Parameter identification problem

The parameter identification problem is an issue commonly encountered during estimation of simultaneous equation models in econometrics.

Example
In estimating a simultaneous system of equations for supply and demand using price (P) and quantity (Q) data, the specification Q = a + b P is not sufficient to identify the supply and the demand parameters separately. Additional variables such as X and Z are necessary so that the two equations may be separately expressed and thus identified as:


 * Supply: $$ \mbox{Q} = \mbox{a}_{S}\, {+}\, \mbox{b}_S\mbox{P}\, {+}\, \mbox{cX} \!$$


 * Demand: $$ \mbox{Q} = \mbox{a}_{D}\, {+}\, \mbox{b}_{D}\mbox{P}\, {+}\, \mbox{dZ} $$

In this example, the variable X might be the wage rate, a determinant of cost and hence of supply; the variable Z might be disposable income, a determinant of demand. In this example, P and Q are the endogenous variables and X and Z are the exogenous variables. In general, for each additional equation to be identified in the system, having at least one additional exogenous variable is necessary.