Granger 2003 - Cointegration and the Error-Correction Representation

Summary

Two integrated ($I(1)$) series are cointegrated when some linear combination of them is stationary. Granger stresses that this is rare — “a surprise when it occurs” (p. 361) — but common among macro series as economic theory predicts. Cointegration has strong consequences that the lecture lists in words: the series share a hidden common stochastic factor; they admit an error-correction model (ECM) in which each series’ change responds to the lagged gap between them (and data generated by an ECM is always cointegrated); their long-horizon forecasts hold a constant ratio (an “equilibrium”); and (p. 366) at least one must Granger-cause the other. The ECM — due to Sargan — is what made cointegration practical, and it is the cure Granger offers for spurious regression. This note gives the formal versions (Engle-Granger 1987 representation theorem, stated from the literature) next to Granger’s verbal statements.

Overview

The joint analysis of several integrated series “was missing an important feature” (p. 361): the possibility that their difference is stationary. Granger presents cointegration as the “missing piece in our approach to modeling groups of series” (p. 362), which immediately explained the spurious-regression problem (p. 363) and connected to his earlier causality concept (p. 366). The intuition is the magnetised pearl chains (two chains with magnets).

Main Content

Definition

“It turns out that the difference between a pair of integrated series can be stationary, and this property is known as ‘cointegration.‘” (Granger 2003, p. 361)

“For cointegration, a pair of integrated, or smooth series, must have the property that a linear combination of them is stationary. Most pairs of integrated series will not have the property, so that cointegration should be considered as a surprise when it occurs. In practice, many pairs of macroeconomic series seem to have the property, as is suggested by economic theory.” (Granger 2003, p. 361)

Cointegration (formal; Engle & Granger 1987 convention)

Let ${\mathbf{x}}_t=(x_{1t},\dots ,x_{nt}{)}^{\prime }$ have every component $I(1)$ (integration). ${\mathbf{x}}_t$ is cointegrated if there exists a nonzero vector $\mathbf{\beta }$ (the cointegrating vector) such that

$$z_t={\mathbf{\beta }}^{\prime }{\mathbf{x}}_t\sim I(0).$$

$z_t$ is the equilibrium error (distance between the magnetised chains). For two series, normalise $\mathbf{\beta }=(1,-\beta {)}^{\prime }$: $z_t=y_t-\beta x_t$. With $n$ series there can be up to $r\le n-1$ linearly independent cointegrating vectors ($r$ = cointegrating rank). The general Engle-Granger definition is ${\mathbf{x}}_t\sim CI(d,b)$: components $I(d)$, combination $I(d-b)$, $b>0$; the lecture’s case is $CI(1,1)$.

Granger’s informal phrasing (p. 361): “two smooth series, properly scaled, can move and turn, but slowly, in similar but not identical fashions, but the distance between them can be stationary.” “Properly scaled” is the cointegrating coefficient $\beta$.

Consequences listed in the lecture

Granger (p. 361): “Once we know that a pair of variables has the cointegration property it follows that they have a number of other interesting and useful properties.” He lists:

Consequences of cointegration (Granger 2003, pp. 361-362, 366; formal versions from Engle & Granger 1987, citation only)

Let $x_t,y_t\sim I(1)$ be cointegrated with $z_t=y_t-\beta x_t\sim I(0)$. Then:

1. Common factor. “they must both be cointegrated with the same hidden common factor” (p. 361). Formally (common-trends / Stock-Watson form): $x_t=a_1{W}_t+{\tilde{x}}_t$, $y_t=a_2{W}_t+{\tilde{y}}_t$ with $W_t\sim I(1)$ a single common stochastic trend and ${\tilde{x}}_t,{\tilde{y}}_t\sim I(0)$; then $\beta =a_2/a_1$ removes $W_t$.
2. Error-correction representation (Granger Representation Theorem). “they can be considered to be generated by what is know as the ‘error-correction model,’ in which the change of one of the series is explained in terms of the lag of the difference between the series, possibly after scaling, and lags of the differences of each series. The other series will be represented by a similar dynamic equation.” (p. 361, “know” [sic]). Formally:

$$\Delta y_t={\mu }_1+{\gamma }_1{z}_{t-1}+\sum _{j=1}^p{a}_{1j}\Delta y_{t-j}+\sum _{j=1}^p{b}_{1j}\Delta x_{t-j}+{\epsilon }_{1t},$$ $$\Delta x_t={\mu }_2+{\gamma }_2{z}_{t-1}+\sum _{j=1}^p{a}_{2j}\Delta y_{t-j}+\sum _{j=1}^p{b}_{2j}\Delta x_{t-j}+{\epsilon }_{2t},$$

with $z_{t-1}=y_{t-1}-\beta x_{t-1}$ (“the lag of the difference between the series, possibly after scaling”), $({\gamma }_1,{\gamma }_2)\ne (0,0)$, and all terms $I(0)$ so the equations are balanced. The adjustment (loading) coefficients ${\gamma }_i$ measure how fast each series corrects the previous gap; stability requires the gap to shrink (e.g. ${\gamma }_1<0$ and/or ${\gamma }_2>0$ when $\beta >0$). In VAR form: $\Delta {\mathbf{x}}_t=\mathbf{\mu }+\mathbf{\alpha }{\mathbf{\beta }}^{\prime }{\mathbf{x}}_{t-1}+{\sum }_j{\Gamma }_j\Delta {\mathbf{x}}_{t-j}+{\mathbf{\epsilon }}_t$ with reduced-rank $\Pi =\mathbf{\alpha }{\mathbf{\beta }}^{\prime }$. 3. Converse. “Data generated by such a model is sure to be cointegrated.” (pp. 361-362) An ECM with stationary $\epsilon$ and $({\gamma }_1,{\gamma }_2)$ giving a stable gap produces cointegrated $I(1)$ series. 4. Long-run forecasts hold a fixed relationship. “when extended some way ahead, the forecasts of the two series will form a constant ratio, as is expected by some asymptotic economic theory.” (p. 362) Formally $\mathbb{E}[z_{T+h}\mid {\mathcal{F}}_T]\to \mathbb{E}[z]$ as $h\to \infty$, so forecasts of $y$ and $\beta x$ stay a constant distance apart; for log-level series, a constant distance in logs is a constant ratio in levels. In marketing forecasting this is why a VAR/ECM is the preferred long-horizon model when sales and marketing spending are cointegrated (Optimal Marketing Decisions and Forecasting, forecasting-approach 4). 5. Causality. “if a pair of series was cointegrated then at least one of them must cause the other” (p. 366), in the sense of Granger causality — because at least one ${\gamma }_i\ne 0$, the lagged level of the other series enters through $z_{t-1}$. Developed in Granger 2003 - Granger Causality and Its Link to Cointegration.

Proof sketch (2 ⇒ cointegration, the “converse”). Subtract $\beta \times$ (second equation) from the first: $\Delta z_t=({\mu }_1-\beta {\mu }_2)+({\gamma }_1-\beta {\gamma }_2)z_{t-1}+(\text{stationary lagged differences})+({\epsilon }_{1t}-\beta {\epsilon }_{2t})$. This is a stationary AR-type equation in $z_t$ whenever $-2<{\gamma }_1-\beta {\gamma }_2<0$ (in the no-extra-lags case), so $z_t\sim I(0)$ while $x_t,y_t$ each inherit a unit root from the common trend. (Cointegration ⇒ ECM) is the harder direction (Engle & Granger 1987, via the Wold representation of $\Delta {\mathbf{x}}_t$ and the reduced rank of its long-run matrix); the lecture states it without proof: Granger “proved the consequences such as the error-correction representation” (p. 363).

Error-correction model (two-variable, single-equation form)

$$\Delta y_t=\mu +\gamma (y_{t-1}-\beta x_{t-1})+\delta \Delta x_t+\sum _{j\ge 1}(\cdots )+{\epsilon }_t,\gamma <0.$$

$\beta$ = long-run (equilibrium) effect of $x$ on $y$; $\delta$ = short-run (impact) effect; $\gamma$ = speed of adjustment (fraction of last period’s disequilibrium removed per period; half-life $\ln 0.5/\ln (1+\gamma )$). Marketing-notation version and applications in ECM in Multivariate Persistence and Cointegration.

History and origins (pp. 362-363)

• Sargan’s ECM. “The error-correction model has been particularly important in making the idea of cointegration practically useful. It was invented by the well-known econometrician Dennis Sargan, who took some famous equations from the theory of economic growth and made them stochastic.” (p. 362)
• Scandinavian development (p. 362): Johansen and Juselius (Copenhagen) “developed and applied sophisticated testing procedures” — the Johansen likelihood-ratio rank tests; Hylleberg (Århus) “extended the theory to seasonal data” (seasonal cointegration); Jansen and colleagues at the Bank of Norway applied it to a large econometric model of Norway; Timo Teräsvirta, “who is from Finland but now lives in Stockholm”, helped develop models “useful in nonlinear formulations of cointegration”. Granger adds that he was “delighted that they are all here as my guests” (p. 362), i.e. they were present at the lecture as his guests.
• The name. Cointegration was “a name that was my own invention, incidentally” (p. 362).
• How it was discovered (p. 363). “A colleague, David Hendry, stated that the difference between a pair of integrated series could be stationary. My response was that it could be proved that he was wrong, but in attempting to do so, I showed that he was correct, and generalized it to cointegration, and proved the consequences such as the error-correction representation.” Granger connects this to Popper’s view (via Hacohen 2000, p. 244) that “discovery was not a matter of logic” but the application of methodology (p. 363). A failed disproof that turns into a theorem is itself a validation lesson: attempted falsification is a productive way to check a claim.
• Multivariate extension. “These ideas and models are fairly easily extended to many variables.” (p. 362)
• Reception. Once the idea became public “it was quickly picked up and used by many other econometricians and applied economists. There are now numerous citations and applications based on it.” (p. 362)
• Why it mattered to its authors. “Rob Engle and I quickly realized that the concept of cointegration and its extensions could be used to explain and remove a variety of difficulties that we had observed in our own research and that of others. It seemed to be the missing piece in our approach to modeling groups of series.” (p. 362) The first such difficulty he names is spurious regression (p. 363); see Granger 2003 - Spurious Regression from Yule to Granger-Newbold.

Why macro modellers adopted it (p. 362)

Granger describes the purpose of macro models: economists “attempt to build models that will approximate it, that will have similar major properties so that one can conduct simple experiments on them, such as determining the impacts of alternative policies or the long-run implications of some new institution” (p. 362), adding the caveat “All models simply assume that the model is correct and extrapolate from there, but hopefully with an indication of uncertainty around future values.” (p. 362) ECMs with cointegration became “a popular form of macro model” applied to “investment, taxes, consumption, employment, interest rates, government expenditure” (p. 362), and these are the equations that “central banks, the Federal Reserve Bank, and various model builders have found useful for policy simulations and other considerations.” (p. 362)

On equilibrium, Granger is careful: the constant-ratio long-run forecast is what interests theorists “concerned with ‘equilibrium.’ Whether the form of equilibria suggested by the models is related to that discussed by the theorists is unclear.” (p. 362)

Testing (standard practice, not in the lecture)

The lecture names no test statistics. The standard procedures it alludes to:

Engle-Granger two-step procedure

1. Pre-test each series for a unit root (ADF/KPSS); proceed only if both are $I(1)$.
2. Estimate the static levels regression $y_t=\alpha +\beta x_t+z_t$ by OLS. If cointegrated, $\hat{\beta }$ is superconsistent (converges at rate $T$, not $\sqrt{T}$), but its $t$-statistic is non-standard.
3. Test ${\hat{z}}_t$ for a unit root with an ADF regression $\Delta {\hat{z}}_t=\rho {\hat{z}}_{t-1}+{\sum }_j{\varphi }_j\Delta {\hat{z}}_{t-j}+e_t$, using Engle-Granger/MacKinnon critical values (more negative than Dickey-Fuller ones, because $\hat{\beta }$ was chosen to make $\hat{z}$ look stationary; about $-3.34$ at 5% asymptotically for two variables with a constant. Finite-sample values are somewhat more negative. MacKinnon’s response-surface value for $T\approx 200$ is about $-3.37$, which is the value used in Experiment 4. The two figures do not conflict: $-3.34$ is the asymptotic value and $-3.37$ the finite-$T$ one).
4. Reject the unit root ⇒ cointegration; fit the ECM in step 5. Fail to reject ⇒ the levels regression of step 2 is spurious; model in differences.
5. Estimate $\Delta y_t=\mu +\gamma {\hat{z}}_{t-1}+\delta \Delta x_t+\cdots +{\epsilon }_t$ by OLS; all regressors are $I(0)$, so standard inference applies to $\gamma ,\delta$.

With $n>2$ series or several cointegrating vectors, use Johansen’s reduced-rank VECM likelihood-ratio (trace / max-eigenvalue) tests. Both procedures are run on simulated data in Granger 2003 - Spurious Regression Monte Carlo in Python.

Examples

Simulated cointegrated pair and its ECM

Setup: $x_t$ a Gaussian random walk; $y_t=1+2x_t+u_t$ with $u_t=0.5u_{t-1}+{\epsilon }_t$; $T=200$; 2,000 replications (Experiments 4-5 in Granger 2003 - Spurious Regression Monte Carlo in Python).

Implied ECM: $\Delta y_t=2\Delta x_t+\Delta u_t$ and $\Delta u_t=-0.5u_{t-1}+{\epsilon }_t=-0.5(y_{t-1}-1-2x_{t-1})+{\epsilon }_t$, so $\gamma =-0.5$, $\delta =2$, $\beta =2$.

Results: levels OLS gives median $\hat{\beta }=2.00$ with Monte Carlo SD 0.032 (superconsistency — compare SD 0.62 for the spurious pair); the Engle-Granger residual test rejects “no cointegration” in 100% of replications (vs 4% for independent walks); the two-step ECM recovers $\hat{\gamma }=-0.522$ (SD 0.063) and $\hat{\delta }=1.999$ (SD 0.074).

Interpretation: when the magnets are real, the levels regression is meaningful and the ECM separates the long-run effect ($\beta$), the short-run effect ($\delta$) and the adjustment speed ($\gamma$); the gap’s half-life is $\ln 0.5/\ln 0.5=1$ period.

Marketing reading: long-run brand equilibrium

Setup: weekly log sales $y_t$ and log category sales $x_t$ for a mature brand (about 150 weeks). Both drift with the category.

Reading: if $y_t-\beta x_t$ (log market share when $\beta =1$) is stationary, the brand is cointegrated with its category: share fluctuates but returns to an equilibrium — Granger’s “constant ratio” in long-run forecasts. An ECM then lets a marketing variable (price, media) enter as a short-run $\Delta$ term or shift the equilibrium itself. If the share has a unit root (no cointegration), the brand’s position is evolving and a levels model that ties sales to the category is spurious. Marketing applications (persistence modelling, long-run marketing effects) are in Multivariate Persistence and Cointegration and Empirical Causal Ordering.

Connections

• Marketing-notation definitions, Johansen test and persistence modelling: Multivariate Persistence and Cointegration (definition, ECM, VAR).
• Cointegration is the constructive resolution of spurious regression: Granger 2003 - Spurious Regression from Yule to Granger-Newbold.
• VECM vs VAR-in-differences, impulse responses: Empirical Causal Ordering.
• Structural alternatives that model the shared trend directly: Bayesian Structural Time-Series Model, Local Linear Trend and Seasonality — a common latent trend $W_t$ is the state-space version of Granger’s “hidden common factor”.
• Equilibrium vs forecasting purposes of a model: Granger 2003 - Forecasting, Decision Science and Aggregation.
• Long-horizon VAR/ECM sales forecasting when cointegration exists: Optimal Marketing Decisions and Forecasting.
• Long-run (persistent) marketing effects: hysteresis and long-run multipliers in Carryover Effects and Distributed Lags; persistent distribution effects versus transient promotion effects in Price and Distribution Effects. A permanent shift in an $I(1)$ sales level is the stochastic-trend counterpart of hysteresis.

See Also

• Granger 2003 - Spurious Regression and Cointegration - Overview — source map
• Granger 2003 - Granger Causality and Its Link to Cointegration — consequence 5
• Granger 2003 - Spurious Regression Monte Carlo in Python — Engle-Granger test and ECM on simulated data
• Multivariate Persistence and Cointegration — marketing applications