Solved – VECM, positive loading coefficients of EC terms

cointegrationvector-error-correction-model

I am looking at two very similar time series (correlation is about .997) – two commodity price series.
Johansen coint. test indicates 1 coint. relationship for case 1 (no trend and no intercept) and no cointegration for other cases described by Johansen.

Lets denote the series A, B.
If I use the order of series A, B, then in the estimated VECM, the coint. vector is (1, -1.003722)'
The alpha coeff. for EC terms are 0.023158 (in equation for A) and 0.431335 (in eq. for B).

Now, if I estimate the VECM with reversed order, that is B, A (which should only influence the normalization of coint. vector) i get these results:
Coint. vector ( 1, -0.996066)'
The alpha coeff. for EC terms are -0.060448 (in equation for A) and -0.444057 (in eq. for B).

I wonder, why did the coefficients for EC terms changed their signs?
I was under the impression that EC terms should have negative coefficients in order to maintain the coint. relationship.

Data: download CSV
Use from obs. 62 onwards
(a side question is that the cointegration results vary if you use all observations, or if you start from obs. 62)

Please, tell me if I understand it in the following example correctly. 

Let's assume we observe: A=5,  B= 5.01;    

For order A, B:    
coint. vector: (1, -1.003722)';  
5 -1.003722*5.01 = -0.02864722;  
EC with loading in eq. for A:  0.023158 * -0.02864722 -> negative (A should go down?);   
EC with loading in eq. for B:  0.431335* -0.02864722 -> negative (B should go down?);  
for B, A:    
coint. vector: ( 1, -0.996066)';  
5.01-0.996066*5 = +0,02967;  
EC with loading in eq. for A:  -0.060448*+0,02967 -> negative (A should go down?);  
EC with loading in eq. for B:  -0.444057*0.02967 -> negative ( B should go down?);

Best Answer

Let the two cointegrating variables be $x$ and $y$.

The error correction term is $x-by$.

Consider equation 1 in the system (equation 1 or 2 does not matter, any one of them is enough to understand).

The error correction term with its loading is $a(x-by)$.

Now consider what happens when you switch the positions of $x$ and $y$ by renormalizing the cointegration vector. You use the following: $a(x-by)=-ba(y-\frac{1}{b}x)$ and get the new error correction term with its loading $a'(y-b'x)$ where $a'=-ba$ and $b'=\frac{1}{b}$.

That means you actually expect the opposite sign, and it's not a mistake. Your reported coefficient values seem to agree with this.

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