FRM Exam Part I · Regression Diagnostics

Omitted Variable Bias and Model Specification in Regression

Updated 11 October 2026 · Fact-checked

Omitted variable bias occurs when you leave out a variable that affects Y and is correlated with an included regressor. The OLS slope then absorbs part of the missing effect and becomes biased and inconsistent. Adding irrelevant variables does not cause bias but raises variance. Good specification balances bias against variance.

Understand Model Specification and Omitted Variable Bias

Every regression is a simplified model of reality. Model specification is the choice of which variables to include and in what form. A wrong choice can damage your coefficient estimates.

Omitted variable bias (OVB) arises when two conditions hold together. First, the left-out variable truly affects Y. Second, it is correlated with at least one included regressor. Then the included regressor picks up some of the omitted variable's effect. The error term now contains the omitted variable, so the regressor is correlated with the error. This breaks the OLS assumption that errors are uncorrelated with regressors. The estimate is biased, and the bias does not shrink as the sample grows, so it is also inconsistent. If either condition fails, there is no bias. An omitted variable that is uncorrelated with your regressors does not bias the slopes, though it can raise the residual variance.

The opposite error is including an extraneous (irrelevant) variable. Its true coefficient is zero. OLS stays unbiased and consistent. But the estimates become less efficient: standard errors rise, especially if the extra variable is correlated with other regressors. Adjusted R-squared and t-tests help you spot such variables.

This creates the bias-variance trade-off. A small model risks bias from omitted variables. A large model has lower bias but higher variance, and it may overfit the sample, forecasting poorly out of sample. Selection tools such as adjusted R², AIC and BIC penalize extra parameters. BIC penalizes more heavily than AIC, so it tends to choose smaller models.

Key formulas to remember

OVB in a two-variable case: True: Y = β0 + β1·X1 + β2·X2 + ε. Omit X2: E[β1_hat] = β1 + β2 · δ, where δ = Cov(X1, X2) ÷ Var(X1); Bias = β2 · δ. δ is the slope from regressing X2 on X1. Sign of bias = sign of β2 times sign of the correlation.
Conditions for OVB: β2 ≠ 0 AND Corr(X1, X2) ≠ 0; Both are needed. If either is zero, the slope on X1 is unbiased.
Effect of an irrelevant variable: Bias = 0; Var(β1_hat) increases when the extra variable is correlated with X1; Unbiased but less efficient.
Adjusted R²: Adj R² = 1 − [(n − 1) ÷ (n − k − 1)] × (1 − R²); k = number of regressors excluding the intercept. It can fall when a useless variable is added.
Information criteria: AIC = ln(SSR ÷ n) + 2k ÷ n; BIC = ln(SSR ÷ n) + k·ln(n) ÷ n; Lower is better. BIC penalty is larger than AIC when n ≥ 8.

How to solve Model Specification and Omitted Variable Bias questions

Use this routine for any question on omitted, irrelevant or mis-specified variables.

1Identify what the true model contains and what the estimated model contains.
2Classify the problem: omitted relevant variable, included irrelevant variable, or both.
3For an omitted variable, check the two conditions: does it affect Y (β2 ≠ 0), and is it correlated with an included regressor?
4If both hold, compute the bias as β2 × δ and find its sign by multiplying the signs.
5If the variable is irrelevant, state: no bias, consistent, but larger standard errors (less efficient).
6For model selection, compare adjusted R², AIC or BIC and note which penalizes complexity more.
7State the conclusion in terms of bias, consistency and efficiency, and link to the bias-variance trade-off.

Quickest way: Two-question screen

When to use it: Multiple-choice questions asking whether an estimator is biased or which specification error occurred.

Ask: was a relevant variable left out? If no, there is no bias from omission.
If yes, ask: is it correlated with an included regressor? If no, no bias.
If both yes, bias = true coefficient of the omitted variable × its correlation direction with the included variable.
If the issue is an extra variable with zero true effect, choose the answer: unbiased but inefficient.
Eliminate options that say irrelevant variables cause bias or that omitted-variable bias disappears with a larger sample.

Common mistakes in Model Specification and Omitted Variable Bias

Saying any omitted variable causes bias.
Students forget the correlation condition.
Fix: Always check both conditions: the variable affects Y and correlates with an included regressor.
Claiming irrelevant variables bias the coefficients.
Mixing up the two types of specification error.
Fix: Irrelevant variables leave OLS unbiased and consistent; they only inflate variance.
Getting the sign of the bias wrong.
Students look at only one sign, not the product.
Fix: Sign of bias = sign(β2) × sign(Corr(X1, X2)). Positive × negative gives downward bias.
Thinking a bigger sample fixes omitted variable bias.
Confusing sampling error with bias.
Fix: OVB makes the estimator inconsistent. More data converges to the wrong value.
Choosing the model with the highest R².
R² never falls when variables are added.
Fix: Use adjusted R², AIC or BIC, which penalize extra parameters.

Worked examples

Example 1

The true model is Y = 2 + 3·X1 + 4·X2 + ε. Regressing X2 on X1 gives a slope of 0.5. You estimate Y on X1 only. What is the expected value of the estimated slope on X1, and is it biased?

Show the solution

Bias = β2 × δ = 4 × 0.5 = 2.
Expected estimate = β1 + bias = 3 + 2 = 5.
Since 5 ≠ 3, the estimator is biased upward (and inconsistent).

Answer: The expected slope is 5, an upward bias of 2.

Example 2

An analyst regresses fund returns on the market factor and adds a third, irrelevant factor that is highly correlated with the market. Which statement is correct: (A) market beta is biased upward; (B) market beta is unbiased but has a larger standard error; (C) market beta is biased downward; (D) market beta is inconsistent?

Show the solution

The extra factor has a true coefficient of zero, so it is irrelevant, not omitted.
Including an irrelevant variable does not violate the zero-conditional-mean assumption, so OLS remains unbiased and consistent.
High correlation with the market factor increases the variance of the beta estimate.
This rules out A, C and D.

Answer: B: the beta is unbiased but less efficient.

Exam tips

Memorize the two conditions for OVB; many questions test whether you notice that one is missing.
Expect a sign question: multiply the sign of the omitted coefficient by the sign of the correlation.
Remember the pairing: omitted relevant variable gives bias; irrelevant variable gives inefficiency.
If asked which criterion favors smaller models, answer BIC.
Do not pick a model on R² alone; it always rises with more variables.

Practice questions from Regression Diagnostics

Model Specification and Omitted Variable Bias: frequently asked questions

What is omitted variable bias in simple terms?

It is the error in a coefficient that appears when you leave out a relevant variable that is correlated with your included regressor. The included variable gets credit for the missing variable's effect.

How is an extraneous variable different from an omitted variable?

An omitted variable is relevant but left out, which biases estimates. An extraneous variable is included but has no true effect, which leaves estimates unbiased but raises their variance.

What is the bias-variance trade-off in model selection?

Simple models can have high bias because they miss relevant variables. Complex models have low bias but high variance and risk overfitting. You choose a model that balances the two, often using adjusted R², AIC or BIC.

Can a larger sample remove omitted variable bias?

No. The estimator is inconsistent, so it converges to the wrong value as the sample grows. Only adding the missing variable, or using a method that handles it, fixes the problem.