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2. In a demand and supply model, if quantity supplied is not responsive to price, the demand function should be normalized with respect to price. On the other hand, if quantity supplied is highly responsive to price, the demand function should be normalized with respect to quantity. (See Section 9.1 for a graphical discussion and Section 9.3 for an empirical illustration.)

3. Before a simultaneous equations model is estimated, one should check whether each equation is identified or not. In linear simultaneous equations systems, a necessary condition for the identification of an equation is the order condition which says that the number of variables missing from the equation should be greater than or equal to the number of endogenous variables in the equation minus one. This counting rule is only a necessary condition. One also has to check the rank condition which is based on the structure of the missing variables in the other equations. This is illustrated with some examples in Section 9.4.

4. It is customary to classify an equation into the over-identified, exactly identified, and under-identified categories according as the number of variables missing from the equation is, respectively, greater than, equal to, or less than the number of endogenous variables minus one. It is not possible to get consistent estimates of the parameters in the under-identified equations. The difference between over-identified and exactly identified equations is simply that the latter are easier to estimate than the former.

5. We have discussed only single-equation methods, that is, estimation of each equation at a time. The methods we have discussed are:

(a) The instrumental variable (IV) methods (Section 9.5).

(b) The two-stage least squares (2SLS) method (Section 9.6).

(c) The limited-information maximum likelihood (LIML) method (Section 9.8). For an exactly identified equation, all the methods are equivalent and give the same answers. For an over-identified equation the IV method gives different estimates depending on which of the missing exogenous variables are chosen as instruments. The 2SLS method is a weighted instrumental variable method.

6. In over-identified equations, the 2SLS estimates depend on the normalization rule adopted. The LIML estimates do not depend on the choice of normalization. The LIML method is thus a truly simultaneous estimation method, but the 2SLS is not because, strictly speaking, since the endogenous variables are jointly determined, normalization should not matter.

7. It is not always true that we can say nothing about the parameters of an under-identified equation. In some cases the ordinary least squares (OLS) estimates, even if they are not consistent, do give us some information about the parameters. Some examples are given in Section 9.9. It is also interesting to note that the concept of identification as discussed by Working in 1927 is quite different from the current discussion which is concentrated on getting consistent estimators for the parameters (see Section 9.9).

8. There are some cases where the simultaneous equations model can be estimated by using the OLS method. One such example is the recursive model. We also give another example-that of estimation of the Cobb-Douglas pro-



Exercises

1. Explain the meaning of each of the following terms.

(a) Endogenous variables.

(b) Exogenous variables.

(c) Structural equations.

(d) Reduced-form equations.

(e) Order condition for identification.

(f) Rank condition for identification.

(g) Indirect least squares.

(h) Two-stage least squares.

(i) Instrumental variable methods, (j) Normalization.

(k) Simultaneity bias. (1) Recursive systems.

2. Explain concisely what is meant by "the identification problem" in the context of the linear simultaneous equations model.

duction function under uncertainty where the OLS method is the appropriate one (see Section 9.9).

9. In recent years, the usual definition of exogeneity has been questioned. Some new terms have been introduced, such as weak exogeneity. strong exogeneity, and super exogeneity. One important question that has been raised is: "Exogenous for what?" If it is for efficient estimation of the parameters (this is the concept of weak exogeneity), a variable can be treated as endogenous in one equation and exogenous in another (as in a recursive system). Also, whether a variable is exogenous or not depends on the parameter to be estimated (see Pratts example).

10. Strong exogeneity is weak exogeneity plus Granger causality. The term "Granger causality" has nothing to do with causality as it is usually understood. A better term for it is "precedence." Some econometricians have equated the concept of exogeneity with Granger causality. The example in Section 9.10 shows that linking Granger causality to exogeneity has some pitfalls. It is belter to keep the two concepts separate.

11. A variable is considered superexogenous if interventions in that variable leave the parameters in the system unaltered. It is a concept that has to do with Lucass critique of econometric policy evaluation. As a definition, the concept is all right. But from the practical point of view its use is questionable.

12. There have been some tests of exogeneity suggested in the context of simultaneous equations systems. These tests depend on the availabiUty of extra instrumental variables. The tests are easy to apply because they depend on the addition of some constructed variables to the usual models, and testing that the coefficients of these added variables are zero.



3. Consider the three-equation model

= 1 \ + + 72i-i + 722-2 + "2 = + Ui

where „ yj, and , are endogenous, and x,, Xj, and Xj are exogenous. Discuss the identification of each of the equations of the model, based on the order and rank conditions.

Now suppose that you want to estimate the first equation by two-stage least squares, but you have only an ordinary least squares program available. Explain carefully, step by step, how you would estimate p,,, 7,2, and var(M,).

4. Consider the model

y, = 2 + + Uf = + 7- + "2

where x is exogenous, and the error terms w, and Uj have mean zero and are serially uncorrelated.

(a) Write down the equations expressing the reduced-form coefficients in terms of the structural parameters.

(b) Show that if 7 = 0, then p can be identified. Are the parameters a and 6 identified in this case? Why or why not?

(c) In the case of 7 = 0, what formula would you use to estimate p? What is the asymptotic variance of your estimator of p?

5. What is meant by the phrase: "The estimator is invariant to normalization"? Do any problems arise if an estimator is not invariant to normalization? Which of the following estimation methods gives estimators that are invariant to normalization?

(a) Indirect least squares.

(b) 2SLS.

(c) Instrumental variable methods.

(d) LIML

Explain how you would choose the appropriate normalization (with respect to quantity or price) in a demand and supply model.

6. The structure of a model with four endogenous and three exogenous variables is as follows: (1 indicates presence and 0 absence of the variable in the equation)

10 1110 0 1110 0 11 0 0 10 10 0 10 110 10

Which of the four equations are identified?

7. Explain how you would compute in simultaneous equations estimation methods.



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