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
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