Unit Roots, Cointegration, and Error Correction

Persistence is not one phenomenon

Compare:

• If $|\rho|<1$, shocks decay and $x_t$ is stationary under stable conditions.
• If $\rho=1$, $\Delta x_t=u_t$ and shocks permanently change the level.
• If $\rho$ is close to one, finite samples may not reliably distinguish the two.

This distinction affects forecasts, standard errors, and the meaning of a level regression. A unit-root test is evidence about a maintained model, not a mechanical permission slip to difference every series.

Why unrelated trends can look convincing

Let

with independent innovations. Regressing $y_t$ on $x_t$ can produce a high $R^2$ and a conventional-looking t-statistic even though the innovations are unrelated. The regression treats persistent excursions as repeated independent evidence.

Reasonable responses are:

• difference the variables if the question concerns short-run changes;
• model deterministic trends if theory specifies them;
• use cointegration if theory concerns a stable linear combination of nonstationary levels.

Cointegration

For an $n$-vector $\mathbf y_t$ whose components are $I(1)$, cointegration rank $r>0$ means that an $n\times r$ matrix $\beta$ exists such that

is stationary. The columns of $\beta$ define long-run relations; they are not automatically causal parameters.

A VAR in levels can be rewritten as a vector error-correction model:

If $0<\operatorname{rank}(\Pi)=r<n$, factor

where:

• $\beta^\top\mathbf y_{t-1}$ contains the long-run disequilibria;
• $\alpha$ contains the adjustment speeds;
• $\Gamma_i$ describe short-run dynamics.

Johansen's likelihood framework turns the rank question into a reduced-rank multivariate estimation problem.

Bivariate derivation

Suppose

while $x_t$ is $I(1)$. Then the spread $u_t=y_t-\theta x_t$ is stationary. Differencing gives

The coefficient $\rho-1<0$ is error correction: a positive deviation in the previous period predicts downward adjustment in $y_t$, conditional on $\Delta x_t$.

R laboratory: recover equilibrium and adjustment

The estimated long-run slope should be near $1.5$, and the error-correction coefficient near $-0.35$. The test statistic is included only as a persistence contrast; valid unit-root inference uses nonstandard critical values and a carefully specified deterministic component.

Finance use: spread, price discovery, or trading?

Examples include:

• spot and futures prices linked by carry;
• yields at different maturities under a term-structure relation;
• prices of economically linked securities;
• multiple venues contributing to price discovery.

Cointegration supports a statistical long-run relation. A trading claim additionally needs:

• stability after the formation sample;
• adjustment fast enough relative to the horizon;
• tradability of both legs;
• transaction costs, funding, shorting, and execution;
• a rule fixed before the test sample;
• tail exposure when the historical relation breaks.

“Stationary spread” is not synonymous with “arbitrage.”

Economics use: equilibrium and adjustment

Examples include consumption and income, prices and monetary aggregates, or related output measures. Here $\beta$ is usually interpreted through economic theory, and $\alpha$ asks which variables respond when the system departs from the long-run relation.

That interpretation requires care:

• normalising one coefficient to one does not make that variable exogenous;
• rank can be sensitive to lag length, deterministic terms, breaks, and sample span;
• a policy regime change can alter both $\alpha$ and $\beta$;
• cointegration does not by itself identify a structural mechanism.

Levels VAR, VECM, or differences?

Practice

1. If $\beta^\top\mathbf y_t$ is stationary, must every element of $\mathbf y_t$ be stationary?
2. In $\Pi=\alpha\beta^\top$, which matrix describes disequilibrium and which describes response?
3. Why can a structural break resemble failure of cointegration?

Next: VAR, Structural Identification, and Local Projections.