17.5. Consider the model

Suppose Yt-i and vt are correlated. To remove the correlation, suppose we use the following instrumental variable approach: First regress Yt on X1t and X2t and obtain the estimated Yt from this regression. Then regress

Yt = a + 0i Xu + 02 X2t + 03 Yt-i + vt where Yt-i are estimated from the first-stage regression.

a. How does this procedure remove the correlation between Yt-i and vt in the original model?

b. What are the advantages of the recommended procedure over the Liviatan approach?

b. Evaluate the median lag for k = 0.2, 0.4, 0.6, 0.8.

c. Is there any systematic relationship between the value of k and the value of the median lag?

17.7. a. Prove that for the Koyck model, the mean lag is as shown in (17.4.10). b. If k is relatively large, what are its implications?

17.8. Using the formula for the mean lag given in (17.4.9), verify the mean lag of 10.959 quarters reported in the illustration of Table 17.1.

Mt = a + 01Y* + 02 R* + ut where M = demand for real cash balances, Y* = expected real income, and R* = expected interest rate. Assume that expectations are formulated as follows:

where y1 and y2 are coefficients of expectation, both lying between 0 and 1.

a. How would you express Mt in terms of the observable quantities?

b. What estimation problems do you foresee?

Adapted from G. K. Shaw, op. cit., p. 26. ^Optional.

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*17.10. If you estimate (17.7.2) by OLS, can you derive estimates of the original parameters? What problems do you foresee? (For details, see Roger N. Waud.t)

17.11. Serial correlation model. Consider the following model:

Assume that ut follows the Markov first-order autoregressive scheme given in Chapter 12, namely, ut = P ut-1 + St where p is the coefficient of (first-order) autocorrelation and where st satisfies all the assumptions of the classical OLS. Then, as shown in Chapter 12, the model

Yt = a(1 - p) + p(Xt - pXt-1) + p Yt_1 + St will have a serially independent error term, making OLS estimation possible. But this model, called the serial correlation model, very much resembles the Koyck, adaptive expectation, and partial adjustment models. How would you know in any given situation which of the preceding models is appropriate?*

17.12. Consider the Koyck (or for that matter the adaptive expectation) model given in (17.4.7), namely,

Suppose in the original model ut follows the first-order autoregressive scheme ut - pu1-t = st, where p is the coefficient of autocorrelation and where St satisfies all the classical OLS assumptions.

a. If p = X, can the Koyck model be estimated by OLS?

b. Will the estimates thus obtained be unbiased? Consistent? Why or why not?

c. How reasonable is it to assume that p = X?

17.13. Triangular, or arithmetic, distributed-lag model.§ This model assumes that the stimulus (explanatory variable) exerts its greatest impact in the current time period and then declines by equal decrements to zero as one goes into the distant past. Geometrically, it is shown in Figure 17.9. Following this distribution, suppose we run the following

"Optional.

"f'Misspecification in the 'Partial Adjustment' and 'Adaptive Expectations' Models," International Economic Review, vol. 9, no. 2, June 1968, pp. 204-217.

*For a discussion of the serial correlation model, see Zvi Griliches, "Distributed Lags: A Survey," Econometrica, vol. 35, no. 1, January 1967, p. 34.

§This model was proposed by Irving Fisher in "Note on a Short-Cut Method for Calculating Distributed Lags," International Statistical Bulletin, 1937, pp. 323-328.

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succession of regressions:

etc., and choose the regression that gives the highest R2 as the "best'' regression. Comment on this strategy. 17.14. From the quarterly data for the period 1950-1960, F. P. R. Brechling obtained the following demand function for labor for the British economy (the figures in parentheses are standard errors)*:

E = 14.22 + 0.172Qt - 0.028t - 0.0007t2 - 0.297Et-1 (2.61) (0.014) (0.015) (0.0002) (0.033)

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