Editing Simultaneous equations model (section)

== Structural and reduced form ==
Suppose there are ''m'' regression equations of the form
: <math>
     y_{it} = y_{-i,t}'\gamma_i + x_{it}'\;\!\beta_i + u_{it}, \quad i=1,\ldots,m,
  </math>

where ''i'' is the equation number, and {{nowrap|''t'' {{=}} 1, ..., ''T''}} is the observation index. In these equations ''x<sub>it</sub>'' is the ''k<sub>i</sub>×''1 vector of exogenous variables, ''y<sub>it</sub>'' is the dependent variable, ''y<sub>−i,t</sub>'' is the ''n<sub>i</sub>×''1 vector of all other endogenous variables which enter the ''i''<sup>th</sup> equation on the right-hand side, and ''u<sub>it</sub>'' are the error terms. The “−''i''” notation indicates that the vector ''y<sub>−i,t</sub>'' may contain any of the ''y''’s except for ''y<sub>it</sub>'' (since it is already present on the left-hand side). The regression coefficients ''β<sub>i</sub>'' and ''γ<sub>i</sub>'' are of dimensions ''k<sub>i</sub>×''1 and ''n<sub>i</sub>×''1 correspondingly. Vertically stacking the ''T'' observations corresponding to the ''i''<sup>th</sup> equation, we can write each equation in vector form as
: <math>
    y_i = Y_{-i}\gamma_i + X_i\beta_i + u_i, \quad i=1,\ldots,m,
  </math>
where ''y<sub>i</sub>'' and ''u<sub>i</sub>'' are ''T×''1 vectors, ''X<sub>i</sub>'' is a ''T×k<sub>i</sub>'' matrix of exogenous regressors, and ''Y<sub>−i</sub>'' is a ''T×n<sub>i</sub>'' matrix of endogenous regressors on the right-hand side of the ''i''<sup>th</sup> equation. Finally, we can move all endogenous variables to the left-hand side and write the ''m'' equations jointly in vector form as
: <math>
    Y\Gamma = X\Beta + U.\,
  </math>
This representation is known as the '''structural form'''. In this equation {{nowrap|''Y'' {{=}} [''y''<sub>1</sub> ''y''<sub>2</sub> ... ''y<sub>m</sub>'']}} is the ''T×m'' matrix of dependent variables. Each of the matrices ''Y<sub>−i</sub>'' is in fact an ''n<sub>i</sub>''-columned submatrix of this ''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 ''−γ<sub>i</sub>'' or zeros, depending on which columns of ''Y'' were included in the matrix ''Y<sub>−i</sub>''. 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 ''X<sub>i</sub>'' is a ''k<sub>i</sub>''-columned submatrix of ''X''. Matrix Β has size ''k×m'', and each of its columns consists of the components of vectors ''β<sub>i</sub>'' and zeros, depending on which of the regressors from ''X'' were included or excluded from ''X<sub>i</sub>''. Finally, {{nowrap|''U'' {{=}} [''u''<sub>1</sub> ''u''<sub>2</sub> ... ''u<sub>m</sub>'']}} is a ''T×m'' matrix of the error terms.

Postmultiplying the structural equation by {{nowrap|Γ<sup> −1</sup>}}, the system can be written in the '''[[reduced form]]''' as
: <math>
    Y = X\Beta\Gamma^{-1} + U\Gamma^{-1} = X\Pi + V.\,
  </math>
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 <math style="vertical-align:0">\scriptstyle\hat\Pi</math> into the individual factors Β and {{nowrap|Γ<sup> −1</sup>}} 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 {{nowrap|''T'' → ∞}} (this later requirement means that in the limit the expression <math style="vertical-align:-.4em">\scriptstyle \frac1TX'\!X</math> 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 ''t''<sup>th</sup> row of matrix ''U'' is denoted by ''u''<sub>(''t'')</sub>, then the sequence of vectors {''u''<sub>(''t'')</sub>} should be iid, with zero mean and some covariance matrix Σ (which is unknown). In particular, this implies that {{nowrap|E[''U''] {{=}} 0}}, and {{nowrap|E[''U′U''] {{=}} ''T'' Σ}}.

Lastly, assumptions are required for identification.