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Type of statistical model
Simultaneous equations models are a type of statistical model in which the dependent variables are functions of other dependent variables, rather than just independent variables. This means some of the explanatory variables are jointly determined with the dependent variable, which in economics usually is the consequence of some underlying equilibrium mechanism. Take the typical supply and demand model: whilst typically one would determine the quantity supplied and demanded to be a function of the price set by the market, it is also possible for the reverse to be true, where producers observe the quantity that consumers demand and then set the price. Simultaneity poses challenges for the estimation of the statistical parameters of interest, because the Gauss–Markov assumption of strict exogeneity of the regressors is violated. And while it would be natural to estimate all simultaneous equations at once, this often leads to a computationally costly non linear optimization problem even for the simplest system of linear equations. This situation prompted the development, spearheaded by the Cowles Commission in the 1940s and 1950s, of various techniques that estimate each equation in the model seriatim, most notably limited information maximum likelihood and two stage least squares.

Firstly, the rank of the matrix X of exogenous regressors must be equal to k, both in finite samples and in the limit as T → ∞ (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.

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 ki + ni ≤ k, 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 Πi0 equals ni, where Πi0 is a (k − ki)×ni matrix which is obtained from Π by crossing out those columns which correspond to the excluded endogenous variables, and those rows which correspond to the included exogenous variables.

In simultaneous equations models, the most common method to achieve identification is by imposing within equation parameter restrictions. Yet, identification is also possible using cross equation restrictions. To illustrate how cross equation restrictions can be used for identification, consider the following example from Wooldridge developed independently by and It is an equation by equation technique, where the endogenous regressors on the right hand side of each equation are being instrumented with the regressors X from all other equations. The method is called “two stage” because it conducts estimation in two steps: For this, the structural system of equations is transformed into the reduced form first. Once the coefficients are estimated the model is put back into the structural form.

Across fields and disciplines simultaneous equation models are applied to various observational phenomena. These equations are applied when phenomena are assumed to be reciprocally causal. The classic example is supply and demand in economics. In other disciplines there are examples such as candidate evaluations and party identification or public opinion and social policy in political science; road investment and travel demand in geography; and educational attainment and parenthood entry in sociology or demography. The simultaneous equation model requires a theory of reciprocal causality that includes special features if the causal effects are to be estimated as simultaneous feedback as opposed to one sided 'blocks' of an equation where a researcher is interested in the causal effect of X on Y while holding the causal effect of Y on X constant, or when the researcher knows the exact amount of time it takes for each causal effect to take place, i. e., the length of the causal lags. Instead of lagged effects, simultaneous feedback means estimating the simultaneous and perpetual impact of X and Y on each other. This requires a theory that causal effects are simultaneous in time, or so complex that they appear to behave simultaneously; a common example are the moods of roommates. To estimate simultaneous feedback models a theory of equilibrium is also necessary – that X and Y are in relatively steady states or are part of a system (society, market, classroom) that is in a relatively stable state.