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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
2. ORDINARY LEAST SQUARES 
2.1. Assumbtions and Proberties 
Ordinary least squares has a number of desirable properties 
when appropriate assumptions are satisfied. Briefly, if the 
explanatory variables in the equation to be estimated are either 
non-stochastic or distributed independently of all past, present, 
and future values of the disturbance term in that equation, if 
the disturbance term is serially uncorrelated and homoscedastic, 
and if there are no a priori restrictions on the parameters to be 
estimated, then ordinary least squares is the best linear unbiased 
estimator. In addition, if the disturbances are normally distri- 
buted, then ordinary least squares is the maximum likelihood 
estimator. 
These assumptions can be weakened in several ways. First, 
if the explanatory variables are not independent of the disturb- 
ance term but are uncorrelated with it in the probability limit, 
then ordinary least squares ceases to be unbiased but is con- 
sistent. If the disturbances are serially correlated, ordinary 
least squares loses efficiency but retains consistency provided 
that such serial correlation does not affect the validity of as- 
sumptions concerning the correlation of the current disturbance 
term and the explanatory variables (a matter to which we shall 
return) (*). Finally, ordinary least squares presents no partic- 
lar difficulties of computation. 
As is well known, however, the minimum assumption for 
the consistency of ordinary least squares — that the explanatory 
variables are uncorrelated with the disturbance term — cannot 
be maintained if the equation to be estimated is one of a system 
of simultaneous structural equations. In this case, ordinary 
least squares loses even consistency when used as an estimator 
(*) See THEIL [32, pp. 219-225] or JOHNSTON [15, pp. 192-195] for a 
discussion of this case. 
61 Fisher - pag. 4