1.1.2 Cointegration, the Error-Correction Form & the Ornstein–Uhlenbeck Spread

Basic Prerequisites: Econometrics & Time Series and 01 · From Zero.

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1. Intuition & Practical Objective

This is the measurement page: given two candidate assets, (i) test whether they are cointegrated, (ii) extract the spread, and (iii) summarise its mean reversion with one speed parameter and one half-life. The practical objective is a pipeline: regress → test the residual → fit an AR(1) → read off $\theta$ and ${\tau }_{1/2}$. Everything downstream (entry/exit thresholds, holding horizon, whether to trade the pair at all) is built on these four numbers.

The key insight is that cointegration is not a property of either series alone but of the combination, and that the same relationship can be written two equivalent ways:

• Static (Engle–Granger): $y_t=\mu +\beta x_t+z_t$ with $z_t$ stationary - the “long-run equilibrium” the spread reverts to.
• Dynamic (Error-Correction Model): the short-run changes of both legs are pulled by the previous period’s disequilibrium $z_{t-1}$. This is the Granger representation theorem: cointegration $⟺$ an error-correction representation exists.

The error-correction picture is what makes pairs trading mechanical: if the spread is above equilibrium, either $y$ falls, $x$ rises, or both - and the ECM coefficients ${\alpha }_1,{\alpha }_2$ tell you which leg does the adjusting (the “leader/follower” structure).

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2. Mathematical Ground Truth & Derivations

2.1 The Engle–Granger two-step method (Tsay §8.5–8.6)

Step 1 - the cointegrating regression. Estimate by OLS the static long-run relation

$$y_t=\mu +\beta x_t+z_t,\hat{\beta }=\frac{\widehat{\mathrm{Cov}}(y,x)}{\widehat{\mathrm{Var}}(x)},\hat{\mu }=\bar{y}-\hat{\beta }\bar{x},$$

and form the residual ${\hat{z}}_t=y_t-\hat{\mu }-\hat{\beta }{x}_t$.

Step 2 - test the residual for a unit root. Run an augmented Dickey–Fuller regression on the residual:

$$\Delta {\hat{z}}_t=c+\gamma {\hat{z}}_{t-1}+\sum _{i=1}^p{\phi }_i\Delta {\hat{z}}_{t-i}+e_t,$$

and reject “no cointegration” if the $t$-statistic on $\gamma$ is sufficiently negative. Because ${\hat{z}}_t$ is estimated (a generated regressor), the null distribution is not the usual Dickey–Fuller distribution; it is more negative. For $N=2$ variables with a constant and no trend, the asymptotic 5% critical value is about $-3.34$ (Engle–Granger 1987; MacKinnon 1991), versus $-2.86$ for an ordinary ADF. Using the wrong table is a classic error.

2.2 The error-correction form (Granger representation)

If $y_t,x_t$ are $I(1)$ and cointegrated, then (Tsay Eq. 8.33, in the bivariate pair case)

$$\begin{array}{rl}\Delta y_t& ={\alpha }_1(z_{t-1}-{\mu }_z)+\sum _i{\gamma }_i^y\Delta y_{t-i}+\sum _j{\delta }_j^y\Delta x_{t-j}+{\epsilon }_t^y,\\ \Delta x_t& ={\alpha }_2(z_{t-1}-{\mu }_z)+\sum _i{\gamma }_i^x\Delta y_{t-i}+\sum _j{\delta }_j^x\Delta x_{t-j}+{\epsilon }_t^x,\end{array}$$

where $z_t=y_t-\beta x_t$. Cointegration requires ${\alpha }_1$ and ${\alpha }_2$ to be opposite in sign (at least one adjusts toward equilibrium; Tsay §8.8 Eq. 8.45). The size of $|{\alpha }_i|$ is the speed at which leg $i$ corrects - the econometric analogue of price discovery / leadership (Hasbrouck Ch 10). If ${\alpha }_1={\alpha }_2=0$, no error correction and hence no cointegration.

2.3 The Ornstein–Uhlenbeck spread

The continuous-time model of the spread is the Ornstein–Uhlenbeck (OU) process:

$$d{z}_t=\theta (\mu -z_t)dt+\sigma d{W}_t,\theta >0.$$

It is the continuous-time AR(1): the increment has conditional mean $\theta (\mu -z_t)dt$ - positive when $z_t<\mu$ (spread too low, expect a rise), negative when $z_t>\mu$. The stationary (equilibrium) distribution is Gaussian with

$$\mathbb{E}[z_t]=\mu ,\mathrm{Var}[z_t]=\frac{{\sigma }^2}{2\theta }.$$

Discretisation (exact). Solving the OU SDE over a step $\Delta t$ gives

$$z_{t+1}=m(1-e^{-\theta \Delta t})+e^{-\theta \Delta t}{z}_t+{\eta }_{t+1},\eta \sim N(0,\frac{{\sigma }^2}{2\theta }(1-e^{-2\theta \Delta t})).$$

So a 1-lag OLS identifies the persistence directly, but which coefficient depends on which regression you run:

• Level regression $z_{t+1}=a+\phi z_t+{\eta }_{t+1}$: the slope is the AR(1) persistence $\phi =e^{-\theta \Delta t}$, so $\boxed{\theta =-\frac{\ln \phi }{\Delta t}}$.
• Δ (change) regression $\Delta z_{t+1}=c+\beta z_t+{\eta }_{t+1}$ with $\Delta z_{t+1}=z_{t+1}-z_t$: the slope is $\beta =e^{-\theta \Delta t}-1$, so $\theta =-\frac{\ln (1+\beta )}{\Delta t}$.

The two are algebraically identical ($\phi =1+\beta$); for small $\theta \Delta t$ the Euler approximation $\theta \approx (1-\phi )/\Delta t=-\beta /\Delta t$ holds to first order. The mean level is $\mu =\frac{a}{1-\phi }$ (level form).

Half-life. The expected time to close half the gap to equilibrium is

$$\boxed{{\tau }_{1/2}=\frac{\ln 2}{\theta }}.$$

(Equivalently, for the discrete AR(1), ${\tau }_{1/2}=\ln (2)/\ln (1/b)$ - the Tsay Ch 2 mean-reversion half-life.) A spread with ${\tau }_{1/2}$ of a few days is tradable with daily data; one with ${\tau }_{1/2}$ of a year is not.

Avellaneda–Lee estimation (Appendix). Using the cumulative residual $X_k={\sum }_{j\le k}{\tilde{R}}_j$ over a 60-day window and the regression $X_{n+1}=a+b{X}_n+{\zeta }_{n+1}$:

$$\kappa =-\log (b)\cdot 252,m=\frac{a}{1-b},{\sigma }_{\text{eq}}=\sqrt{\frac{\mathrm{Var}(\zeta )}{1-b^2}},$$

with the acceptance filter $\kappa >252/30$ (half-life under ~30 trading days, i.e. $0<b<0.9672$).

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3. Computational Implementation - the full pipeline

Stdlib only. We simulate a known OU spread ($\theta =0.20$/day, ${\tau }_{1/2}=3.47$ days) on top of a common random walk, then run Engle–Granger, the ADF test on the residual, and the OU fit - and check that we recover the truth.

All three stages recover the truth: $\hat{\beta }=0.9955$ (true $1.0$), an ADF $t=-10.255$ far below the $-3.34$ threshold (strong rejection of the unit root), and $\hat{\theta }=0.1990$ vs the simulated $0.20$, giving a half-life of $3.48$ vs $3.47$ days.

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4. Failure Modes & First-Principles Breakdowns

1. Wrong critical values. Testing the estimated residual with ordinary ADF tables over-rejects cointegration (the true CV is $\approx -3.34$, not $-2.86$). Use Engle–Granger / Phillips–Ouliaris / MacKinnon tables, or Johansen (Ch 6).
2. Low power in short windows. The test has weak power against slowly mean-reverting alternatives: a pair with a half-life of 6 months looks like a random walk in a 250-day sample. Cointegration fails to be rejected is not evidence of a tradeable pair.
3. Deterministic-specification sensitivity. Whether you include a constant or a trend changes both $\hat{\beta }$ and the critical values (Tsay §8.6.1 lists five cases). Choosing the spec after seeing the result is data snooping.
4. OLS $\beta$ is biased in finite samples and not necessarily the fastest-reverting combination. Engle–Granger OLS minimises the residual variance of one leg; it is super-consistent but can be inefficient. Johansen (Ch 6) or a symmetric (total-least-squares) estimator can recover the true vector better.
5. Half-life estimated on the wrong scale or the wrong regression. Mixing the level regression ($\theta =-\ln \phi /\Delta t$) with the change regression ($\theta =-\ln (1+\beta )/\Delta t$), or using the Euler shortcut $\theta \approx -\beta /\Delta t$, introduces error at large $\theta \Delta t$; at daily frequency with $\theta \le 0.2$ the difference is negligible, but be consistent about which regression you fit and about $\Delta t$ (252 vs 365).

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5. References

• Tsay, Analysis of Financial Time Series
• Tsay
• Engle, R. F. & Granger, C. W. J., Econometrica 55(2), 1987.
• Avellaneda, M. & Lee, J.-H., Quantitative Finance 10(7), 2010
• Hasbrouck, Empirical Market Microstructure

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Pairs Selection & Hedge · 04 · Trading Rules & Backtest
• Base: Econometrics & Time Series · Statistics & Inference
• Advanced: 06 · Advanced Extensions (Johansen multivariate test)