Examples
Worked examples
- Is an instance
A regression of wages on years of education, omitting cognitive ability. Ability plausibly affects wages directly and is correlated with education (higher-ability people tend to complete more schooling), so the estimated return to education absorbs part of ability's effect, typically biasing it upward.
- Is an instance
A regression of a city's crime rate on its number of police officers, omitting population density and poverty rate. Both omitted factors are plausibly correlated with staffing decisions and independently affect crime rate, so the estimated police-crime relationship is confounded by both.
Counter-examples
Looks similar, but isn't
- Not an instance
A regression estimating the effect of a randomly assigned scholarship on graduation rates, omitting family income. Income independently affects graduation, but because the scholarship was randomly assigned, income is uncorrelated with scholarship receipt -- condition (2) fails, so omitting it does not bias the scholarship's estimated effect, even though income is a real and omitted determinant of the outcome.
Editorial commentary
Omitted variable bias (OVB) is the bias in a regression coefficient that results from leaving a relevant variable out of the model. It is the regression-specific name for the same problem epidemiologists call confounding: an unmodeled factor is quietly doing part of the work your included variable is getting credit for.
OVB requires both of the following to hold for the omitted variable, not just one:
- It has its own effect on the outcome (dependent variable), independent of the variable you included.
- It is correlated with an included independent variable.
If either condition fails, leaving the variable out does not bias your coefficient — it may still be a worse model in other ways (lower explanatory power, wider standard errors), but the coefficient on the variable you did include stays an unbiased estimate of that variable’s own effect. This is the detail that gets skipped in casual explanations of OVB, and it is exactly what the worked counter-example below is built to show.
Why It Happens Mechanically
If the true relationship is y = β0 + β1x + β2w + ε and a regression is run leaving w out, w‘s influence does not vanish — it gets absorbed into the error term of the short regression. If x and w are correlated, the regressor is now correlated with that error term, which is exactly the condition ordinary least squares needs to be violated for its coefficient to come out biased. The bias term is β2 × the correlation between x and w: its sign is the product of two signs (the omitted variable’s effect on the outcome, and its correlation with the included variable), which means the direction of the bias can often be reasoned about even when the omitted variable itself can’t be measured or included.
Detecting and Addressing It
Because the omitted variable is, by definition, not in the dataset, OVB does not show up in ordinary diagnostics — R-squared, residual plots, and significance tests all look normal even when a coefficient is badly biased. The practical responses are: include the omitted variable if it can be measured at all (even a rough proxy), use a research design that removes it structurally (panel data with fixed effects for anything time-invariant, an instrumental variable for anything that varies over time, or a natural experiment), or reason explicitly about the likely direction and size of the bias rather than ignoring it. A full derivation, a worked numerical simulation, and the fixes that do and do not actually work are covered in endogeneity: the three sources and the remedy that matches each, where omitted-variable bias is one of three related problems (alongside reverse causality/simultaneity and measurement error) that all produce the same statistical symptom — a regressor correlated with the error term — through different mechanisms.
Machine-readable encodings
Use in your systems
<role vocab="credit"
vocab-identifier="https://casrai.org/dictionary/"
vocab-term="Omitted Variable Bias"
vocab-term-identifier="https://casrai.org/dictionary/term/omitted-variable-bias" />{
"@context": "https://schema.org",
"@type": "DefinedTerm",
"@id": "https://casrai.org/dictionary/term/omitted-variable-bias",
"name": "Omitted Variable Bias",
"identifier": "https://casrai.org/dictionary/term/omitted-variable-bias",
"description": "Omitted variable bias (OVB) is the bias in a regression coefficient that occurs when a variable is left out of the model that (1) has its own effect on the dependent variable, independent of the variable(s) included, and (2) is correlated with an included independent variable. Because the omitted variable's influence does not disappear, it is absorbed into the error term; since that error term is now correlated with the regressor, the coefficient on the included variable no longer isolates that variable's own effect on the outcome -- part of what it measures is really the omitted variable's effect, passed through the correlation between the two. Both conditions must hold: a variable correlated with the regressor but with no independent effect on the outcome produces no bias, and a variable that affects the outcome but is uncorrelated with the regressor (for example, because the regressor was randomly assigned) produces no bias either, no matter how important that variable otherwise is.",
"inDefinedTermSet": "https://casrai.org/dictionary/domain/research-outputs#set",
"url": "https://casrai.org/dictionary/term/omitted-variable-bias",
"sameAs": [],
"license": "https://creativecommons.org/licenses/by/4.0/",
"publisher": {
"@id": "https://casrai.org/#organization"
},
"author": {
"@id": "https://casrai.org/#editorial-team"
},
"datePublished": "2026-08-31T20:09:32",
"dateModified": "2026-08-31T20:09:32",
"inLanguage": "en-GB",
"isAccessibleForFree": true
}
