Estimate Factor Returns and Residual Returns

# 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? ### What a Strong Answer Covers - The weighted least-squares objective and factor-return estimator - Rank, conditioning, constraints, and regularization - Universe residuals computed with the same exposure definitions - Point-in-time data alignment, diagnostics, and implementation complexity ### Follow-up Questions 1. How would you handle a nearly singular weighted exposure matrix? 2. Why might large-cap weights improve one factor estimate but hurt representativeness? 3. What changes when returns are observed across many dates? ```hint Express the estimate as weighted least squares Fit the cross section on the estimation sample, then apply the fitted factor returns to the larger universe before subtracting. ```