In statistics, omittedvariable bias (OVB) is the bias that appears in estimates of parameters in a regression analysis when the assumed specification is incorrect, in that it omits an independent variable (possibly nondelineated) that should be in the model.
Two conditions must hold true for omittedvariable bias to exist in linear regression:
As an example, consider a linear model of the form
where
We let
and
Then through the usual least squares calculation, the estimated parameter vector based only on the observed xvalues but omitting the observed z values, is given by:
(where the "prime" notation means the transpose of a matrix).
Substituting for Y based on the assumed linear model,
Taking expectations, the final term
falls out by the assumption that U has zero expectation. Simplifying the remaining terms:
The second term above is the omittedvariable bias in this case. Note that the bias is equal to the weighted portion of z_{i} which is "explained" by x_{i}.

