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The identification problem is a deductive, logical issue that must be solved before estimating an economic model. In a demand and supply model, the equilibrium point belongs to both curves, and many presumptive curves can be drawn through such a point. We need prior information on the slopes, intercepts, and error terms to identify the true from the presumptive demand and supply curves. Such prior information will give a set of structural equations. If the equations are linear, and the error terms are normally distributed with zero mean and constant variance, then a model is formed for estimation.
A typical identification process may fix the demand curve and shift the supply curve, cutting the demand curve at many points to trace it out. By the zero mean assumption of the error term, half the observations are expected above and half below the demand curve. In the same way, the supply curve can be identified. This method originated with Ragnar Frisch (1938) and Trygve Haavelemo (1944). Tjalling Koopmans evolved the order and rank conditions for identifying linear models (1949). Franklin Fisher’s work was the first major textbook on the subject (1966), and Charles Manski extended it to the social sciences (1995).
The order condition is the most used technique for identifying a model. Each equation in the model has a predetermined variable that is either given from outside of the model (exogenous) or determined in the model (endogenous) but fitted with a lag. A standard exogenous variable for the supply curve is technology, T, and for the demand curve it is income, Y. A two-function model with Q = f (P, Y ) for the demand and Q = f (P, T ) for the supply, where P is price and Q is quantity, meets the order condition for identification by excluding technology from the demand curve, and by excluding income from the supply curve. The number of excluded variables EV is one in each equation. The order condition requires that EV equal the number of exogenous variables M, less one. An equation is exactly identified if EV = M – 1, overidentified if EV > M – 1, and underidentified otherwise. In the twofunction model above, the equations are exactly identified because the exogenous variables P and Q yield M – 1 = 1, and each equation has only one excluded variable, Y or T. The rank condition guarantees that the equations can be solved. Econometric texts often create a spreadsheet to demonstrate the rank condition. For the model above, the column is labeled with the variables Q, P, Y, T, and the rows contain information on the equations. Each cell has either a 0 for an excluded variable or a 1 for an included variable. For the demand function above, the entry for the row vector is [1, 1, 1, 0] and for the supply function [1, 1, 0, 1]. To identify the demand curve for the order condition, first locate the zero in its vector, then pick up the corresponding number in the supply vector. The pickedup number, which is 1, should be equal to M – 1, which is also 1. With many equations, the numbers that we pick up will array into many rows and columns. The general rank test requires one to find M – 1 rows and M – 1 columns in that array whose elements are not all zeros, because such a (M – 1)(M – 1) spreadsheet will make the model solvable.
- Fisher, Franklin M. 1966. The Identification Problem in Econometrics. New York: McGraw-Hill.
- Frisch, Ragnar. 1938. Statistical Versus Theoretical Relations in Economic Macrodynamics. League of Nations memorandum. Geneva, Switzerland: League of Nations.
- Haavelmo, Trygve. 1944. The Probability Approach in Econometrics. Econometrica 12 (July): 1–115.
- Koopmans, Tjalling C. 1949. Identification Problem in Economic Model Construction. Econometrica 17: 15–144.
- Koopmans, Tjalling C., Herman Rubin, and R. B. Leipnik. 1950. Measuring the Equation Systems of Dynamic Economics. In Statistical Inference in Dynamic Economic Models, ed. Tjalling C. Koopmans, 53–237. New York: Wiley.
- Manski, Charles F. 1995. Identification Problems in the Social Sciences. Cambridge, MA: Harvard University Press.
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