Y. The m×m matrix Γ, which describes the relation between the dependent variables, has a complicated structure. It has ones on the diagonal, and all other elements of each column i are either the components of the vector −γi or zeros, depending on which columns of Y were included in the matrix Y−i. The T×k matrix X contains all exogenous regressors from all equations, but without repetitions (that is, matrix X should be of full rank). Thus, each Xi is a ki-columned submatrix of X. Matrix Β has size k×m, and each of its columns consists of the components of vectors βi and zeros, depending on which of the regressors from X were included or excluded from Xi. Finally, is a T×m matrix of the error terms. Postmultiplying the structural equation by , the system can be written in the reduced form as This is already a simple general linear model, and it can be estimated for example by ordinary least squares. Unfortunately, the task of decomposing the estimated matrix into the individual factors Β and is quite complicated, and therefore the reduced form is more suitable for prediction but not inference. Assumptions Firstly, the rank of the matrix X of exogenous regressors must be equal to k, both in finite samples and in the limit as (this later requirement means that in the limit the expression should converge to a nondegenerate k×k matrix). Matrix Γ is also assumed to be non-degenerate. Secondly, error terms are assumed to be serially independent and identically distributed. That is, if the tth row of matrix U is denoted by u(t), then the sequence of vectors {u(t)} should be iid, with zero mean and some covariance matrix Σ (which is unknown). In particular, this implies that , and . Lastly, assumptions are required for identification. Identification The identification conditions require that the system of linear equations be solvable for the unknown parameters. More specifically, the order condition, a necessary condition for identification, is that for each equation , which can be phrased as “the number of excluded exogenous variables is greater or equal to the number of included endogenous variables”. The rank condition, a stronger condition which is necessary and sufficient, is that the rank of equals , where is a matrix which is obtained from by crossing out those columns which correspond to the excluded endogenous variables, and those rows which correspond to