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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.
## Omitted-variable bias in linear regression

Two conditions must hold true for omitted variable bias to exist in linear regression:## References

- 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:

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

Substituting for *y* based on the assumed linear model,

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

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

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

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 $x\_i$.

- Greene, WH
*Econometric Analysis, 2nd ed.*. Macmillan.

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Last updated on Thursday June 12, 2008 at 10:44:05 PDT (GMT -0700)

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Last updated on Thursday June 12, 2008 at 10:44:05 PDT (GMT -0700)

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