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Omitted-variable bias
 
 
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Omitted-variable bias (OVB) is the bias that appears in estimates of parameter
Parameter

In mathematics, statistics, and the mathematical sciences, a parameter is a quantity that defines certain characteristics of systems or function s....
s in a regression analysis
Regression analysis

In statistics, regression analysis is a collective name for techniques for the modeling and analysis of numerical data consisting of values of a dependent variable and of one or more independent variables ....
 when the assumed specification
Specification (regression)

In regression analysis and related fields such as econometrics, specification is the process of converting a theory into a regression model. This process consists of selecting an appropriate function for the model and choosing which variables to include....
 is incorrect, in that it omits an independent variable that should be in the model.

conditions must hold true for omitted variable bias to exist in linear regression:

As an example, consider a linear model of the form , where is treated as a vector and is a scalar.






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Omitted-variable bias (OVB) is the bias that appears in estimates of parameter
Parameter

In mathematics, statistics, and the mathematical sciences, a parameter is a quantity that defines certain characteristics of systems or function s....
s in a regression analysis
Regression analysis

In statistics, regression analysis is a collective name for techniques for the modeling and analysis of numerical data consisting of values of a dependent variable and of one or more independent variables ....
 when the assumed specification
Specification (regression)

In regression analysis and related fields such as econometrics, specification is the process of converting a theory into a regression model. This process consists of selecting an appropriate function for the model and choosing which variables to include....
 is incorrect, in that it omits an independent variable that should be in the model.

Omitted-variable bias in linear regression
Linear regression

In statistics, linear regression is used for two things;Linear regression is a form of regression analysis in which the relationship between one or more independent variables and another variable, called the dependent variable, is modeled by a least squares function, called linear regression equation....

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 variable
    Independent variable

    The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects....
    s.


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 .