Error correction model: Difference between revisions

Revision as of 03:36, 14 February 2021

Revision as of 15:39, 4 April 2021

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===An example of ECM===

The idea of cointegration may be demonstrated in a simple macroeconomic setting. Suppose, consumption <math>C_t</math> and disposable income <math>Y_t</math> 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 <math>C_t = 0.9 Y_t</math>. From the econometrician's point of view, this long run relationship (aka cointegration) exists if errors from the regression <math>C_t = \beta Y_t+\varepsilon_t</math> are a [[Stationary process|stationary]] series, although <math>Y_t</math> and <math>C_t</math> are non-stationary. Suppose also that if <math>Y_t</math> suddenly changes by <math>\Delta Y_t</math>, then <math>C_t</math> changes by <math>\Delta C_t = 0.5 \, \Delta Y_t</math>, that is, [[marginal propensity to consume]] equals 50%. Our lastfinal assumption is that the gap between current and equilibrium consumption decreases each period by 20%.

In this setting a change <math>\Delta C_t = C_t - C_{t-1}</math> in consumption level can be modelled as <math>\Delta C_t = 0.5 \, \Delta Y_t - 0.2 (C_{t-1}-0.9 Y_{t-1}) +\varepsilon_t</math>. The first term in the RHS describes short-run impact of change in <math>Y_t</math> on <math>C_t</math>, 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 <math>\varepsilon_t</math> be zero for all t. Suppose in period ''t''&nbsp;−&nbsp;1 the system is in equilibrium, i.e. <math>C_{t-1} = 0.9 Y_{t-1}</math>. Suppose that in the period t <math>Y_t</math> increases by 10 and then returns to its previous level. Then <math>C_t</math> first (in period t) increases by 5 (half of 10), but after the second period <math>C_t</math> begins to decrease and converges to its initial level. In contrast, if the shock to <math>Y_t</math> is permanent, then <math>C_t</math> slowly converges to a value that exceeds the initial <math>C_{t-1}</math> by&nbsp;9.