1.1.5 Failure Modes & Practice

Basic Prerequisites: 04 · Trading Rules & Backtest and Backtesting Hygiene & Deflated Sharpe.

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1. Intuition & Practical Objective

Statistical arbitrage is mathematically clean and empirically fragile in three specific ways. This page names them precisely so a practitioner knows which assumptions to distrust and how the failures show up in money terms:

1. Structural break. The equilibrium ($\beta$, ${\mu }_z$, or the very existence of cointegration) moves. The spread no longer reverts.
2. Data snooping / selection bias. The pair was chosen because it looked good in the past; that choice inflates the backtest.
3. Costs, borrow and crowding. Two legs of spread cost, scarce short borrow, and the fact that every quant fund runs the same screen - so the trades unwind together.

The discipline is not cynicism; it is knowing exactly where the model is an approximation so the residual risk can be measured before it is realised.

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2. Mathematical Ground Truth & Derivations

2.1 Structural break in the cointegrating vector

Suppose the true relationship is $y_t={\mu }_t+{\beta }_t{x}_t+z_t$ where ${\beta }_t$ or ${\mu }_t$ changes at time $\tau$. The estimated residual using the pre-break ${\hat{\beta }}_0$ is

$${\hat{z}}_t=y_t-{\hat{\beta }}_0{x}_t=\underset{\text{stationary}}{\underbrace{(\mu +z_t)}}+\underset{\text{I(1) if }{\beta }_{\tau }\ne {\hat{\beta }}_0}{\underbrace{({\beta }_{\tau }-{\hat{\beta }}_0)x_t}}.$$

A break in $\beta$ injects a nonstationary component into the “spread” - the residual is no longer $I(0)$, the test loses its meaning, and the z-score grows without bound. The test battery:

• Chow test for a known break date,
• CUSUM / CUSUMSQ on the residuals (recursive),
• rolling / recursive cointegration ($\hat{\beta }$ over a moving window),
• Bai–Perron for an unknown break date.

Empirically, one of the most damaging breaks is a merger or index reconstitution: the target’s price converges to the acquirer’s offer and stops trading as an independent series; the cointegration is destroyed by construction.

2.2 The decline of pairs trading

GGR (2006) found average annualised excess returns up to $11\%$ over 1962–2002. Do & Faff (2010) extended the sample and documented a steady, structural decline:

• mean excess return on employed capital fell (delayed-trading rule) from $0.86\%$/month (1962–88) to $0.37\%$ (1989–2002), a $57\%$ decline, and further to $0.24\%$ (2003–09); the no-delay rule runs higher but lower, $1.24\%\to 0.56\%\to 0.33\%$;
• the strategy remained statistically significant but economically thin after costs;
• profitability is state-dependent - it spiked in 2001–02 (the dot-com bust) and again in 2008–09.

Why it decays. Arbitrage capital competes away the mispricing; the mechanism is the same as any alpha decay. Krauss (2017) confirms the pattern across the literature and notes that the distance approach is the most robust to data snooping while the cointegration approach tends to survive costs better in some studies.

2.3 Data snooping in pair selection

Screening $N$ stocks yields $M=N(N-1)/2$ pairs; the best in-sample pair has an in-sample statistic that is an extreme order statistic of $M$ draws. Under the null of no cointegration anywhere, the best pair still shows a strongly negative ADF $t$-stat purely by chance. Formally, with $M$ independent tests the expected minimum $t$-statistic grows roughly like $-{\Phi }^{-1}(1/M)$ in magnitude; for $M=200$, ${\Phi }^{-1}(1-1/200)={\Phi }^{-1}(0.995)\approx 2.576$, i.e. a spurious "$-2.58$" ADF is expected. Two corrections:

• Out-of-sample protocol (GGR): choose pairs on a formation window, trade a disjoint trading window - never re-tune on the trading window.
• Deflated Sharpe Ratio / multiple-testing haircut (Bailey & López de Prado): deflate the reported Sharpe by the number of trials.

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3. Computational Implementation - the failures in numbers

Stdlib only. Part A builds a pair that is genuinely cointegrated for 300 days, then has the residual acquire a permanent drift (the economic link breaks), and trades it with the same rule. Part B searches 200 independent random-walk pairs for the “best” in-sample cointegration and shows its out-of-sample collapse.

Part A is the catastrophic case: the pair passes the cointegration test in-sample ($t=-5.56^{\ast }$), then the spread marches to $+16\sigma$ (and $+33\sigma$) out-of-sample. A stop at $3.5\sigma$ fires and the position is closed for a loss; a strategy without a stop is ruinous. The ADF test on the full sample would have been polluted by the $I(1)$ break term - you only see the break by re-testing recursively.

Part B is the subtle case: no pair is cointegrated (all are independent random walks), yet the best of $200$ shows ADF $t=-4.49$ in-sample - convincingly “cointegrated.” On a fresh window the same pair gives $t=-0.85$: nothing. The in-sample statistic was pure selection.

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4. Failure Modes & First-Principles Breakdowns (numbered)

1. Structural break in $\beta$ or drift (the divergence trap). Mergers, index changes, dividend cuts, technological substitution. Symptom: the spread widens $2\sigma \to 4\sigma \to 10\sigma$ and stops reverting. First-principle check: rolling/recursive cointegration and CUSUM tests on the residual; a hard statistical stop.
2. Data snooping / selection bias. The pair was picked ex post; the reported Sharpe is an extreme order statistic. Symptom: great backtest, dead live. Fix: out-of-sample formation/trading split, deflated Sharpe, purged CV.
3. Transaction costs and turnover. Two legs of spread cost each time the position changes; turnover scales as $1/{\tau }_{1/2}$. Symptom: gross Sharpe $2.5\to$ net $-0.4$ (the flat-file’s own illustration). Fix: cost-aware thresholds, longer half-life screen, portfolio netting (Avellaneda–Lee ETF netting).
4. Short borrow and asymmetric costs. The short leg may be unborrowable or expensive precisely when the divergence is largest - a hidden, one-sided risk.
5. Crowding and unwinding. When many funds run the same residual-reversion book, a deleveraging event forces a correlated unwind and a sharp, temporary loss. The August 2007 quant quake: Avellaneda & Lee (2010) and Khandani & Lo (2007) attribute the drawdown to simultaneous unwinding, not a model failure.
6. Regime dependence of profitability. Returns concentrate in high-volatility periods (2001–02, 2008–09); a strategy calibrated on calm markets under-estimates the tail moves that produce both the profits and the drawdowns.
7. Non-synchronous pricing and microstructure. Stale closes, bid-ask bounce and nonsynchronous legs fabricate convergence on paper that cannot be captured live.

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5. References

• Do, B. & Faff, R., “Does Simple Pairs Trading Still Work?”, Financial Analysts Journal 66(4), 2010
• Do, B. & Faff, R., “Are Pairs Trading Profits Robust to Trading Costs?”, Journal of Financial Research 35(2), 2012.
• Avellaneda, M. & Lee, J.-H., Quantitative Finance 10(7), 2010
• Krauss, C., J. Economic Surveys 31(2), 2017
• Hasbrouck, Empirical Market Microstructure
• Bailey, D. H. & López de Prado, M.

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6. Connected Graph Bridges

• Back: 04 · Trading Rules & Backtest · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling: Backtesting Hygiene & Deflated Sharpe
• Bridges: Adverse Selection & Glosten–Milgrom · Transaction Costs & Turnover