## Exercises

19.1. Show that the two definitions of the order condition of identification are equivalent.

19.2. Deduce the structural coefficients from the reduced-form coefficients given in (19.2.25) and (19.2.27).

19.3. Obtain the reduced form of the following models and determine in each case whether the structural equations are unidentified, just identified, or overidentified:

19.4. Check the identifiability of the models of exercise 19.3 by applying both the order and rank conditions of identification.

19.5. In the model (19.2.22) and (19.2.28) of the text it was shown that the supply equation was overidentified. What restrictions, if any, on the structural parameters will make this equation just identified? Justify the restrictions you impose.

19.6. From the model

Y1t = 010 + 012 Y2t + Y11 X1t + U1t Y2t = 020 + 021 Y1t + Y22 X2t + U2t the following reduced-form equations are obtained:

Y1t = n1o + nn X1t + n12 X2t + w, Y2t = n2o + n21 X1t + n22 X2t + v, a. Are the structural equations identified?

b. What happens to identification if it is known a priori that y11 = 0?

19.7. Refer to exercise 19.6. The estimated reduced-form equations are as follows:

Y1t = 4 + 3 X1t + 8 X2t Y2t = 2 + 6 X1t + 10 X2t a. Obtain the values of the structural parameters.

b. How would you test the null hypothesis that y11 = 0?

19.8. The model

Y1t = 010 + 012 Y2t + Y11 X1t + U1t Y2t = 020 + 021 Y1t + U2t

CHAPTER NINETEEN: THE IDENTIFICATION PROBLEM 759

produces the following reduced-form equations:

Y1t = 4 + 8 X1t Y2t = 2 + 12 X1t a. Which structural coefficients, if any, can be estimated from the reduced-form coefficients? Demonstrate your contention.

b. How does the answer to (a) change if it is known a priori that (1) P12 = 0 and (2) P10 = 0?

19.9. Determine whether the structural equations of the model given in exercise 18.8 are identified.

19.10. Refer to exercise 18.7 and find out which structural equations can be identified.

19.11. Table 19.3 is a model in five equations with five endogenous variables Y and four exogenous variables X: 