In regression analysis, the **omitted-variable-bias** is the error that is incurred on **partial-effects-coefficients **of other explanatory variables in a **restricted regression model**. Assume a simple regression model, where Variable $y_i$ is explained by the Variable $x_{1i}$ and the error term $e_i$ for $i=[1,2,3, … , n]$ observations:

Then consider the hypothesis, that a Variable $x_{2i}$ explains the dependent variable $y_i$ and can be depicted by the following extended regression model:

Setting both equations equal and solving for the error term of the simple:

The error term $e_i$ in the simple regression model includes the deviation of $\tilde\beta_0$ and $\tilde\beta_2$ of the extended regression model from the former coefficients. The partial effects of the omitted variables $\tilde\beta_2$ and the error term of the extended regression model $v_i$ are also included in the error term of the simple regression model. Two factors play a role in the quantification of the omitted-variable-bias:

**Partial effects**of the omitted-variable on the explained variable.**Correlation and Covariance**of the omitted variable with the rest of the explanatory variables

## Partial effects of Omitted Variable and Correlation with Other Explanatory Variables

Two outcomes are possible: either there is **no bias** or there is a **positive bias** or **negative bias** on the partial effects of other explanatory variables in the restricted model.

### A. No Bias Scenario

If the omitted-variable has **zero partial effects** in the unrestricted model or **zero correlation/covariance** (independence between explanatory variables) there is no bias incurred on other partial effects in the restricted model.

### B. Negative Bias Scenario

**Negative (positive) partial effects** of omitted-variable and **positive (negative) correlation with other explanatory variables** ** simultaneously** leads to a negative bias on the partial effects of other partial effects of explanatory variables in the restricted model. In this case the signs are in opposite terms (+ and – ).

### C. Positive Bias Scenario

**Positive partial effects** of omitted-variable and **positive correlation with other explanatory variables** ** simultaneously** lead to a negative bias on the partial effects of other partial effects of explanatory variables in the restricted model. Similarly, if we simultaneously have negative signs. In this case we have two possible constellations ( + and +) or (- and – ).