uppose the quantities
are connected by the
relationships
where the numbers
are unknown, the
are Gaussian variables with zero mean and unknown deterministic variances.
Such relationships may represent a simple supply-demand equilibrium model with
being the supply-demand at the equilibrium and each of the equations
representing the influence of price on supply (for first equation) or demand
(second equation). The variable
is not correlated with
Note that the simple linear regression should not be used to estimate each of
the parameters
because it is neither a maximal likelihood nor unbiased estimate in such
situation. This simple fact is called the ''simultaneous equations bias''.
Intuitively, one has to remove the influence of second equation to apply the
regression to the first equation. This is the idea of the two stage least
squares procedure.
