Estimate Factor Returns and Residual Returns

At one date, a weighted estimation sample contains stock returns, factor exposures, and estimation weights. A larger trading universe contains returns and exposures to the same factors. Derive the factor-return estimate and show how to compute residual returns for every stock in the trading universe. Give formulas or pseudocode and state the conditions needed for a stable estimate.

Constraints & Assumptions

• Let r_s be the sample return vector, B_s its exposure matrix, and W a diagonal matrix of nonnegative estimation weights.
• Let r_u and B_u describe the larger universe with identical factor definitions and column order.
• Returns and exposures are measured at compatible times; no look-ahead data may be used.
• Industry or categorical factor columns can be collinear with an intercept unless an identification constraint is imposed.

Clarifying Questions to Ask

• Are returns raw, excess, or already adjusted for market and corporate actions?
• Do weights represent inverse variance, market capitalization, liquidity, or another choice?
• Which intercept, industry, and style-factor constraints define an identifiable model?
• How should missing exposures and stocks outside the estimation sample be handled?