Omitted-variable Bias - Omitted-variable Bias in Linear Regression

Omitted-variable Bias in Linear Regression

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

• x_i is a 1 × p row vector, and is part of the observed data;
• β is a p × 1 column vector of unobservable parameters to be estimated;
• z_i is a scalar and is part of the observed data;
• δ is a scalar and is an unobservable parameter to be estimated;
• the error terms u_i are unobservable random variables having expected value 0 (conditionally on x_i and z_i);
• the dependent variables y_i are part of the observed data.

We let

and

Then through the usual least squares calculation, the estimated parameter vector based only on the observed x-values 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,

On taking expectations, the contribution of the final term is zero; this follows from the assumption that U has zero expectation. On simplifying the remaining terms:

The second term above is the omitted-variable bias in this case. Note that the bias is equal to the weighted portion of z_i which is "explained" by x_i.