In simultaneous equations models the classification of variables as endogenous and exogenous is quite arbitrary. Often, current values of some variables are treated as endogenous and lagged values of these same variables are treated as exogenous even when the current and lagged values are very highly correlated. In addition, some equations are regarded as identified (over- or exactly-) and others are not identified by merely looking at how many variables are missing from the equation, and the under-identified equations are regarded as nonestimable. The discussion above illustrates that least squares estimation of under-identified equations can still be worthwhile. Also, instead of simply counting the number of exogenous variables, one should also look at how high their inter-correlations are.

Recursive Systems

Not all simultaneous equations models give biased estimates when estimated by OLS. An important class of models for which OLS estimation is valid is that of recursive models. These are models in which the errors from the different equations are independent and the coefficients of the endogenous variables show a triangular pattern. For example, consider a model with three endogenous variables: y,, , and , and three exogenous variables, Zu Z2, and Zj, which has the following structure:

1 + 12 2 + + a,z, = ,

2 + + «22 = «2

-I- =

where ,, Mj. are independent. The coefficients of the endogenous variables are

1 12 Pl3 1 P23 1

which form a triangular structure. In such systems, each equation can be estimated by ordinary least squares. Suppose that in the example above «2 and «3 are correlated but they are independent of ,; then the model is said to be block-recursive. The second and third equations have to be estimated jointly, but the first equation can be estimated by OLS. Note that 2 and depend on «2 and 3 only and hence are independent of the error term ,.

Estimation of Cobb-Douglas Production Functions

In the estimation of production functions, output, labor input, and capital input are usually considered the endogenous variables and wage rate and price of capital as the exogenous variables.

"A. Zellner, J. Kmenta, and J. Dreze, "Specification and Estimation of Cobb-Douglas Production Function Models," Econometrica, October 1966, pp. 786-795.

Suppose that we use the Cobb-Douglas production function

X, = ALf Kf e"" (9.18)

where ,, is the error term; then taking logs we can write it as

- 2, - , = Ci + „ (9.19)

where „ = log „ , = log L„ ,, = log „ and , = log .

Since 2, and , are endogenous variables, we cannot estimate this equation by OLS. We have to add the equations for 2, and y,, and these are obtained from the marginal productivity conditions. However, Zellner, Kmenta, and Dreze" argue that in this case we can estimate equation (9.19) by OLS. Their argument is that since output given (9.18) is stochastic, the firm should be maximizing expected profits (not profits). If ,, ~ 1N(0, cfi), then

E{e"") = exp (V)

Hence

£(Z,) = ALTKf exp (V)

Expected profits = /?, = , { ,) - w,L, - r,K„ where p is the price of output, w the price of labor input, and r the price of capital input. Maximization of expected profits gives us the following conditions:

= 0 = exp(«„-b) dL; L, ap,

BKi K, ,

Taking logs and adding error terms t/j, and 3, to these equations, we get (the errors depict errors in maximization of expected profits)

1, - 2, = + t/„ + U2, 1, - , = , + „ + ,

where

2, = log () - (9.20)

Now we can solve equations (9.19) and (9.20) to get the reduced forms for y,„ 2,. We can easily see that when we substitute for y,, from equation (9.19)

into equations (9.20), the ,, term cancels. Thus , and y,, involve only the error terms «2, and ,,.

It is reasonable to assume that ,, are independent of Mj, and Uy because ,,-are errors due to "acts of nature" like weather and ,, and «3, are "human errors." Under these assumptions yj, and , (which depend on «2, and ,, only) are independent of ,,. Hence OLS estimation of the production function (9.19) yields consistent estimates of the parameters a and p. We thus regress y,, on 2, and ,..

Here is an example where there is no simultaneity bias by using the OLS method. The model is not recursive either, but from the peculiar way the error terms entered the equations we could show that in the first equation the included endogenous variables are independent of the error term in that equation.

*9.10 Exogeneity and Causality

The approach to simultaneous equations models that we have discussed until now is called the Cowles Foundation approach. Its name derives from the fact that it was developed during the late 1940s and early 1950s by the econometricians at the Cowles Foundation at the University of Chicago. The basic premise of this approach is that the data are assumed to have been generated by a system of simultaneous equations. The classification of variables into "endogenous" and "exogenous," and the causal structure of the model are both given a priori and are untestable. The main emphasis is on the estimation of the unknown parameters for which the Cowles Foundation devised several methods (limited-information and full-information methods). This approach has, in recent years, been criticized on several grounds:

1. The classification of variables into endogenous and exogenous is sometimes arbitrary.

2. There are usually many variables that should be included in the equation that are excluded to achieve identification. This argument was made by T. C. Liu- in 1960 but did not receive much attention. It is known as the Liu critique.

3. One of the main purposes of simultaneous equations estimation is to forecast the effect of changes in the exogenous variables on the endogenous variables. However, if the exogenous variables are changed and profit-maximizing agents see the change coming, they would modify their behavior accordingly. Thus the coefficients in the simultaneous equations models cannot be assumed to be independent of changes in the exogenous variables. This is now called the Lucas critique.

-T. C. Liu, "Under-Identification, Structural Estimation, and Forecasting," Econometrica, Vol. 28, 1960, pp. 855-865.

"R. E. Lucas, "Econometric Policy Evaluation: A Critique," in Karl L. Brunner (ed.), Tlie Phillips Curve and Labor Markets (supplement to the Journal of Monetary Economics), 1976, pp. 19-46.