chapter 11 simultaneous equations models

Principles of Econometrics, 4th Edition Chapter 11: Simultaneous Equations Models

Chapter 11Simultaneous Equations Models

Walter R. Paczkowski Rutgers University


11.1 A Supply and Demand Model11.2 The Reduced-Form Equations11.3 The Failure of Least Squares Estimation11.4 The Identification Problem11.5 Two-Stage Least Squares Estimation11.6 An Example of Two-Stage Least Squares

Estimation11.7 Supply and Demand at the Fulton Fish

Market

Chapter Contents


We will consider econometric models for data that are jointly determined by two or more economic relations– 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 from most of the econometric models we have considered so far, because they consist of a set of equations


11.1 A Supply and Demand Model


A very simple supply and demand model might look like:

11.1A Supply and

Demand Model

1 2Demand: dQ P X e

1Supply: sQ P e

Eq. 11.1

Eq. 11.2


11.1A Supply and

Demand ModelFIGURE 11.1 Supply and demand equilibrium


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

11.1A Supply and

Demand Model


The endogenous variables P and Q are dependent variables and both are random variablesThe 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

11.1A Supply and

Demand Model


For the error terms, we have:

11.1A Supply and

Demand Model

2

2

( ) 0, var( )

( ) 0, var( )

cov( , ) 0

d d d

s s s

d s

E e e

E e e

e e

Eq. 11.3


An ‘‘influence diagram’’ is a graphical representation of relationships between model components

11.1A Supply and

Demand Model


11.1A Supply and

Demand Model FIGURE 11.2 Influence diagrams for two regression models


11.1A Supply and

Demand Model FIGURE 11.3 Influence diagram for a simultaneous equations 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

11.1A Supply and

Demand Model


11.2 The Reduced-Form Equations


The two structural equations Eqs. 11.1 and 11.2 can be solved to express the endogenous variables P and Q as functions of the exogenous variable X.– This reformulation of the model is called the

reduced form of the structural equation system

11.2The Reduced-Form

Equations


To solve for P, set Q in the demand and supply equations to be equal:

– Solve for P:


Equations

1 1 2s dP e P X e

2

1 1 1 1

1 1

d se eP X

X v

Eq. 11.4

(11.4)


Solving for Q:

– The parameters π1 and π2 in Eqs. 11.4 and 11.5 are called reduced-form parameters. The error terms v1 and v2 are called reduced-form errors


Equations

Eq. 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

(11.5)


11.3 The Failure of Least Squares

Estimation


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

11.3The Failure of Least Squares Estimation


11.4 The Identification Problem


In the supply and demand model given by Eqs. 11.1 and 11.2:– 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

11.4The Identification

Problem



Problem FIGURE 11.4 The effect of changing income


It is the absence of variables in one equation that are present in another equation that makes parameter estimation possible


Problem


A general rule, which is called a necessary condition for identification of an equation, is: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 fewer than M - 1variables are omitted from an equation, then it is said to be unidentified, and its parameters cannot be consistently estimated


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


Problem


11.5 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

11.5Two-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.

11.5Two-Stage Least

Squares Estimation


The reduced-form model is:

Suppose we know π1.– Then through substitution:

11.5Two-Stage Least

Squares Estimation

1 1 1( )P E P v X v Eq. 11.6

1 1

1 1 1

s

s

Q E P v e

E P v e

Eq. 11.7


We can estimate π1 using from the reduced-form equation for P – A consistent estimator for E(P) is:

– Then:

11.5Two-Stage Least

Squares Estimation

Eq. 11.8

1̂

1ˆ ˆ P X

1 *ˆQ P e


Estimating Eq. 11.8 by least squares generates the so-called two-stage least squares estimator of β1, which is consistent and normally distributed in large samples

11.5Two-Stage Least

Squares Estimation


The two stages of the estimation procedure are:1. Least squares estimation of the reduced-form

equation for P and the calculation of its predicted value

2. Least squares estimation of the structural equation in which the right-hand-side endogenous variable P is replaced by its predicted value

11.5Two-Stage Least

Squares Estimation

P̂

P̂


Suppose the first structural equation in a system of M simultaneous equations is:

11.5Two-Stage Least

Squares Estimation

11.5.1The General Two-

Stage Least Squares Estimation Procedure

1 2 2 3 3 1 1 2 2 1y y y x x e Eq. 11.9


If this equation is identified, then its parameters can be estimated in the two steps:1. Estimate the parameters of the reduced-form

equations

by least squares and obtain the predicted values

11.5Two-Stage Least

Squares Estimation



2 12 1 22 2 2 2

3 13 1 23 2 3 3

K K

K K

y x x x vy x x x v

2 12 1 22 2 2

3 13 1 23 2 3

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ

K K

K K

y x x xy x x x

Eq. 11.10


If this equation is identified, then its parameters can be estimated in the two steps (Continued):2. Replace the endogenous variables, y2 and y3,

on the right-hand side of the structural Eqs. 11.9 by their predicted values from Eqs. 11.10:

• Estimate the parameters of this equation by least squares

11.5Two-Stage Least

Squares Estimation



*1 2 2 3 3 1 1 2 2 1ˆ ˆy y y x x e


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

11.5Two-Stage Least

Squares Estimation

11.5.2The 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

11.5Two-Stage Least

Squares Estimation

11.5.2The Properties of the

Two-Stage Least Squares Estimators


11.6 An Example of Two-Stage Least

Squares Estimation


Consider a supply and demand model for truffles:

11.6An Example of Two-Stage Least Squares

Estimation

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

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

Eq. 11.11

Eq. 11.12


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


Estimation

11.6.1Identification


The reduced-form equations are:


Estimation

11.6.2The Reduced-Form

Equations

11 21 31 41 1

12 22 32 42 2

i i i i i

i i i i i

Q PS DI PF vP PS DI PF v



Estimation


Equations

Table 11.1 Representative Truffle Data



Estimation


Equations

Table 11.2a Reduced Form for Quantity of Truffles (Q)



Estimation


Equations

Table 11.2b Reduced Form for Price of Truffles (P)


From Table 11.2b we have:


Estimation

11.6.3The Structural

Equations

12 22 32 42ˆ ˆ ˆ ˆ ˆ

32.512 1.708 7.603 1.354P PS DI PF

PS DI PF



Estimation


Equations

Table 11.3a 2SLS Estimates for Truffle Demand



Estimation


Equations

Table 11.3b 2SLS Estimates for Truffle Supply


11.7 Supply and Demand at the Fulton

Fish Market


11.7Supply and Demand

at the Fulton Fish Market

If supply is fixed, with a vertical supply curve, then price is demand-determined– Higher demand leads to higher prices but no

increase in the quantity supplied– If this is true, then the feedback between prices

and quantities is eliminated– Such models are said to be recursive and the

demand equation can be estimated by ordinary least squares rather than the more complicated two-stage least squares procedure




A key point is that ‘‘simultaneity’’ does not require that events occur at a simultaneous moment in time– Specify the demand equation for this market as:

• α2 is the price elasticity of demand– The supply equation is:

• β2 is the price elasticity of supply

1 2 3 4 5

6

ln ln

t t t t t

dt t

QUAN PRICE MON TUE WED

THU e

Eq. 11.13

t 1 2 3ln ln st t tQUAN PRICE STORMY e Eq. 11.14




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, and it does appear in the supply equation

– In the supply equation, the four daily indicator variables that are included in the demand equation are omitted

11.7.1Identification




The reduced-form equations specify each endogenous variable as a function of all exogenous variables:


Equations

11 21 31 41 51

61 1

ln

t t t t t

t t

QUAN MON TUE WED THU

STORMY v

12 22 32 42 52

62 2

ln

t t t t t

t t

PRICE MON TUE WED THU

STORMY v

Eq. 11.15

Eq. 11.16





Equations

Table 11.4a Reduced Form for ln(Quantity) Fish





Equations

Table 11.4b Reduced Form for ln(Price) Fish




The key reduced-form equation is Eq. 11.16 for ln(PRICE):– To identify the supply curve, the daily indicator

variables must be jointly significant– To identify the demand curve, the variable

STORMY must be statistically significant• The supply has a significant shift variable, so

that we can reliably estimate the demand equation• The two-stage least squares estimator

performs very poorly if the shift variables are not strongly significant


Equations




To implement two-stage least squares, we take the predicted value from the reduced-form regression and include it in the structural equations in place of the right-hand-side endogenous variable:


Equations

12 22 32 42 52

62

ˆ ˆ ˆ ˆ ˆlnˆ

t t t t t

t

PRICE MON TUE WED THUSTORMY




If the coefficients on the daily indicator variables are all identically zero, then:

– If we replace ln(PRICE) in the supply equation (Eq. 11.14) with this predicted value, there will be exact collinearity between and the variable STORMY, which is already in the supply equation

– Two-stage least squares will fail


Equations

12 62ˆ ˆln t tPRICE STORMY

ln PRICE




11.7.3Two-Stage Least

Squares Estimation of Fish Demand

Table 11.5 2SLS Estimates for Fish Demand


Key Words


endogenous variablesexogenous variablesidentification

Keywords

reduced-form equationreduced-form errorsreduced-form parameters

simultaneous equationsstructural parameterstwo-stage least squares


Appendices


The covariance between P and es for the supply and demand model is:

11AAn Algebraic

Explanation of the Failure of

Least Squares

Eq. 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

sd 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


The least squares estimator of the supply equation is:

– Substituting, we get:

where

11AAn Algebraic


Least Squares

Eq. 11A.2 1 2i i

i

PQb

P

11 1 12 2

i i si isi i si

i i

P P e Pb e h eP P

Eq. 11A.3

2i i ih P P


The expected value of the least squares estimator is:

– Also:

11AAn Algebraic


Least Squares

1 1 1i siE b E h e

21

21

1 2 2

s

s

s

PQ P Pe

E PQ E P E Pe

E PQ E PeE P E P


In large samples, as N→∞,

11AAn Algebraic


Least Squares

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


There has always been great interest in alternatives to the standard IV/2SLS estimator– The search for better alternatives was energized

by the discovery of the problems weak instruments pose for the usual IV/2SLS estimator

11B2SLS

Alternatives


Suppose the first structural equation within a system is:

– If this equation is identified, then its parameters can be estimated

11B.1The k-Class of

Estimators

1 2 2 1 1 2 2 1α β βy y x x e Eq. 11B.1

11B2SLS

Alternatives


The parameters of the reduced-form equation are consistently estimated by least squares, so that:

11B.1The k-Class of

Estimators

2 12 1 22 2 2ˆ ˆ ˆπ π πK KE y x x x Eq. 11B.2

11B2SLS

Alternatives


The reduced form residuals are:

– Or:

– The two-stage least squares estimator is an IV estimator using as an instrument

11B.1The k-Class of

Estimators

2 2 2v̂ y E y

Eq. 11B.3 2 2 2ˆE y y v

2E y

11B2SLS

Alternatives


The k-class of estimators is a unifying framework.– A k-class estimator is an IV estimator using

instrumental variable – It is called a class of estimators because it

represents the least squares estimator if k = 0 and the 2SLS estimator if k = 1

11B.1The k-Class of

Estimators

2 2ˆy kv

11B2SLS

Alternatives


The LIML estimator is one of the oldest estimators for an equation within a system of simultaneous equations, or any equation with an endogenous variable on the righthand side

11B.2The LIML Estimator

11B2SLS

Alternatives


The equation

is in normalized form, meaning that we have chosen one variable to appear as the dependent variable– In general the first equation can be written in

implicit form as

– There is no rule that says y1has to be the dependent variable in the first equation.

– Normalization amounts to setting α1or α2 to the value -1


1 2 2 1 1 2 2 1α β βy y x x e

1 1 2 2 1 1 2 2 1α α β β 0y y x x e

11B2SLS

Alternatives


We can write:

– The exogenous variables x3, ... , xK were omitted

– If included, then:


*1 1 2 2 1 1 1 2 2β β θ θy x x e x x Eq. 11B.4

*1 1θ θK Ky x x Eq. 11B.5

11B2SLS

Alternatives


The least variance ratio estimator chooses α1and α2 so that the ratio of the sum of squared residuals from Eq. 11B.4 relative to the sum of squared residuals from Eq. 11B.5 is as small as possible


11B2SLS

Alternatives


Define:

– The minimum value of , call it , results in the LIML estimator when used as k in the k-class estimator

– That is, use when forming the instrument and the resulting IV estimator is the

LIML estimator


*1 2

*1

from regression of on , 1 from regression of on , , K

SSE y x xlSSE y x x

Eq. 11B.6

l l̂

ˆk l

2 2ˆy kv

11B2SLS

Alternatives


A uses the k-class value:

– The value a is a constant

11B.2.1Fuller’s Modified

LIML

Eq. 11B.7 ˆ ak lN K

11B2SLS

Alternatives


A value a = 4 leads to an estimator that minimizes the ‘‘mean square error’’ of estimation– If we are estimating some parameter δ using an

estimator , then the mean square error of estimation is

11B.2.1Fuller’s Modified

LIML

2

2

2

ˆ ˆδ δ δ

ˆ ˆvar δ δ δ

ˆ ˆvar δ δ

MSE E

E

bias

δ̂

11B2SLS

Alternatives


Some findings are:– For the 2SLS estimator the amount of bias is an

increasing function of the degree of over identification. • The distributions of the 2SLS and least

squares estimators tend to become similar when overidentification is large.• LIML has the advantage over 2SLS when

there are a large number of instruments.– The LIML estimator converges to normality

faster than the 2SLS estimator and is generally more symmetric

11B.2.2Advantages of

LIML

11B2SLS

Alternatives


11B.2.3Stock-Yogo Weak IV Tests for LIML

Table 11B.1 Critical Values for the Weak Instrument Test Based on LIML Test Size (5% level of significance)

11B2SLS

Alternatives


11B.2.3Stock-Yogo Weak IV Tests for LIML

Table 11B.2 Critical Values for the Weak Instrument Test Based on Fuller-k Relative Bias (5% level of significance)

11B2SLS

Alternatives


We estimate the HOURS supply equation:

The LIML estimates are given in Table 11B.3– The models we consider are:

11B.2.3aTesting for Weak Instruments with

LIML

1 2 3 4 5β β β β 6 βHOURS MTR EDUC KIDSL NWIFEINC e Eq. 11B.8

2

Model 1: endogenous: ; :

Model 2: endogenous: ; : , ,Model 3: endogenous: , ; : ,Model 4: endogenous: , ; : , ,

MTR IV EXPER

MTR IV EXPER EXPER LARGECITYMTR EDUC IV MOTHEREDUC FATHEREDUCMTR EDUC IV MOTHEREDUC FATHEREDUC EXPER

11B2SLS

Alternatives


11B.2.3aTesting for Weak Instruments with

LIML

Table 11B.3 LIML Estimations11B2SLS

Alternatives


11B.2.3bTesting for Weak Instruments with Fuller Modified

LIML

Table 11B.4 Fuller (a = 1) Estimations11B2SLS

Alternatives


We previously carried out a Monte Carlo simulation to explore the properties of the IV/2SLS estimators– Now we employ the same experiment, adding

aspects of the new estimators we have introduced

11B.3Monte Carlo Simulation

Results

11B2SLS

Alternatives



Results

Table 11B.5 Monte Carlo Simulation Results11B2SLS

Alternatives


The mean square error measures how close the estimates are to the true parameter value– For the IV/2SLS estimator, the empirical mean

square error is:


Results

2100002 2 21

ˆ ˆβ β β 10,000mmMSE

11B2SLS

Alternatives


We can conclude that the Fuller modified LIML has lower mean square error than the IV/2SLS estimator in each experiment, and when the instruments are weak, the improvement is large


Results

11B2SLS

Alternatives

chapter 11 simultaneous equations models

Documents

demand equations

demand modelfigure

simple supply

demand equilibriumthe

demand model11

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systemsuch variables