Recursive Models And Ordinary Least Squares

We saw in Chapter 18 that because of the interdependence between the stochastic disturbance term and the endogenous explanatory variable(s), the OLS method is inappropriate for the estimation of an equation in a system of simultaneous equations. If applied erroneously, then, as we saw in Section 18.3, the estimators are not only biased (in small samples) but also inconsistent; that is, the bias does not disappear no matter how large the sample size. There is, however, one situation where OLS can be applied appropriately even in the context of simultaneous equations. This is the case of the recursive, triangular, or causal models. To see the nature of these models, consider the following three-equation system:

Y3t = P30 + @31Y1t + @32Y2t + Y31X1t + Y32X2t + u3t where, as usual, the Y's and the Xs are, respectively, the endogenous and exogenous variables. The disturbances are such that cov(uit, U2t) = cov(uit, U3t) = cov(U2t, U3t) = 0

that is, the same-period disturbances in different equations are uncorrelated (technically, this is the assumption of zero contemporaneous correlation).

Now consider the first equation of (20.2.1). Since it contains only the exogenous variables on the right-hand side and since by assumption they are uncorrelated with the disturbance term U1t, this equation satisfies the critical assumption of the classical OLS, namely, uncorrelatedness between the

4Lawrence R. Klein, A Textbook of Econometrics, 2d ed., Prentice Hall, Englewood Cliffs, N.J., 1974, p. 150.


explanatory variables and the stochastic disturbances. Hence, OLS can be applied straightforwardly to this equation. Next consider the second equation of (20.2.1), which contains the endogenous variable Y1 as an explanatory variable along with the nonstochastic Xs. Now OLS can also be applied to this equation, provided Y1t and u2t are uncorrelated. Is this so? The answer is yes because ui, which affects Yi, is by assumption uncorrelated with u2. Therefore, for all practical purposes, Y1 is a predetermined variable insofar as Y2 is concerned. Hence, one can proceed with OLS estimation of this equation. Carrying this argument a step further, we can also apply OLS to the third equation in (19.2.1) because both Y1 and Y2 are uncorrelated with u3.

Thus, in the recursive system OLS can be applied to each equation separately. Actually, we do not have a simultaneous-equation problem in this situation. From the structure of such systems, it is clear that there is no interdependence among the endogenous variables. Thus, Y1 affects Y2, but Y2 does not affect Y1. Similarly, Y1 and Y2 influence Y3 without, in turn, being influenced by Y3. In other words, each equation exhibits a unilateral causal dependence, hence the name causal models.5 Schematically, we have Figure 20.1.

As an example of a recursive system, one may postulate the following model of wage and price determination:

Price equation: Pt = 010 + 011 Wt-1 + 012Rt + 013M + 014Lt + uu Wage equation: Wt = 020 + 021UNt + 032 Pt + u2t (20.2.2)

5The alternative name triangular stems from the fact that if we form the matrix of the coefficients of the endogenous variables given in (20.2.1), we obtain the following triangular matrix:

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