# Error-Correction Mechanisms

# Error-Correction Mechanisms

Consider an econometric model of a dynamic relationship between a variable *y _{t}* and explanatory variables

*x*

_{1t }, …,

*x*, taking the form

_{Kt}where *u _{t}* is a random error term. An alternative representation of this model is

where Δ*y _{t}* =

*y*–

_{t}*y*and Δ

_{t-1}*x*=

_{it}*x*–

_{it}*x*for

_{i},_{t-1}*i*= 1, …,

*k*, and the relationships between the coefficients are γ

_{0}= β

_{0}, γ

_{1i }= β

_{1i }, γ

_{2}= β

_{3}– 1, and δ

_{i}= (β

_{1i }+ β

_{2i })/(1 – β

_{3}). Equation (1) is sometimes called the autoregressive distributed lag (ARDL) representation, while (2) is the error correction mechanism (ECM) representation. Generalizations to higher orders of lag are easily obtained. The ECM representation has the attractive features of representing an economic agent’s decision in terms of a rule-of-thumb response to current changes, according to parameters γ

_{1i }and corrections to deviations from a desired long-run equilibrium relation with parameters δ

_{i }. For this reason the acronym ECM is sometimes taken to stand for equilibrium correction mechanism. Equation has the disadvantage of being nonlinear in parameters, so that estimation and inference are less straightforward than in the case of equation.

The ECM form was first proposed by Denis Sargan (1924-1996) for a model of wages and prices (Sargan 1964). It was subsequently popularized by the work of David Hendry and others in the context of modeling applications in macroeconomics, such as the consumption function and the demand for money. The seminal contribution is J. E. H. Davidson, D. F. Hendry, F. Srba, and J. S. Yeo (1978), commonly referred to in the literature as DHSY.

The approach later acquired special prominence due to the work of Clive W. J. Granger, who independently explored the implications of modeling economic time series as integrated (I(1)) processes; in other words, processes generated as the partial sums of stationary, weakly dependent increments. (A random walk is a simple example.) This type of model, also called a stochastic trend model, describes many series observed in economics and finance. If in the driving processes *x _{1t}*, …,

*x*are I(1), and γ

_{kt}_{2}< 0, then

*y*~ I(1) also, but

_{t}is I(0) (i.e., a stationary, weakly dependent process). The variables are then said to be cointegrated and *z _{t}* is called the cointegrating residual. Cointegration (i.e., combining the twin modeling concepts of stochastic trend representations for economic series and cointegrated relations characterizing long-run interactions over the economic cycle) has been a profoundly influential idea in macroeconomics, earning Robert F. Engle and Granger (1987) the Nobel Prize for economics in 2003.

In practice, such models are often generalized to a system of dynamic equations, explaining several variables in terms of a common set of cointegrating relations. In reduced form the resulting models are called vector error correction models (VECMs) or reduced rank vector autoregressions (VARs). Following the work of Søren Johansen, a closed VECM for an *m* -vector of variables *x _{t}* is commonly represented in matrix notation as

where *z _{t}* = β′x

_{t }(

*s*× 1) is the vector of cointegrating residuals. The rank of the

*m*×

*s*matrices β and α is called the cointegrating rank of the system. In a cointegrated system the inequalities

*0 < s < m*must hold. The Granger representation theorem states that a linear dynamic model generates cointegrating relations if and only if it has a VECM representation.

**SEE ALSO** *Cointegration; Lags, Distributed; Least Squares, Two-Stage*

## BIBLIOGRAPHY

Davidson, J. E. H., D. F. Hendry, F. Srba, and J. S. Yeo. 1978. Econometric Modelling of the Aggregate Time-Series Relationship between Consumers’ Expenditure and Income in the United Kingdom. *Economic Journal* 88: 661-692.

Engle, Robert F., and Clive W. J. Granger. 1987. Cointegration and Error Correction: Representation, Estimation, and Testing. *Econometrica* 55 (2): 251-276.

Johansen, Søren. 1988. Statistical Analysis of Cointegrating Vectors. *Journal of Economic Dynamics and Control* 12: 231-254.

Nelson, Charles R., and Charles I. Plosser. 1982. Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications. *Journal of Monetary Economics* 10: 139-162.

Sargan, J. D. 1964. Wages and Prices in the United Kingdom: A Study in Econometric Methodology (with Discussion). In *Econometric Analysis for National Economic Planning*. Vol. 16 of *Colston Papers*, eds. Peter Edward Hart, Gordon Mills, and John King Whitaker, 25-63. London: Butterworth.

*James Davidson*

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