Question

which error have both positive and negative sign​

Answer 1

Explanation:

There is nothing wrong with the error correction term having both positive and negative loadings.

In fact, you could construct an example yourself.

Take y1,t=∑tτ=1ε1,τ, y2,t=∑tτ=1ε2,τ, …, yk,t=∑tτ=1ε1,τ where εi,τ are i.i.d. across all {i,τ}. Clearly, yi,ts are integrated processes.

Define, for example, yk+1,t=y1,t−y2,t+y3,t−y4,t+y5,t−…+(−1)kyk,t+εk+1,t where εk+1,t is a stationary variable.

Then a linear combination

yk+1,t−y1,t+y2,t−y3,t+y4,t−y5,t+…−(−1)kyk,t=εk+1,t

is stationary. So (y1,y2…,yk,yk+1) are cointegrated. The latter sum can serve as an error correction term. Note that it has altering signs. By changing the construction of yk+1,t (in the second bullet point) you can get whatever loadings you like (positive or negative) in the error correction term.

Answer 2

Explanation:

neutralised error have both positive and negative