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Chapter 11 Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

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Page 1: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

Chapter 11

Simultaneous Equations Models

Prepared by Vera Tabakova, East Carolina University

Page 2: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

Chapter 11: Simultaneous Equations Models

11.1 A Supply and Demand Model

11.2 The Reduced Form Equations

11.3 The Failure of Least Squares

11.4 The Identification Problem

11.5 Two-Stage Least Squares Estimation

11.6 An Example of Two-Stage Least Squares Estimation

11.7 Supply and Demand at the Fulton Fish Market

Slide 11-2Principles of Econometrics, 3rd Edition

Page 3: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.1 A Supply and Demand Model

Figure 11.1 Supply and demand equilibrium

Slide11-3Principles of Econometrics, 3rd Edition

Page 4: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.1 A Supply and Demand Model

Slide11-4Principles of Econometrics, 3rd Edition

(11.1)

(11.2)

1 2Demand: dQ P X e

1Supply: sQ P e

Page 5: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.1 A Supply and Demand Model

Slide11-5Principles of Econometrics, 3rd Edition

(11.3)

2

2

( ) 0, var( )

( ) 0, var( )

cov( , ) 0

d d d

s s s

d s

E e e

E e e

e e

Page 6: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.1 A Supply and Demand Model

Figure 11.2 Influence diagrams for two regression models

Slide11-6Principles of Econometrics, 3rd Edition

Page 7: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.1 A Supply and Demand Model

Figure 11.3 Influence diagram for a simultaneous equations model

Slide11-7Principles of Econometrics, 3rd Edition

Page 8: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.2 The Reduced Form Equations

Slide11-8Principles of Econometrics, 3rd Edition

(11.4)

1 1 2s dP e P X e

2

1 1 1 1

1 1

d se eP X

X v

Page 9: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.2 The Reduced Form Equations

Slide11-9Principles of Econometrics, 3rd Edition

(11.5)

1

21

1 1 1 1

1 2 1 1

1 1 1 1

2 2

s

d ss

d s

Q P e

e eX e

e eX

X v

Page 10: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.3 The Failure of Least Squares

Slide 11-10Principles of Econometrics, 3rd Edition

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.

Page 11: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.4 The Identification Problem

In the supply and demand model given by (11.1) and (11.2)

the parameters of the demand equation, 1 and 2, cannot be

consistently estimated by any estimation method, but

the slope of the supply equation, 1, can be consistently estimated.

Slide 11-11Principles of Econometrics, 3rd Edition

Page 12: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.4 The Identification Problem

Figure 11.4 The effect of changing income

Slide11-12Principles of Econometrics, 3rd Edition

Page 13: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.4 The Identification Problem

Slide 11-13Principles of Econometrics, 3rd Edition

A Necessary Condition for Identification: 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 less than M–1 variables are omitted

from an equation, then it is said to be unidentified and its

parameters can not be consistently estimated.

Page 14: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.4 The Identification Problem

Slide 11-14Principles of Econometrics, 3rd Edition

Remark: The two-stage least squares estimation procedure is

developed in Chapter 10 and shown to be an instrumental

variables estimator. The number of instrumental variables required

for estimation of an equation within a simultaneous equations

model is equal to the number of right-hand-side endogenous

variables. Consequently, identification requires that the number of

excluded exogenous variables in an equation be at least as large as

the number of included right-hand-side endogenous variables. This

ensures an adequate number of instrumental variables.

Page 15: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.5 Two-Stage Least Squares Estimation

Slide 11-15Principles of Econometrics, 3rd Edition

(11.6)

(11.7)

1 1 1( )P E P v X v

1 1

1 1 1

1 *

s

s

Q E P v e

E P v e

E P e

Page 16: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.5 Two-Stage Least Squares Estimation

Slide 11-16Principles of Econometrics, 3rd Edition

(11.8)

1ˆ ˆ P X

1 *ˆ ˆQ P e

Page 17: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.5 Two-Stage Least Squares Estimation

Estimating the (11.8) by least squares generates the so-called two-

stage least squares estimator of β1, which is consistent and

asymptotically normal. To summarize, 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

Slide 11-17Principles of Econometrics, 3rd Edition

ˆ.P

ˆ.P

Page 18: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.5.1 The General Two-Stage Least Squares Estimation Procedure

1. Estimate the parameters of the reduced form equations

by least squares and obtain the predicted values.

Slide 11-18Principles of Econometrics, 3rd Edition

(11.9)1 2 2 3 3 1 1 2 2 1y y y x x e

2 12 1 22 2 2 2

3 13 1 23 2 3 3

K K

K K

y x x x v

y x x x v

Page 19: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.5.1 The General Two-Stage Least Squares Estimation Procedure

Slide 11-19Principles of Econometrics, 3rd Edition

(11.10)2 12 1 22 2 2

3 13 1 23 2 3

ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ

K K

K K

y x x x

y x x x

Page 20: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.5.1 The General Two-Stage Least Squares Estimation Procedure

2. Replace the endogenous variables, y2 and y3, on the right-hand side

of the structural (11.9) by their predicted values from (11.10)

Estimate the parameters of this equation by least squares.

Slide 11-20Principles of Econometrics, 3rd Edition

*1 2 2 3 3 1 1 2 2 1ˆ ˆy y y x x e

Page 21: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.5.2 The Properties of the Two-Stage Least Squares Estimator

The 2SLS estimator is a biased estimator, but it is consistent.

In large samples the 2SLS estimator is approximately normally

distributed.

Slide 11-21Principles of Econometrics, 3rd Edition

Page 22: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.5.2 The Properties of the Two-Stage Least Squares Estimator

The variances and covariances of the 2SLS estimator are unknown in

small samples, but for large samples we have expressions for them

which we can use as approximations. These formulas are built into

econometric software packages, which report standard errors, and t-

values, just like an ordinary least squares regression program.

Slide 11-22Principles of Econometrics, 3rd Edition

Page 23: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.5.2 The Properties of the Two-Stage Least Squares Estimator

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. Always use specialized 2SLS or

instrumental variables software when obtaining estimates of

structural equations.

Slide 11-23Principles of Econometrics, 3rd Edition

Page 24: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.6 An Example of Two-Stage Least Squares Estimation

Slide11-24Principles of Econometrics, 3rd Edition

(11.11)

(11.12)

1 2 3 4Demand: di i i i iQ P PS DI e

1 2 3Supply: si i i iQ P PF e

Page 25: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.6.1 Identification

The rule for identifying an equation is that 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 and 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.

Slide11-25Principles of Econometrics, 3rd Edition

Page 26: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.6.2 The Reduced Form Equations

Slide11-26Principles of Econometrics, 3rd Edition

11 21 31 41 1

12 22 32 42 2

i i i i i

i i i i i

Q PS DI PF v

P PS DI PF v

Page 27: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.6.2 The Reduced Form Equations

Slide11-27Principles of Econometrics, 3rd Edition

Page 28: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.6.2 The Reduced Form Equations

Slide11-28Principles of Econometrics, 3rd Edition

Page 29: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.6.2 The Reduced Form Equations

Slide11-29Principles of Econometrics, 3rd Edition

Page 30: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.6.3 The Structural Equations

Slide11-30Principles of Econometrics, 3rd Edition

12 22 32 42ˆ ˆ ˆ ˆ ˆ

32.512 1.708 7.602 1.354

i i i i

i i i

P PS DI PF

PS DI PF

Page 31: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.6.3 The Structural Equations

Slide11-31Principles of Econometrics, 3rd Edition

Page 32: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.6.3 The Structural Equations

Slide11-32Principles of Econometrics, 3rd Edition

Page 33: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.7 Supply and Demand at the Fulton Fish Market

Slide11-33Principles of Econometrics, 3rd Edition

(11.13)

(11.14)

1 2 3 4 5 6ln ln dt t t t t t tQUAN PRICE MON TUE WED THU e

t 1 2 3ln ln st t tQUAN PRICE STORMY e

Page 34: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.7.1 Identification

The necessary condition for an equation to be identified is that in this

system of M = 2 equations, it must be true that at least M – 1 = 1

variable must be omitted from each equation. In the demand equation

the weather variable STORMY is omitted, while it does appear in the

supply equation. In the supply equation, the four daily dummy

variables that are included in the demand equation are omitted.

Slide11-34Principles of Econometrics, 3rd Edition

Page 35: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.7.2 The Reduced Form Equations

Slide11-35Principles of Econometrics, 3rd Edition

(11.15)

(11.16)

11 21 31 41 51 61 1ln t t t t t t tQUAN MON TUE WED THU STORMY v

12 22 32 42 52 62 2ln t t t t t t tPRICE MON TUE WED THU STORMY v

Page 36: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.7.2 The Reduced Form Equations

Slide11-36Principles of Econometrics, 3rd Edition

Page 37: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.7.2 The Reduced Form Equations

Slide11-37Principles of Econometrics, 3rd Edition

Page 38: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.7.2 The Reduced Form Equations

To identify the supply curve the daily dummy variables must be jointly significant. This implies that at least one of their coefficients is statistically different from zero, meaning that there is at least one significant shift variable in the demand equation, which permits us to reliably estimate the supply equation.

To identify the demand curve the variable STORMY must be statistically significant, meaning that supply has a significant shift variable, so that we can reliably estimate the demand equation.

Principles of Econometrics, 3rd Edition Slide11-38

Page 39: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.7.2 The Reduced Form Equations

Principles of Econometrics, 3rd Edition Slide11-39

12 22 32 42 52 62ˆ ˆ ˆ ˆ ˆ ˆln t t t t t tPRICE MON TUE WED THU STORMY

12 62ˆ ˆln t tPRICE STORMY

Page 40: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

11.7.3 Two-Stage Least Squares Estimation of Fish Demand

Principles of Econometrics, 3rd Edition Slide11-40

Page 41: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

Keywords

Slide 11-41Principles of Econometrics, 3rd Edition

endogenous variables exogenous variables Fulton Fish Market identification reduced form equation reduced form errors reduced form parameters simultaneous equations structural parameters two-stage least squares

Page 42: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

Chapter 11 Appendix

Slide 11-42Principles of Econometrics, 3rd Edition

Appendix 11A An Algebraic Explanation of the

Failure of Least Squares

Page 43: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

Appendix 11A An Algebraic Explanation of the Failure of Least Squares

Principles of Econometrics, 3rd Edition Slide 11-43

(11A.1)

1 1

11 1

2

1 1

cov ,

[since 0]

[substitute for ]

[since is exogenous]

[since , assumed uncorre

s s s

s s

s

d ss

s

d s

P e E P E P e E e

E Pe E e

E X v e P

e eE e X

E ee e

2

1 1

lated]

0s

Page 44: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

Appendix 11A An Algebraic Explanation of the Failure of Least Squares

Principles of Econometrics, 3rd Edition Slide 11-44

(11A.2)

(11A.3)

1 2i i

i

PQb

P

11 1 12 2

i i si isi i si

i i

P P e Pb e h e

P P

1 1 1i siE b E h e

Page 45: Simultaneous Equations Models Prepared by Vera Tabakova, East Carolina University

Appendix 11A An Algebraic Explanation of the Failure of Least Squares

Principles of Econometrics, 3rd Edition Slide 11-45

21

21

1 2 2

s

s

s

PQ P Pe

E PQ E P E Pe

E PQ E Pe

E P E P

21 1

1 1 1 12 2 2 2

/

/s si i

i

E PQ E PeQ P Nb

P N E P E P E P