9.2 ENDOGENOUS AND EXOGENOUS VARIABLES 357

completely price inelastic), a shift in the demand curve produces a change in price only, and if the supply curve is vertical (infinite price elasticity), a shift in the demand curve produces a change in quantity only.

Thus in equation (9.1) the error term w is correlated with p when the supply curve is upward sloping or perfectly horizontal. Hence an estimation of the equation by ordinary least squares produces inconsistent estimates of the parameters.

We could have written the demand function as

p = a + fiq + u (9.1)

But, again, u will be correlated with q if the supply function is upward sloping or perfectly vertical. There is the question of whether the demand function should be written as in equation (9.1) or (9. )- If it is written as in (9.1), we say that the equation is normalized with respect ro «7 (i.e., the coefficient of q is unity). If it is written as in (9.1), we say that the equation is normalized with respect to p. Since price and quantity are determined by the interaction of demand and supply, it should not matter whether we normalize the equations with respect to p or with respect to q.

The estimation methods we use should produce identical estimates whatever normalization we adopt. However, the preceding discussion and Figure 9.1 suggests that if quantity supplied is not responsive to price, the demand function should be normalized with respect to p, that is, it should be written as in (9.1). On the other hand, if quantity supplied is highly responsive to price, the demand function should be normalized with respect to q as in (9.1). An empirical example will be given in Section 9.3 to illustrate this point. The question of normalization is discussed in greater detail in Section 9.7.

Returning to the demand and supply model, the problem is that we cannot consider the demand function in isolation when we are studying the relationship between quantity and price as in (9.1). The solution is to bring the supply function into the picture, and estimate the demand and supply functions together. Such models are known as simultaneous equations models.

9.2 Endogenous and Exogenous Variables

In simultaneous equations models variables are classified as endogenous and exogenous. The traditional definition of these terms is that endogenous variables are variables that are determined by the economic model and exogenous variables are those determined from outside.

Endogenous variables are also called jointly determined and exogenous variables are called predetermined. (It is customary to include past values of endogenous variables in the predetermined group.) Since the exogenous variables are predetermined, they are independent of the error terms in the model. They thus satisfy the assumptions that the xs satisfy in the usual regression model of on xs.

The foregoing definition of exogeneity has recently been questioned in recent

econometric literature, but we have to postpone this discussion to a later section (Section 9.10). We will discuss this criticism and its implications after going through the conventional methods of estimation for simultaneous equation models.

Consider now the demand and supply model

(7 = u, + + Cjy + M, demand function 2)

q = a2 + biP + CiR + Ui supply function

q is the quantity, p the price, the income, R the rainfall, and , and are the error terms. Here p and q are the endogenous variables and and R are the exogenous variables. Since the exogenous variables are independent of the error terms M and and satisfy the usual requirements for ordinary least squares estimation, we can estimate regressions of p and qony and R by ordinary least squares, although we cannot estimate equations (9.2) by ordinary least squares. We will show presently that from these regressions of p and q ony and R we can recover the parameters in the original demand and supply equations (9.2). This method is called indirect least squares-it is indirect because we do not apply least squares to equations (9.2). The indirect least squares method does not always work, so we will first discuss the conditions under which it works and how the method can be simplified. To discuss this issue, we first have to clarify the concept of identification.

9.3 The Identification Problem:

Identification Through Reduced Form

We have argued that the error terms , and are correlated with p in equations (9.2), and hence if we estimate the equation by ordinary least squares, the parameter estimates are inconsistent. Roughly speaking, the concept of identification is related to consistent estimation of the parameters. Thus if we can somehow obtain consistent estimates of the parameters in the demand function, we say that the demand function is identified. Similarly, if we can somehow get consistent estimates of the parameters in the supply function, we say that the supply function is identified. Getting consistent estimates is just a necessary condition for identification, not a sufficient condition, as we show in the next section.

If we solve the two equations in (9.2) for q and p in terms of and R, we get

a,b2 - «21 c,b2 < „

= -IT-+ -ry - Tf + error

02 - by b2 - t>y O2 - by (jj

ay - flj C C2 „

p =--- + 7--y - 7--R + an error

&2 -by b2 - by b2 - by

These equations are called the reduced-form equations. Equations (9.2) are called the structural equations because they describe the structure of the economic system. We can write equations (9.3) as

9.3 THE IDENTIFICATION PROBLEM: IDENTIFICATION THROUGH REDUCED FORM 359

(? = TT, + + TTji? + v, 4j

p = TTji + + TTji? + V2

where v, and V2 are error terms and

The its are called reduced-form parameters. The estimation of the equations (9.4) by ordinary least squares gives us consistent estimates of the reduced form paramters. From these we have to obtain consistent estimates of the parameters in equations (9.2). These parameters are called structural parameters. Comparing (9.3) with (9.4) we get

- i - "2

C2 = (1 - 2) , = - f>2)

a, = TT, - ,1 4 = , - {>2ft

Since „ U2, Bf, B2, ˆx, ˆ2 are all single-valued functions of the , they are consistent estimates of the corresponding structural parameters. As mentioned earlier, this method is known as the indirect least squares method.

It may not be always possible to get estimates of the structural coefficients from the estimates of the reduced-form coefficients, and sometimes we get multiple estimates and we have the problem of choosing between them. For example, suppose that the demand and supply model is written as

q = Ui + biP + c,y + demand function ,„

= «2 + 2 + U2 supply function

Then the reduced form is

a.bi - «21

t>2 - Ol b2 - 0,

g, - 2 c, 2 - bf 12 - bi

= TT, + 2 + V, P = -I- 4 + V2

In this case 62 - 72/1x4 and 02 = ir, - 2 3. But there is no way of getting estimates of fl,, fc,, and c,. Thus the supply function is identified but the demand function is not. On the other hand, suppose that we have the model

q = a, + bip + M, demand function

= fl2 + + C2R + U2 supply function