Most macroeconomic series wander. Income, consumption and price levels drift with no fixed mean, and a regression between two of them can post a high R² that means nothing. You saw why in unit roots and the Dickey–Fuller test: once a series is I(1), standard inference breaks down. This page asks the sequel question — when are two drifting series genuinely tied together, and how do you model the tie? That is cointegration, where an econometrics tutor in Cardiff spends much of intermediate teaching.
1 · Why two trending series can fool you
Take two independent random walks — series with nothing in common, each last period’s value plus a fresh shock. Regress one on the other. You expect a slope near zero and an insignificant t-statistic. You get the opposite.
Granger and Newbold showed in 1974 that this regression returns a large t-statistic and a high R² most of the time, purely because both series trend — least squares reads their accidental alignment as a relationship. This is the spurious regression problem, and it does not fade as data accumulate — the t-statistic only grows more misleading. A strong regression between two I(1) variables means nothing on its own; what separates a real relationship from a coincidence is cointegration.
2 · What cointegration actually means
Sometimes two I(1) series are not independent. Driven by the same underlying force, they wander together. Formally, y and x are cointegrated when each is I(1) on its own, yet some linear combination y − βx is stationary, I(0).
The intuition is a shared stochastic trend. Both series carry the same wandering component, and y − βx subtracts it out, leaving something that reverts to a mean — the equilibrium error. Far from its mean, the pair has drifted apart; near zero, it sits at equilibrium. Because the combination is stationary, the two never stray arbitrarily far.
Real pairs behave this way: consumption and income trend together yet their gap stays bounded; spot and futures prices move as a unit, their difference — the basis — reverting to a stable level. Two I(1) levels, one I(0) combination — the diagram makes it visible.
3 · The Engle–Granger two-step
The Engle–Granger method tests for it in two steps, both of which you can already do.
Step one — estimate the long-run relation. Regress y on x by ordinary least squares and keep the residuals, êt = yt − β̂xt. If the pair is cointegrated, that residual is the estimated equilibrium error.
Step two — test the residual for a unit root. Run an ADF-type test on êt. If it is stationary, y − βx is I(0) and the series are cointegrated. If it still has a unit root, the regression was spurious.
One honest caveat. Least squares chose β̂ to minimise the residual’s variance, so it looks more stationary than real data would. The ordinary Dickey–Fuller critical values are therefore too generous — you use the more negative Engle–Granger (MacKinnon) values instead, stated here but not derived.
4 · The error-correction model
Cointegration says two series share a long-run relationship. The error-correction model (ECM) says how they return to it after a shock. For the change in y:
Δyt = α(y − βx)t−1 + γΔxt + εt
Read the pieces. The bracket (y − βx)t−1 is last period’s equilibrium error. Its coefficient α is the whole point: the speed of adjustment, the fraction of that disequilibrium corrected this period. The γΔxt term is short-run comovement, irrelevant to the long run.
For stability, α must be negative: if y sat above its long-run level, the error was positive, and a negative α makes Δy negative, pulling y back. Were α zero, there would be no correction and the “relationship” would be a fiction. A larger |α| means a faster return.
The Granger representation theorem (Engle and Granger, 1987) closes the loop: two series are cointegrated if and only if an ECM exists for them.
Worked example — consumption adjusting back to income
Take consumption y and income x, both I(1) and cointegrated, with long-run relation y = 0.9x. The equilibrium error is z = y − 0.9x.
Step 1 — Write the rule. The ECM is Δy = αz + short-run terms, with z last period’s error. Set α = −0.4.
Step 2 — Isolate the correction. Hold income steady and switch off the short-run terms and ε. Then this period’s error is (1 + α) times last period’s — that is 0.6z.
Step 3 — Start out of equilibrium. A one-off spending spree leaves consumption 10 units above what income justifies: the starting error is z = 10.
Step 4 — First step. Δy = −0.4 × 10 = −4. Consumption falls by 4, so the error drops to 6 — forty percent of the gap closed in one period.
Step 5 — Iterate. Each period keeps sixty percent of the gap: it runs 10 → 6 → 3.6 → 2.16 → 1.296 → 0.7776, i.e. 0.6t × 10. The disequilibrium halves in about 1.4 periods and is all but gone within seven.
Step 6 — Interpret. α = −0.4 is the engine. Its negative sign makes the pair converge; its size sets the pace at forty percent of the gap each period. Had α been zero, the error would never shrink — no error correction, and no cointegration.
Given two drifting series and a cointegrating relation, could you write the error-correction model — and read the speed of adjustment straight off the sign and size of α? Deriving how a pair is pulled back to its long-run equilibrium, rather than defining cointegration, is exactly what time-series questions reward. A one-on-one econometrics tutor works the Engle–Granger steps and the ECM with you until the speed of adjustment is a result you derive, not a term you recite. Book a trial session.
Practice
Q1. An ECM has α = −0.25 and last period’s equilibrium error was 8. With x fixed and no new shock, find the change in y this period and the new error.
Q2. With α = −0.4 and a starting error of 10 (so the error follows 0.6t × 10), find the first whole period t at which it falls below 1.
Q3. Two cointegrated pairs: pair A has α = −0.2, pair B has α = −0.6. (a) Which returns faster? (b) For each, what fraction of a disequilibrium remains after two periods?
Answers. Q1: Δy = −0.25 × 8 = −2, so the new error is 0.75 × 8 = 6 — a quarter of the gap closes. Q2: t = 5 (at t = 4 the error is 1.296, above 1; at t = 5 it is 0.7776). Q3: (a) Pair B, since |−0.6| > |−0.2|. (b) The fraction left after two periods is (1 + α)²: pair A, 0.8² = 0.64; pair B, 0.4² = 0.16.
Key takeaways
- A high R² between two I(1) series proves nothing — independent random walks look significant by accident (spurious regression).
- Cointegration is a shared stochastic trend: two I(1) series whose combination y − βx is stationary — the equilibrium error.
- Engle–Granger is two steps: estimate the long-run relation, then test its residual for a unit root, with the larger cointegration critical values.
- The ECM puts the relationship in motion: the speed of adjustment α < 0 is the fraction of last period's gap corrected now.
- Cointegration ⇔ an ECM exists — the Granger representation theorem.
Why Cardiff students choose our econometrics tutoring
- Built from the relation up: sessions derive the ECM from the cointegrating equation, so you can reconstruct the speed-of-adjustment result under exam pressure, not memorise it.
- The distinctions examiners test, drilled: spurious versus genuine regression, why residual tests need their own critical values, and the sign and size of α.
- One-on-one and matched to your module: a tutor works from your own past papers and software output, whether your course runs on EViews, Stata or R.
FAQ
Q: What is the difference between correlation and cointegration?
A: Correlation is short-run comovement and can be spurious between two trending series. Cointegration is a long-run property: the series share a stochastic trend, so a combination of them is stationary.
Q: Can two stationary series be cointegrated?
A: No — it is defined only for series that are individually I(1). If they are already stationary, ordinary regression is valid.
Q: What does the speed-of-adjustment coefficient tell me?
A: The fraction of last period’s disequilibrium corrected this period. An α of −0.4 closes forty percent of the gap each period, and must be negative for convergence.
Q: Why not use ordinary Dickey–Fuller critical values on the residual?
A: The residual is fitted to look as stationary as possible, so it passes too easily. Use the larger, more negative Engle–Granger (MacKinnon) values.
Q: Is a two-variable ECM the same as a VECM?
A: A vector error-correction model is the multivariate version — several cointegrated series estimated together by Johansen’s method. The two-variable model here is the entry point.
Book an econometrics tutor in Cardiff
Cointegration and error-correction models reward students who can derive the speed of adjustment, not just define it — the spurious-regression trap, the Engle–Granger steps, and the sign and size of α. One-on-one sessions build that fluency on your own past papers and datasets. Tell us your university and module, and we will match you with the right tutor this week.
