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124

is not response to price. This is also confirmed by the structural estimate of bj, which shows a wrong sign for 62 as well but a coefficient close to zero.

Dropping P from the supply function and using OLS, we get the supply function as

Q = 0.0025 + 0.6841Z R = 0.898

(0.0857)

Thus we normalize the demand function with respect to P, drop the variable P from the supply function, and estimate both equations by OLS.

9.4 Necessary and Sufficient Conditions for Identification

In Section 9.3 our discussion was in terms of obtaining estimates for the structural parameters from estimates of the reduced-form parameters. An alternative way of looking at the identification problem is to see whether the equation under consideration can be obtained as a linear combination of the other equations.

Consider, for instance, equations (9.5). Take a weighted average of the two equations with weights w and (1 - w). Then we get

q = w(o, + bxp + c,y) -I- (1 - w)(o2 + *2P) + w 1

= a\ + b\p + cjy + u, (9.7)

where a, = wa, + (1 - w)a2 bl = wb, -I- (1 - w)b2 cl = wc, and , = wui -I- (1 - w)u2

Equation (9.7) looks like the first equation in (9.5). Thus when we estimate the parameters of the demand function, we do not know whether we are getting estimates of the parameters in the demand function or in some weighted average of the demand and supply functions. Thus the parameters in the demand function are not identified. The same cannot be said about the supply function because the only way that equation (9.7) can look like the supply function in (9.5) is if cJ = 0 (i.e., wc, = 0). But since c, # 0, we must have w = 0. Tiat is, the weighted average gives zero weight to the demand function. Hence when we estimate the supply function, we are sure that the estimates we have are of the supply function. Thus the parameters of the supply function are identified.

Checking this condition has been easy with two equations. But when we have many equations, we need a more systematic way of checking this condition. For illustrative purposes consider the case of three endogenous variables y„ 2, and three exogenous variables ,, Z2, z. We will mark with a cross x if a variable occurs in an equation and a 0 if not. Suppose that the equation system is the following:



Equation

The rule for identification of any equation is as follows

1. Delete the particular row.

2. Pick up the columns corresponding to the elements that have zeros in that row.

3. If from this array of columns we can find (g - 1) rows and columns that are not all zeros, where g is the number of endogenous variables and no column (or row) is proportional to another column (or row) for all parameter values, then the equation is identified. Otherwise, not. If there is one such column (or row) delete that column (or row).

This condition is called the rank condition for identification and is a necessary and sufficient condition.

We will now apply the order and rank conditions to the illustrative example we have. The number of equations is g = 3. Hence the number of missing variables is 2 for the first equation, 3 for the second, and 2 for the third. Thus, by the order condition, the first and third equations are exactly identified and the second equation is over-identified. We will see that the rank condition gives us a different answer.

To check the rank condition for equation 1, we delete the first row and pick up the columns corresponding to the missing variables and Zi- The columns are

0 0

Note that we have only one row with not all elements zero. Thus, by the rank condition the equation is not identified. For equation 2, we delete row 2, and pick up the columns corresponding to , , and Z2- We get

We now have two rows (and two columns) with not all elements zero. Thus the equation is identified. Similarly, for the third equation deleting the third row, the columns for y, and give

X X

-The array of columns is called a matrix and the condition that we have stated is that the rank of this matrix be (g ~ 1): hence the use of the term ranic condition. We have avoided the use of matrix notation and stated the condition in an alternative fashion. A derivation using matrix notation is presented in the appendix to this chapter.



9 4 NECESSARY AND SUFFICIENT CONDITIONS FOR IDENTIFICATION

Again we have two rows with not all elements zero. Hence the equation is identified. Note that the rank condition stastes whether the equation is identified or not. From the order condition we know whether it is exactly identified or over-idemified.

In summary the second and third equations are estimable. The first one is not, and the order condition misleads us into thinking that it is so. There are many estimation methods for simultaneous equations models that break down if the order condition is not satisfied but do give us estimates of the parameters if the order condition is satisfied even if the rank condition is not. In such cases these estimates are meaningless. Thus it is desirable to check the rank condition. In our example, for equation 1, the rank condition is not satisfied. What this means is that the estimates we obtain for the parameters in equation 1 are actually estimates of some linear combinations of the parameters in all the equations and thus have no special economic interpretation. This is what we mean when we say that equation 1 is not estimable.

Illustrative Example

As an illustration, consider the following macroeconomic model with seven endogenous variables and three exogenous variables. The endogenous and exogenous variables are:

Endogenous

Exogenous

real consumption

real government purchases

real investment

real tax receipts

TV =

employment

nominal money stock

price level

interest rate

real income

money wage rate

The equations are:

(1) = + bJ - c,r + diR +

(2) I = a-i + bjY + C2R + U2

(3) Y = + I + G

(4) M = + bjY + CjR + dyP +

(5) Y = 04 + bN + m4

(6) N = + bsW c,P + Us

(7) / = fle + VV + CftP +

(consumption function) (investment function) (identhy)

(liquidity preference function) (production function) (labor demand) (labor supply)



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