Chapter 9

1. Chapter 9 Simultaneous Equations Models (聯立方程式模型)

2. What is in this Chapter? • How do we detect this problem? • What are the consequences? • What are the solutions?

3. 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 • Predetermined vs. jointly determined • Exogenous vs. Endogenous

4. 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" • Finally, this chapter also discusses causality

5. 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

6. 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

9. 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

10. 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

11. 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

12. 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

13. 9.2 Endogenous and Exogenous Variables • Consider now the demand and supply mode q = a1 + b1p + c1y + 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

14. 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)

15. 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

16. 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

17. 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

18. 9.3 The Identification Problem: Identification Through Reduced Form • If we solve the two equations in(9.2) for q and p in terms of y and R, we get • These equations are called the reduced-form equations. • Equation (9.2) are called the structural equations because they describe the structure of the economic system.

19. 9.3 The Identification Problem: Identification Through Reduced Form • We can write equations (9.3) as where v1 and v2 are error terms and

20. 9.3 The Identification Problem: Identification Through Reduced Form • The π’s are called reduced-form parameters. • The estimation of the equations (9.4) by ordinary least squares gives us consistent estimates of the reduced form parameters. • From these we have to obtain consistent estimates of the parameters in

21. 9.3 The Identification Problem: Identification Through Reduced Form • Since are all single-valued function of the ,they are consistent estimates of the corresponding structural parameters. • As mentioned earlier, this method is known as the indirect least squares method.

22. 9.3 The Identification Problem: Identification Through Reduced Form • It may not be always possible to get estimates of the structural coefficients from the estimates of the reduced-form coefficients, and sometimes we get multiple estimates and we have the problem of choosing between them. • For example, suppose that the demand and supply model is written as

23. 9.3 The Identification Problem: Identification Through Reduced Form • Then the reduced from is

24. 9.3 The Identification Problem: Identification Through Reduced Form or • In this case and . • But these is no way of getting estimates of a1, b1, and c1. • Thus the supply function is identified but the demand function is not.

25. 9.3 The Identification Problem: Identification Through Reduced Form • On the other hand, suppose that we have the model • Now we can check that the demand function is identified but the supply function is not.

26. 9.3 The Identification Problem: Identification Through Reduced Form • Finally, suppose that we have the system

27. 9.3 The Identification Problem: Identification Through Reduced Form or • Now we get two estimates of b2. • One is and the other is , and these need not be equal. • For each of these we get an estimate of a2, which is .

28. 9.3 The Identification Problem: Identification Through Reduced Form • On the other hand, we get no estimate for the parameters a1 , b1, c1, and d1 of the demand function. • Here we say that the supply function is overidentified and the demand function is underidentified. • When we get unique estimates for the structural parameters of an equation fro, the reduced-form parameters, we say that the equation is exactly identified.

29. 9.3 The Identification Problem: Identification Through Reduced Form • When we get multiple estimates, we say that the equation is overidentified, and when we get no estimates, we say that the equation is underidentified (or not identified). • There is a simple counting rule available in the linear systems that we have been considering. • This counting rule is also known as the order condition for identification.

30. 9.3 The Identification Problem: Identification Through Reduced Form • This rule is as follows: Let g be the number of endogenous variables in the system and k the total number of variables (endogenous and exogenous) missing from the equation under consideration. • Then

31. 9.3 The Identification Problem: Identification Through Reduced Form • This condition is only necessary but not sufficient. • Let us apply this rule to the equation systems we are considering. • In equations (9.2), g, the number of endogenous variable, is 2 and there is only one variable missing from each equation (i.e., k=1). • Both equations are identified exactly.

32. 9.3 The Identification Problem: Identification Through Reduced Form • In equations (9.5), again g=2. • There is no variable missing from the first equation (i.e., k=0); hence it is underidentified. • There is one variable missing in the second equation (i.e., k=1); hence it is exactly identifies. • In equation (9.6) • there is no variable missing in the first equation; hence it is not identified. • In the second equation there are two variables missing; thus k>g-1 and the equation is overidentified.

33. 9.3 The Identification Problem: Identification Through Reduced Form • Illustrative Example • In Table 9.1 data are presented for demand and supply of pork in the United States for 1922-1941

36. 9.3 The Identification Problem: Identification Through Reduced Form • Pt, retail price of pork (cents per pound) • Qt, consumption of pork (pounds per capita) • Yt, disposable personal income (dollars per capital) • Zt, “predetermined elements in pork production.”

37. 9.3 The Identification Problem: Identification Through Reduced Form • The coefficient of Y in the second equation is very close to zero and the variable Y can be dropped from this equation. • This would imply that b2=0, or supply is not responsive to price. • In any case, solving from the reduced from to the structural from, we get the estimates of the structural equation as

38. 9.3 The Identification Problem: Identification Through Reduced Form • The least squares estimates of the demand function are: • Normalized with respect to Q • Normalized with respect to P

39. 9.3 The Identification Problem: Identification Through Reduced Form • The structural demand function can also be written in the two forms: • Normalized with respect to Q • Normalized with respect to P • The estimates of the parameters in the demand function are almost the same with the direct least squares method as with the indirect least squares method when the demand function is normalized with respect to P.

40. 9.3 The Identification Problem: Identification Through Reduced Form • Which is the correct normalization? • We argued in Section 9.1 that if quantity supplied is not responsive to price, the demand function should be normalized with respect to P. • We saw that fact the coefficient of Y in the reduced-form equation for Q was close to zero implied that b2=0 or quantity supplied is not responsive to price.

41. 9.3 The Identification Problem: Identification Through Reduced Form • This is also confirmed by the structural estimate of b2, which show a wrong sign for b2 as well but a coefficient close to zero. • Dropping P from the supply function and using OLS, we get the supply function as

42. 9.5 Methods of Estimation: The Instrumental Variable Method • In previous sections we discussed the indirect least squares method • However, this method is very cumbersome if there are many equations and hence it is not often used • Identification problem • Here we discuss some methods that are more generally applicable • The Instrumental Variable Method

43. 9.5 Methods of Estimation: 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

44. 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

45. 9.5 Methods of Estimation: The Instrumental Variable Method • This gives • But can be written as

46. 9.5 Methods of Estimation: The Instrumental Variable Method • The probability limit of this expression is since cov (z, x) ≠0. • Hence plim ,thus proving that is a consistent estimator for β. • Note that we require z to be correlated with x so that cov (z, x) ≠0.

47. 9.5 Methods of Estimation: The Instrumental Variable Method • Now consider the simultaneous equations model where y1, y2 are endogenous variables, z1, z2, z3 are exogenous variables, and u1, u2 are error term. • Since z1 and z2 are independent of u1, • cov (z1, u1) =0 , cov (z2, u1) =0 • However, y2 is not independent of u1 • cov (y2, u1) ≠0.

48. 9.5 Methods of Estimation: The Instrumental Variable Method • Since we have three coefficients to estimate, we have to find a variable that is independent of u1. • Fortunately, in this case we have z3 and cov(z3,u1)=0. • z3 is the instrumental variable for y2. • Thus, writing the sample counterparts of these three covariances, we have three equations

49. 9.5 Methods of Estimation: The Instrumental Variable Method • The difference between the normal equation for the ordinary least squares method and the instrumental variable method is only in the last equation.

50. 9.5 Methods of Estimation: The Instrumental Variable Method • 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