1. Chapter 9 Simultaneous Equations Models

2. What is in this Chapter? In Chapter 4 we mentioned that one of the assumptions in the basic regression model is that the explanatory variables are uncorrelated with the error term In this chapter we relax that assumption and consider the case where several variables are jointly determined

3. What is in this Chapter? This chapter first discusses the conditions under which equations are estimable in the case of jointly determined variables (the "identification problem") and methods of estimation One major method is that of "instrumental variables," a method we shall also discuss in Chapter 11 Finally, this chapter also discusses recent work on exogeneity and causality

4. 9.1 Introduction In the usual regression model y is the dependent or determined variable and x1, x2, x3... Are the independent or determining variables The crucial assumption we make is that the x's are independent of the error term u Sometimes, this assumption is violated: for example, in demand and supply models

5. 9.1 Introduction Suppose that we write the demand function as: where q is the quantity demanded, p the price, and u the disturbance term which denotes random shifts in the demand function In Figure 9.1 we see that a shift in the demand function produces a change in both price and quantity if the supply curve has an upward slope

6. 9.1 Introduction If the supply curve is horizontal (i.e., completely price inelastic), a shift in the demand curve produces a change in price only If the supply curve is vertical (infinite price elasticity), a shift in the demand curve produces a change in quantity only

7. 9.1 Introduction Thus in equation (9.1) the error term u is correlated with p when the supply curve is upward sloping or perfectly horizontal Hence an estimation of the equation by ordinary least squares produces inconsistent estimates of the parameters

8. 9.2 Endogenous and Exogenous Variables In simultaneous equations models variables are classified as endogenous and exogenous The traditional definition of these terms is that endogenous variables are variables that are determined by the economic model and exogenous variables are those determined from outside

9. 9.2 Endogenous and Exogenous Variables Endogenous variables are also called jointly determined and exogenous variables are called predetermined. (It is customary to include past values of endogenous variables in the predetermined group.) Since the exogenous variables are predetermined, they are independent of the error terms in the model They thus satisfy the assumptions that the x's satisfy in the usual regression model of y on x's

10. 9.2 Endogenous and Exogenous Variables Consider now the demand and supply mode q = a1 + b1p + c1 y + u1 demand function q = a2 + b2p + c2R + u2 supply function (9.2) q is the quantity, p the price, y the income, R the rainfall, and u1 and u2 are the error terms Here p and q are the endogenous variables and y and R are the exogenous variables

11. 9.2 Endogenous and Exogenous Variables Since the exogenous variables are independent of the error terms u1 and u2 and satisfy the usual requirements for ordinary least squares estimation, we can estimate regressions of p and q on y and R by ordinary least squares, although we cannot estimate equations (9.2)by ordinary least squares We will show presently that from these regressions of p and q on y and R we can recover the parameters in the original demand and supply equations (9.2)

12. 9.2 Endogenous and Exogenous Variables This method is called indirect least squares—it is indirect because we do not apply least squares to equations (9.2) The indirect least squares method does not always work, so we will first discuss the conditions under which it works and how the method can be simplified. To discuss this issue, we first have to clarify the concept of identification

13. 9.3 The Identification Problem: Identification Through Reduced Form We have argued that the error terms u1 and u2 are correlated with p in equations (9.2),and hence if we estimate the equation by ordinary least squares, the parameter estimates are inconsistent Roughly speaking, the concept of identification is related to consistent estimation of the parameters Thus if we can somehow obtain consistent estimates of the parameters in the demand function, we say that the demand function is identified

14. 9.3 The Identification Problem: Identification Through Reduced Form Similarly, if we can somehow get consistent estimates of the parameters in the supply function, we say that the supply function is identified Getting consistent estimates is just a necessary condition for identification, not a sufficient condition, as we show in the next section

15. 9.3 The Identification Problem: Identification Through Reduced Form

24. Illustrative Example The indirect least squares method we have described is rarely used In the following sections we describe a more popular method of estimating simultaneous equation models This is the method of two-stage least squares (2SLS) However, if some coefficients in the reduced- form equations are close to zero, this gives us some information about what variables to omit in the structural equations

25. 9.3 The Identification Problem: Identification Through Reduced Form We will provide a simple example of a two-equation demand and supply model where the estimates from OLS, reduced-form least squares, and indirect least squares provide information on how to formulate the model The example also illustrates some points we have raised in Section 9.1 regarding normalization The model is from Merrill and Fox. In Table 9.1 data are presented for demand and supply of pork in the United States for 1922-1941 The model estimated by Merrill and Fox is

26. 9.3 The Identification Problem: Identification Through Reduced Form

31. 9.5 Methods of Estimation: The Instrumental Variable Method In previous sections we discussed the identification problem Now we discuss some methods of estimation for simultaneous equations models Actually, we have already discussed one method of estimation: the indirect least squares method However, this method is very cumbersome if there are many equations and hence it is not often used Here we discuss some methods that are more generally applicable

32. 9.5 Methods of Estimation: The Instrumental Variable Method These methods of estimation can be classified into two categories: –1. Single-equation methods (also called "limited-information methods") –2. System methods (also called "full- information methods").

33. 9.5 Methods of Estimation: The Instrumental Variable Method A general method of obtaining consistent estimates of the parameters in simultaneous equations models is the instrumental variable method Broadly speaking, an instrumental variable is a variable that is uncorrelated with the error term but correlated with the explanatory variables in the equation For instance, suppose that we have the equation y = ßx + u

34. 9.5 Methods of Estimation: The Instrumental Variable Method where x is correlated with u Then we cannot estimate this equation by ordinary least squares The estimate of ß is inconsistent because of the correlation between x and u If we can find a variable z that is uncorrelated with u, we can get a consistent estimator for ß We replace the condition cov (z, u) = 0 by its sample counterpart

35. 9.5 Methods of Estimation: The Instrumental Variable Method

38. Consider the second equation of our model Now we have to find an instrumental variable for y1 but we have a choice of z1 and z2 This is because this equation is overidentified (by the order condition) Note that the order condition (counting rule) is related to the question of whether or not we have enough exogenous variables elsewhere in the system to use as instruments for the endogenous variables in the equation with unknown coefficients

39. 9.5 Methods of Estimation: The Instrumental Variable Method If the equation is underidentified we do not have enough instrumental variables If it is exactly identified, we have just enough instrumental variables If it is overidentified, we have more than enough instrumental variables In this case we have to use weighted averages of the instrumental variables available We compute these weighted averages so that we get the most efficient (minimum asymptotic variance) estimator

40. 9.5 Methods of Estimation: The Instrumental Variable Method

48. 9.6 Methods of Estimation: The Two-Stage Least Squares Method

62. 9.10 Exogeneity and Causality There are two concepts of exogeneity that are usually distinguished: –1. Predeterminedness. A variable is predetermined in a particular equation if it is independent of the contemporaneous and future errors in that equation. –2. Strict exogeneity. A variable is strictly exogenous if it is independent of the contemporaneous, future, and past errors in the relevant equation.

63. 9.10 Exogeneity and Causality

64. Engle, Hendry, and Richard are not satisfied with the foregoing definitions of exogeneity and suggest three more concepts: –1. Weak exogeneity. –2. Superexogeneity –3. Strong exogeneity

65. 9.10 Exogeneity and Causality Since these concepts are often used and they are not difficult to understand anyway, we will discuss them briefly. The concept of strong exogeneity is linked to another concept: "Granger causality." There is a proliferation of terms here, but they occur frequently in recent econometric literature One important point to note is that whether a variable is exogenous or not depends on the parameter under consideration Consider the equation

66. 9.10 Exogeneity and Causality

70. So which of Xt and Yt, is exogenous, if at all? The answer depends on the parameters of interest If we are interested in the parameters α, ß, σ, then Xt is exogenous and equations (9.22) are the ones to estimate If we are interested in the parameters γ,…, then Yt is exogenous and equations (9.23) are the ones to estimate

71. 9.10 Exogeneity and Causality

75. Granger Causality Granger starts from the premise that the future cannot cause the present or the past. If event A occurs after event B, we know that A cannot cause B. At the same time, if A occurs before B, it does not necessarily imply that A causes B. For instance, the weatherman's prediction occurs before the rain. This does not mean that the weatherman causes the rain.

76. 9.10 Exogeneity and Causality In practice, we observe A and B as time series and we would like to know whether A precedes B, or B precedes A, or they are contemporaneous For instance, do movements in prices precede movements in interest rates, or is it the opposite, or are the movements contemporaneous? This is the purpose of Granger causality It is not causality as it is usually understood

77. 9.10 Exogeneity and Causality

78. As mentioned earlier. Learner suggests using the simple word "precedence" instead of the complicated words Granger causality since all we are testing is whether a certain variable precedes another and we are not testing causality as it is usually understood However, it is too late to complain about the term since it has already been well established in the econometrics literature. Hence it is important to understand what it means