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a. Compare these regression results with those given in Eq. (8.2.1). What changes do you see? And how do you account for them?

b. Is it worth adding the variable TFR to the model? Why?

c. Since all the individual t coefficients are statistically significant, can we say that we do not have a collinearity problem in the present case?

10.4. If the relation X1i + k2X2i + k3X3i = 0 holds true for all values of X1t k2, and k3, estimate r12.3, r13.2, and r231. Also find R 23, Rf 13, and R| 12-What is the degree of multicollinearity in this situation? Note: R2 2 3 is the coefficient of determination in the regression of Yon X2 and X3. Other R2 values are to be interpreted similarly.

10.5. Consider the following model:

Yt = 01 + 02 Xt + fa Xt-1 + 04 Xt-2 + 05 Xt-3 + 06 Xt-4 + U

where Y = consumption, X = income, and t = time. The preceding model postulates that consumption expenditure at time t is a function not only

CHAPTER TEN: MULTICOLLINEARITY 377

of income at time t but also of income through previous periods. Thus, consumption expenditure in the first quarter of 2000 is a function of income in that quarter and the four quarters of 1999. Such models are called distributed lag models, and we shall discuss them in a later chapter.

a. Would you expect multicollinearity in such models and why?

b. If collinearity is expected, how would you resolve the problem?

10.6. Consider the illustrative example of Section 10.6. How would you reconcile the difference in the marginal propensity to consume obtained from (10.6.1) and (10.6.4)?

10.7. In data involving economic time series such as GNP, money supply, prices, income, unemployment, etc., multicollinearity is usually suspected. Why?

### 10.8. Suppose in the model

Yi = 01 + 02 + 03 X3i + Ui that r23, the coefficient of correlation between X2 and X3, is zero. Therefore, someone suggests that you run the following regressions:

Yi = a + a2 Xn + U1i Yi = Y1 + Y3 X3i + U2i a. Will a2 = 02 and Y3 = fa3? Why?

b. Will equal a1 or Y1 or some combination thereof?

c. Will var (f}2) = var (a2) and var (/?3) = var (Y3)?

10.9. Refer to the illustrative example of Chapter 7 where we fitted the Cobb-Douglas production function to the Taiwanese agricultural sector. The results of the regression given in (7.9.4) show that both the labor and capital coefficients are individually statistically significant.

a. Find out whether the variables labor and capital are highly correlated.

b. If your answer to (a) is affirmative, would you drop, say, the labor variable from the model and regress the output variable on capital input only?

c. If you do so, what kind of specification bias is committed? Find out the nature of this bias.

10.10. Refer to Example 7.4. For this problem the correlation matrix is as follows: