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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 distributed,
 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 disturbance
 term but are uncorrelated with it in the probability limit,
then ordinary least squares ceases to be unbiased but is consistent.
 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 assumptions
 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 particlar
 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.

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