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Suppose prices in a market are set by supply and demand. We would like to know the equations for the demand function and the supply function. But the equation we get by regressing quantity on market price cannot generally be identified as specifically the demand function or the supply function. In special cases we can use regression to get the demand function or the supply function but not both. For example, suppose the supply function is subject to random shifts but the demand function remains fixed. Then equilibrium points of price and output will lie on the demand curve and consequently a regression of quantity on price will give us the demand function. On the other hand, if demand is subject to random fluctuations and the supply curve remains fixed then each market equilibrium combination of price and output will lie on the supply curve and consequently a regression of output on price will give us the supply curve. But when both the demand and supply cuves are subject to random fluctuations the regression of output on price is neither the demand curve nor the supply curve. The situation is shown in the three graphs below. In the diagrams the green lines represent the true market supply schedules, the blue lines the true market demand schedules and the red line the regression line for price and quantity.
The Identification Problem in econometrics has to do with being able to solve for unique values of the parameters of the structural model from the values of the parameters of the reduced form of the model. The reduced form of a model is the one in which the endogenous variables are expressed as functions of the exogenous variables. If there are time lagged relationships in which current values of endogenous variables depend upon past values of endogenous variables as well as exogenous variables then the reduced form expresses current endogenous variables as functions of exogenous variables and past endogenous variables. The combination of exogenous and past endogenous variables can be called predetermined variables.
If the reduced form coefficients are compatible with many different values for the structural coefficients then the model is said to be underidentified. If generally it not possible to find any values of structural coefficients that are compatible with the reduced form coefficients then the model is said to be over identified. If one and only one value of each structural coefficient is compatible with the reduced form coefficients the model is said to be just identified or exactly identified.
For more on this topic go to Identification Problem.
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