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# 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 $E\left[u_i|x_i,z_i\right]=0$. 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 $hat\left\{beta\right\}$ is given by:

$hat\left\{beta\right\} = \left(x\text{'}x\right)^\left\{-1\right\}x\text{'}y.,$

Substituting for y based on the assumed linear model,

$hat\left\{beta\right\} = \left(x\text{'}x\right)^\left\{-1\right\}x\text{'}\left(xbeta+zdelta+u\right)=\left(x\text{'}x\right)^\left\{-1\right\}x\text{'}xbeta + \left(x\text{'}x\right)^\left\{-1\right\}x\text{'}zdelta + \left(x\text{'}x\right)^\left\{-1\right\}x\text{'}u.,$

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

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

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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