Error correction model

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An error correction model is a dynamical system with the characteristics that the deviation of the current state from its long-run relationship will be fed into its short-run dynamics.

An error correction model is not a model that corrects the error in another model. Error Correction Models (ECMs) are a category of multiple time series models that directly estimate the speed at which a dependent variable—Y—returns to equilibrium after a change in an independent variable—X. ECMs are a theoretically-driven approach useful for estimating both short term and long term effects of one time series on another. Thus, they often mesh well with our theories of political and social processes. ECMs are useful models when dealing with integrated data, but can also be used with stationary data.

Long-run relationship[edit]

A rough long-run relationship can be determined by the cointegration vector,[1] and then this relationship can be utilized to develop a refined dynamic model which can have a focus on long-run or transitory aspect such as the two VECM of a usual VAR in Johansen test.

An example of ECM[edit]

The idea of cointegration may be demonstrated in a simple macroeconomic setting. Suppose, consumption C_t and disposable income Y_t are macroeconomic time series that are related in the long run (see Permanent income hypothesis). Specifically, let average propensity to consume be 90%, that is, in the long run C_t = 0.9 Y_t. From the econometrician's point of view, this long run relationship (aka cointegration) exists if errors from the regression C_t = \beta Y_t+\epsilon_t are a stationary series, although Y_t and C_t are non-stationary. Suppose also that if Y_t suddenly changes by \Delta Y_t, then C_t changes by \Delta C_t = 0.5 \Delta Y_t, that is, marginal propensity to consume equals 50%. Our last assumption is that the gap between current and equilibrium consumption decreases each period by 20%.

In this setting a change \Delta C_t = C_t - C_{t-1} in consumption level can be modelled as \Delta C_t = 0.5 \Delta Y_t - 0.2 (C_{t-1}-0.9 Y_{t-1}) +\epsilon_t. The first term in the RHS describes long-run impact of change in Y_t on C_t, the second term explains long-run gravitation towards the equilibrium relationship between the variables, and the third term reflects random shocks that the system receives (e.g. shocks of consumer confidence that affect consumption). To see how the model works, consider two kinds of shocks: permanent and transitory (temporary). For simplicity, let \epsilon_t be zero for all t. Suppose in period t-1 the system is in equilibrium, i.e. C_{t-1} = 0.9 Y_{t-1}. Suppose that in the period t Y_t increases by 10 and then returns to its previous level. Then C_t first (in period t) increases by 5 (half of 10), but after the second period C_t begins to decrease and converges to its initial level. In contrast, if the shock to Y_t is permanent, then C_t slowly converges to a value that exceeds the initial C_{t-1} by 9.

This structure is common to all ECM models. In practice, econometricians often first estimate the cointegration relationship (equation in levels), and then insert it into the main model (equation in differences).

VECM[edit]

A vector error correction model (VECM) adds error correction features to a multi-factor model such as a vector autoregression model.

Notes[edit]

  1. ^ Engle, Robert F.; Granger, Clive W. J. (1987). "Co-integration and error correction: Representation, estimation and testing". Econometrica 55 (2): 251–276. JSTOR 1913236. 

Further reading[edit]