Omitted-variable 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 that should be in the model.
Two conditions must hold true for omitted variable bias to exist in linear regression:
- the omitted variable must be a determinant of the dependent variable (i.e., its true regression coefficient is not zero); and
- the omitted variable must be correlated with one or more of the included independent variables.
As an example, consider a linear model of the form , where is treated as a vector and is a scalar. For simplicity suppose that . Now consider what happens if one were to regress on only . Through the usual least squares calculus, the estimated parameter vector is given by:
Substituting for y based on the assumed linear model,
Taking expectations, the final term falls out by the assumed conditional expectation above. Simplifying the remaining terms:
The above is an expression for the omitted variable bias in this case. Note that the bias is equal to the weighted portion of which is "explained" by .
- Greene, WH Econometric Analysis, 2nd ed.. Macmillan.