Cross-Sectional Factor Model Estimation and Diagnostics

Daily stock excess returns $r_t \in \mathbb{R}^N$ follow a linear factor model: $r_t = X_t \beta_t + \varepsilon_t$ where $X_t$ is an $N \times K$ matrix of standardized exposures to style factors (including Value), $\beta_t$ is the $K \times 1$ vector of factor returns on day $t$, and $\varepsilon_t$ is the vector of idiosyncratic errors. 1. Describe how to estimate $\beta_t$ via cross-sectional OLS each day and how to compute $t$-statistics for the factor return estimates over a multi-day horizon. 2. Propose a concrete procedure to detect multicollinearity among the factor exposures and to identify unstable factor loadings over time. 3. Identify one structural drawback of a purely linear factor model and describe how you would test for the presence of nonlinearity in the return-factor relationship.