Gold Miners vs Gold Beta-Hedged Residual Reversion: fade a >=2-sigma 10-session GDX-vs-GLD idiosyncratic move, dollar/beta-neutral long-short, 15-session time stop (USEQ 1-DAY)

Change the mechanism rather than retune parameters. 76 pair events are too few to optimize on, and the holdout would see only about 15. (1) Size the position continuously from z: target GDX notional = -clip(z/2, -1, 1) × G/(1+β), hedged with GLD at the estimated beta. Rebalance only when the target moves by more than 0.25 of full size. Aim for at least 60% time in market and more than 300 rebalance events. (2) Add a veto that sets the target to zero when the 60-day GDX/GLD return correlation is below 0.5, or when |z| is above 3.5 (the residual is trending rather than reverting). (3) Keep beta and z computed from prior sessions only, the 6% pair stop, and gross exposure at or below 0.8 × equity. (4) Set these targets in advance: full-history Sharpe at least 0.6, no single year above 40% of profit, and calm and normal regimes both non-negative. If the IC t-stat is still below 2, abandon the premise as falsified. ## Library refinements (from the knowledge library; test them, do not assume them) The library covers GLD-GDX directly. Hudson & Thames / ArbitrageLab use this exact pair to show stochastic-control pair trading, where the optimal spread weight is linear in the mispricing and is cut back outside a 'stabilization region'. That supports replacing the 2-sigma on/off trade with continuous z-proportional sizing plus a taper at extreme z. Pairs-trading reviews add a half-life-based time stop, a veto on idiosyncratic-news jumps and a dynamic (Kalman) hedge ratio. These all address the current weak result (Sharpe 0.23, PF 1.19, 76 pair events). 1. [sizing] Continuous z-proportional pair sizing instead of 2-sigma on/off: Drop the entry_z/exit_z state machine. Each session, after z is formed from prior-session stats, set target GDX notional = -clip(z/2.0, -1, 1) * G/(1+beta), with G = equity * 0.8, and hedge GLD at +beta times the GDX notional with the opposite sign. Rebalance both legs only when |target_frac - current_frac| > 0.25 of full size, or when the target crosses zero (go flat). Keep the 6% pair stop on GDX-leg entry notional (after a stop, the target is 0 for reentry_cooldown=3 sessions). Tunables: full_size_z in [1.5, 3.0] (default 2.0) and rebalance_band in [0.15, 0.35] (default 0.25). — In the Jurek-Yang and Mudchanatongsuk models (both shown on GLD/GDX), the optimal spread weight h*(t,x) contains the term -k(x-theta)/eta^2, which is linear in the mispricing. The 'optimal portfolio weights are dynamic', so the position scales with divergence rather than switching at one threshold. Our binary 2-sigma entry produced only 76 pair events, too few to optimize or holdout-test. A graded position puts capital to work at |z| 0.5-2 and multiplies the decision count, which is what the analyst's ≥300 rebalance target needs. Fees: 0.25-of-size steps on 0.8x gross cost little per step on USEQ, which has zero commission and only spread/impact costs. The library's equity curves are claims; our backtest decides. (source: Pairs Trading with Stochastic Control and OU process - Hudson & Thames p.1; ou model mudchanatongsuk (ArbitrageLab docs) p.1; ou model jurek (ArbitrageLab docs) p.1) 2. [regime] Stabilization-region taper: shrink the position when divergence is extreme: Multiply the target from refinement 1 by taper(|z|): 1.0 for |z| <= 3.0, linear down to 0 at |z| = 4.0, and 0 above that. Once the taper has cut the position to 0, do not re-open until |z| has fallen back below 3.0. All z values come from prior-session statistics only. — Jurek-Yang's result, as summarised in the article: 'there is a critical level of the mispricing beyond which further divergence ... precipitates a reduction in the allocation'. Outside the stabilization region 'the wealth effect dominates, leading the agent to curb his position'. That is a principled version of the analyst's |z|>3.5 veto. A residual that has already moved more than 3 sigma is more likely a regime break than noise, and that is where the 6% pair stop gets hit. The article illustrates stabilization bounds on GLD-GDX in Fig. 4, but that figure is not stored in the library, so I rely on the text only. (source: Pairs Trading with Stochastic Control and OU process - Hudson & Thames p.1; ou model jurek (ArbitrageLab docs) p.1) 3. [filter] Idiosyncratic-jump veto (news-driven divergence continues): Each session, compute sigma_d = std of daily residuals over the prior beta_lookback=60 sessions. If any single-day residual inside the current resid_window has |resid| > 4.0 * sigma_d, set the target to 0 and keep it at 0 until that day leaves the window. This does not stop an open position by itself; the target simply goes flat at the next rebalance. — Robot Wealth's literature review reports that pair-trading profit 'is related to idiosyncratic news. News affecting one of the stocks in isolation is predictive of a divergence that continues, rather than converges.' Our residual sum S cannot tell ten small drifts (flow pressure, which should revert) from one large one-day gap (a miner-sector shock, M&A or a guidance change, which may not revert). Vetoing windows that are dominated by one jump targets the losing trades directly. The library gives no threshold; 4 sigma is our pre-registered choice. (source: Pairs Trading Literature Review - Robot Wealth p.1) 4. [exit] Half-life-matched horizon: z window and time stop from the OU half-life: Each session, fit AR(1) to the cumulative residual S over the prior 250 sessions (dS_t = a + b*S_{t-1}) and compute HL = -ln(2)/ln(1+b). If b >= 0, treat the spread as non-reverting and set the target to 0. Set the time stop to max_hold = clip(round(2*HL), 5, 30) sessions: if the target has kept the same sign for longer than max_hold, force it to 0 and apply the 3-session cooldown. Replace the fixed max_hold=15 with this rule and log HL so the analyst can see it. — The QuantInsti stat-arb project estimates the half-life from a lagged regression of the spread, then builds its z-score over 'half-life' intervals. Robot Wealth reports that 'return potential decreases significantly with time after divergence', hence the rule to 'puke trades that haven't converged after a period of time'. Our fixed 15-session stop ignores how fast the GDX/GLD residual actually reverts in each era. A b >= 0 check also gives an in-strategy test that the premise still holds, which complements the analyst's IC t-stat test. (source: Kalman Filter Techniques And Statistical Arbitrage In China's Futures Market In Python p.1; Pairs Trading Literature Review - Robot Wealth p.1; Introduction to Pairs Trading (Lecture 42) p.1) 5. [parameter] Kalman-filter hedge ratio instead of 60-day rolling OLS: Estimate (alpha, beta) of r_gdx on r_gld with a 2-state random-walk Kalman filter. Use transition covariance delta/(1-delta)*I with delta=1e-4, and observation variance equal to the residual variance of the first 60 sessions; after that, update it with an EWMA of squared innovations over a 60-session span. Use the PRIOR-step state estimate to form today's residual and hedge, then update. Keep the clamp beta in [0.3, 4.0]. Use the same beta for the GLD hedge leg. — The QuantInsti project replaces fixed-window regression with a Kalman filter for the hedge ratio: 'there's no window length that we need to specify' and it 'weigh[s] recent observations more'. ArbitrageLab's hedge-ratio module notes that OLS hedge ratios depend on which leg is the dependent variable (TLS, Johansen and minimum-half-life are alternatives). Its GLD/GDX example shows OLS with GLD as the dependent variable. Robot Wealth notes the regression slope 'tends to be quite unstable'. Each jump in a 60-day OLS beta leaks gold beta into the residual, which creates fake z extremes. Confidence is low because the library only claims this is better; our backtest decides. (source: Kalman Filter Techniques And Statistical Arbitrage In China's Futures Market In Python p.1; hedge ratios (ArbitrageLab docs) p.1; Pairs Trading Literature Review - Robot Wealth p.1)

Verification failed (Layer 4 — QA review) [class=hypothesis_mismatch]: - [edge_concern] The sandbox produced 328 closed positions with negative net returns and profit factor 0.798. Full-history analysis should assess convergence after spread, impact and borrow costs; these results do not determine this correctness verdict. - [critical] The supplied hypothesis requires fading residual moves of at least 2 sigma. The code instead continuously targets exposure proportional to z, without consulting entry_z. With rebalance_band=0.25 and full_size_z=2, a flat book can enter when |z| exceeds approximately 0.5, subject to whole-share sizing. The rationale refers to iteration feedback authorizing this replacement, but that feedback is absent from this task's hypothesis. (line 237) — fix: Implement the supplied >=2-sigma entry gate and the authorized exit structure, or obtain a revised task-bound hypothesis explicitly specifying continuous exposure and its vetoes. - [critical] The rebalance early return checks only the GDX exposure fraction and total gross notional. It never checks GLD's deviation from the current beta-weighted hedge. A change in beta can therefore leave a materially incorrect hedge while the GDX difference remains within the band and gross remains within budget. For example, after beta falls, the existing GLD short can remain oversized indefinitely despite the stated beta-neutral sizing. (line 307) — fix: Compute desired whole-share quantities for both legs before the early return. Require both legs' target deviations and the residual beta exposure to satisfy explicit tolerances before skipping a rebalance. - [warning] The attached optimization plan tunes entry_z and exit_z, but neither parameter affects this implementation. Trials varying those dimensions would measure identical behavior and would not test the declared entry/exit parameterization. (line 53) — fix: Reconcile the implementation and optimization plan with the authorized hypothesis before optimization; do not substitute new parameter permissions implicitly.