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Given that 0 < p1 < 1 and that a2 and a| are both positive, it is obvious from Eq. (18.3.10) that plim (P0 will always be greater than p1; that is, p1 will overestimate the true p1.9 In other words, fa is a biased estimator, and the bias will not disappear no matter how large the sample size.

18.4 THE SIMULTANEOUS-EQUATION BIAS: A NUMERICAL EXAMPLE

To demonstrate some of the points made in the preceding section, let us return to the simple Keynesian model of income determination given in Example 18.2 and carry out the following Monte Carlo study.10 Assume that

8As stated in App. A, the plim of a constant (for example, p1) is the same constant and the plim of (A/B) = plim (A)/plim (B). Note, however, that E(A/B) = E(A)/E(B).

9In general, however, the direction of the bias will depend on the structure of the particular model and the true values of the regression coefficients.

10This is borrowed from Kenneth J. White, Nancy G. Horsman, and Justin B. Wyatt, SHAZAM: Computer Handbook for Econometrics for Use with Basic Econometrics, McGraw-Hill, New York, 1985, pp. 131-134.

728 PART FOUR: SIMULTANEOUS-EQUATION MODELS

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