Financial Risk Manager Part 1

Explanation

When an important variable is omitted from a regression model, the assumption that E(εᵢ|Xᵢ) = 0 is violated. This is one of the key assumptions of the multiple linear regression model, which states that the conditional distribution of the error term εᵢ given the independent variables Xᵢ has a mean of zero.

Why this matters:

1. Violation of the zero conditional mean assumption: When an important explanatory variable is omitted, the error term εᵢ now contains the effect of that omitted variable. Since the omitted variable is likely correlated with the included variables, the error term is no longer independent of the included X variables.

2. Consequences:

   • Bias in OLS estimator: The OLS estimator becomes biased and inconsistent. The bias occurs because the omitted variable's effect gets absorbed into the error term, and if that omitted variable is correlated with the included variables, the estimated coefficients will be biased.
   • Endogeneity problem: The error term becomes correlated with the explanatory variables, violating the exogeneity assumption.

3. Why other options are incorrect:

   • Option B: The product of residuals and independent variables being zero is a property of OLS estimation, not an assumption about the population model.
   • Option C: The sum of residuals being zero is a property of OLS estimation with an intercept, not a population assumption.
   • Option D: The coefficient of determination (R²) being zero is unrelated to omitted variable bias; R² measures goodness of fit.

Key Insight:

The violation of E(εᵢ|Xᵢ) = 0 due to omitted variable bias means that the OLS estimator no longer provides unbiased estimates of the true population parameters. This is a fundamental problem in regression analysis that can lead to incorrect inferences about relationships between variables.