XLE/USO energy equity-vs-crude OU log-spread reversion (USEQ daily, dollar-neutral, point-in-time OU fit)

Translation of Mahajan et al. (2019), 'Stochastic Spread Pairs Trading in the Indian Commodity Market' (arXiv:1907.08397, abstract: cointegrated commodity pairs, single-factor stochastic (OU) model on the LOG spread, trading-rule parameters fitted on training data). MCX spot commodities are not on our venues, so the closest backtestable commodity pair with long history is used: XLE.USEQ (integrated oil and gas equities) against USO.USEQ (front-month WTI crude ETF), both on 1-DAY bars since 2006 (~20 years, ~5,100 aligned sessions). Economic tie: XLE constituents' cash flows are driven by crude, so the log spread log(XLE) - beta*log(USO) is mean-reverting in the medium run. Short-run gaps open from equity-specific flows, crude futures roll and contango shocks, and these close as the equity market re-prices oil exposure. Point-in-time implementation of the paper's model: (1) hedge ratio beta = rolling 252-session OLS slope of daily log returns of XLE on USO, lagged one bar. (2) spread s_t = log(XLE_t) - beta_{t-1}*log(USO_t). (3) the OU parameters (theta = long-run mean, sigma_eq = stationary std, kappa from the AR(1) coefficient) are fitted by AR(1) regression of s on its lag over the last `z_window` sessions, using completed bars only. Closed-form AR(1) is used instead of the paper's differential-evolution fit; it is equivalent for a single-factor OU and adds no compute. (4) z = (s_t - theta)/sigma_eq. Long-short, dollar-neutral, 2 legs, flat most of the time. Crypto alternatives were checked first and rejected on data. Binance/OKX gold and silver perps (XAU/XAG/XPT/COPPER) have only ~7-10 months of 1h history. Gold-token spreads (XAU vs PAXG vs XAUT on Binance, XAG Binance vs OKX) have hourly spread std of 3-6 bps and lost money after fees in every variant (avg trade -0.01% to -0.06%). The pre-study on USEQ daily (rolling beta, 60-session z, entry |z|>1.5-2.0, exit at the mean, 20-session cap, 6 bps cost) gives XLE/USO about 100-158 trades, avg trade +0.51% to +0.58%, win rate ~0.6, Sharpe ~0.30-0.33, positive in 60-70% of years. HONEST CAVEAT: XLE/USO was the best of 21 commodity-ETF pairs scanned (GLD, SLV, GDX, XLE, XOP, USO, UNG). GLD/SLV and GLD/GDX, the 'textbook' commodity pairs, had no edge, so selection bias is material and the expected OOS Sharpe is modest. What the full paper would have to confirm: the exact trading-rule form (entry/exit bands in OU-sigma units, any stop-loss band), whether the bands were fixed or optimized per pair, and whether the reported Sharpe > 1.4 is net of costs. The abstract says the parameters were 'optimized by backtesting on the training period', so the 1.4 is likely an in-sample-tuned figure. source_ref: arXiv:1907.08397 abstract (method paragraph: Johansen cointegration selection + single-factor stochastic spread on log spread + trading rule tuned on the 80% training split).