terms are fine.
✗Thinking MLR.3 rules out any correlation among regressors. It bans only exact linear dependence, not ordinary multicollinearity.
✗Believing MLR.4 is just about heteroskedasticity. Zero conditional mean concerns the mean of $u$ ; the variance of $u$ is a separate assumption.
✗Assuming these four assumptions deliver normal t and F distributions. Exact normal inference needs the extra normality assumption, not just MLR.1 through MLR.4.

Revision bullets

•MLR.1 linear in parameters, MLR.2 random sampling, MLR.3 no perfect collinearity, MLR.4 zero conditional mean.
•MLR.1 to MLR.4 deliver unbiased OLS slopes.
•MLR.4, $E(u\mid x)=0$ , is the assumption that OVB and endogeneity break.
•MLR.3 bans only exact collinearity, not strong correlation.
•Adding homoskedasticity (MLR.5) gives the Gauss-Markov result.

Quick check

Which assumption is violated by omitted variable bias and endogeneity?

MLR.3 (no perfect collinearity) rules out which situation?

Connected topics

E(u|x)=0 MLR model Omitted var bias Gauss-Markov Multicollin.Heteroskedastic

Sources

Wooldridge, Introductory Econometrics, Ch. 3
Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Sec. 3.3
Wooldridge, Introductory Econometrics, Ch. 4
Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Ch. 4 (CLM assumptions)

How to cite this page

Dr. Phil's Quant Lab. (2026). Assumptions of Multiple Regression. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-mlr-assumptions

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