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 .
Correcting for Omitted-Variables and Measurement-Error Bias in Regression with an Application to the Effect of Lead on IQ
Jun 01, 1998; The conceptual and practical obstacles to characterizing cause and effect in nonexperimental as well as experimental data through...
Comment: Problems with Using Auxiliary Information to Correct Omitted Variables When Estimating the Effect of Lead on IQ
Jun 01, 1998; 1. INTRODUCTION It is well known that omitting relevant variables when using ordinary least squares (OLS) regression leads to...