Simultaneous Equations Models Chapter 14 Simultaneous Equations Models Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 1

14.2 A Supply and Demand Model 14.3 The Reduced-Form Equations Chapter Contents 14.1 Introduction 14.2 A Supply and Demand Model 14.3 The Reduced-Form Equations 14.4 The Failure of Least Squares Estimation 14.5 The Identification Problem 14.6 Two-Stage Least Squares Estimation 14.7 An Example of Two-Stage Least Squares Estimation Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 2

14.1 Introduction Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 3

14.1 Introduction These simultaneous equations models differ from those previously studied because in each model there are two or more dependent variables rather than just one Simultaneous equations models also differ form most of the econometric models we have considered so far because they consist of a set of equations. The least squares estimation procedure is not appropriate in these models Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 4

A Supply and Demand Model 14.2 A Supply and Demand Model Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 5

14.2 A Supply and Demand Model Supply and demand jointly determine the market price of a good and the amount of it that is sold. A very simple supply and demand model might look like: Eq. 14.2.1 Eq. 14.2.2 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 6

FIGURE 14.1 Supply and demand equilibrium 14.2 A Supply and Demand Model FIGURE 14.1 Supply and demand equilibrium Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 7

It takes two equations to describe the supply and demand equilibrium 14.2 A Supply and Demand Model It takes two equations to describe the supply and demand equilibrium The two equilibrium values, for price and quantity, p* and q*, respectively, are determined at the same time In this model the variables p and q are called endogenous variables because their values are determined within the system we have created Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 8

14.2 A Supply and Demand Model The income variable x has a value that is determined outside this system Such variables are said to be exogenous, and these variables are treated like usual ‘‘x’’ explanatory variables Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 9

For the error terms, we have: 14.2 A Supply and Demand Model For the error terms, we have: Eq. 14.2.3 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 10

14.2 A Supply and Demand Model An ‘‘influence diagram’’ is a graphical representation of relationships between model components Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 11

FIGURE 14.2 Influence diagrams for two regression models A Supply and Demand Model FIGURE 14.2 Influence diagrams for two regression models Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 12

FIGURE 14.3 Influence diagram for a simultaneous equations model 14.2 A Supply and Demand Model FIGURE 14.3 Influence diagram for a simultaneous equations model Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 13

14.2 A Supply and Demand Model The fact that p is an endogenous variable on the right-hand side of the supply and demand equations means that we have an explanatory variable that is random This is contrary to the usual assumption of ‘‘fixed explanatory variables’’ The problem is that the endogenous regressor P is correlated with the random errors, ed and es, which has a devastating impact on our usual least squares estimation procedure, making the least squares estimator biased and inconsistent Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 14

The Reduced-Form Equations 14.3 The Reduced-Form Equations Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 15

14.3 The Reduced-Form Equations The two structural equations Eqs. 14.2.1 and 14.2.2 can be solved to express the endogenous variables p and q as functions of the exogenous variable y. This reformulation of the model is called the reduced form of the structural equation system Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 16

To solve for P, set Q in the demand and supply equations to be equal: 14.3 The Reduced-Form Equations To solve for P, set Q in the demand and supply equations to be equal: Solve for P: Eq. 14.3.1 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 17

14.3 The Reduced-Form Equations Solving for q: The parameters p1 and p2 in Eqs. 11.4 and 11.5 are called reduced-form parameters. The error terms v1 and v2 are called reduced-form errors Eq. 14.3.2 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 18

The Failure of Least Squares Estimation 14.4 The Failure of Least Squares Estimation Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 19

14.4 The Failure of Least Squares Estimation 14.4.1 An Intuitive Explanation Of The Failure Of Least Squares The least squares estimator of parameters in a structural simultaneous equation is biased and inconsistent because of the correlation between the random error and the endogenous variables on the right-hand side of the equation Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 20

First, let obtain the covariance 14.4 The Failure of Least Squares Estimation 14.4.2 An Algebraic Explanation Of The Failure Of Least Squares First, let obtain the covariance The least squares estimator in (14.2.2) is Substitute for q from (14.3.2) Eq. 14.4.1 Eq. 14.4.2 Eq. 14.4.3 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 21

The expected value of the least squares estimator is 14.4 The Failure of Least Squares Estimation 14.4.2 An Algebraic Explanation Of The Failure Of Least Squares The expected value of the least squares estimator is The because es and p are correlated. Eq. 14.4.4 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 22

The Identification Problem 14.5 The Identification Problem Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 23

In the supply and demand model given by Eqs. 14.2.1 and 14.2.2: 14.5 The Identification Problem In the supply and demand model given by Eqs. 14.2.1 and 14.2.2: The parameters of the demand equation, α1and α2, cannot be consistently estimated by any estimation method The slope of the supply equation, β1, can be consistently estimated Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 24

FIGURE 14.4 The effect of changing income 14.5 The Identification Problem FIGURE 14.4 The effect of changing income Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 25

14.5 The Identification Problem It is the absence of variables in one equation that are present in another equation that makes parameter estimation possible Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 26

14.5 The Identification Problem A general rule, which is called a necessary condition for identification of an equation, is: In a system of M simultaneous equations, which jointly determine the values of M endogenous variables, at least M - 1 variables must be absent from an equation for estimation of its parameters to be possible When estimation of an equation’s parameters is possible, then the equation is said to be identified, and its parameters can be estimated consistently. If fewer than M - 1variables are omitted from an equation, then it is said to be unidentified, and its parameters cannot be consistently estimated Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 27

14.5 The Identification Problem The identification condition must be checked before trying to estimate an equation If an equation is not identified, then changing the model must be considered before it is estimated Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 28

Two-Stage Least Squares Estimation 14.6 Two-Stage Least Squares Estimation Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 29

This is often abbreviated as 2SLS 14.6 Two-Stage Least Squares Estimation The most widely used method for estimating the parameters of an identified structural equation is called two-stage least squares This is often abbreviated as 2SLS The name comes from the fact that it can be calculated using two least squares regressions Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 30

Consider the supply equation discussed previously 14.6 Two-Stage Least Squares Estimation Consider the supply equation discussed previously We cannot apply the usual least squares procedure to estimate β1 in this equation because the endogenous variable P on the right-hand side of the equation is correlated with the error term es. Eq. 14.2.2 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 31

The reduced-form model is: Suppose we know π1. 14.6 Two-Stage Least Squares Estimation The reduced-form model is: Suppose we know π1. Then through substitution: Eq. 14.6.1 Eq. 14.6.2 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 32

We can estimate π1 using from the reduced-form equation for P 14.6 Two-Stage Least Squares Estimation We can estimate π1 using from the reduced-form equation for P A consistent estimator for E(P) is: Then: Eq. 14.6.3 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 33

14.6 Two-Stage Least Squares Estimation Estimating Eq. 14.6.3 by least squares generates the so-called two-stage least squares estimator of β1, which is consistent and normally distributed in large samples Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 34

The two stages of the estimation procedure are: 14.6 Two-Stage Least Squares Estimation The two stages of the estimation procedure are: Least squares estimation of the reduced-form equation for p and the calculation of its predicted value Least squares estimation of the structural equation in which the right-hand-side endogenous variable p is replaced by its predicted value Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 35

14.6 Two-Stage Least Squares Estimation 14.6.1 The General Two-Stage Least Squares Estimation Procedure Suppose the first structural equation in a system of M simultaneous equations is: Eq. 14.6.4 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 36

Estimate the parameters of the reduced-form equations 14.6 Two-Stage Least Squares Estimation 14.6.1 The General Two-Stage Least Squares Estimation Procedure If this equation is identified, then its parameters can be estimated in the two steps: Estimate the parameters of the reduced-form equations by least squares and obtain the predicted values Eq. 14.6.5 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 37

Estimate the parameters of this equation by least squares 14.6 Two-Stage Least Squares Estimation 14.6.1 The General Two-Stage Least Squares Estimation Procedure If this equation is identified, then its parameters can be estimated in the two steps (Continued): Replace the endogenous variables, y2 and y3, on the right-hand side of the structural Eqs. 14.6.4 by their predicted values from Eqs. 14.6.5: Estimate the parameters of this equation by least squares Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 38

The 2SLS estimator is a biased estimator, but it is consistent 14.6 Two-Stage Least Squares Estimation 14.6.2 The Properties of the Two-Stage Least Squares Estimators The properties of the two-stage least squares estimator are as follows: The 2SLS estimator is a biased estimator, but it is consistent In large samples the 2SLS estimator is approximately normally distributed Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 39

14.6 Two-Stage Least Squares Estimation 14.6.2 The Properties of the Two-Stage Least Squares Estimators The properties of the two-stage least squares estimator are as follows (Continued): The variances and covariances of the 2SLS estimator are unknown in small samples, but for large samples we have expressions for them that we can use as approximations If you obtain 2SLS estimates by applying two least squares regressions using ordinary least squares regression software, the standard errors and t-values reported in the second regression are not correct for the 2SLS estimator Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 40

An Example of Two-Stage Least Squares Estimation 14.7 An Example of Two-Stage Least Squares Estimation Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 41

Consider a supply and demand model for truffles: 14.7 An Example of Two-Stage Least Squares Estimation Consider a supply and demand model for truffles: Eq. 14.7.1 Eq. 14.7.2 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 42

The rule for identifying an equation is: 14.7 An Example of Two-Stage Least Squares Estimation 14.7.1 Identification The rule for identifying an equation is: In a system of M equations at least M - 1 variables must be omitted from each equation in order for it to be identified In the demand equation the variable PF is not included; thus the necessary M – 1 = 1 variable is omitted In the supply equation both PS and DI are absent; more than enough to satisfy the identification condition Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 43

The reduced-form equations are: 14.7 An Example of Two-Stage Least Squares Estimation 14.7.2 The Reduced-Form Equations The reduced-form equations are: Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 44

Table 14.1 Representative Truffle Data And Demand Data 14.7 An Example of Two-Stage Least Squares Estimation Table 14.1 Representative Truffle Data And Demand Data 14.7.2 The Reduced-Form Equations OBS p q ps di pf 1 9.88 19.89 19.97 21.03 10.52 2 13.41 13.04 18.04 20.43 19.67 3 11.57 19.61 22.36 18.70 13.74 4 13.81 17.13 20.87 15.25 17.95 5 17.79 22.55 19.79 27.09 13.71 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 45

Table 14.2a Reduced Form for Quantity of Truffles (q) 14.7 An Example of Two-Stage Least Squares Estimation Table 14.2a Reduced Form for Quantity of Truffles (q) 14.7.2 The Reduced-Form Equations Variable Estimate Std.Error t-value p-value INTERCEP 7.895099 3.243422 2.434 0.0221 PS 0.656402 0.142538 4.605 0.0001 DI 0.216716 0.070047 3.094 0.0047 PF -0.506982 0.121262 -4.181 0.0003 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 46

Table 14.2b Reduced Form for Price of Truffles (p) 14.7 An Example of Two-Stage Least Squares Estimation Table 14.2b Reduced Form for Price of Truffles (p) 14.7.2 The Reduced-Form Equations Variable Estimate Std.Error t-value p-value INTERCEP -10.837473 2.661412 -4.072 0.0004 PS 0.569382 0.116960 4.868 0.0001 DI 0.253416 0.057478 4.409 0.0002 PF 0.451302 0.099502 4.536 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 47

14.7 An Example of Two-Stage Least Squares Estimation 14.7.2 The Reduced-Form Equations From Table 11.2b we have: Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 48

Table 14.3a 2SLS Estimates for Truffle Demand 14.7 An Example of Two-Stage Least Squares Estimation Table 14.3a 2SLS Estimates for Truffle Demand 14.7.2 The Reduced-Form Equations Variable Estimate Std.Error t-value p-value INTERCEP -4.27947 5.543884 -0.772 0.4471 P -1.12338 0.494255 -2.273 0.0315 PS 1.296033 0.355193 3.649 0.0012 DI 0.501398 0.228356 2.196 0.0372 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 49

Table 14.3b 2SLS Estimates for Truffle Supply 14.7 An Example of Two-Stage Least Squares Estimation Table 14.3b 2SLS Estimates for Truffle Supply 14.7.2 The Reduced-Form Equations Variable Estimate Std.Error t-value p-value INTERCEP -4.27947 5.543884 -0.772 0.4471 P -1.12338 0.494255 -2.273 0.0315 PS 1.296033 0.355193 3.649 0.0012 DI 0.501398 0.228356 2.196 0.0372 Undergraduated Econometrics Chapter 14：Simultaneous Equations Models Page 50