Omitted Variables

Motivation. What if there are external variables that are related to $Y$? We want to characterize how off we might be if we omitted a variable.

Omitting a Variable from a True 2-var Model §

Assume the true model should have the following:

$$Y_i={\beta }_0+{\beta }_1{X}_i+{\beta }_2\underset{\text{ Omitted }}{\underbrace{Z_i}}+{ϵ}_i$$

We didn’t take into account $Z$, and we mistakenly used the following model:

$$Y_i^{\prime }={\beta }_0^{\text{omit}}+{\beta }_1^{\text{omit}}{X}_{1,i}+{ϵ}_i$$

Then, our estimators ${\widehat{{\beta }_0}}^{\text{omit}}$ and ${\widehat{{\beta }_1}}^{omit}$ will be different from $\widehat{{\beta }_0},\widehat{{\beta }_1}$. We can state precisely how off they will be (=Omitted Variable Bias):

$${\widehat{{\beta }_1}}^{\text{omit}}=\widehat{{\beta }_1}+\underset{\text{ Bias }}{\underbrace{\widehat{{\beta }_2}\widehat{{\delta }_1}}}$$

where $\widehat{{\delta }_1}$ is from the auxiliary regression

$$Z_i=\widehat{{\delta }_0}+\widehat{{\delta }_1}{X}_i+{\tau }_i$$

Omitting a Variable from a 3-var Model §

• True model: $\widehat{Y_i}=\widehat{{\beta }_0}+\widehat{{\beta }_1}{X}_{1,i}+\widehat{{\beta }_2}{X}_{2,i}+\widehat{{\beta }_3}{X}_{3,i}+\widehat{{ϵ}_i}$
• Omitted model: $\widehat{Y_i^{\prime }}=\widehat{{\beta }_0^{\prime }}{X}_{1,i}+\widehat{{\beta }_2^{\prime }}{X}_{2,i}+\widehat{{ϵ}_i^{\prime }}$
• Estimator relationships:
  • $\widehat{{\beta }_1^{\prime }}=\widehat{{\beta }_1}+\widehat{{\beta }_3}\widehat{{\delta }_1}$
  • $\widehat{{\beta }_2^{\prime }}=\widehat{{\beta }_2}+\widehat{{\beta }_3}\widehat{{\delta }_2}$
• Auxiliary regression: $X_3=\widehat{{\delta }_0}+\widehat{{\delta }_1}{X}_1+\widehat{{\delta }_2}{X}_2$

General Form §

For true model:

$$\widehat{Y_i}=\widehat{{\beta }_0}+\widehat{{\beta }_1}{X}_{1,i}+\cdots +\widehat{{\beta }_{n-1}}{X}_{n-1,i}+\widehat{{\beta }_n}{X}_n$$

where $X_n$ is the omitted variable. Our omitted model:

$$\widehat{Y_i}$$ $$\widehat{{\beta }_i^{\prime }}=\widehat{{\beta }_i}+{\hat{\beta }}_{\text{omit}}\widehat{{\delta }_i}$$

where

$$X_{\text{omit}}=\widehat{{\delta }_0}+\widehat{{\delta }_1}{X}_2+\cdots +{\hat{\delta }}_{n-1}{X}_{n-1}$$