Lesson M03.L03: Gauss–Markov Assumptions and BLUE

Module: M03 — Multiple Regression Analysis
Level: Graduate (ECMT5001, USYD S1 2027)
Duration: 35 min
Learning Objective: State and interpret the six MLR assumptions; articulate the Gauss–Markov theorem and what BLUE means; identify which assumption is most likely violated in applied work and what consequence follows.
Provenance: Wooldridge, Introductory Econometrics, Chapter 3 (Section 3.3), Chapter 3 Appendix

Explanation

The Six MLR Assumptions

The properties of OLS — unbiasedness, efficiency, and valid inference — depend on which assumptions hold. Wooldridge labels these MLR.1–MLR.6.

MLR.1 — Linear in Parameters

\[y = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k + u\]

The model is linear in the $\beta_j$'s. The $x_j$'s can be nonlinear functions of underlying variables (e.g., $\ln(\text{wage})$, $\text{age}^2$, $\text{educ} \times \text{female}$) — as long as each enters the equation multiplied by a single $\beta_j$.

Violation: A truly nonlinear parameter structure, e.g., $y = \beta_0 + \beta_1^{\beta_2} x$. Such models require nonlinear least squares (NLLS).

MLR.2 — Random Sampling

The observed data $\{(x_{i1},\ldots,x_{ik},y_i) : i=1,\ldots,n\}$ constitute a random sample from the population described by MLR.1.

Consequence of violation: Survey non-response bias, cluster sampling without correction, or time-series data (where observations are autocorrelated) can all violate MLR.2. Results still hold under weaker conditions (e.g., stationarity in time-series), but require more care.

MLR.3 — No Perfect Collinearity

In the sample, no independent variable is an exact linear combination of the others. Equivalently, $\mathbf{X}'\mathbf{X}$ has full rank $k+1$.

Example violation: Including both "male" and "female" dummies (each 0/1) when those are the only two groups — they sum to 1, exactly equal to the intercept column. OLS cannot be computed.

Important: MLR.3 is about perfect collinearity only. High (imperfect) multicollinearity is permitted but inflates standard errors — see M03.L05.

MLR.4 — Zero Conditional Mean (Exogeneity)

\[E(u \mid x_1, \ldots, x_k) = 0\]

The error term has zero mean conditional on all values of all included regressors. This is the most important assumption for unbiasedness of OLS.

It rules out: - Omitted variables that correlate with any $x_j$ (M03.L04) - Simultaneity / reverse causality (e.g., wages affecting education choices) - Measurement error in regressors (M10)

When MLR.4 fails, OLS is biased and inconsistent — the gold-standard fix is instrumental variables (M10).

MLR.5 — Homoskedasticity

\[\text{Var}(u \mid x_1, \ldots, x_k) = \sigma^2\]

The variance of the error is constant across all covariate values. Combined with MLR.4, this gives $E(u^2 \mid \mathbf{x}) = \sigma^2$.

Consequence of violation (heteroskedasticity): OLS remains unbiased (MLR.4 intact) but is no longer BLUE — a weighted least squares (WLS) estimator has smaller variance. The standard errors reported by default OLS are also incorrect; use heteroskedasticity-robust standard errors (M06).

MLR.6 — Normality of the Error

\[u \mid x_1,\ldots,x_k \sim \mathcal{N}(0, \sigma^2)\]

Implies MLR.4 and MLR.5. Needed for exact finite-sample $t$- and $F$-tests under OLS.

In large samples: By the Central Limit Theorem, $t$-statistics are asymptotically standard normal even without MLR.6, so this assumption becomes less critical as $n$ grows. For small samples (e.g., $n < 30$), violation of normality can distort inference.

The Gauss–Markov Theorem

Gauss–Markov Theorem

Under MLR.1–MLR.5, the OLS estimator $\hat{\boldsymbol{\beta}}$ is BLUE:
Best Linear Unbiased Estimator.

Breaking down BLUE:

Letter	Meaning
B (Best)	Among all linear unbiased estimators, OLS has the smallest variance for every $\beta_j$
L (Linear)	The estimator is a linear function of $y_1,\ldots,y_n$
U (Unbiased)	$E(\hat{\beta}_j) = \beta_j$ for all $j$
E (Estimator)	It is a specific estimator, not a guess

"Best" in the BLUE sense means: for any other linear unbiased estimator $\tilde{\beta}_j$,

\[\text{Var}(\hat{\beta}_j) \leq \text{Var}(\tilde{\beta}_j)\]

This is a minimum variance result — OLS squeezes the most information out of the data, given the constraints.

Which Assumptions Drive What?

Result	Assumptions Required
OLS is unbiased: $E(\hat{\boldsymbol{\beta}}) = \boldsymbol{\beta}$	MLR.1–MLR.4
OLS is efficient (BLUE)	MLR.1–MLR.5
OLS $t$-tests are exact in finite samples	MLR.1–MLR.6
OLS $t$-tests are asymptotically valid	MLR.1–MLR.4 + CLT (large $n$)

Worked Example: Economic Violations of Each Assumption

Context: A researcher estimates the effect of rainfall on crop yield for 200 Australian wheat farms:

\[\text{yield}_i = \beta_0 + \beta_1\,\text{rain}_i + \beta_2\,\text{fertiliser}_i + \beta_3\,\text{soil quality}_i + u_i\]

Assumption	Potential Violation	Consequence
MLR.1	True relationship is $\text{yield} = \beta_0 \cdot \text{rain}^{\beta_1}$ (Cobb-Douglas)	OLS fitted model is misspecified; must use NLLS or log-transform
MLR.2	Farms selected because they're large (size-biased sample)	Estimates may not generalise to all farms; selection bias
MLR.3	Researcher includes both `sand_fraction` and `clay_fraction` where they sum to 1 (together with `silt_fraction` already included)	OLS fails: singular matrix; drop one category
MLR.4	Fertiliser use is chosen by the farmer partly in response to expected yield (reverse causality)	OLS estimate of $\beta_2$ is biased; need IV
MLR.5	Variance of yield is much larger for farms with high rainfall (heteroskedasticity)	OLS is unbiased but standard errors are wrong; use robust SEs
MLR.6	Yield has a left-skewed distribution (bounded below at zero; catastrophic failures create a fat left tail)	$t$-tests are approximate in small samples; large-sample CLT still applies

Common Misconception

OLS is the best estimator (full stop)

The Gauss–Markov theorem says OLS is best among linear unbiased estimators — not among all possible estimators. If MLR.5 fails (heteroskedasticity), weighted least squares (WLS) is a linear estimator that is also unbiased and beats OLS on variance. If the error is known to be non-normal, some maximum likelihood estimators can dominate OLS in small samples. BLUE is not "best estimator ever" — it is "best in a specific class under specific conditions."

Multicollinearity violates MLR.3

Only perfect collinearity violates MLR.3. Imperfect multicollinearity (high $R_j^2$ from regressing $x_j$ on other regressors) is perfectly consistent with MLR.3 — OLS remains BLUE. The problem with imperfect multicollinearity is large standard errors, not bias or breakdown of the estimator.

Practice Prompts

Assumption mapping. For each of the following scenarios, state which MLR assumption is violated and whether OLS is (a) biased, (b) inefficient, or (c) invalid for inference:
(i) The error variance is three times larger for firms with revenues above $1 billion.
(ii) A model of house prices omits suburb fixed effects, and suburb correlates with both price and included regressors (number of bedrooms, bathrooms).
(iii) A researcher includes male (0/1) and female (0/1) as the only two gender dummies with a constant.
BLUE theorem. An alternative linear estimator for $\beta_1$ is the "split-sample estimator" $\check{\beta}_1 = (\bar{y}_H - \bar{y}_L)/({\bar{x}_H - \bar{x}_L})$, where $H$ and $L$ denote the top and bottom halves of the $x_1$ distribution. This estimator is linear in $y$ and unbiased under MLR.1–MLR.4. What does the Gauss–Markov theorem imply about its variance relative to $\hat{\beta}_1^{OLS}$?
MLR.4 vs MLR.5. Explain in one paragraph why MLR.4 (zero conditional mean) is more damaging to violate than MLR.5 (homoskedasticity). In what sense does heteroskedasticity "only" affect efficiency and standard errors, while endogeneity affects the coefficients themselves?

Further Resources

Wooldridge, Ch 3.3–3.4, Appendix 3A (Gauss–Markov proof sketch)
Davidson & MacKinnon, Econometric Theory and Methods, Ch 2 (rigorous BLUE proof)
Stock & Watson, Introduction to Econometrics, Ch 6.6 (BLUE intuition)

Result	Assumptions Required
OLS is unbiased: \(E(\hat{\boldsymbol{\beta}}) = \boldsymbol{\beta}\)	MLR.1–MLR.4
OLS is efficient (BLUE)	MLR.1–MLR.5
OLS \(t\)-tests are exact in finite samples	MLR.1–MLR.6
OLS \(t\)-tests are asymptotically valid	MLR.1–MLR.4 + CLT (large \(n\))

Assumption	Potential Violation	Consequence
MLR.1	True relationship is \(\text{yield} = \beta_0 \cdot \text{rain}^{\beta_1}\) (Cobb-Douglas)	OLS fitted model is misspecified; must use NLLS or log-transform
MLR.2	Farms selected because they're large (size-biased sample)	Estimates may not generalise to all farms; selection bias
MLR.3	Researcher includes both `sand_fraction` and `clay_fraction` where they sum to 1 (together with `silt_fraction` already included)	OLS fails: singular matrix; drop one category
MLR.4	Fertiliser use is chosen by the farmer partly in response to expected yield (reverse causality)	OLS estimate of \(\beta_2\) is biased; need IV
MLR.5	Variance of yield is much larger for farms with high rainfall (heteroskedasticity)	OLS is unbiased but standard errors are wrong; use robust SEs
MLR.6	Yield has a left-skewed distribution (bounded below at zero; catastrophic failures create a fat left tail)	\(t\)-tests are approximate in small samples; large-sample CLT still applies