Big O in probability notation

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

A rough long-run relationship can be determined by the cointegration vector, 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

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 $C_t=0.9Y_t$. From the econometrician's point of view, this long run relationship (aka cointegration) exists if errors from the regression $C_t=\beta Y_t+{ϵ}_t$ are a stationary series, although $Y_t$ and $C_t$ are non-stationary. Suppose also that if $Y_t$ suddenly changes by $\mathrm{\Delta }{Y}_t$, then $C_t$ changes by $\mathrm{\Delta }{C}_t=0.5\mathrm{\Delta }{Y}_t$, 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 $\mathrm{\Delta }{C}_t=C_t-C_{t-1}$ in consumption level can be modelled as $\mathrm{\Delta }{C}_t=0.5\mathrm{\Delta }{Y}_t-0.2(C_{t-1}-0.9Y_{t-1})+{ϵ}_t$. 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 ${ϵ}_t$ be zero for all t. Suppose in period t-1 the system is in equilibrium, i.e. $C_{t-1}=0.9Y_{t-1}$. 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, econometrists often first estimate the cointegration relationship (equation in levels), and then insert it into the main model (equation in differences).

VECM

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

Notes

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Further reading

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