where u1, u2, and u3 are stochastic disturbances.

In the preceding model there are three equations in three endogenous variables Q, L, and K. P, R, and W are exogenous.

a. What problems do you encounter in estimating the model if a + 0 = 1, that is, when there are constant returns to scale?

b. Even if a + 0 = 1, can you estimate the equations? Answer by considering the identifiability of the system.

c. If the system is not identified, what can be done to make it identifiable?

Note: Equations (2) and (3) are obtained by differentiating Q with respect to labor and capital, respectively, setting them equal to W/P and R/P, transforming the resulting expressions into logarithms, and adding (the logarithm of) the disturbance terms.

20.6. Consider the following demand-and-supply model for money:

Demand for money: Mf = 00 + 01Y1 + 02 Rt + Pt + u1t Supply of money: Ms = a0 + a1 Yt + u2t where M = money Y = income R = rate of interest P = price

Assume that R and P are predetermined.

a. Is the demand function identified?

b. Is the supply function identified?

c. Which method would you use to estimate the parameters of the identified equation(s)? Why?

d. Suppose we modify the supply function by adding the explanatory variables Yt-1 and Mt-1. What happens to the identification problem? Would you still use the method you used in c? Why or why not?

20.7. Refer to exercise 18.10. For the two-equation system there obtain the reduced-form equations and estimate their parameters. Estimate the

788 PART FOUR: SIMULTANEOUS-EQUATION MODELS

Problems indirect least-squares regression of consumption on income and compare your results with the OLS regression.

20.8. Consider the following model:

Yt = a0 + a1 Rt + Ut where Mt (money supply) is exogenous, Rt is the interest rate, and Yt is GDP.

a. How would you justify the model?

b. Are the equations identified?

c. Using the data given in Table 20.2, estimate the parameters of the identified equations. Justify the method(s) you use.

20.9. Suppose we change the model in exercise 20.8 as follows:

Rt = 00 + 01 Mt + 02 Yt + 03 Yt_1 + u1t Yt = a0 + a1 Rt + u2t a. Find out if the system is identified.

b. Using the data given in Table 20.2, estimate the parameters of the identified equation(s).

20.10. Consider the following model:

Yt = a0 + a1 Rt + a2 It + ut where the variables are as defined in exercise 20.8. Treating I (domestic investment) and M exogenously, determine the identification of the system. Using the data of Table 20.2, estimate the parameters of the identified equation(s).

20.11. Suppose we change the model of exercise 20.10 as follows:

Rt = 00 + 01 Mt + 02 Yt + mt Yt = a0 + a1 Rt + a2 It + ut It = Y0 + Y1 Rt + Ut

Assume that M is determined exogenously.

a. Find out which of the equations are identified.

b. Estimate the parameters of the identified equation(s) using the data given in Table 20.2. Justify your method(s).

20.12. Verify the standard errors reported in (20.5.3).

20.13. Return to the demand-supply model given in Eqs. (20.3.1) and (20.3.2). Suppose the supply function is altered as follows:

Qt = 00 + 01 Pt-1 + Ut where Pt-1 is the price prevailing in the previous period. a. If X (expenditure) and Pt-1 are predetermined, is there a simultaneity problem?

CHAPTER TWENTY: SIMULTANEOUS-EQUATION METHODS 789

b. If there is, are the demand and supply functions each identified? If they are, obtain their reduced-form equations and estimate them from the data given in Table 20.1.

c. From the reduced-form coefficients, can you derive the structural coefficients? Show the necessary computations.

20.14. Class Exercise: Consider the following simple macroeconomic model for the U.S. economy, say, for the period 1960-1999.*

Private consumption function:

Ct = a0 + a1 Yt + a2Ct-1 + u1t a1 > 0, 0 < a2 < 1

Private gross investment function:

It = A) + A Yt + 02Rt + A3 It-1 + u2t A > 0, 02 < 0, 0 <03 < 1

A money demand function:

Rt = ¿0 + ¿1 Yt + k2 Mt-1 + ¿3 Pt + ¿4 Rt-1 + u3t k1 > 0, k2 < 0, k3 > 0,0 < k4 < 1

Income identity:

Yt = Ct + It + Gt where C = real private consumption; I = real gross private investment, G = real government expenditure, Y = real GDP, M = M2 money supply at current prices, R = long-term interest rate (%), and P = Consumer Price Index. The endogenous variables are C, I, R, and Y. The predetermined variables are: Ct-1, It-1, Mt-1, Pt, Rt-1, and Gt plus the intercept term. The us are the error terms.

a. Using the order condition of identification, determine which of the four equations are identified, either exact or over.

b. Which method(s) do you use to estimate the identified equations?

c. Obtain suitable data from government and/or private sources, estimate the model, and comment on your results.

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