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134

Strong Exogeneity

If X, is weakly exogenous and x, is not preceded by any of the endogenous variables in the system, x, is defined to be strongly exogenous. As an example, consider the model

y, = fix, + Ul,

X, = a,x, , + 2 ,„, + U2,

(«,„ «2/) have a bivariate normal distribution and are serially independent; var(M,) = CT,,, var(w2,) = CT22, and cov(w,„ «2») = 012- If 012 = 0» then x, is weakly exogenous because the marginal distribution of x, does not involve fi and CT,,. However, the second equation shows that y, precedes x,. (x, depends on y, ,.) Hence x, is not strongly exogenous.

We have used the word "precedence" following Leamer but the definition by Engle, Hendry, and Richard is in terms of a concept called Granger causality and is as follows. If jc, is weakly exogenous and x, is not caused in the sense of Granger by any of the endogenous variables in the system, then x, is defined to be strongly exogenous. Simply stated the term "Granger causality" means "precedence" but we will discuss it in greater detail.

Granger Causality

Granger starts from the premise that the future cannot cause the present or the past. If event A occurs after event B, we know that A cannot cause B. At the same time, if A occurs before B, it does not necessarily imply that A causes B. For instance, the weathermans prediction occurs before the rain. This does not mean that the weatherman causes the rain. In practice, we observe A and as time series and we would like to know whether A precedes B, or precedes A, or they are contemporaneous. For instance, do movements in prices precede movements in interest rates, or is it the opposite, or are the movements contemporaneous? This is the purpose of Granger causality. It is not causality as it is usually understood.

Granger" devised some tests for causality (in the limited sense discussed above) which proceed as follows. Consider two time series, {y,} and {x,}. The series x, fails to Granger cause y, if in a regression of y, on lagged ys and lagged jcs, the coefficients of the latter are zero. That is, consider

, = S + 2 -,- + "/ 1=1 /=1

Then if (3, = (/ = 1, 2, . . . , ), , fails to cause ,. The lag length is, to some extent, arbitrary.

"C. W. J. Granger, "Investigating Causal Relations by Econometric Models and Cross-Spectral Methods," Econometrica, Vol. 37, January 1969, pp. 24-36.



1 - aa2

Thus TT2, = 0 implies that ajP,, + P21 = 0. From this it does not follow that «2 = 0. Thus Granger noncausality does not necessarily imply that x, is pre-

-"C. A. Sims, "Money, Income and Causality," American Economic Review. Vol. 62, 1972. pp. 450-552.

-G. Chamberlain, "The General Equivalence of Granger and Sims Causality," Econometrica, Vol. 50, 1982, pp. 569-582. Sims, "Money," p. 550.

"T. F. Cooley and S. F. LeRoy, " A-theoretical Macroeconometrics," Journal of Monetary Economics, Vol. 16, No. 3, November 1985, pp. 283-308, see Sec. 5 on Granger causality and Cowles causality.

An alternative test provided by Sims** is as follows: x, fails to cause y, in the Granger sense if in a regression of y, on lagged, current, and future xs, the latter coefficients are zero. Consider the regression

/ = Z + ,

Test = 0 0 = 1. 2, . . . , A:,). What this says is that the prediction of from current and past jts would not be improved if future values of x are included. There are some econometric differences between the two tests, but the two tests basically test the same hypothesis.2

As mentioned earlier, Leamer suggests using the simple word "precedence" instead of the complicated word Granger causality since all we are testing is whether a certain variable precedes another and we are not testing causaHty as it is usually understood. However, it is too late to complain about the term since it has already been well established in the econometrics literature. Hence it is important to understand what it means.

Granger Causality and Exogeneity

As we defined earlier. Granger noncausality is necessary for strong exogeneity as defined by Engle, Hendry, and Richard. Sims aiso regards tests for Granger causality as tests for exogeneity.However, Granger noncausality is neither necessary nor sufficient for exogeneity as understood in the usual simultaneous equations literature.This point can be illustrated with the example in equations (9.21). We said that x, is predetermined for y, in the first equation if = 0, and X, is strictly exogenous for y, if = 0 and P21 = 0.

Now to see what the Granger test does, write the reduced forms for y, and jc,:

y, = /-1 + nx,-i + v„ X, = TT2,y, , + ,- + Vj, For Granger noncausality, we have to have ttj) = 0. But

« + P21



Tests for Exogeneity

The Cowles Foundation approach to simultaneous equations held the view that causality and exogeneity cannot be tested. These are things that have to be specified a priori. In recent years it has been argued that if some variables are specified as exogenous and the equation is identified, one can test whether some other variables considered endogenous are indeed endogenous or not.

As an illustration, consider the following. We have a simultaneous equations model with three endogenous variables y,, 2, and , and three exogenous variables z,, Zl, and Zi- Suppose that the first equation of the model is

1 = 2 2 + + a,z, +

We want to test whether 2 and can be treated as exogenous for the estimation of this equation. To test this hypothesis, we obtain the predicted values 2 and of 2 and , respectively, from the reduced form equations for 2 and y,. We then estimate the model

y, = fiiyi + + a,z, + 72:2 + 735*3 + «1

by OLS and test the hypothesis: 72 = 7 = 0 (using the F-test described in Chapter 4). If the hypothesis is rejected, 2 and cannot be treated as exogenous. If it is not rejected, 2 and can be treated as exogenous.

Summary

1. In simultaneous equations models, each equation is normalized with respect to one endogenous variable. Strictly speaking, since the endogenous variables are all jointly determined, it should not matter which variable is chosen for normalization. However, some commonly used methods of estimation (like the two-stage least squares) do depend on the normalization rule adopted. In practice, normalization is determined by the way economic theories are formulated.

The test procedure described here is Hausmans test discussed in greater detail in Section 12.10. It is equivalent to some other tests suggested in the literature. Two references are D. Wu, "Alternative Tests of Independence Between Stochastic Regressors and Disturbances," Econometrica, Vol. 41, 1973, pp. 733-750, and N. S. Revankar, "Asymptotic Relative Efficiency Analysis of Certain Tests of Independence in Structural Systems." International Economic Review, Vol. 19, 1978, pp. 165-179.

determined. Conversely, «2 = 0 does not imply that 2 = 0. However, a2 = 0 and 21 = 0 implies that tt2i = 0, although the converse is not true.

Thus a test for Granger noncausality is not useful as a test for exogeneity. Some argue that it is, nevertheless, useful as a descriptive device for time-series data.



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