1.1.6 Advanced Extensions

Basic Prerequisites: 02 · Cointegration & the Spread and 04 · Trading Rules & Backtest.

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1. Intuition & Practical Objective

Three extensions take StatArb beyond the two-asset t-test:

1. Johansen’s multivariate test - instead of a pair, allow a basket and ask “how many* independent equilibrium relationships exist?” (the cointegrating rank). This is the correct tool when several assets share the same factors and you do not want to pre-pick a pair.
2. Optimal stopping / stochastic control of the OU spread - instead of fixed $2\sigma$ thresholds, solve for the thresholds that maximise expected return per unit time net of costs.
3. Dynamic hedging (Kalman filter) - replace the static OLS $\beta$ with a time-varying ${\beta }_t$ estimated recursively, so the hedge adapts when the relationship drifts (the bridge to Signal Processing & Kalman Filtering).

All three keep the OU spread at the centre; they differ in how many assets and how much structure they assume.

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2. Mathematical Ground Truth & Derivations

2.1 The Johansen procedure (Tsay §8.6.2–8.6.3; Johansen 1988/1991)

Start from a VAR($p$) in levels and rewrite it in error-correction form for a $k$-dimensional $I(1)$ vector $x_t$:

$$\Delta x_t=\Pi x_{t-1}+\sum _{i=1}^{p-1}{\Gamma }_i\Delta x_{t-i}+a_t,\Pi =\sum _{i=1}^p{\Phi }_i-I=-\Phi (1).$$

The rank of $\Pi$ is the cointegrating rank $m$: $\Pi =\alpha {\beta }^{\prime }$, with $\beta$ ($k\times m$) the cointegrating vectors and $\alpha$ the adjustment speeds.

Estimation (the reduced-rank regression). Run two auxiliary regressions and collect residuals:

$$R_{0t}=\Delta x_t-\text{proj on }(\Delta x_{t-1},\dots ),R_{1t}=x_{t-1}-\text{proj on }(\Delta x_{t-1},\dots ).$$

Form the moment matrices $S_{ij}=\frac{1}{T}{\sum }_t{R}_{it}{R}_{jt}^{\prime }$, and solve the generalised eigenvalue problem

$$|\lambda S_{11}-S_{10}{S}_{00}^{-1}{S}_{01}|=0⟹{\hat{\lambda }}_1\ge \cdots \ge {\hat{\lambda }}_k.$$

The cointegrating vectors are the eigenvectors, normalised so $e^{\prime }{S}_{11}e=I$.

Tests (H0: rank $=m$). Two standard statistics, with nonstandard (Brownian-motion) critical values:

$$L{R}_{\text{tr}}(m)=-(T-p)\sum _{i=m+1}^k\ln (1-{\hat{\lambda }}_i)\text{(rank}=m\text{ vs }>m),$$ $$L{R}_{\max }(m)=-(T-p)\ln (1-{\hat{\lambda }}_{m+1})\text{(rank}=m\text{ vs }m+1).$$

For $k=2$ with a restricted constant, the 95% critical values are $\approx 15.41$ (rank $\le 0$) and $\approx 3.76$ (rank $\le 1$): reject “no cointegration” if the first exceeds $15.41$, and fail to reject rank 1 if the second is below $3.76$.

2.2 Multivariate StatArb (Avellaneda–Lee “generalized pairs trading”)

Trade a stock against a weighted portfolio (basket) rather than a single peer. With factors $F_j$ and residuals ${\tilde{R}}_i=R_i-{\sum }_j{\beta }_{ij}{F}_j$, the “generalized” spread is the idiosyncratic residual ${\tilde{R}}_i$, and the same OU/z-score machinery applies per stock. Market-neutrality ${\sum }_i{\beta }_{ij}{Q}_i=0$ holds at the book level, so the net factor exposure cancels.

2.3 Optimal stopping / thresholds (Elliott, van der Hoek & Malcolm 2005; Bertram 2010)

Model the spread as an OU process and the trade as a first-passage problem: enter at level $a$, exit at level $m$ ($a<m$ for a long-spread trade). The cycle time $T=T_1+T_2$ (entry→exit + exit→next entry) is random; the return per cycle is deterministic, $r(a,m,c)=m-a-c$ with cost $c$. By renewal theory the expected return and variance per unit time are

$$\mu (a,m,c)=\frac{r(a,m,c)}{\mathbb{E}[T]},{\sigma }^2(a,m,c)=\frac{r^2(a,m,c)\mathrm{Var}(T)}{{\mathbb{E}}^3[T]},$$

and $\mathbb{E}[T],\mathrm{Var}(T)$ come from the first-passage-time density of the OU process (Itô-transformed to a dimensionless system). Maximising a Sharpe-type objective over $(a,m)$ yields the optimal thresholds - recovering the fixed $2\sigma$ rule as a special (suboptimal-in-general) case.

2.4 Dynamic hedge ratio (Kalman filter)

Let ${\beta }_t$ follow a random walk ${\beta }_t={\beta }_{t-1}+w_t$ with observation $y_t=\alpha +{\beta }_t{x}_t+v_t$. The Kalman filter gives the one-step-ahead estimate ${\hat{\beta }}_{t|t-1}$; use it as the hedge ratio so the spread is $z_t=y_t-{\hat{\beta }}_{t|t-1}{x}_t$. This turns the static cointegration regression into an adaptive one - essential when $\beta$ drifts slowly (see the sibling topic for the full state-space derivation).

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3. Computational Implementation - the Johansen trace test

Stdlib only. We simulate a genuine cointegrated bivariate system (random walk $x_1$, and $x_2=0.9x_1+$ stationary) so the true rank is 1, then run the full Johansen reduced-rank regression with a 2×2 analytic generalised-eigenvalue solve.

The procedure recovers the simulated rank exactly: the first trace statistic ($103.14$) is far above its critical value (reject “no cointegration”), while the second ($3.12$) sits below $3.76$ (a second vector is not supported). The estimated ${\hat{\lambda }}_1=0.0983$ is a measure of how strongly the equilibrium is restored by the error-correction term.

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4. Failure Modes & First-Principles Breakdowns

1. Johansen critical values depend on the deterministic spec. The five cases (no constant, restricted constant, unrestricted constant, restricted/unrestricted trend; Tsay §8.6.1) have different tables. Resembling the wrong table invalidates the rank conclusion.
2. Rank ≠ profitability. A system can have rank 1 with an unfavourably slow adjustment ($\alpha$ small) or a spread whose ${\sigma }_{\text{eq}}$ is tiny relative to costs - econometrically cointegrated, economically untradeable.
3. Basket legibility. Trading a stock against a basket multiplies transaction costs and creates netting and borrow complexity; Avellaneda–Lee rely on the net ETF position being small, which need not hold in stress.
4. Optimal-stopping thresholds assume stationarity. The Elliott/Bertram solution is only optimal under a fixed OU; the moment the parameters drift or the equilibrium breaks, the “optimal” thresholds are worse than a plain stop.
5. Kalman $\beta$ can chase noise. If the state-noise variance is set too high, the hedge ratio over-fits recent co-movement and the spread becomes white noise by construction (a filter artefact, not cointegration).
6. The nested risk. These extensions add parameters to a problem whose core risk (structural break, Ch 5) they do not fix. More machinery is not robustness.

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5. References

• Tsay, Analysis of Financial Time Series
• Johansen, S., “Statistical Analysis of Cointegration Vectors”, Journal of Economic Dynamics and Control 12(2–3), 1988; and Econometrica 59(6), 1991.
• Avellaneda, M. & Lee, J.-H., Quantitative Finance 10(7), 2010
• Elliott, R. J., van der Hoek, J. & Malcolm, W. P., “Pairs Trading”, Quantitative Finance 5(3), 2005
• Krauss, C., J. Economic Surveys 31(2), 2017
• Harvey, A. C., Forecasting, Structural Time Series Models and the Kalman Filter

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6. Connected Graph Bridges

• Back: 05 · Failure Modes · Index Hub
• Sibling: Signal Processing & Kalman Filtering (dynamic hedge ratio) · Fundamental Multi-Factor Models (factor choice for residuals)
• Foundations: Econometrics & Time Series · Linear Algebra & Matrices · Stochastic Calculus & Itô (for the OU first-passage solution)
• Cross-pillar: Transaction Costs & Turnover