The math behind every screen.

A single asset price is, to a first approximation, a martingale. The right thing to bet on is not its level but a deviation from a relationship. Engle and Granger (1987) formalised the idea: two processes $x_t,y_t$ that are individually $I(1)$ (integrated of order 1) can have a linear combination $y_t-\beta x_t$ that is $I(0)$ (stationary). The vector $(1,-\beta )$ is then called the cointegration vector and the residual $u_t=y_t-\alpha -\beta x_t$ is the spread we trade.

Sector neutrality is the practitioner's lever for keeping the relationship stable: if we pick A and B from inside the same industry, both prices share a sector factor and most macroeconomic shocks cancel. What is left to bet on is the idiosyncratic disagreement between A and B — usually a small, mean-reverting process.

Step 1 — OLS regression in levels:

Under cointegration the OLS estimator $\hat{\beta }$ is super-consistent: it converges to the true long-run coefficient at rate $T$ rather than the usual $\sqrt{T}$, even when the regressors are I(1). That is what makes a simple rolling regression a viable real-time estimator.

Step 2 — ADF on the residuals:

Under $H_0:\gamma =0$ the residual has a unit root and A and B are not cointegrated. The test statistic is the t-ratio on $\hat{\gamma }$. Critical values come from MacKinnon (2010) response surfaces — different from the standard Dickey-Fuller table because $\hat{u}$ is estimated, not observed:

For two variables with a constant (case "c"): ${\tau }_{0.05}(\infty )=-3.336$ versus the plain ADF ${\tau }_{0.05}(\infty )=-2.862$. The shift is real and ignoring it inflates false-positive cointegration calls. The Lab uses the cointegration CVs in cointegration.ts and the standard ADF CVs in adf.ts.

Consider an AR(1) process $u_t=\rho u_{t-1}+{\epsilon }_t$. Subtract $u_{t-1}$ from both sides:

If $|\rho |<1$ then $\gamma <0$: a high $u_{t-1}$ implies a negative expected change next bar — the spring pulls. If $\rho =1$ then $\gamma =0$: no pull, random walk. The augmented form adds lagged differences to soak up serial correlation in the noise:

The Lab uses MacKinnon's (1996) sample-size adjusted critical values, so a 60-bar window and a 1200-bar window are not held to the same threshold.

Static OLS. The all-sample point estimate. Useful as a baseline, but assumes the cointegration coefficient is constant — which is rarely true once you cross months of data:

Rolling OLS. A window of length $w$ slides forward; we recompute ${\hat{\beta }}_t$ using only observations in the window $(t-w+1,t)$. Easy to reason about, breaks gracefully when the relationship shifts, and is what the Lab uses by default.

Kalman filter. Treat the hedge ratio itself as a state that evolves slowly. Let ${\theta }_t=\left(\begin{array}{c}{\alpha }_t\\ {\beta }_t\end{array}\right)$ with state and observation equations:

The standard recursion produces a posterior mean and covariance for the state at every bar. The process-noise scale $Q=\delta I$ controls adaptivity: a small $\delta$ keeps β nearly constant; a larger $\delta$ lets it follow regime shifts but at the cost of estimation noise. Predict-then-update:

Elliott, van der Hoek & Malcolm (2005) is the textbook starting point for a state-space approach to mean-reverting spreads; Chan (Algorithmic Trading, 2013) gives the implementation that this Lab follows.

A long-spread entry triggers when $z_t\le -z_{\text{entry}}$ and unwinds when $z_t\ge -z_{\text{exit}}$; the short-spread rule is the mirror image. Hard stops cap the worst case: a stop-loss when $|z_t|\ge z_{\text{stop}}$ (the relationship has probably broken) and a time-stop after $H$ bars (the mean reversion was supposed to have happened by now). The Backtest Studio surfaces all four levers as sliders.

The continuous-time OU process is the canonical mean-reverting SDE:

Discretise at unit step and run an AR(1) regression on the spread: $\Delta S_t=a+b{S}_{t-1}+{\epsilon }_t$. Then $b=-\theta$ and the half-life of mean reversion is

A spread with $t_{1/2}=5$ bars closes half its gap in a week and is highly tradable; a spread with $t_{1/2}=200$ bars takes nearly a year and dies on costs. The Pair Lab plots the half-life and the Portfolio module screens out pairs whose half-life exceeds a user threshold.

Bertram (2010) derives a closed-form expression for expected return per unit time as a function of the entry level $a$ for a symmetric OU strategy:

The expected first-passage time $E[T(a)]$ involves imaginary error functions, but the qualitative result is intuitive: if costs are small, the optimum is a tight band of about $\pm 0.75\sigma$; as costs rise the optimum widens toward $\pm 2\sigma$ and beyond. The Lab plots a numerical scan instead of the special-function form so the intuition is preserved without machinery.

Maillard, Roncalli & Teiletche (2010) define the equal-risk-contribution (ERC) portfolio as the $w⪰0$ with $\sum w_i=1$ satisfying:

For uncorrelated strategies this collapses to inverse-volatility weighting $w_i\propto 1/{\sigma }_i$; for correlated strategies the iterative solver in risk/sizing.ts finds the fixed point. The Lab reports both.

Dollar-neutral: equal dollar long, equal dollar short. Gross exposure is 100%, net dollar exposure is zero. Simple and symmetric, but the residual is exposed to whatever β-loading the two legs have to the broader market.

β-hedged: for every $1 long in A, short $β in B, where β is the cointegration coefficient (often ≈ 1 for sector pairs but not always). Net market exposure is approximately zero, even if dollar exposure is asymmetric. Preferable when the two legs have different market betas.

Both modes are available in the Backtest Studio. The Portfolio page reports residual β-to-market for the full book of pairs and flags drift away from sector neutrality.

Higher ILLIQ means a one-dollar trade moves the price more — i.e., a thinner book. The Portfolio module screens out pairs whose worse leg sits above a percentile threshold, on the principle that any edge gets eaten by impact and slippage in illiquid names.

Pure in-sample testing is worthless because every parameter — the z-entry, the lookback, the stop-loss — is implicitly chosen with knowledge of the future. Walk-forward fixes this by splitting the sample into a sequence of (training, test) windows. Models are fit on training only; performance is recorded on test only; the window rolls forward.

A second-best, lighter alternative — and the default in the Lab — is rolling estimation: the hedge ratio and z-score moments are re-estimated on every bar using only data up to that bar. This produces an honest equity curve at the cost of some statistical inefficiency in the early part of the sample.

Do & Faff (2010, 2012) showed that pairs strategies identified by Gatev et al. delivered shrinking returns post-2002 once realistic costs and walk-forward selection were imposed. The Lab enforces both by default.

Stack the levels into $Y_t\in {\mathbb{R}}^k$ and write the vector autoregression in error-correction form (VECM):

The rank of $\Pi$ equals the number of cointegration relations. Decomposing $\Pi =\alpha {\beta }^{\prime }$, the columns of $\beta$ are cointegration vectors; the rows of $\alpha$ are loadings: how fast each variable adjusts back toward equilibrium. The latter is the practical pay-off — Johansen tells you which leg leads and which leg lags.

Johansen's trace test compares

to Osterwald-Lenum (1992) critical values. The eigenvalues come from a generalised eigenvalue problem on residual second-moment matrices; the Lab implements the 2-variable case in johansen.ts and reports both the trace and max-eigenvalue statistics on the Methods page.

Kwiatkowski, Phillips, Schmidt & Shin (1992) construct a Lagrange-multiplier statistic

where ${\hat{\sigma }}_{\infty }^2$ is the long-run variance estimated with a Newey-West kernel. Under the null of stationarity, ${\eta }_{\mu }$ is bounded; under a unit-root alternative, the cumulative sum drifts and the statistic explodes. Critical values: ${\eta }_{\mu ,5\%}=0.463$. The cleanest pair-trading workflow uses ADF and KPSS together: a pair worth trading both rejects the unit-root null (ADF) and fails to reject the stationarity null (KPSS).

Lo & MacKinlay (1988) propose

Under a random walk, $\mathrm{VR}(q)\to 1$. A value below 1 implies negative serial correlation — what we want from a tradable spread. The Lab reports the heteroskedasticity-robust z-statistic across multiple horizons $q\in \{2,4,8,16,32,64\}$, because a single horizon can be misleading when the spread mean-reverts at a specific timescale.

Take the spread, split it into non-overlapping chunks of length $n$, compute the rescaled range $R(n)/S(n)$ for each. Then $E[R(n)/S(n)]\sim c\cdot n^H$, and $H$ is the slope of $\log (R/S)$ on $\log n$:

• $H=0.5$: random walk.
• $H<0.5$: anti-persistent / mean-reverting (what we want).
• $H>0.5$: trending — bad news for a pairs strategy.

Hurst is best used as a sanity check: tradable spreads almost always show $H\in [0.30,0.45]$; if your "cointegrated" pair returns 0.55, something is wrong.

Compute the standardised cumulative sum of recursive residuals, $W_t={\sigma }^{-1}{\sum }_j{w}_j$. Under the null of stable parameters, $W_t$ is approximately Brownian motion; bands $\pm a\sqrt{T}$ grow linearly. An excursion outside the bands means the parameter you assumed was constant has shifted. The Lab plots the CUSUM with bands at the 5% level on the Methodspage — the "Broken" pair (PYRE-VALE) is constructed specifically so the bands are crossed at the regime break.

Take a 12-month formation window, normalise both prices to start at 1, compute the sum of squared distances $\mathrm{SSD}={\sum }_t({\hat{A}}_t-{\hat{B}}_t{)}^2$. Pick the minimum-SSD pair (or use a ranked list) and trade in the next 6-month window. Open when the normalised spread diverges by $\pm 2{\sigma }_{\mathrm{formation}}$ and close when it returns to zero.

The strength of the method is robustness — there are no estimated parameters to over-fit. Its weakness, traced cleanly by Do & Faff (2010, 2012), is that it implicitly assumes $\beta =1$; it under-performs when α ≠ 1 and is sensitive to the choice of normalisation reference point. The Lab runs it head-to-head with the cointegration approach on the Strategies page.

Fit an Ornstein-Uhlenbeck process to the spread, recover $({\mu }_{\mathrm{OU}},{\sigma }_{\mathrm{OU}})$, and define

Avellaneda & Lee (2010) propose $s_{\mathrm{open}}=1.25$ and $s_{\mathrm{close}}=0.5$. The big advantage: signals are calibrated to the SDE, not to short-window noise. The big disadvantage: when $({\mu }_{\mathrm{OU}},{\sigma }_{\mathrm{OU}})$ drift, the s-score keeps using stale parameters and either over- or under-trades. The Strategies page runs this side-by-side with the rolling z-score on the same pair.

VaR / CVaR. ${\mathrm{VaR}}_{\alpha }$ is the α-quantile of the return distribution; ${\mathrm{CVaR}}_{\alpha }=E[r|r\le {\mathrm{VaR}}_{\alpha }]$ (Rockafellar-Uryasev, 2002) is the average loss in the tail beyond it. CVaR is a coherent risk measure; VaR is not.

Ulcer Index (Martin & McCann 1989) measures depth and duration of drawdowns:

Pain ratio = annual return / Ulcer Index. Sterling ratio = annual return divided by the average annual maximum drawdown — smoother than Calmar, which depends on a single worst-DD point.

The Risk Lab computes all of these alongside the standard Sharpe / Sortino / Calmar trio and a stationary-bootstrap CI on the Sharpe.

Politis & Romano (1994) draw block lengths from a geometric distribution with mean $1/p$, paste blocks together to length $T$, and re-compute the statistic of interest on each resample. Random block lengths preserve the strong-mixing property of the original series so the implied distribution of the Sharpe is asymptotically valid even when returns are not iid.

A 95% bootstrap CI on the Sharpe is the simplest, most under-used antidote to data-mined backtests. A "Sharpe of 1.2" with a 95% CI of [−0.4, +2.5] is not what it looks like.

Real execution costs scale as $\Delta P/P\sim \eta \sigma \sqrt{X/V_t}$, where $X$ is the order size, $V_t$ is daily volume, $\sigma$ is daily volatility and $\eta$is a market constant near 1 (Almgren et al. 2005). The Lab's default flat-bps cost is a first approximation; the Portfolio page surfaces a per-pair capacity proxy of 1% of mean dollar volume on the worse leg, and the Theory page recommends imposing a square-root impact term once the strategy is sized to a real book.

Half-life and capacity interact: a fast-mean-reverting pair lets you turn over inventory inside the impact-decay window, so realised costs converge toward the spread crossing rather than the full impact curve.

The Lab covers, end to end:

• Single-pair cointegration via Engle-Granger, ADF and Johansen.
• Three hedge-ratio estimators (static OLS, rolling OLS, Kalman).
• Two complementary stationarity tests (KPSS) and two persistence diagnostics (VR, Hurst).
• Two regime-stability tools (rolling ADF p-value, Brown-Durbin-Evans CUSUM).
• Three trading recipes (cointegration z-score, distance method, OU s-score).
• Walk-forward backtesting with realistic costs, drawdown halt, time-stops.
• Risk-parity, β-hedged, dollar-neutral sizing.
• Full extended-risk panel + stationary-bootstrap CIs on Sharpe.

It explicitly does not yet cover:

• Copula-based pair dependence (Liew & Wu 2013, Krauss & Stübinger 2017).
• Hidden Markov / regime-switching models for breakdown detection.
• Cross-sectional PCA on a real equity universe (Avellaneda-Lee in full).
• Multi-leg baskets via Johansen with k ≥ 3 (the trace test scales, the rest needs UI).
• EM-based Kalman tuning for δ.
• Live Almgren-Chriss execution scheduling.

Each of these is a natural next module — the Lab's structure separates math, data and UI cleanly so a pull request adding e.g. copula pair-trading is local to two files.

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