Time Series

5.7.2

ECM in Lag Operator Form

We can write the error correction model in lag operator form as well. Start with (9).

∆Y t  =  β 0 ∆X t  + γ [Y t

1

−

− β 3 X t

1 ] + ν t

−

Y t − Y t−1  =  β 0 X t − β 0 X t−1 + γY t−1 − γβ 3 X t−1  + ν t Y t − Y t−1 − γY t−1  =  β 0 X t − β 0 X t−1 + γβ 3 X t−1  + ν t Y t (1 − (1 + γ )L) = ( β 0 − (β 0  + γβ 3 )L)X t  + ν t Since  γ  =  ρ − 1,  β 3  =

β 2 γ 

−

and  β 2  =  β 0  + β 1  , this can be rewritten as:

Y t (1 − (1 + ρ − 1)L) =

  β 0 −

β 0  + γ 

−

β 0 − β 1 γ 

 

L X t  + ν t

Y t (1 − ρL) = (β 0  + β 1 L)X t  + ν t Y t

=



β 0  + β 1 L 1 − ρL



X t  +

ν t (1 − ρL)

 

(82)

Note that this is exactly the same as the ADL model written in lag operator form shown in Eq. (68). This is as one would expect given that we have just seen that the ADL model and error correction model are exactly equivalent. 5.7.3

Estimating an ECM Model

There are two ways to estimate an ECM model. 1. Engle-Granger Two Step Procedure

•  Estimate the following regression:  Y t  =  α + γX t  + t . In STATA, type regress Y X

•  From these estimates, generate the residuals i.e. et = Y t  − α − γX t . This is how much the system is out of equilibrium. In STATA, type predict e,resid

•  Now include the lag of the residuals from the initial regression, so that we have  ∆ Y t = β 0  + β 1 ∆X t 1  + et 1 . In STATA, type −

−

regress d.Y L.d.X L.e

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