Omitted-variable bias

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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.

[edit] 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 $y i = x i β + z i δ + u i$ , 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,

$\hat{\beta} = (x'x)^{-1}x'(x\beta+z\delta+u)=(x'x)^{-1}x'x\beta + (x'x)^{-1}x'z\delta + (x'x)^{-1}x'u.\,$

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 $x i$ .