The lag order in first differences equals the lag order in levels minus one. Consider the following two examples (ignoring deterministic components), starting with the level representation:
1) y = c1 * L.y + d0 * x + d1 * L.x + u
2) y = c1 * L.y + c2 * L2.y + d0 * x + d1 * L.x + d2 * L2.x + u
The corresponding error-correction representation is as follows:
D.y = a * (L.y - b * L.x) + e1 * LD.y + f0 * D.x + f1 * LD.x + u
By analytical reformulations of models 1) and 2) above you can compute the following parameter restrictions for both cases:
1) a = c1 - 1 ; b = (d0 + d1) / (1 - c1) ; e1 = 0 ; f0 = d0 ; f1 = 0
2) a = c2 + c1 - 1 ; b = (d0 + d1 + d2) / (1 - c1 - c2) ; e1 = - c2 ; f0 = d0 ; f1 = - d2
As you can see, in model 1) the coefficient e1 of LD.y in the error-correction representation equals zero. It is only non-zero, when there are at least two lags of y in the level representation, as in model 2). The reason is simply that LD.y = L.y - L2.y but there is no L2.y in the level representation of model 1) that can be used to form LD.y in the corresponding error-correction representation.
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