Definitions

omitted

Omitted-variable bias

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.

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 y_i = x_i beta + z_i delta + u_i, where x_i is treated as a vector and z_i is a scalar. For simplicity suppose that E[u_i|x_i,z_i]=0. Now consider what happens if one were to regress y_i on only x_i. Through the usual least squares calculus, the estimated parameter vector hat{beta} is given by:

hat{beta} = (x'x)^{-1}x'y.,

Substituting for y based on the assumed linear model,

hat{beta} = (x'x)^{-1}x'(xbeta+zdelta+u)=(x'x)^{-1}x'xbeta + (x'x)^{-1}x'zdelta + (x'x)^{-1}x'u.,

Taking expectations, the final term (x'x)^{-1}x'u falls out by the assumed conditional expectation above. Simplifying the remaining terms:

E[hat{beta} ] = beta + delta (x'x)^{-1}x'z.,

The above is an expression for 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.

References

  • Greene, WH Econometric Analysis, 2nd ed.. Macmillan.

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