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Partial cointegration

Separate transient mean reversion from permanent random-walk shifts, and see why a partial-cointegration residual need not return to zero.

Partial cointegration models a linear combination of nonstationary time series as the sum of a mean-reverting component and a random walk. In pairs research, this lets a temporary relative-price dislocation coexist with permanent shifts. A large fitted spread therefore need not be a gap that will close completely.
The meaning here is the Clegg–Krauss model used in pairs trading. Ordinary cointegration requires a stationary linear combination of integrated series; partial cointegration allows a stochastic trend to remain in that combination. The word “partial” refers to this decomposition, not a probability that two prices will converge. Clegg and Krauss, §2.1

Two kinds of shock in one residual

For two level series XtX_t and YtY_t, each integrated of order one, write a fixed-coefficient relation

Yt=α+βXt+ut,ut=Mt+Rt,Mt=ρMt−1+εM,t,∣ρ∣<1,Rt=Rt−1+εR,t.\begin{aligned} Y_t &= \alpha+\beta X_t+u_t,\\ u_t &= M_t+R_t,\\ M_t &= \rho M_{t-1}+\varepsilon_{M,t},\qquad |\rho|<1,\\ R_t &= R_{t-1}+\varepsilon_{R,t}. \end{aligned}
Here α\alpha is a fixed intercept and β\beta is a fixed loading. The autoregressive component MtM_t is transient: the effect of a shock decays with powers of ρ\rho. The random-walk component RtR_t is permanent: its shocks have no built-in reversal. The basic Gaussian specification takes the two innovation sequences to be mutually independent, zero-mean white noise with variances σM2\sigma_M^2 and σR2\sigma_R^2. Model definition and intercept convention
With positive random-walk innovation variance, the whole residual is nonstationary, even when MtM_t is stationary. If σR2=0\sigma_R^2=0, RtR_t is constant and the residual reduces to a stationary component around a fixed level, given stationary initialization of MtM_t. This is the full-cointegration boundary. It does not require that fixed level to be zero.

Seven units does not mean seven units will unwind

Consider a fictional current state with Mt=2M_t=2, Rt=5R_t=5 and ρ=0.5\rho=0.5. The residual is ut=7u_t=7 spread units. These component states are stipulated as known for illustration; they are not ordinarily observed separately.

With future innovations having conditional mean zero,

E[ut+h∣Mt=2,Rt=5]=2(0.5)h+5.\mathbb{E}[u_{t+h}\mid M_t=2,R_t=5] =2(0.5)^h+5.
HorizonExpected transient componentExpected random-walk componentExpected residual
Now257
One period156
Two periods0.555.5
Three periods0.2555.25
Long-horizon limit055
The transient component has a half-life of log⁡(0.5)/log⁡(0.5)=1\log(0.5)/\log(0.5)=1 period. This is not a half-life for the whole residual. Its conditional expected level tends to 5, not zero. Nor does that forecast mean the realised residual settles at 5: future random-walk shocks continue to accumulate.
Fictional known-state illustration: M starts at 2 and halves each period, R has conditional mean 5, and the residual conditional mean falls from 7 to 6 and tends to 5 rather than zero.
Fictional known-state illustration: M starts at 2 and halves each period, R has conditional mean 5, and the residual conditional mean falls from 7 to 6 and tends to 5 rather than zero.
Known-state conditional means with zero-mean future innovations. The chart omits realised shocks and forecast uncertainty; it is not a simulated price path.

A converging conditional mean can coexist with growing uncertainty. Conditional on known current states and fixed parameters, independence gives

Var(ut+h∣Mt,Rt)=σM21−ρ2h1−ρ2+hσR2,h≥1.\text{Var}(u_{t+h}\mid M_t,R_t) =\sigma_M^2\frac{1-\rho^{2h}}{1-\rho^2}+h\sigma_R^2, \qquad h\ge 1.

The transient contribution is bounded; the random-walk contribution grows with the horizon. This follows by summing the squared innovation weights in the two state recursions. Uncertainty about the current states or fitted parameters requires additional treatment.

The split must be estimated

Observing a residual of 7 does not reveal how much is transient. The authors' approach uses a state-space model, a Kalman filter and maximum likelihood to estimate parameters and hidden components. Filtered states carry estimation uncertainty. A historical smoothed estimate using later observations cannot be treated as a signal available at that earlier date. Authors' estimation and hidden-state documentation
Fitting a decomposition is also different from finding evidence for both components. The authors' documented test framework compares the unrestricted model with pure-random-walk and pure-AR(1) alternatives. An ordinary residual unit-root rejection is not the definition of partial cointegration; a random-walk component is allowed. Boundary hypotheses require appropriate calibration rather than an automatic ordinary likelihood-ratio reference distribution. Authors' testing documentation
For a trade, the relevant candidate dislocation is the estimated transient component. Even if it decays, changes in RtR_t can overwhelm that movement. Trading results also depend on hedge implementation, transaction costs, borrow availability and stability outside the estimation sample. The model supplies neither zero convergence nor a profit guarantee.
For the contrasting stationary-residual framework, Cointegration and Error Correction develops residual tests and equilibrium adjustment. The preview is public; the full lesson requires membership.

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  • partial cointegration
  • pairs trading
  • mean reversion
  • state space models
  • permanent shocks
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