390 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 
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while A has zeros everywhere on its principal diagonal. The 
assumption that there are no terms in y,_, for 8>1 involves 
no loss of generality in the present discussion, since it can 
always be accomplished by redefinition of y, and expansion of 
the equation system and will be used only for convenience in 
dealing with the solution of (2.1) regarded as a system of stoch- 
astic difference equations. 
If: 
(R.1) A is triangular; 
(R.2) The variance-covariance matrix of the current dis- 
turbances is diagonal; 
(R.3) No current disturbance is correlated with any past 
disturbance; 
then the model is recursive and does not violate the assumption 
that in each equation the disturbance term is uncorrelated with 
the variables which appear therein other than the one to be 
explained by that equation. Ordinary least squares is then 
a consistent estimator and is the maximum likelihood estimator 
if each element of , is normally distributed and homoscedastic. 
To see that the no-correlation assumption is not violated, 
we solve the system for y,, obtaining: 
(2.2) y,=(1-A)'By, ;+(I1-A)Cz,+ (I-A) lu. 
fo wv bl 
Denote (I- A)~! by D and note that it is triangular by (R.1). 
We may take the zero elements to lie above the principal dia- 
gonal. Assuming that DB is stable, we have: (7) 
(2.3) 
æ 6 
y, = 2 (DB) (DC z,  * Du, 
(7) We shall not discuss the assumption of the stability of DB in any 
detail at this point. If it is not stable, then it suffices to assume that the 
model begins with non-stochastic initial conditions. Obviously, if stability 
fails the assumption of no serial correlation becomes of even greater im- 
portance than if stability holds. We shall return to this and shall discuss 
the question of stability in general in a later section. 
6] Fisher - pag. 6