dependent of other conditions. Confining attention to the OC’s relating to the x’s,
we have6
(α) Assume that (B1) and (C1) are satisfied. Then: (i) All OC’s (32) are linearly
dependent on all admissible OC’s relating to equations differenced over one
period and a subset of the OC’s relating to two-period differences. (ii) All
OC’s (34) are linearly dependent on all admissible OC’s relating to IV’s dif-
ferenced over one period and a subset of the IV’s differenced over two periods.
(β) Assume that (B2) and (C2) are satisfied. Then: (i) All OC’s (33) are linearly
dependent on all admissible OC’s relating to equations differenced over one
period and a subset of the OC’s relating to differences over 2(τ +1) periods.
(ii) All OC’s (35) are linearly dependent on all admissible OC’s relating to
IV’s differenced over one period and a subset of the IV’s differenced over
2(τ +1) periods.
We denote the non-redundant conditions defined by (α) - (β) as essential OC’s.
The following propositions are shown in Bi0rn (2000, section 2.d):
Proposition 1: Assume that (B1) and (C1) are satisfied. Then
(a) E[xi0p(∆it,t-1)] = 0K,1 forp =1,...,t-2,t+1,...,T; t =2,...,T are
K(T - 1)(T -2) essential OC’s for equations differenced over one period.
(b) E[xi0t(∆it+1,t-1)] = 0K,1 for t =2,...,T- 1 are K(T -2) essential OC’s for
equations differenced over two periods.
(c) The other OC’s are redundant: among the 1KT(T — 1)(T — 2) conditions in
(32) when T>2, only KT(T- 2) are essential.
Proposition 2: Assume that (B1) and (C1) are satisfied. Then
(a) E[(∆xip,p-1)0it]=0K,1 for t = 1,...,p—2,p+1,...,T; p = 2,...,T are
K(T— 1)(T — 2) essential OC’s for equations in levels, with IV’s differenced
over one period.
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