Spatial Econometric Analysis Using GAUSS 8 Kuan-Pin Lin Portland State University.

Spatial Econometric Analysis Using GAUSS

8

Kuan-Pin LinPortland State University

Panel Data AnalysisA Review

Model Representation N-first or T-first representation

Pooled Model Fixed Effects Model Random Effects Model

Asymptotic Theory N→∞, or T→∞ N→∞, T→∞ Panel-Robust Inference


The Model

One-Way (Individual) Effects: Unobserved Heterogeneity Cross Section and Time Series Correlation

''it it it

it it i t itit i t it

yy u v e

u v e

xx

'it it i ity u e x

( , ) 0, ( , ) 0,

( , ) 0,i j it jt

it i

Cov u u Cov e e i j

Cov e e t


N-first Representation

Dummy Variables Representation

T-first Representation'

1,2,..., ; 1, 2,...,

( )

it it i it

i i i T i

N T

y u ei N t T

u

x β

y X β i e

y Xβ I i u e

'

1,2,..., ; 1,2,...,

( )

ti ti i ti

t t t

T N

y u et T i N

x β

y X β u e

y Xβ i I u e

N T T Nor

y Xβ Du eD I i D i I


Notations'

1, 1 2, 1 , 11 1 11'

1, 2 2, 2 , 22 2 22

'1, 2, ,

1

2

, , ,

,

i i K ii ii

i i K ii iii i i

iT iT K iTiT iT KiT

t t

tt t

tN

x x xy ex x xy e

x x xy e

yy

y

xx

y X e β

x

x

y X

'1, 1 2, 1 , 1 1 11

'1, 2 2, 2 , 2 2 22

'1, 2, ,

, ,

t t K t t

t t K t ttt

tN tN K tN tN NtN

x x x e ux x x e u

x x x e u

xe u

x

Pooled (Constant Effects) Model

'

'

'

2

( 1,2,..., ; 1, 2,..., )

assuming

1 ,

( | ) , ( | )

it it i it

i

it it it

it it

it it it

e

y u e i N t T

u u i

y u e

uy e

E Var

x β

x ββ

w x δ

w δ y Wδ e

e X 0 e X I

Fixed Effects Model

ui is fixed, independent of eit, and may be correlated with xit.

' ( 1, 2,..., ; 1, 2,..., )it it i ity u e i N t T x β

( , ) 0, ( , ) 0i it i itCov u e Cov u x

,

, 1, 2,...,1,2,...,

i i i T i

t t t

u i i Nt T

y X ey X u e

Fixed Effects Model Fixed Effects Model

Classical Assumptions Strict Exogeneity: Homoschedasticity: No cross section and time series correlation:

Extensions: Panel Robust Variance-Covariance Matrix

( | , ) 0itE e u X2( | , )it eVar e u X

2( | , ) e NTVar e u X I

( | , )Var e u X

Random Effects Model Error Components

ui is random, independent of eit and xit.

Define the error components as it = ui + eit

'

( 1, 2,..., ; 1,2,..., )it it it

it i it

yu e i N t T

x β

( , ) 0, ( , ) 0, ( , ) 0i it i it it itCov u e Cov u Cov e x x

( ), 1, 2,...,( ), 1, 2,...,

i i i T i

t t t

u i i Nt T

y X ey X u e

Random Effects Model

Random Effects Model Classical Assumptions

Strict Exogeneity

X includes a constant term, otherwise E(ui|X)=u.Homoschedasticity

Constant Auto-covariance (within panels)

( | ) 0, ( | ) 0 ( | ) 0it i itE e E u E X X X

2 2 '( | )i e T u T TVar ε X I i i

2 2

2 2 2

( | ) , ( | ) , ( , ) 0

( | )it e i u i it

it e u

Var e Var u Cov u e

Var

X X

X

Random Effects Model

Random Effects Model Classical Assumptions (Continued)

Cross Section Independence

Extensions:Panel Robust Variance-Covariance Matrix

2 2 '( | )( | )

i e T u T T

N

VarVar

ε X I i iε X Ω I

Fixed Effects Model Estimation

Within Model Representation'

'

' '

'

( ) ( )

it it i it

i i i i

it i it i it i

it it it

y u e

y u e

y y e e

y e

x β

x β

x x β

x β

'1 , ( 0, ' )

i i i

i i i

T T T T

orQ Q Q

where Q Q Q Q QT

y X β ey X β e

I i i i

Fixed Effects Model Estimation

Model Assumptions

2

2

2 2 '

( | ) 0

( | ) (1 1/ )

( , | , ) ( 1/ ) 0,

1( | ) ( )

( | )

it it

it it e

it is it is e

i i e e T T T

N

E e

Var e T

Cov e e T t s

Var QT

Var

x

x

x x

e X I i i

e X Ω I

Fixed Effects Model Estimation: OLS

Within Estimator: OLS

1' 1 ' ' '

1 1

' 1 ' ' 1

1 12 ' ' '

1 1 1

12 '

1

2

ˆ ( )

ˆˆ ( ) ( ) ( )

ˆ

ˆ

ˆˆ '

i i i

N NOLS i i i ii i

OLS

N N Ne i i i i i ii i i

Ne i ii

e

Var

Q

y X β e y Xβ e

β XX Xy X X X y

β XX XΩX XX

X X X X X X

X X

e

ˆ ˆ/ ( ),NT N K e e y Xβ

Fixed Effects Model Estimation: ML

Normality Assumption'

2

'

2 2

( 1, 2,..., )( 1,2,..., )

~ ( , )

, , ,1

~ (0, ), '

i

it it i it

i i i T i

i e T

i i i i i i i i i

T T T

i e e

y u e t Tu i N

normal iid

with Q Q Q

QT

normal where QQ Q

x βy X β i e

e 0 I

y X β e y y X X e e

I i i

e


Log-Likelihood Function

Since Q is singular and |Q|=0, we maximize

2 ' 1

2 ' 12

1 1( , | , ) ln 2 ln2 2 2

1 1ln 2 ln( ) ln2 2 2 2

i e i i i i

e i ie

Tll

T T Q Q

β y X e e

e e

2 2 '2

1( , | , ) ln 2 ln( )2 2 2i e i i e i i

e

T Tll

β y X e e


ML Estimator2 2

1

'2 21

2 2

ˆ( , ) argmax ( , | , )

ˆ ˆ 1 ˆ ˆˆ ˆ1 ,

ˆ ˆ'ˆ ˆ1 ( 1)

Ne ML i e i ii

Ni ii

e e i i i

e e

ll

NT T

TT N T

β β y X

e ee y X β

e e

Fixed Effects ModelHypothesis Testing

Pool or Not Pool F-Test based on dummy

variable model: constant or zero coefficients for D w.r.t F(N-1,NT-N-K)

F-test based on fixed effects (unrestricted) model vs. pooled (restricted) model

'

'

. ( , )it it i it

i

it it it

y u evs u u i

y u e

x β

x β

' '

( ) / 1 ~ ( 1, )/ ( )

ˆ ˆ ˆ ˆ,

R UR

UR

UR FE FE R PO PO

RSS RSS NF F N NT N KRSS NT N K

RSS RSS

e e e e

Fixed Effects ModelHypothesis Testing

Based on estimated residuals of the fixed effects model: Heteroscedasticity

Breusch and Pagan (1980) Autocorrelation: AR(1)

Breusch and Godfrey (1981)

' ˆ , 1,...,i i i i N e y x β

2221' ~ (1)

1 'NTLMT

e ee e

Random Effects Model Estimation: GLS

The Model

2 2 '

2 22

2

' '

,( | )

( | )

1 1,

i i i i i T i

i i

i i e T u T T

e ue T

e

T T T T T T

uE

Var

TQ Q

where Q QT T

y X β ε ε i eε X 0

ε X I i i

I

I i i I i i


GLS

11 1 1 1 1

1 1

11 1 1

1

2 21 '

2 2 2 2 2 2

1 22

2 2

ˆ ( )

ˆ( ) ( )

1 1

1

N NGLS i i i ii i

NGLS i ii

u eT T T T

e e u e e u

eT

e e u

Var

where Q QT T

and Q QT

β XΩ X XΩ y X X X y

β XΩ X X X

I i i I

I


Feasible GLS Based on estimated residuals of fixed effects

model

1 1 1

1 1

1 2 2 212 2

1

ˆ ˆ ˆ( )ˆ ˆ( ) ( )

1 1ˆ ˆ ˆ ˆ,ˆ ˆ

GLS

GLS

T e ue

Var

where Q Q T

β XΩ X XΩ y

β XΩ X

I

2

2 2 21 1

ˆ ˆˆ ' / ( 1)1ˆ ˆ ˆ ˆˆ ˆ ˆ ' / ,

e

Tu e i itt

N T

T T N where e eT

e e

e e

Random Effects Model Estimation: ML

Log-Likelihood Function

' '

2 2 1

( ) ( 1, 2,..., )( 1, 2,..., )

~ ( , )

1 1( , , | , ) ln 2 ln2 2 2

it it i it it it

i i i

i

i e u i i i i

y u e t Ti N

normal iid

Tll

x β x βy X β εε 0

β y X ε ε


where2 2

2 2 '2

2 21 '

2 2 2 2 2 2

2 22 ' 2

2 2

( )

1 1 ( )

| | ( ) ( ) 1

e ue T u T T T

e

u eT T T T

e u e e u e

T Tu ue T T T e

e e

TQ Q

Q QT T

T

I i i I

I i i I

I i i


ML Estimator

2 2 2 21

2 2 1

2 22

2

2 2' 2 '

2 2 21 1

ˆ ˆ ˆ( , , ) argmax ( , , | , )

1 1( , , | , ) ln 2 ln2 2 2

1ln 2 ln2 2

1 ( ) ( )2

Ne u ML i e u i ii

i e u i i i i

e ue

e

T Tuit it it itt t

e e u

ll

whereTll

TT

y yT

β β y X

β y X ε ε

x β x β

Random Effects ModelHypothesis Testing

Pool or Not Pool Test for Var(ui) = 0, that is

For balanced panel data, the Lagrange-multiplier test statistic (Breusch-Pagan, 1980) is:

, , ,( ) ( ) ( )it is i it i is it isCov Cov u e u e Cov e e


Pool or Not Pool (Cont.)

2

22

1 1

21 1

'

ˆ ˆ'( ) 1 ~ (1)ˆ ˆ2 1 '

ˆ1

2 1 ˆ

ˆˆ 1

ˆ

T N

N Titi t

N Titi t

it it it

Pooled

NTLMT

eNTT e

where e yu

e J I ee e

βx


Fixed Effects vs. Random Effects '

0

'1

: ( , ) 0 ( )

: ( , ) 0 ( )i it

i it

H Cov u random effects

H Cov u fixed effects

x

x

Estimator Random EffectsE(ui|Xi) = 0

Fixed EffectsE(ui|Xi) =/= 0

GLS or RE-OLS(Random Effects)

Consistent and Efficient

Inconsistent

LSDV or FE-OLS(Fixed Effects)

ConsistentInefficient

ConsistentPossibly Efficient


Fixed effects estimator is consistent under H0 and H1; Random effects estimator is efficient under H0, but it is inconsistent under H1.

Hausman Test Statistic ' 1

2

ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )

ˆ ˆ ˆ~ (# ), # # ( )

RE FE RE FE RE FE

FE FE RE

H Var Var

provided no intercept

β β β β β β

β β β


Alternative Hausman Test Estimate the random effects model

F Test that = 0

' ' ' '( ) ( ) ( )it i it i it i ity y e x x β x x γ

0 0: 0 : ( , ) 0i itH H Cov u γ x


Heteroscedasticity H0: θ2=0 | θ1=0 H0: θ1=0 | θ2=0 H0: θ2=0, θ1=0

'

2

2 2 '1

2 2 '1

2

2 2 '2

~ (0, )

( ), 1,..., , 1,...,

( ), 1,..., ,

~ (0, )

( ), 1,...,

it

it

it

i

i

it it it

it i it

it e

e e it

e e i

i u

u u i

yu e

e

h i N t T

or h i N t

u

h i N

x β

z

h

f


Heteroscedasticity (Cont.) Based on random effects model with

homoscedasticity:2 2 2 2 2

1ˆˆ ˆ ˆ ˆ ˆ ˆ, 1,..., ; , ,i i i u e u ei N T e y X β

2 1

1 20| 4

1

' '

' '

1 ' ( ' ) ' ~ (# )ˆ2

[ , 1,... ], ( / )ˆ ˆ[ , 1,..., ], ( / )

i N N N

i i i T T i

LM S F F F F S F

F i N F N F

S S i N S T

f I i i

e i i e


Heteroscedasticity (Cont.)

1 2

1 20|

' '

4 414 4 4 41 1

' '

' '

1 ' ( ' ) ' ~ (# )2

[ , 1,... ], ( / )

ˆ ˆ ( 1) 1 1,ˆ ˆ ˆ ˆ

ˆ ˆ[ , 1,..., ], ( / )ˆ ˆ[ , 1,..., ], ( / )

i N N N

e

e e

i i i T T i

i i i T T T i

LM S H H H H S Ha

H i N H N H

Ta S S S

S S i N S T

S S i N S T

h I i i

e i i e

e I i i e


Heteroscedasticity (Cont.)

Baltagi, B., Bresson, G., Pirotte, A. (2006) Joint LM test for homoscedasticity in a one-way error component model. Journal of Econometrics, 134, 401-417.

1 2 2 1

20, 0 0 0 ~ (# # )LM LM LM F H


Autocorrelation: AR(1) Based on random effects model with no

autocorrelation:

LM test statistic is tedious, see Baltagi, B., Li, Q. (1995) Testing AR(1) against

MA(1) disturbances in an error component model. Journal of Econometrics, 68, 133-151.

2 2 2 2 21

ˆˆ , 1,...,

ˆ ˆ ˆ ˆ ˆ, ,i i i

u e u e

i N

T

e y X β


Joint Test for AR(1) and Random Effects Based on OLS residuals:

Marginal Test for AR(1) & Random Effects

2

22 2 2

0, 0

'1

4 2 ~ (2)2( 1)( 2)

ˆ ˆ ˆ ˆ'( ) '1,ˆ ˆ ˆ ˆ' '

u

N T T

NTLM A AB TBT T

A B

ε I i i ε ε ε

ε ε ε ε

ˆˆ ε y - Xβ

2

22 2 2 2

00~ (1); ~ (1)

2( 1) 1u

NT NTLM A LM BT T


Robust LM Tests for AR(1) and Random Effects Because

2 2 2* *0 00, 0 0 0u u u

LM LM LM LM LM

2* 2 2

0

2* 2 20

(2 ) ~ (1)2( 1)(1 2 / )

( / ) ~ (1)( 1)(1 2 / )

u

NTLM B AT TNTLM B A T

T T

Panel Data AnalysisAn Example: U. S. Productivity

The Model (Munnell [1988]):

0 1 2

3 4

ln( ) ln( ) ln( )ln( ) ( )

it it it

it it i it

it it it it

gsp public privateemp unemp u e

public hwy water util

Panel Data AnalysisAn Example: U. S. Productivity

Productivity Data 48 Continental U.S. States, 17 Years:1970-1986

STATE = State name, ST_ABB = State abbreviation, YR = Year, 1970, . . . ,1986, PCAP = Public capital, HWY = Highway capital, WATER = Water utility capital, UTIL = Utility capital, PC = Private capital, GSP = Gross state product, EMP = Employment, UNEMP = Unemployment rate

U. S. ProductivityBaltagi (2008) [munnell.1, munnell.2]

Panel Data Model ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) + 4(Unemp) +

FixedEffects s.e

RandomEffects s.e

-0.026 0.029 0.003 0.024

0.292 0.025 0.310 0.020

3 0.768 0.030 0.731 0.026

4 -0.005 0.001 -0.006 0.001

0 2.144 0.137

F(47,764) =75.82 LM(1) = 4135

Hausman LM(4) = 905.1

http://web.pdx.edu/~crkl/WISE/SEAUG/examples/baltagi/munnell.1

http://web.pdx.edu/~crkl/WISE/SEAUG/examples/baltagi/munnell.2

Panel Data AnalysisAnother Example: China Provincial Productivity

Cobb-Douglass Production Function ln(GDP) = + ln(L) + ln(K) +

Fixed Effects s.e.

Random Effects s.e

0.30204 0.078 0.4925 0.078

0.04236 0.0178 0.0121 0.0176

2.6714 0.6254

F(29,298) = 158.81 LM(1) = 771.45

Hausman LM(2) = 48.4

References B. H. Baltagi, Econometric Analysis of Panel Data, 4th

ed., John Wiley, New York, 2008. W. H. Greene, Econometric Analysis, 6th ed., Chapter 9:

Models for Panel Data, Prentice Hall, 2008. C. Hsiao, Analysis of Panel Data, 2nd ed., Cambridge

University Press, 2003. J. M. Wooldridge, Econometric Analysis of Cross Section

and Panel Data, The MIT Press, 2002.

Spatial Econometric Analysis Using GAUSS 8 Kuan-Pin Lin Portland State University.

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