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

Figure 11.1 Supply and demand equilibrium

1 2Demand: dQ P X e= α +α +

1Supply: sQ P e= β +

2

2

( ) 0, var( )

( ) 0, var( )

cov( , ) 0

d d d

s s s

d s

E e e

E e e

e e

= = σ

= = σ

=

Figure 11.2 Influence diagrams for two regression models

Figure 11.3 Influence diagram for a simultaneous equations model

1 1 2s dP e P X eβ + = α +α +

( ) ( )2

1 1 1 1

1 1

d se eP X

X v

α −= +

β −α β −α

= π +

( ) ( )

( ) ( )

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

= β +

⎡ ⎤α −= β + +⎢ ⎥β −α β −α⎣ ⎦

β α β −α= +

β −α β −α

= π +

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.

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

the parameters of the demand equation, α1 and α2, cannot be

consistently estimated by any estimation method, but

the slope of the supply equation, β1, can be consistently estimated.

Figure 11.4 The effect of changing income

A Necessary Condition for Identification: In a system of Msimultaneous equations, which jointly determine the values of Mendogenous 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.

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.

1 1 1( )P E P v X v= + = π +

( )

( ) ( )

( )

1 1

1 1 1

1 *

s

s

Q E P v e

E P v e

E P e

= β ⎡ + ⎤ +⎣ ⎦

= β + β +

= β +

1ˆ ˆP X= π

1 *ˆ ˆQ P e= β +

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

ˆ.P

ˆ.P

1. Estimate the parameters of the reduced form equations

by least squares and obtain the predicted values.

1 2 2 3 3 1 1 2 2 1y y y x x e= α +α +β +β +

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

= π + π + + π +

= π + π + + π +

L

L

2 12 1 22 2 2

3 13 1 23 2 3

垐 垐

垐 垐

K K

K K

y x x x

y x x x

= π + π + + π

= π + π + + π

L

L

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.

*1 2 2 3 3 1 1 2 2 1垐y y y x x e= α +α +β +β +

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

In large samples the 2SLS estimator is approximately normally

distributed.

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.

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.

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

1 2 3Supply: si i i iQ P PF e= β +β +β +

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.

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

= π + π + π + π +

= π + π + π + π +

12 22 32 42ˆ 垐 垐

32.512 1.708 7.602 1.354

i i i i

i i i

P PS DI PF

PS DI PF

= π + π + π + π

= − + + +

( ) ( )1 2 3 4 5 6ln ln dt t t t t t tQUAN PRICE MON TUE WED THU e= α +α +α +α +α +α +

( ) ( )t 1 2 3ln ln st t tQUAN PRICE STORMY e= β +β +β +

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.

( ) 11 21 31 41 51 61 1ln t t t t t t tQUAN MON TUE WED THU STORMY v= π + π + π + π + π + π +

( ) 12 22 32 42 52 62 2ln t t t t t t tPRICE MON TUE WED THU STORMY v= π + π + π + π + π + π +

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.

( ) 12 22 32 42 52 62垐 垐 垐ln t t t t t tPRICE MON TUE WED THU STORMY= π + π + π + π + π + π

( ) 12 62垐ln t tPRICE STORMY= π + π

Slide 11-41Principles of Econometrics, 3rd Edition

Slide 11-42Principles of Econometrics, 3rd Edition

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

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

= ⎡ − ⎤ ⎡ − ⎤⎣ ⎦ ⎣ ⎦

= =

= π +

⎡ ⎤−= π⎢ ⎥β −α⎣ ⎦

−=β −α

2

1 1

lated]

0s−σ= <β −α

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 eP P

⎛ ⎞β += = β + = β +⎜ ⎟

⎝ ⎠

∑ ∑ ∑∑ ∑

( ) ( )1 1 1i siE b E h e= β + ≠ β∑

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 PeE P E P

= β +

= β +

β = −

( )( )

( )( )

( )( )

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 Pσ β −α

= → =β + = β − < β∑∑