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Chapter 9

Chapter 9. Simultaneous Equations Models. 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

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Chapter 9

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  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

  16. 9.3 The Identification Problem: Identification Through Reduced Form

  17. 9.3 The Identification Problem: Identification Through Reduced Form

  18. 9.3 The Identification Problem: Identification Through Reduced Form

  19. 9.3 The Identification Problem: Identification Through Reduced Form

  20. 9.3 The Identification Problem: Identification Through Reduced Form

  21. 9.3 The Identification Problem: Identification Through Reduced Form

  22. 9.3 The Identification Problem: Identification Through Reduced Form

  23. 9.3 The Identification Problem: Identification Through Reduced Form

  24. 9.3 The Identification Problem: Identification Through Reduced Form • 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

  27. 9.3 The Identification Problem: Identification Through Reduced Form

  28. 9.3 The Identification Problem: Identification Through Reduced Form

  29. 9.3 The Identification Problem: Identification Through Reduced Form

  30. 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

  36. 9.5 Methods of Estimation: The Instrumental Variable Method

  37. 9.5 Methods of Estimation: The Instrumental Variable Method

  38. 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

  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

  41. 9.5 Methods of Estimation: The Instrumental Variable Method

  42. 9.5 Methods of Estimation: The Instrumental Variable Method

  43. 9.5 Methods of Estimation: The Instrumental Variable Method

  44. 9.5 Methods of Estimation: The Instrumental Variable Method

  45. 9.5 Methods of Estimation: The Instrumental Variable Method

  46. 9.5 Methods of Estimation: The Instrumental Variable Method

  47. 9.5 Methods of Estimation: The Instrumental Variable Method

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

  49. 9.6 Methods of Estimation: The Two-Stage Least Squares Method

  50. 9.6 Methods of Estimation: The Two-Stage Least Squares Method

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