before(see the previous slide).Warwick Business School 18
Error Correction Model (ECM)The ECM is a popular representation of the ADL model which incorporates bothlong-run equilibrium and short run dynamics in the system. ()()()()(ECM) 111RHS) on the add and(subtract 1Ysides)both from (subtract 1(ADL) 110101011100t111101110⎥⎦⎤⎢⎣⎡−+−−−−+Δ=Δ⇒−+++Δ+=Δ⇒−+++=Δ⇒+++=−−−−−−−−−−tttttttttttttttttXYXYXYXXYYXXYYXXYφδδφμφδδφδδδμφδδμφδδμThe Error Correction Term(ECT) This is the equilibrium errorin the previous period:The speed of adjustment to dis-equilibrium is measured byShort run dynamicsFrom ADL to ECM1−φ1−tεWarwick Business School 19
Comments on ECM1. The ECM incorporates both long-run and short-run effects assuming φ<1 (i.e., assuming cointegration).2. If φ=1(no cointegration) then only the differenced variables appear in the model (corresponding to short run effects).3. If there is cointegrationφ−1<0 parameterizes the speed of adjustment of Yto dis-equilibrium in the previous period.–If Y is abovethe long-run equilibrium in the previous period then Y will fallin the following period (and vice-versa)()()001000101111>Δ⇒>−⇒<<Δ⇒<−⇒>−−−−ttttttYYεφεεφε()01>⇒−tεFor example, if φ−1=−0.5 then 50% of thedis-equilibrium in period t-1 is corrected inperiod t. A further (φ−1) φ(=25%) of this dis-equilibrium is corrected in period t+1……a further (φ−1) φnis corrected in period t+n. The cumulative sum of the adjustments(φ−1)+(φ−1)φ+(φ−1)φ2…is -1 ⇒100% adjustment back to equilibrium(in the long-run).Warwick Business School 20
Comments on ECM
4. All the variables in the ECM are stationary
–If Y and X are I(1) then ΔY are ΔX are I(0)
–If Y and X are cointegratedthen ε~I(0)
–If Y and X are not cointegratedthen only the differenced (stationary) variables appear in the model (it’s no longer an ECM but a differenced model of Y).
Since all the variables are stationary, the ECM can be estimated by OLS with classical t and F tests being valid for inferences.Warwick Business School 21
Granger Representation Theorem
This is a fundamental theorem in cointegrationanalysis. It states that:
–If there exists a linear combination z of I(1) variables such that z~CI(1,1) then there must exist an ECM for the data.
–If there exists an ECM for a group of I(1) variables then they must be cointegratedCI(1,1).
In other words, cointegrationis both a necessary and sufficientcondition for the existence of an ECM amongst I(1) variables:
ECM for I(1) variables exists ⇔z~CI(1,1)
Given the ubiquity of ECMsin applied work the theorem has important empirical implications.
Notably, we need to test for cointegrationbefore we can validly estimate an ECM for variables which are I(1) in levels.Warwick Business School 22
Estimating CointegratedSystems
Engle and Granger Two Step EstimatorEG 2 step estimator embodies the principle of the GRT: –Firstly test for cointegration(estimate the long-run); –Then, if there is cointegration, estimate the short-run dynamics in the ECM.Step 1 (Estimate the long-run parameters)a)Test the variables individually for unit roots (see seminar 7)b)Estimate the cointegratingregression using OLS e.g
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