The General Linear Case
Let y be a column vector of M endogenous variables. In the case above with Q and P, we have M = 2. Let x be a column vector of exogenous variables; in the case above x consists only of Z. The structural linear model (without error terms, as above) is:
where A and B are matrices; A is a square M × M matrix. The reduced form of the system is:
Again, each endogenous variable depends on each exogenous variable. It is easily verified that:
Without restrictions on the A and B, the coefficients of A and B can not be identified from data on y and x: each row of the structural model is just a linear relation between y and z with unknown coefficients. (Again the parameter identification problem.) The M reduced form equations (the rows of the matrix equation y = Π x above) can be identified from the data because each of them contains only one endogenous variable.
Read more about this topic: Reduced Form
Famous quotes containing the words general and/or case:
“I never saw any people who appeared to live so much without amusement as the Cincinnatians.... Were it not for the churches,... I think there might be a general bonfire of best bonnets, for I never could discover any other use for them.”
—Frances Trollope (1780–1863)
“Instructing in cures, therapists always recommend that “each case be individualized.” If this advice is followed, one becomes persuaded that those means recommended in textbooks as the best, means perfectly appropriate for the template case, turn out to be completely unsuitable in individual cases.”
—Anton Pavlovich Chekhov (1860–1904)