Co-Integration and Error Correction: Representation, Estimation, and Testing

Summary

The foundational paper establishing the theory, estimation, and testing of co-integrated systems. Introduces the formal CI(d,b) definition, proves the Granger Representation Theorem (cointegration ↔ error-correction ↔ moving-average equivalence), establishes ordinary least squares (OLS) superconsistency for the cointegrating vector, develops the two-step estimator, and proposes seven statistics for testing the null of no cointegration — with critical values from Monte Carlo simulation. Applied to consumption–income, wages–prices, interest rate spreads, and monetary aggregates.

Key Claims

• CI(d,b) definition: Components of $x_t \sim I(d)$ are co-integrated of order $(d,b)$ if there exists $\alpha \neq 0$ such that $\alpha'x_t \sim I(d-b)$, $b>0$. Typically $d=b=1$.
• Granger Representation Theorem: For $x_t \sim CI(1,1)$, error-correction representation, Wold moving-average (MA) representation (with $C(1)$ of reduced rank $N-r$), and autoregressive representation are all equivalent and mutually imply one another.
• Superconsistency (Stock 1987): OLS of $y_t$ on $x_t$ converges at rate $T$ (not $\sqrt T$); pre-estimation error in $\hat\beta$ has no first-order effect on the second-stage error-correction model (ECM) estimator.
• Two-step estimation: (1) OLS for $\hat\beta$; (2) maximum likelihood (ML) of ECM with $\hat z_{t-1} = \hat\beta' x_{t-1}$ fixed — second-stage estimators are consistent and asymptotically normal.
• Seven cointegration tests: cointegrating-regression Durbin-Watson (CRDW), Dickey-Fuller (DF), augmented Dickey-Fuller (ADF; recommended), restricted vector autoregression (RVAR), unrestricted VAR (UVAR), augmented restricted VAR (ARVAR), differenced VAR (DVAR) — all applied to $\hat z_t$ residuals; critical values are non-standard (Monte Carlo, Tables I–III) because $\hat z_t$ are estimated.
• ADF dominates in power: The augmented Dickey-Fuller test on residuals has the best power among the seven statistics proposed.
• Empirical findings: Consumption–income: cointegrated (ADF ≈ −3.11); wages–prices: cointegrated; short–long interest rates: cointegrated (ADF ≈ −3.42, adjustment ≈ −0.46/period); monetary aggregates (M1, M2, M3) and gross national product (GNP): not cointegrated.

Concepts Introduced or Extended

• Cointegration
• Spurious Regression

Entities Mentioned

• Robert Engle
• Clive Granger

Quotes

"If each element of a vector of time series $x_t$ achieves stationarity after differencing, but a linear combination $\alpha'x_t$ is already stationary, the time series are said to be co-integrated with co-integrating vector $\alpha$."

"It is shown that co-integration of $x_t$ is equivalent to the existence of an error correction formulation."

My Take

This paper is the bedrock of the modern cointegration literature. The Granger Representation Theorem is the result that every subsequent development builds on — Johansen's (1988, 1991) maximum-likelihood estimator (MLE) is essentially the efficient version of what EG initiated. The seven test statistics proposed here have largely been superseded by Johansen's trace and max-eigenvalue tests, but the ADF on residuals remains a standard diagnostic. The superconsistency result is the key insight that makes the two-step approach work: the first stage is so accurate that you can treat $\hat\beta$ as known for all second-stage purposes. The empirical finding that monetary aggregates and GNP are not cointegrated was influential and somewhat controversial.