FRM Exam Part I · Linear Regression

OLS Assumptions and Properties of Estimators for FRM Part I

Updated 11 October 2026 · Fact-checked

OLS gives the best linear unbiased estimators (BLUE) when the classical assumptions hold: linearity, zero conditional mean of errors, no perfect multicollinearity, constant error variance, uncorrelated errors and a random sample. The Gauss-Markov theorem says OLS then has the smallest variance among linear unbiased estimators.

Understand OLS Assumptions and Properties of Estimators

Ordinary least squares (OLS) picks the intercept and slopes that make the sum of squared residuals as small as possible. The estimates are random, because they depend on the sample you drew. So you need to ask how good the estimator is, not just what number it gave.

Two properties matter most. An estimator is unbiased if its expected value equals the true parameter: E(β̂) = β. This is a statement about the average across many samples, at a fixed sample size. An estimator is consistent if it converges in probability to the true value as the sample size grows. An estimator can be biased but consistent (for example, the sample variance dividing by n). It can also be unbiased but not consistent (for example, using only the first observation as the estimate of a mean).

The classical assumptions are what make OLS work well. In the multiple regression setting you must know them:

Linearity: the model is linear in the parameters, Y = β0 + β1X1 + ... + βkXk + ε.
Zero conditional mean: E(ε | X) = 0. The errors are not correlated with the regressors. This is the key assumption for unbiasedness.
No perfect multicollinearity: no regressor is an exact linear combination of the others.
Homoskedasticity: Var(ε | X) is constant.
No serial correlation: Cov(εi, εj) = 0 for i ≠ j.
Random sampling / well-behaved data: observations are i.i.d. (or at least the data are not dominated by outliers, so large outliers are unlikely).

The Gauss-Markov theorem says: if linearity, zero conditional mean, no perfect multicollinearity, homoskedasticity and no serial correlation hold, OLS is BLUE. Best means lowest variance. Linear means the estimator is a linear function of Y. Unbiased means E(β̂) = β. Normality of errors is not needed for BLUE. Normality is only needed for exact t and F distributions in small samples. In large samples, the central limit theorem makes the estimators approximately normal.

When assumptions fail, different things break. Violating zero conditional mean (omitted variable, measurement error in X) causes bias and usually inconsistency. Heteroskedasticity or serial correlation leaves OLS unbiased and consistent but no longer best, and the usual standard errors are wrong, so t-tests mislead. Perfect multicollinearity makes OLS impossible to compute.

Key formulas to remember

Unbiasedness: E(β̂) = β; Holds at any sample size if zero conditional mean holds and the model is correctly specified.
Consistency: plim β̂ = β, i.e. β̂ → β in probability as n → ∞; Large-sample property. Needs the regressors to be uncorrelated with the error.
Zero conditional mean: E(ε | X) = 0; Violation causes bias and inconsistency.
Homoskedasticity and no serial correlation: Var(εi | X) = σ² and Cov(εi, εj | X) = 0 for i ≠ j; Needed for BLUE and valid usual standard errors.
Gauss-Markov theorem: Under the classical assumptions, Var(β̂OLS) ≤ Var(β̃) for any other linear unbiased β̃; Normality is not required.
Simple regression slope and its variance: β̂1 = Cov(X,Y) ÷ Var(X); Var(β̂1) = σ² ÷ Σ(Xi − X̄)²; More spread in X and more observations reduce variance.
Standard error of regression: s² = SSR ÷ (n − k − 1); Unbiased estimate of σ² with k slope coefficients.

How to solve OLS Assumptions and Properties of Estimators questions

Most questions ask which assumption is violated, what property is lost, or what a term means. Use this order.

1Identify what the question asks: a definition (unbiased, consistent, BLUE), a violated assumption, or a consequence.
2If it is a scenario, name the problem: omitted variable, measurement error, non-constant variance, correlated residuals, or near-collinear regressors.
3Link the problem to the assumption it breaks: omitted or mismeasured regressors break E(ε | X) = 0; changing variance breaks homoskedasticity; correlated residuals break no serial correlation.
4State the consequence: broken zero conditional mean means bias and inconsistency; broken homoskedasticity or no serial correlation means unbiased and consistent but not efficient, with invalid standard errors.
5Check whether the question says small sample or large sample. Unbiased is a finite-sample idea; consistent is a large-sample idea.
6Eliminate options that claim normality is needed for BLUE, or that heteroskedasticity biases the coefficients.
7For calculations, apply the formula given (for example β̂1 = Cov ÷ Var, or s² = SSR ÷ (n − k − 1)) and check units.

Quickest way: Assumption-to-consequence map

When to use it: Use it for any conceptual MCQ on OLS properties when time is short.

Ask: does the problem involve the error being correlated with X? If yes, answer bias and inconsistency.
If not, ask: is it about variance or correlation of errors? If yes, answer unbiased and consistent but not BLUE, with wrong standard errors.
If it is about collinearity between regressors, answer still unbiased, but large standard errors; perfect collinearity means no solution.
If the option says OLS needs normal errors to be BLUE, reject it.

Common mistakes in OLS Assumptions and Properties of Estimators

Saying heteroskedasticity makes OLS coefficients biased.
Students mix up bias with wrong standard errors.
Fix: Heteroskedasticity leaves β̂ unbiased and consistent. It loses efficiency and makes the usual standard errors invalid.
Thinking normal errors are required for BLUE.
Normality shows up in t-tests and is remembered as an OLS assumption.
Fix: Gauss-Markov does not use normality. Normality only gives exact small-sample t and F distributions.
Treating unbiased and consistent as the same.
Both mean the estimator is right in some sense.
Fix: Unbiased: expected value equals the truth at a given n. Consistent: converges to the truth as n grows. Neither implies the other.
Believing multicollinearity biases the estimates.
It sounds like a serious violation.
Fix: Imperfect multicollinearity keeps OLS unbiased but inflates variances. Only perfect multicollinearity breaks OLS.
Reading BLUE as the best of all estimators.
The word best is taken literally.
Fix: BLUE means best only among linear unbiased estimators. A biased or nonlinear estimator can have lower variance.
Using n − k instead of n − k − 1 for the error variance.
Confusion over whether k counts the intercept.
Fix: With k slope coefficients plus an intercept, divide SSR by n − k − 1.

Worked examples

Example 1

A regression of fund return on a market factor omits a second factor that is correlated with the market factor and also affects returns. Which statement about the OLS estimate of the market coefficient is correct? A) Unbiased and consistent, but inefficient. B) Biased and inconsistent. C) Unbiased but inconsistent. D) Biased but consistent.

Show the solution

The omitted factor is part of the error term and is correlated with the included regressor.
So E(ε | X) = 0 is violated.
A violated zero conditional mean assumption gives bias that does not vanish as n grows.
So the estimator is both biased and inconsistent.
Option A describes heteroskedasticity. Options C and D wrongly separate the two properties.

Answer: B) Biased and inconsistent.

Example 2

In a regression with an intercept and 3 slope coefficients estimated on 54 observations, the sum of squared residuals is 460. Estimate the error variance s² and the standard error of the regression.

Show the solution

Degrees of freedom = n − k − 1 = 54 − 3 − 1 = 50.
s² = SSR ÷ 50 = 460 ÷ 50 = 9.2.
Standard error of regression = √9.2 ≈ 3.033.

Answer: s² = 9.2 and the standard error of the regression is about 3.03.

Exam tips

Memorize the consequence table: bias comes from correlation between X and the error; inefficiency comes from heteroskedasticity or serial correlation.
When an option says normality is required for BLUE, it is almost always a distractor.
Read for the words small sample versus large sample to separate unbiased from consistent.
For degrees of freedom, count slope coefficients, then subtract one more for the intercept.
Be able to define each letter of BLUE in one phrase.

Practice questions from Linear Regression

OLS Assumptions and Properties of Estimators: frequently asked questions

What does BLUE mean in regression?

BLUE stands for best linear unbiased estimator. Among all estimators that are linear in Y and unbiased, OLS has the smallest variance. This holds under the Gauss-Markov assumptions.

What is the difference between an unbiased and a consistent estimator?

Unbiased means the average of the estimates across repeated samples equals the true parameter at a given sample size. Consistent means the estimate gets arbitrarily close to the true value as the sample grows. An estimator can have one without the other.

Does the Gauss-Markov theorem require normally distributed errors?

No. It needs linearity, zero conditional mean, no perfect multicollinearity, homoskedasticity and no serial correlation. Normality is only used for exact small-sample inference.

What happens to OLS if the errors are heteroskedastic?

The coefficient estimates remain unbiased and consistent, but OLS is no longer the minimum variance linear estimator. The usual standard errors are wrong, so t-tests and confidence intervals can mislead. Robust standard errors are the common fix.