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Concept# Parameter identification problem

Summary

In economics and econometrics, the parameter identification problem arises when the value of one or more parameters in an economic model cannot be determined from observable variables. It is closely related to non-identifiability in statistics and econometrics, which occurs when a statistical model has more than one set of parameters that generate the same distribution of observations, meaning that multiple parameterizations are observationally equivalent.
For example, this problem can occur in the estimation of multiple-equation econometric models where the equations have variables in common.
Consider a linear model for the supply and demand of some specific good. The quantity demanded varies negatively with the price: a higher price decreases the quantity demanded. The quantity supplied varies directly with the price: a higher price increases the quantity supplied.
Assume that, say for several years, we have data on both the price and the traded quantity of this good. Unfortunately this is not enough to identify the two equations (demand and supply) using regression analysis on observations of Q and P: one cannot estimate a downward slope and an upward slope with one linear regression line involving only two variables. Additional variables can make it possible to identify the individual relations.
In the graph shown here, the supply curve (red line, upward sloping) shows the quantity supplied depending positively on the price, while the demand curve (black lines, downward sloping) shows quantity depending negatively on the price and also on some additional variable Z, which affects the location of the demand curve in quantity-price space. This Z might be consumers' income, with a rise in income shifting the demand curve outwards. This is symbolically indicated with the values 1, 2 and 3 for Z.
With the quantities supplied and demanded being equal, the observations on quantity and price are the three white points in the graph: they reveal the supply curve. Hence the effect of Z on demand makes it possible to identify the (positive) slope of the supply equation.

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