Design Features for Residual Volatility

Problem Statement

You are tasked with building a feature pipeline to predict the residual (idiosyncratic) volatility of a set of 500 equities. We define residual volatility as the volatility of an asset's returns after removing the impact of systematic market factors (e.g., market beta, sector exposure).

Suppose you have the following data for a universe of stocks N over time T:

• R_{i,t}: Daily return of stock i at time t.
• F_t: A vector of market factors (e.g., market index, sector indices).

Task

1. Propose a methodology to extract the 'residual return' epsilon_{i,t} for each stock.
2. Design a feature set that could predict the k-day forward-looking realized residual volatility.
3. Discuss how you would handle the 'Curse of Dimensionality' if you decide to incorporate cross-sectional correlations into your feature set.

Constraints

• Your residual calculation should account for non-stationary beta (time-varying sensitivity to market factors).
• The feature engineering pipeline must be robust to the 'look-ahead bias' inherent in financial time-series.

Example

If R_{i,t} is the observed return and B_i * F_t is the predicted return from a factor model, your feature generator should take epsilon_{i,t} = R_{i,t} - B_i * F_t as input to compute rolling windows of standard deviation or other distributional signatures (skew, kurtosis).