—0.2588852E-02

Source: Kenneth J. White, Nancy G. Horsman, and Justin B. Wyatt, SHAZAM Computer Handbook for Econometrics for Use with Damodar Gujarati: Basic Econometrics, September 1985, p. 132.

Source: Kenneth J. White, Nancy G. Horsman, and Justin B. Wyatt, SHAZAM Computer Handbook for Econometrics for Use with Damodar Gujarati: Basic Econometrics, September 1985, p. 132.

the values of investment I are as shown in column 3 of Table 18.1. Further assume that

The ut thus generated are shown in column (4).

For the consumption function (18.2.3) assume that the values of the true parameters are known and are p0 = 2 and p1 = 0.8.

From the assumed values of p0 and p1 and the generated values of ut we can generate the values of income Yt from (18.3.1), which are shown in column 1 of Table 18.1. Once Yt are known, and knowing p0, fa, and ut, one can easily generate the values of consumption Ct from (18.2.3). The C's thus generated are given in column 2.

Since the true fa and fa are known, and since our sample errors are exactly the same as the "true" errors (because of the way we designed the Monte Carlo study), if we use the data of Table 18.1 to regress Ct on Yt we should obtain fa = 2 and fa = 0.8, if OLS were unbiased. But from (18.3.7) we know that this will not be the case if the regressor Yt and the disturbance ut are correlated. Now it is not too difficult to verify from our data that the (sample) covariance between Yt and ut is Y ytu = 3.8 and that J]yt2 = 184.

CHAPTER EIGHTEEN: SIMULTANEOUS-EQUATION MODELS 729

That is, ft is upward-biased by 0.02065.

Now let us regress Ct on Yt, using the data given in Table 18.1. The regression results are

C t = 1.4940 + 0.82065Yt se = (0.35413) (0.01434) (18.4.2)

As expected, the estimated ft is precisely the one predicted by (18.4.1). In passing, note that the estimated ft too is biased.

In general the amount of the bias in ft depends on ft, a2 and var (Y) and, in particular, on the degree of covariance between Y and u.11 As Kenneth White et al. note, "This is what simultaneous equation bias is all about. In contrast to single equation models, we can no longer assume that variables on the right hand side of the equation are uncorrelated with the error term.''12 Bear in mind that this bias remains even in large samples.

In view of the potentially serious consequences of applying OLS in simultaneous-equation models, is there a test of simultaneity that can tell us whether in a given instance we have the simultaneity problem? One version of the Hausman specification test can be used for this purpose, which we discuss in Chapter 19.

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