Reduced Form - The General Linear Case

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.