Omitted Variable Bias Formula in Multiple Regression

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Econometrics / Linear Regression.

Your Task
Suppose the true data-generating process is $y = X_1 \beta_1 + X_2 \beta_2 + \epsilon$, with $\mathbb{E}[\epsilon \mid X_1, X_2] = 0$. However, an analyst omits $X_2$ and runs the short regression $y = X_1 \tilde{\beta}_1 + u$.
(1) Derive the Omitted Variable Bias (OVB) formula for $\mathbb{E}[\tilde{\beta}_1 \mid X_1, X_2]$.
(2) State the two distinct conditions under which the omitted variable bias is exactly zero.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.