Now we can check that the demand function is identified but the supply function is not.

Finally, suppose that we have the system

(J = + b,p + + dyR + demand function ,

q = 02 + b2P + «2 supply function

Rainfall affects demand (if there is rain, people do not go shopping), but not supply. The reduced-form equations are

b2 - by h2 - bi b2 - by -b + R +

/?2 ~ bf 2 ~ by b2 ~ by

(7 = TT, + + TiyR + v,

p = -714 -t- TTsy -t- ( -t- V2

Now we get two estimates of &2- One is 2 = 2/ 5 and the other is B2 = / , and these need not be equal. For each of these we get an estimate of «2, which is a2 = - Bir.

On the other hand, we get no estimates for the parameters by, c,, and d, of the demand function. Here we say that the supply function is over-identified and the demand function is under-identified. When we get unique estimates for the structural parameters of an equation from the reduced-form parameters, we say that the equation is exactly identified. When we get multiple estimates, we say that the equation if over-identified, and when we get no estimates, we say that the equation is under-identified (or not identified).

There is a simple counting rule available in the linear systems that we have been considering. This counting rule is also known as the order condition for identification. This rule is as follows: Let g be the number of endogenous variables in the system and the total number of variables (endogenous and exogenous) missing from the equation under consideration. Then:

1. lfk = g- 1, the equation is exactly identified.

2. If k> g - 1, the equation is over-identified.

3. If A; < g - 1, the equation is under-identified.

This condition is only necessary but not sufficient. In Section 9.4 we will give a necessary and sufficient condition.

Let us apply this rule to the equation systems we are considering. In equations (9.2). g, the number of endogenous variables, is 2, and there is only one variable missing from each equation. Hence both equations are identified exactly. In equations (9.5), again g = 2. There is no variable missing from the first equation; hence it is under-identified. There is one variable missing in the second equation; hence it is exactly identified. In equation (9.6), there is no variable missing in the first equation; hence it is not identified. In the second

equation there are two variables missing; thus k> g - I and the equation is over-identified.

Illustrative Example

The indirect least squares method we have described is rarely used. In the following sections we describe a more popular method of estimating simultaneous equation models. This is the method of two-stage least squares (2SLS). However, if some coefficients in the reduced-form equations are close to zero, this gives us some information about what variables to omit in the structural equations. We will provide a simple example of a two-equation demand and supply model where the estimates from OLS, reduced-form least squares, and indirect least squares provide information on how to formulate the model. The example also illustrates some points we have raised in Section 9.1 regarding normalization.

The model is from Merrill and Fox. In Table 9.1 data are presented for demand and supply of pork in the United States for 1922-1941. The model estimated by Merrill and Fox is

= o, + biP + CF + M demand function = 02 + biP + C2Z + V supply function

The equations were fitted for 1922-1941. The reduced-form equations were (standard errors in parentheses)

P = -0.0101 + 1.0813 F - 0.8320Z = 0.893

(0.1339) (0.1159)

Q = 0.0026 - 0.0018F + 0.6839Z R = 0.898

(0.0673) (0.0582)

The coefficient of Fin the second equation is very close to zero and the variable Y can be dropped from this equation. This would imply that 62 = 0, or supply is not responsive to price. In any case, solving from the reduced form to the structural form, we get the estimates of the structural equation as

Q = -0.0063 - 0.8220F + 0.8870F demand function

Q = 0.0026 - 0.0017F + 0.6825Z supply function

The least squares estimates of the demand function are: Normalized with respect to Q:

Q = - 0.0049 - 0.7205F + 0.7646F R = 0.903

(0.0594) (0.0967)

Normalized with respect to P:

P = -0.0070 - 1.25186 + 1.0754F 7? = 0.956

, (0.1032) (0.0861)

William C. Merrill and Karl A. Fox, Introduction to Economic Statistics (New York: Wiley, 1971).

"P,, retail price of pork (cents per pound); Q„ consumption of pork (pounds per capita); Y„ disposable personal income (dollars per capita); Z„ "predetermined elements in pork production."

Source: William C. Merrill and Karl A. Fox, Introduction to Economic Statistics (New York: Wiley, 1971), p. 539.

The structural demand function can also be written in the two forms: Normalized with respect to Q:

Q = -0.0063 - 0.8220 + 0.8870

Normalized with respect to P:

P - -0.0077 - 1.21656 + 1.0791 r

The estimates of the parameters in the demand function are almost the same with the direct least squares method as with the indirect least squares method when the demand function is normalized with respect to P.

Which is the correct normalization? We argued in Section 9.1 that if quantity supplied is not responsive to price, the demand function should be normalized with respect to P. We saw that the fact that the coefficient of Y in the reduced-form equation for Q was close to zero implied that 62 = 0 or quantity supplied