Notes: Y1 = GDP = gross domestic product ($, billions, seasonally adjusted) Y2 = M2 = M2 money supply ($, billions, seasonally adjusted) X1 = GPDI = gross private domestic investment ($, billions, seasonally adjusted) X2 = FEDEXP = Federal government expenditure ($, billions, seasonally adjusted) X3 = TB6 = 6-month treasury bill rate (%). Source: Economic Report of the President, 2001. Tables B-2, B-69, B-73, B-84.

Notes: Y1 = GDP = gross domestic product ($, billions, seasonally adjusted) Y2 = M2 = M2 money supply ($, billions, seasonally adjusted) X1 = GPDI = gross private domestic investment ($, billions, seasonally adjusted) X2 = FEDEXP = Federal government expenditure ($, billions, seasonally adjusted) X3 = TB6 = 6-month treasury bill rate (%). Source: Economic Report of the President, 2001. Tables B-2, B-69, B-73, B-84.

776 PART FOUR: SIMULTANEOUS-EQUATION MODELS

investment X1 and government expenditure X2, obtaining the following results:

Yit = 2587.351 + 1.6707XU + 1.9693Xzt se = (72.0011) (0.1646) (0.0983) (20.5.1)

Stage 2 Regression. We now estimate the money supply function

(20.4.2), replacing the endogenous variable Y1 by Y\ estimated from (20.5.1) (= Yi). The results are as follows:

As we pointed out previously, the estimated standard errors given in (20.5.2) need to be corrected in the manner suggested in Appendix 20.A, Section 20A.2. Effecting this correction (most econometric packages can do it now), we obtain the following results:

As noted in Appendix 20A, Section 20A.2, the standard errors given in

(20.5.3) do not differ much from those given in (20.5.2) because the R2 in Stage 1 regression is very high.

OLS Regression. For comparison, we give the regression of money stock on income as shown in (20.4.2) without "purging" the stochastic Y1t of the influence of the stochastic disturbance term.

Y2t = -2195.468 + 0.7911 Yu se = (126.6460) (0.0211) (20.5.4)

Comparing the "inappropriate" OLS results with the Stage 2 regression, we see that the two regressions are virtually the same. Does this mean that

CHAPTER TWENTY: SIMULTANEOUS-EQUATION METHODS 777

the 2SLS procedure is worthless? Not at all. That in the present situation the two results are practically identical should not be surprising because, as noted previously, the R2 value in the first stage is very high, thus making the estimated Y1t virtually identical with the actual Y1t. Therefore, in this case the OLS and second-stage regressions will be more or less similar. But there is no guarantee that this will happen in every application. An implication, then, is that in overidentified equations one should not accept the classical OLS procedure without checking the second-stage regression(s).

Simultaneity between GDP and Money Supply. Let us find out if GDP (Y1) and money supply (Y2) are mutually dependent. For this purpose we use the Hausman test of simultaneity discussed in Chapter 19.

First we regress GDP on X1 (investment expenditure) and X2 (government expenditure), the exogenous variables in the system (i.e., we estimate the reduced-form regression.) From this regression we obtain the estimated GDP and the residuals Vt, as suggested in Eq. (19.4.7). Then we regress money supply on estimated GDP and vt to obtain the following results:

Yzt = -2198.297 + 0.7915Yt + 0.6984Vt se = (129.0548) (0.0215) (0.2970) (20.5.5)

Since the t value of Vt is statistically significant (the p value is 0.0263), we cannot reject the hypothesis of simultaneity between money supply and GDP, which should not be surprising. (Note: Strictly speaking, this conclusion is valid only in large samples; technically, it is only valid as the sample size increases indefinitely.)

Hypothesis Testing. Suppose we want to test the hypothesis that income has no effect on money demand. Can we test this hypothesis with the usual t test from the estimated regression (20.5.2)? Yes, provided the sample is large and provided we correct the standard errors as shown in (20.5.3), we can use the t test to test the significance of an individual coefficient and the F test to test joint significance of two or more coefficients, using formula (8.5.7).17

What happens if the error term in a structural equation is autocorrelated and or correlated with the error term in another structural equation in the system? A full answer to this question will take us beyond the scope of

17But take this precaution: The restricted and unrestricted RSS in the numerator must be calculated using predicted Y (as in Stage 2 of 2SLS) and the RSS in the denominator is calculated using actual rather than predicted values of the regressors. For an accessible discussion of this point, see T. Dudley Wallace and J. Lew Silver, Econometrics: An Introduction, Addison-Wesley, Reading, Mass., 1988, Sec. 8.5.

778 PART FOUR: SIMULTANEOUS-EQUATION MODELS

the book and is better left for the references (see the reference given in footnote 7). Nevertheless, estimation techniques (such as Zellner's SURE technique) do exist to handle these complications.

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