Econometric Analysis of Panel Data

• Fixed Effects and Random Effects: Extensions– Time-invariant Variables– Two-way Effects– Nested Random Effects

Time Invariant Variables

• The Model

• Fixed Effects

– 2 can not be identified, thus the individual effects ui can not be estimated.

' ' ' 11 2

2it it i it it i i ity u e u e

βx β x x

β

' '' '

' '

1 1 2 2( ) ( 1 1 ) 1 ( )

1 1 2 2it it i i it

it i it i it i

i i i i i

y u ey y e e

y u e

x β x βx x β

x β x β

Time Invariant Variables

• Fixed Effects: Two-Step Approach' '

'

2 '

' '

ˆˆ(1) 1 1 1 1

ˆˆ(2) 2 2 2

ˆˆ( ) 0, ( ) ( ) / 1 ( 1) 1

( )

ˆ ˆˆ 1 1 2 2

it it i it i i i FE

i i i OLS

i i i e i i i

i i i FE i OLS

y e y

w

E w Var w Var T Var

heteroscedasticity

u y

x β x β

x β β

x β x

x β x β

Time Invariant Variables

• Random Effects

– Mundlak’s Approach

• Estimate random effects model including group means:

' '

' 2

( , ) 0 1 1 2 2 ,

assume 1 1 2 2 , ( | ) 0, ( | )

i it it it i i it

i i i i i it i it w

If Cov u and y u e

u w E w Var w

x x β x β

x γ x γ x x

' , ( , ) 0it it i it i ity u e requires Cov u x β x

' '

'

1 1 1 1 2 2 ( : 2 2 2)

ˆˆ ˆ1 1

ˆ ˆ1 1 2 2

it it i i i it

i i i

i i

y w e Note

u w

y

x β x γ x δ δ β γ

x γ

x β x δ

Example: Returns to Schooling• Cornwell and Rupert Model (1988)

• Data (575 individuals over 7 ears)

– Dependent Variable yit:• LWAGE = log of wage

– Explanatory Variables xit:• Time-Variant Variables x1it:

– EXP = work experienceWKS = weeks workedOCC = occupation, 1 if blue collar, IND = 1 if manufacturing industrySOUTH = 1 if resides in southSMSA = 1 if resides in a city (SMSA)MS = 1 if marriedUNION = 1 if wage set by union contract

• Time-Invariant Variables x2i:– ED = years of education

FEM = 1 if femaleBLK = 1 if individual is black

' '1 1 2 2it it i i ity u e x β x β

Two Way Effects

• The Model

• Assumptions

' ( 1,2,..., )it it i t it iy u v e t T x β

2

2

2

( | ) 0

( | ) ( | ) 0 ( )

( | ) , ( , | , ) 0

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

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

it it

i it t it

it it e it js it js

i it u i j it jt i jt it jt

t it v t s it is t is it is

E e

E u E v random effects only

Var e Cov e e

Var u Cov u u Cov u e

Var v Cov v v Cov v e

x

x x

x x x

x x x x x

x x x x x ) 0

( , | ) 0i t itCov u v

x

Two-Way Effects

• Dummy Variable Representation

'

1 1

2 2

( 1,2,..., )

( 1,2,..., )

1 1,2,...,, ,

0

i i

i i

it it i t it i

i i i i T i i i i T i

i

i ii ij

T iT

y u v e t T

u u

i N

v d

v d if j Twhere d

otherwise

v d

x β

βy X d i e y Wδ i e

v

v d

Two-Way Effects

• Using one-way fixed effects or random effects model to estimate the dummy variable representation of two-way effects model.

( 1,2,..., )ii i i T iu i N

y Wδ i e

y Wδ u e

Two-Way Effects

• Two-Way Fixed Effects Model– Between Estimator

– Within Estimator (Group Means Deviations)'

' ' ' '

'

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

( ) ( )

it it i t it i

it i t it i t it i t

it it it

y u v e t T i N

y y y y e e e e

y e

x β

x x x x β

x β

'

'

'

i i i i

t t t t

y u v e

y u v e

y u v e

x β

x β

x β

Two-Way Effects

• Two-Way Fixed Effects Model– OLS

– Estimated Individual and Time Effects

'

' 1 ' 2 ' 1

2 '

ˆ ˆˆ ˆ( ) , ( ) ( )

ˆ ˆˆ / ( 1 )

ˆ ˆ

it it it i i i

OLS OLS e

e

y e

Var

NT N T K

x β y X β e y Xβ e

β XX Xy β XX

ee

e y Xβ

' '

' '

ˆˆ ( ) ( )

ˆ( ) ( )

i i i

t t t

u y y u

v y y v

x x β

x x β

Two-Way Effects

• Two-Way Random Effects Model– Partial Group Means Deviations

'

' ' ' '

2 2

2 2 2 2

2

2 2 2

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

( ) ( )

1 , 1 ,

1

it it i t it i

it i i t t it

it i i t t it it i i t t it

e ei t

e i u e t v

eit i t

e i u t v

y u v e t T i N

y y y y

e e e e

whereT N

T N

x β

x x x x β

Two-Way Effects

• Two-Way Random Effects Model– Consistent estimates of s are derived from:

• e2 asym. var. of two-way fixed effects model

• u2 asym. var. of between (individual) effects model or

one-way fixed (individual) effects model• v

2 asym. var. of between (time) effects model or one-way fixed (time) effects model

– For improved efficiency, iterate the consistent estimation until convergence.

Nested Random Effects• Three-Level Model

• Assumptions– Each successive component of error term is imbedded or

nested within the preceding component

• Model Estimation– GLS, ML, etc.

' ' ( )

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

ijt ijt ijt ijt i ij ijt

i ij

y u w e

i M j N t T

x β x β

2 2 2

( | ) ( | ) ( | ) 0

( | ) , ( | ) , ( | )

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

ijt ij i

ijt e ij w i u

ijt ij ijt i ij i

E e E w E u

Var e Var w Var u

Cov e w Cov e u Cov w u

X X X

X X X

X X X

Example: U. S. Productivity

• The Model (Munnell [1988]) – Two-level model

– Three-level model

– See, B.H. Baltagi, S.H. Song, and B.C. Jung, The Unbalanced Nested Error Component Regression Model, Journal of Econometrics, 101, 2001, 357-381.

0 1 2 3

4 5 6

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

ln( ) ln( ) ln( )

ijt ijt ijt ijt

ijt ijt ijt i ij ijt

gsp cap hwy water

util emp unemp u w e

0 1 2 3

4 5 6

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

ln( ) ln( ) ln( )

jt jt jt jt

jt jt jt j jt

gsp cap hwy water

util emp unemp u e

Example: U. S. Productivity

• Description– i=1,…,9 regions; j=Ni states

• 6. Gulf: AL, FL, LA, MO• Mid West: IL, IN, KY, MI, MN, OH, WI• Mid Atlantic: DE, MD, NJ, NY, PA, VA• 8. Mountain: CO, ID, MT, ND, SD, WY• 1. New England: CD, ME, MA, NH, RI, VT• South: GA, NC, SC, TN, WV• 7. Southwest: AZ, NV, NM, TX, UT• Tornado Alley: AK, IA, KS, MS, NE, OK• 9. West: CA, OR, WA

– t=1970-1986 (17 years)

Example: U. S. Productivity

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

• STATE = State name, • ST_ABB = State abbreviation (Region = 1, . . . , 9),• 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