Thayer Watkins
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Illustrations of the Identification Problem
in Econometrics

First consider a model of the market for bread in which the supply of bread depends upon the price of bread p and the price of flour f; i.e.,

Q = a - bp - ef

The demand for bread depends upon the price of bread p and the level of consumer income I;

Q = -c + dp + gI

These are the structural equations of the model. There are six parameters in the structural form of the model.

In the model price and quantity, p and Q, are endogenous; i.e., they are determined within the model. The price of flour f and consumer income I are exogenous variables. They are determined outside of the model.

The solution of the model for the endogenous variables gives

p = (a+c)/(b+d) - [e/(b+d)]f + [g/(b+d)]I
Q = a - [b(a+c)/(b+d)] + [be/(b+d)]f - [bg/(b+d)]I

These two equations are the reduced form of the model. The statistical estimation of the reduced form equations will give

p = h0 + h1f + h2I
Q = h3 + h4f + h5I

These six reduced form coefficients, the h's, are functions of the six structural parameters; i.e.,

h0 = (a+c)/(b+d)
h1 = - [e/(b+d)
h2 = [g/(b+d)]
h3 = a - [b(a+c)/(b+d)]
h4 = [be/(b+d)]
h5 = - [bg/(b+d)].

From a knowledge of the values of the reduced form coefficients, the h's, one can determine unique values for the structural parameters {a, b, c, d, e, g}. The model is then said to be exactly identified.

Now contrast the above model with the simple supply and demand model:

Q = a - bp
Q = -c + dp

In this model there are no exogenous variables. There are four structural parameters. The reduced form of this model is:

p = (a+c)/(b+d)
Q = a - [b(a+c)/(b+d)]

There are only two reduced form coefficients to determine four structural parameters. There will be many combinations of values of the four structural parameters there are consistent with the two values of the reduced form coefficients. The model is said to be under identified.

An over identified model is one in which there are more reduced form coefficients than there are structural parameters. This would mean that for arbitrarily given reduced form coeffients there is no solution for the structural parameters. If the model correctly describes the empirical market then not just any values of the reduced form coefficients can arise. The only reduced form coefficients that can arise are ones consistent with a set of values of the structral parameters.

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