One solution to the Lucas critique is to make the coefficients of the simultaneous equations system depend on the exogenous poHcy variables. Since this makes the model a varying parameter model, which is beyond the scope of this book, we will not pursue it here."

Leamer suggests redefining the concept of exogeneity. He suggests defining the variable x as exogenous if the conditional distribution of given x is invariant to modifications that alter the process generating x. What this says is that a variable is defined as exogenous if the Lucas critique does not apply to it. It is, however, not clear whether such a redefinition solves the problem raised by Lucas.

There are two concepts of exogeneity that are usually distinguished:

1. Predeterminedness. A variable is predetermined in a particular equation if it is independent of the contemporaeneous and future errors in that equation.

2. Strict exogeneity. A variable is strictly exogenous if it is independent of the contemporaneous, future, and past errors in the relevant equation.

To explain these concepts we have to consider a model with lagged variables. Consider*

y, = a,x, + p„y, , + nx,, + , p 21)

X, = OLjy, + 21 ,-1 + P22,-1 + «2,

with „ and Uj, mutually and serially independent. If = 0, x, is predetermined for y, in the first equation. On the other hand, x, is strictly exogenous for y, only if a2 = 0 and P21 = 0. Because if P2, 5 0. x, depends on ,through y,„,. That is, in the first equation x, is not independent of past errors.

In nondynamic models, and models with no serial correlation in the errors, we do not have to make this distinction. For instance, in Section 9.9 we considered an example of estimation of the Cobb-Douglas production function under the hypothesis of maximization of expected profit. We saw that yj and y, were independent of the error term ,. Thus these variables are exogenous for the estimation of the parameters in the production function. Similarly, in a recursive system, the (nonnormalized) endogenous variables in each equation can be treated as exogenous for the purpose of estimation of the parameters in that equation.

Engle, Hendry, and Richard" are not satisfied with the foregoing definitions of exogeneity and suggest three more concepts:

"This is the solution suggested in Maddala, Econometrics, Chap. 17, "Varying Parameter Models."

"E. E. Leamer, "Vector Autoregressions for Causal Inference," in K. Brunner and A. Meltzer (eds.). Understanding Monetary Regimes (supplement to Journal of Monetary Economics), 1985, pp. 255-304.

"This example and the discussion that follows is from R. L. Jacobs, E. E. Leamer, and M. P.

Ward, "Difficulties with Testing for Causation," Economic Inquiry, 1979, pp. 401-413.

"R. F. Engle, D. F. Hendry, and J. F. Richard, "Exogeneity," Econometrica, Vol. 51, March

1983.

"The example is from J. W. Pratt and R. Schlaifer, "On the Nature and Discovery of Structure," Journal of the American Statistical Association, Vol. 79, March 1984, pp. 9-21. It is also discussed in Leamer, "Vector Autoregressions."

1. Weak exogeneity.

2. Strong exogeneity.

3. Super exogeneity.

Since these concepts are often used and they are not difficult to understand anyway, we will discuss them briefly. The concept of strong exogeneity is linked to another concept: "Granger causality." There is a proliferation of terms here but since they occur frequently in recent econometric literature and are not difficuh to understand, we will go through them.

One important point to note is that whether a variable is exogenous or not depends on the parameter under consideration. Consider the equation*

= bx +

where b is the unknown parameter and is a variable that is unknown and unnamed. This equation can also be written as

= b*x + V

where b* = b + \ and v = - x. Again, b* is an unknown parameter and v is an unknown variable. If E{u\x) = 0, it cannot be true that £(vx) = 0. So is X exogenous or not? It clearly depends on the parameter value. This points to the importance of the question: "Exogenous for what?"

As yet another example, consider the case where y, and x, have a bivariate normal distribution with means E{y,) = p-,, E{x,) = \i2 and variances and co-variance given by var(y,) = a,,, var(x,) = CT22, and cov(y„ x,) = ,2. The conditional distribution of y, given x, is

y,\x, ~ IN(a + fix,, CT)

where fi = ctiJctii, a = p-, - fi\X2, and = or,, - 22. We can write the joint distribution of y, and x, as

/(y,> X,) = g{y,\x,)h{x,)

and we can write the model as

y, = a + fix, + M„ Ml, - IN(0, a) (9.22)

X, = fJL2 + V2, V2, ~ IN(0, CT22)

where cov(x„ «„) = 0 and cov(m,„ v,,) = 0 by construction. If we consider this set of equations, x, is "exogenous." On the other hand, we can similarly write

f{y„ X,) = h(x,\y,)g(y,)

and write the model as

X, = + by, + U2, U2, ~ (IN)(0, 0)2) 23)

y, = M-i + v„ v„ ~ IN(0, CT,,)

where »,, m,,) = covCm,,, v„) = 0 by construction and 8 = / ,,, 7 = - 8 -1 = / - Now , is "exogenous" in this model. So which of X, and y, is exogenous, if at all? The answer depends on the parameters of interest. If we are interested in the parameters a, (3. a-, then x, is exogenous and equations (9.22) are the ones to estimate. If we are interested in the parameters 7, 8, -, then y, is exogenous and equations (9.23) are the ones to estimate.

The considerations above led to the following definition of weak exogeneity by Engle, Hendry, and Richard:

Weak Exogeneity

A variable x, is said to be weakly exogenous for estimating a set of parameters \ if inference on X conditional on x, involves no loss of information. That is, if we write

fiy, X,) = g{y,\x,)hix,)

where g(y,\x,) involves the parameters X, weak exogeneity implies that the marginal distribution h{x,) does not involve the parameters X. Essentially, the parameters in h(x,) are nuisance parameters.

In the example where y, and x, have a bivariate normal distribution, there are five parameters: ,, [2, -,,, , 12- These can be transformed by a one-to-one transformation into (a, (3, ) and (p.2, 022)- The two sets are separate. Thus for the estimation of (a, (3, a--) we do not need information on ([ij, 0-22). Hence x, is weakly exogenous for the estimation of (a, (3, ).

Superexogeneity

The concept of superexogeneity is related to the Lucas critique. If x, is weakly exogenous and the parameters in /(y,jr,) remain invariant to changes in the marginal distribution of x„ then x, is said to be superexogenous. In the example we have been considering, namely that of y, and x, having a bivariate normal distribution, x, is weakly exogenous for the estimation of (a, (3, o) in (9.22). But it is not superexogenous because if we change p., and CT22, the parameters in the marginal distribution of x„ this will produce changes in (a, (3, ct-). Note that weak exogeneity is a condition required for efficient estimation. Superexogeneity is a condition required for policy purposes.

Leamer finds it unnecessary to require weak exogeneity as a condition for superexogeneity. He argues that this confounds the problem of efficient estimation with that of policy analysis. His definition of exogeneity is the same as the definition of superexogeneity by Engle, Hendry, and Richard without the requirement of weak exogeneity.