Winsorization Impact on OLS

A return $R$ is drawn from a two-component normal mixture: with probability $0.99$, $R \sim N(0, 1)$, and with probability $0.01$, $R \sim N(0, 25)$. The two components are independent. Define the winsorized return as $R^{(c)} = \text{sign}(R) \cdot \min(|R|, c)$ with threshold $c = 3$. (a) Express $E[R^{(c)}]$ and $\text{Var}(R^{(c)})$ in terms of normal CDFs, densities, and the mixture weights. You may leave standard normal tail integrals unevaluated (e.g., $\Phi(\cdot)$, $\phi(\cdot)$). (b) Qualitatively but precisely, explain how winsorization changes: - The estimated Sharpe ratio when $R$ is used as the return series. - The stability of OLS coefficient estimates when $R$ is the target (dependent) variable.