1. Simultaneous Equations Models
Chapter 14
Simultaneous Equations Models
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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2. 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
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3. 14.1 Introduction Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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4. 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
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5. A Supply and Demand Model
14.2
A Supply and Demand Model
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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6. 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 Eq
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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7. 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
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8. 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
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9. 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
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10. For the error terms, we have:
14.2
A Supply and Demand Model
For the error terms, we have: Eq
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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11. 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
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12. 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
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13. 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
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14. 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

15. The Reduced-Form Equations
14.3
The Reduced-Form Equations
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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16. 14.3
The Reduced-Form Equations
The two structural equations Eqs and 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
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17. 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
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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18. 14.3
The Reduced-Form Equations
Solving for q:
The parameters p1 and p2 in Eqs and 11.5 are called reduced-form parameters. The error terms v1 and v2 are called reduced-form errors Eq
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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19. The Failure of Least Squares Estimation
14.4
The Failure of Least Squares Estimation
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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20. 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
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21. 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 Eq Eq
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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22. 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
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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23. The Identification Problem
14.5
The Identification Problem
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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24. 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 and :
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
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25. FIGURE 14.4 The effect of changing income
14.5
The Identification Problem
FIGURE 14.4 The effect of changing income
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Chapter 14：Simultaneous Equations Models
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26. 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
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27. 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
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28. 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
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Chapter 14：Simultaneous Equations Models
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29. Two-Stage Least Squares Estimation
14.6
Two-Stage Least Squares Estimation
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Chapter 14：Simultaneous Equations Models
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30. 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

31. 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
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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32. 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 Eq
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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33. 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
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
Page 33

34. 14.6
Two-Stage Least Squares Estimation
Estimating Eq 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
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35. 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
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36. 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
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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37. 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
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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38. 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 by their predicted values from Eqs :
Estimate the parameters of this equation by least squares
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
Page 38

39. 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
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40. 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
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41. An Example of Two-Stage Least Squares Estimation
14.7
An Example of Two-Stage Least Squares Estimation
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Chapter 14：Simultaneous Equations Models
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42. 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 Eq
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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43. 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
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44. 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
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45. 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
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Chapter 14：Simultaneous Equations Models
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46. 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
2.434
0.0221 PS
4.605
0.0001 DI
3.094
0.0047 PF
-4.181
0.0003
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Chapter 14：Simultaneous Equations Models
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47. 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
-4.072
0.0004 PS
4.868
0.0001 DI
4.409
0.0002 PF
4.536
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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48. 14.7
An Example of Two-Stage Least Squares Estimation
14.7.2
The Reduced-Form Equations
From Table 11.2b we have:
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Chapter 14：Simultaneous Equations Models
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49. 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
-0.772
0.4471 P
-2.273
0.0315 PS
3.649
0.0012 DI
2.196
0.0372
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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50. 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
-0.772
0.4471 P
-2.273
0.0315 PS
3.649
0.0012 DI
2.196
0.0372
Undergraduated Econometrics
Chapter 14：Simultaneous Equations Models
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