What is Omitted Variable Bias, How it Happens and How to Determine Its Direction

What Is Omitted Variable Bias?

Formally, if the true model is:

\[ Y = \beta_0 + \beta_1 X + \beta_2 Z + u \]

but you estimate:

\[ Y = \beta_0 + \tilde{\beta}_1 X + u \]

then the estimated coefficient is:

\[ \tilde{\beta}_1 = \beta_1 + \beta_2 \cdot \frac{\text{Cov}(X,Z)}{\text{Var}(X)} \]

The second term is the bias.

Why Omitted Variable Bias Happens

1. The omitted variable affects the outcome

If \(Z\) influences \(Y\), leaving it out means its effect gets absorbed into the error term.

2. The omitted variable is correlated with the included regressor

If \(Z\) is correlated with \(X\), then part of Z’s effect is incorrectly attributed to X.

3. The error term becomes correlated with X

This violates the zero conditional mean assumption:

\[ E[u \mid X] = 0 \]

Once this assumption fails, OLS is biased and inconsistent.

4. The regression “blames” X for Z’s effect

The coefficient on X shifts to compensate for the missing variable.

5. The direction of bias is predictable

The sign of the bias depends on:

• the sign of \( \beta_2 \) (Z’s effect on Y)
• the sign of \( \text{Cov}(X,Z) \)

How to Determine the Direction of Omitted Variable Bias

Step 1: Identify the omitted variable Z

Ask: “What variable is missing that affects Y and is correlated with X?”

Step 2: Determine the sign of Z’s effect on Y

Is \( \beta_2 \) positive or negative? Does Z increase or decrease Y?

Step 3: Determine the sign of the correlation between X and Z

Is X positively or negatively related to Z?

Step 4: Multiply the signs

The direction of bias is:

\[ \text{Bias} = \beta_2 \cdot \text{sign}(\text{Cov}(X,Z)) \]

If the product is:

• positive → upward bias
• negative → downward bias

Step 5: Interpret the result

If the bias is upward, the estimated coefficient is too large. If downward, it is too small.

Step 6: Compare the true and estimated coefficients

\[ \tilde{\beta}_1 > \beta_1 \quad \text{(upward bias)} \] \[ \tilde{\beta}_1 < \beta_1 \quad \text{(downward bias)} \]

Numerical Example

Suppose the true model is:

\[ Y = \beta_1 X + \beta_2 Z + u \]

Let:

• \(\beta_2 = +4\) (Z increases Y)
• \(\text{Cov}(X,Z) > 0\) (X and Z are positively correlated)

Then:

\[ \text{Bias} = (+4) \cdot (+) = \text{positive} \]

The estimated coefficient on X will be too large.

If instead \(\beta_2 > 0\) but \(\text{Cov}(X,Z) < 0\), the bias would be negative.