**4. Back-Testing Framework**

The empirical back-testing study was performed from January 1998–December 2015 at intraday prices for the S&P 500 index components (see Section 4.1). According to Gatev et al. (2006) and Nakajima (2019), we divided the dataset into overlapping study periods, which were shifted by one day each. Each study period consisted of two consecutive phases. In the formation period (Section 4.2), the most appropriate stocks were selected using predefined models and criteria. In the subsequent out-of-sample trading period (Section 4.3), the top stocks were traded using rule-based entry and exit signals; this procedure avoids any look-ahead bias. Summarizing, we developed a full-fledged statistical arbitrage framework based on a jump–diffusion model (JDS), which is able to capture intraday and overnight high-frequency price dynamics.
