M.6.5 Cointegration & Multivariate Models

Basic Prerequisites: 03 · Forecasting & Unit Roots (unit roots) and Linear Algebra & Matrices (matrix decompositions).

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1. Intuition & Practical Objective

Individual asset prices are unit-root processes - but combinations of them need not be. Two prices can each wander without bound while their spread stays in a tight band, pulled back to equilibrium by arbitrage. That shared long-run equilibrium is cointegration, and it is the statistical engine behind pairs trading, index arbitrage, and any strategy that bets on relative value. This page’s objective: model multivariate systems, detect cointegration, and exploit it through the error-correction mechanism (ECM).

The intuition in three steps:

1. Two $I(1)$ series $Y_t,X_t$ are cointegrated if a linear combination $z_t=Y_t-\beta X_t$ is $I(0)$ (stationary). The number of such independent combinations is the cointegrating rank.
2. The Granger Representation Theorem says cointegrated series must move to correct the spread - the ECM $\Delta Y_t=\gamma (Y_{t-1}-\beta X_{t-1})+\dots$ with $\gamma <0$. Long-run equilibrium pins down a relation that nothing in the individual unit-root processes can see.
3. Detection is a unit-root test on the spread. Regress $Y$ on $X$, take the residual, ADF-test it: reject ⇒ cointegrated; fail to reject ⇒ spurious (the trap of 03 · Unit Roots). The Johansen procedure generalizes this to $k$ series and estimates the full cointegrating space.

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2. Mathematical Ground Truth & Derivations

VAR(p) (Tsay eq. 8.13): $x_t={\varphi }_0+{\Phi }_1{x}_{t-1}+\cdots +{\Phi }_p{x}_{t-p}+a_t$, $a_t\sim (0,\Sigma )$. Stationary iff all eigenvalues of the companion matrix are inside the unit circle. Estimation by OLS (each equation) / ML; order selection by AIC/BIC/HQ; forecasting and orthogonalized impulse-response (Cholesky, ordering-dependent - Tsay §8.2.5).

Cointegration (Tsay §8.5). For a $k$-dimensional $I(1)$ series with $h$ unit roots: cointegration exists iff $0<h<k$, with $k-h$ cointegrating factors; the cointegrating vectors are the columns of $\beta$ with ${\beta }^{\prime }{x}_t$ stationary. Example: $y_{1t}=x_{1t}-2x_{2t}$ stationary while each component is $I(1)$.

Error-correction form (Engle–Granger, Tsay eq. 8.33–8.34).

$$\nabla x_t=\alpha {\beta }^{\prime }{x}_{t-1}+\sum {\Phi }_i^{\ast }\nabla x_{t-i}+a_t-\sum {\Theta }_j{a}_{t-j},$$

with $\alpha {\beta }^{\prime }={\Phi }_p+\cdots +{\Phi }_1-I=-\Phi (1)$. The term ${\beta }^{\prime }{x}_{t-1}$ (the lagged spread) is the “compensation” that avoids over-differencing. Rank of $\Pi =\alpha {\beta }^{\prime }$: $0$ ⇒ no cointegration; $k$ ⇒ $x_t$ is $I(0)$; $m$ with $0<m<k$ ⇒ $m$ cointegrating vectors and $k-m$ common stochastic trends $y_t={\alpha }_{\perp }^{\prime }{x}_t$.

Engle–Granger two-step. (1) Regress $Y_t$ on $X_t$, get $\hat{\beta }$ and residual ${\hat{z}}_t=Y_t-\hat{\beta }{X}_t$. (2) ADF-test ${\hat{z}}_t$: stationary ⇒ cointegrated. Then the ECM regresses $\Delta Y_t$ on the lagged residual with a negative “speed-of-adjustment” coefficient $\gamma$ (mean reversion toward $\beta X+\mu$).

Johansen MLE (Tsay §8.6.2). From two auxiliary regressions get residual covariances $S_{00},S_{01},S_{11}$, then solve the generalized eigenproblem $|\lambda S_{11}-S_{10}{S}_{00}^{-1}{S}_{01}|=0$; cointegrating vectors = leading eigenvectors. Tests:

• Trace: $L{R}_{tr}(m)=-(T-p){\sum }_{i=m+1}^k\ln (1-{\hat{\lambda }}_i)$ - $H_0:$ rank $=m$ vs $>m$;
• Max-eigenvalue: $L{R}_{max}(m)=-(T-p)\ln (1-{\hat{\lambda }}_{m+1})$ - rank $=m$ vs $m+1$.

Both have nonstandard (Brownian-motion-functional) critical values. Tsay’s interest-rate example (TB3m/TB6m weekly): VAR(3) by BIC, trace $83.27$ vs 95% CV $19.96$ - rejects rank 0 strongly, one cointegrating vector $tb3m-1.0124tb6m$, ECM $\alpha =(-0.0949,-0.0211)$.

Threshold cointegration (Tsay §8.7) - corrected 3-regime reading. The S&P 500 futures/cash basis model is a three-regime threshold ECM (regimes $z\le {\gamma }_1$, ${\gamma }_1<z\le {\gamma }_2$, $z>{\gamma }_2$ with ${\gamma }_1<0<{\gamma }_2$): the error-correction/cointegration term is active in the two outer regimes (arbitrage active, past transaction-cost threshold) and insignificant in the middle no-arbitrage band where prices behave like a random walk. (Not “2-regime” - the middle band is regime-free of cointegration.)

Pairs trading (Tsay §8.8). Two similar-risk (e.g. same-industry) stocks are cointegrated; the spread $w_t=p_{1t}-\gamma p_{2t}$ is stationary around mean ${\mu }_w$; the ECM is $[r_1,r_2{]}^{\prime }=[{\alpha }_1,{\alpha }_2{]}^{\prime }(w_{t-1}-{\mu }_w)+{\epsilon }_t$ with ${\alpha }_1,{\alpha }_2$ of opposite signs. Portfolio (long 1 share of stock 1, short $\gamma$ of stock 2) has return $r_{p,t+i}=w_{t+i}-w_t$: enter at ${\mu }_w-\delta$, unwind at ${\mu }_w+\delta$ (needs $2\delta >\eta$ cost), net profit $2\delta -\eta$.

PCA/factors (Tsay Ch 9) - the statistical factor side: eigenvalues of ${\Sigma }_r$ (or correlation ${\rho }_r$); PCs $y_i=e_i^{\prime }r$, $\mathrm{Var}(y_i)={\lambda }_i$, cumulative variance $\sum {\lambda }_i/\sum {\lambda }_j$. Factor model $r_{it}={\alpha }_i+{\beta }_{i1}{f}_{1t}+\cdots +{\beta }_{im}{f}_{mt}+{\epsilon }_{it}$, $\mathrm{Cov}(r_t)=\beta {\Sigma }_f{\beta }^{\prime }+D$. (Full treatment: Fundamental Multi-Factor Models.)

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3. Computational Implementation - Engle–Granger cointegration in action

Simulate a cointegrated pair (a common random-walk trend $z$, then $x=z$, $y=\beta z+\mu +$ stationary spread), run the Engle–Granger residual ADF, and estimate the ECM speed-of-adjustment. Contrast with two independent random walks (spurious). Stdlib only.

The Engle–Granger procedure recovers the true hedge ratio $\beta \approx 2$ and rejects the residual’s unit root hard ($-17.0$ vs $-2.86$): the pair is cointegrated. The ECM coefficient is $-0.43$ - each period the spread closes ~43% of its gap toward equilibrium. The control (two independent random walks) fails the residual ADF ($-0.83$), exactly the spurious case - the only difference between “tradeable pair” and “statistical mirage” is this residual test.

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4. Failure Modes & First-Principles Breakdowns

1. Spurious cointegration from data-snooping. If you screen many pairs for the best-trading one using in-sample tests, you overstate cointegration - the same selection bias ESL Ch 7 warns about for CV. Pairs must be chosen by economic logic (similar-risk peers) and validated out-of-sample, not by hunting the biggest in-sample t-stat.
2. Scaling & deterministic-spec sensitivity. Cointegration tests are sensitive to how the series are scaled and to the assumed deterministic terms (no constant / restricted const / drift / trend - Tsay §8.6.1’s five cases). A wrong spec flips the verdict.
3. The 3-regime trap (corrected). Real arbitrage is band-threshold: the ECM coefficient is ~0 in the no-arbitrage middle band and only significant in the outer regimes. Estimating a single linear ECM ignores transaction costs and makes the relationship look weaker than it is.
4. Impulse-response ordering dependence. In a VAR, Cholesky-orthogonalized impulse responses depend on variable ordering (Tsay §8.2.5) - never report one ordering’s “causality” as if it were invariant.
5. Cointegrating vectors are not unique. Any rotation of $\beta$ is also valid (identifying constraint ${\beta }^{\prime }=[I_m,{\beta }_1^{\prime }]$); interpreting “the” cointegrating vector requires economic theory, not the test.

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5. References

• Tsay, Analysis of Financial Time Series
• Engle, R.F. & Granger, C.W.J. (1987), “Co-integration and Error Correction: Representation, Estimation, and Testing,” Econometrica
• Johansen, S. (1991), “Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models,” Econometrica
• Granger, C.W.J. (1987), representation theorem

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6. Connected Graph Bridges

• Back: 03 · Forecasting & Unit Roots · Index Hub
• Forward: 06 · Advanced Extensions (multivariate vol, state-space)
• Applied: Stat-Arb & Pairs Trading · Fundamental Multi-Factor Models · Black–Litterman