Explanation

Omitted variable bias occurs in regression analysis when a relevant variable is excluded from the model, leading to biased and inconsistent estimates of the regression coefficients. For omitted variable bias to occur, two key conditions must be met:

Condition II: At least one of the included regressors must be correlated with the omitted variable

• If the omitted variable is uncorrelated with all included regressors, then omitting it does not cause bias in the coefficient estimates.
• The correlation creates a situation where the included variable is "picking up" some of the effect of the omitted variable.

Condition III: The omitted variable must be a determinant of the dependent variable

• If the omitted variable does not affect the dependent variable, then omitting it doesn't matter.
• The omitted variable must be relevant to explaining variation in the dependent variable.

Why the other conditions are not necessary:

Condition I (R² vs adjusted R²): This is irrelevant to omitted variable bias. R² and adjusted R² are measures of model fit, not conditions for bias.

Condition IV (Homoskedasticity): Homoskedasticity relates to the variance of errors, not omitted variable bias. Heteroskedasticity affects efficiency of estimates, not bias from omitted variables.

Condition V (Number of regressors ≤ 5): This is arbitrary and irrelevant. Omitted variable bias can occur regardless of the number of included regressors.

Key Insight:

Omitted variable bias is essentially a problem of endogeneity - when an explanatory variable is correlated with the error term. When a relevant variable is omitted and correlated with included variables, that omitted variable's effect becomes part of the error term, creating correlation between the included variable and the error term.

Mathematically: If the true model is: $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$ But we estimate: $Y = \beta_0 + \beta_1 X_1 + u$ where $u = \beta_2 X_2 + \epsilon$

Then $\hat{\beta}_1$ will be biased if $X_1$ and $X_2$ are correlated, because $\text{Cov}(X_1, u) = \beta_2 \text{Cov}(X_1, X_2) \neq 0$.