Chapter 10_Models for Panel Data

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Chapter 10

MODELS FOR PANEL DATA

Panel data (longitudinal data): same entities are observed overtime

The basic framework for panel data is a regression model of the form:

(1) ∑

+ +

= k

j

it l l i itj j

Y

1 ( 1 )( 1 )

ε α

it l l i k k

it z

X β + α +ε

=

×

×

×

× )( 1 ) ( 1 )( 1 )

1

(

Where: z i =[1 z i2 z il]

1 2

l

α α α α

 

 

 

=

 

 

 

[ 1 2 ]

it it it itk

X = X X X

There are k regressors in X , not including a constant term it

The heterogeneity or individual effect is (z iα) where z contains a constant term and a i

set of individual or group specific variable, which may be observed, such as race, sex, location,… or unobserved, such as family specific characteristics, individual

heterogeneity in skill or preferences,… All of which are taken to be constant over time t

Therefore: z i =[1 z i2 z il] = constant over time t

If z is observed for all cross-sections (individuals), then the entire model can be treated i

as an ordinary linear model and fit by least squares

When z is not observed (most of the cases), complications arise that leads to main i

objective of the analysis will be consistent and efficient estimation of the partial effects

it it it

E Y X X

∂

Assumption of strict exogeneity: E(εit X i1,X i2, ,X iT)= 0

That is, the current disturbance is uncorrelated with the independent variable in every

period of t

Assumption of mean independence:

i i i iT

l

1 1α

×

=

Or = (fixed effect) = h X( i)= αi

If z i =[1 z i2 z il] =1, which contains only a constant term Then OLS provides consistent and efficient estimates of the intercept α and the slope vector (common

effect model)

Thus equation (1) → Y it it

l l i k k

it z

X β + α +ε

=

×

×

×

× )( 1 ) ( 1 )( 1 ) 1

(

Where X it obs of all k explanatory variables: it



1

( 1)

2

( 1)

( 1)

T

N

T

NT

Y

Y

Y

×

×

×



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

T

k

N

T k NT

X X X

β

×

×

×

×





( 1)

(1 1) ( 1) ( 1)

T

T NT

i i

i

α

×

×

×

×

 

 

  +  

 

 

 





1 ( 1) 2

( 1) ( 1)

T

N T NT

ε ε ε

×

×

×



( 1)

1 1

1

T i

×

 

 

 

=

 

 

 

st country: Y1 =X1β α ε+i + 1

with classical assumptions: 2

( ) 0 ( ) ( , ) 0

it it

it js

E Var Cov

ε

ε

ε ε

=







with (∀ ≠i j) or (t ≠ s)

III FIXED EFFECTS

1 1

( i )

l

E z α

×

1 1

i

α

× = h X ( i) Different across units can be captured in differences in the constant term:

it it i it

Y = X β +zα ε+

it it i it i i

White noise υi =(z iα α− i)=z iα−h X( i)

Because z iα−h X( i)is uncorrelated with Xi, we may absorb it in the disturbance:

i it i

it it i it

Each αiis treated as an unknown parameter to be estimated:

(NT Y1)

×

1 ( 1) (1 1) 2

T

N

i i X

i

α α

α

× ×



1 (1 1) 2

( 1)

N

NT

NT N D

N

α α

α

×

×

×

 

 

 

 

 

 



 

1

1 1

1 0 0

0 0 0

0

d

 

 

 

 

 

 

 

 

 

=  

 

 

 

 

 

 

 

 







0 0

0 1 1

1 0 0

0

i

d

 

 

 

 

 

 

 

 

 

=  

 

 

 

 

 

 

 

 







1

2

( 1)

N

NT

Y

Y

Y

×

 

 

 

 

 

 

 =



1 2

N

NT k

X X

X

×

 

 

 

 

 

 

 (k 1)

β

×

0 0

NT N

i i

i

×

+





   



1 ( 1) 2

( 1) ( 1)

T

N T N

α α α

×

×

×





1 ( 1) 2

( 1) ( 1)

T

N T NT

ε ε ε

×

×

×

(NT Y1)

× (NT k X )(kβ1) (NT N D ) (Nα1) (NTε1)

This model is also called “Least Squares Dummy Variables” (LSDV) because it can be estimated directly with the intercept dummies

There are 3 ways to estimate the pooled regression model:

it it it

Y = +α X β ε+

Using OLS:

( 1)

ˆ

OLS

k

β

×

1

1 1

N T

it it

i t

X X X X

−

= =

1 1

N T

it it

i t

X X Y Y

= =

Note:

(1 )

( it )

k

X X

×

( 1)

( it ) '

k

X X

×

−

(1X k)

× and

(k Y1)

× are overall means:

1

it

NT

1

it

k Y Y k

NT

Using the deviation from the group means

( 1)

k

Y Y α X X β ε ε

×

We get the Within – Group estimator:

ˆβLSDV

( 1)

ˆ

within k

β

×

=

1

1 1

N T

it i it i

i t

X X X X

−

= =

1 1

N T

it i it i

i t

X X Y Y

= =

This is also the LSDV or fixed effect estimator of

Where

.

.

1

1

T

k t T

t

X X T

Y Y T



=







=



∑

∑ group means (i = 1, 2, , N)

We can write the model in terms of the group means:

Y = +α X β ε+ (i = 1, 2, , N)

We use only N observations, (the group means) Apply OLS to N observations, we

get the Between – Group estimator:

ˆβBetween

1

1

N

i

X X X X

−

=

1

N

i

X X Y Y

=

Back to the fixed effects:

X M X − X M Y

=

where M D = −I D D D( ' )−1D'

D

M =

0 0

0

M M

M







'

T

T

= −

[ ]1 2

( FE) ' D

VarCov β =σ X M X −

ˆ Y M X D Y M X D

− − Note: Why pooled estimator, within group estimator and between group estimator are different?

Because:

• Pooled estimator:

2 ˆ

1 1

pooled

N T it

i t

Min e

• Within-group estimator:

2 ˆ

1 1

within

N T

it i

i t

Min e e

• Between-group estimator:

2 ˆ

1 1

( )

between

N T i

i t

Min e

There are 3 different minimum problems, e it are the same from:

it it it

Y = +α X β + e

ˆ

it it it

e =Y − −α X β

Note that in deviation form: ( ) ( ) ˆ

are the same

The LSDV approach can be extended to include a time specific effect as well:

it it i t it

Y X d d d g g g

 

(one of the time effects must be dropped to avoid perfect co linearity – the group effects and time effects both sum to one)

1

0

j any t

it

d

if not

=



= 



1

0

s any i it

g

if not

=



= 



For panel data now we can use

• Pooled

• Fixed effects

o Time effects only

o Group effects only

o Time and group effect

ˆ

LSDV

FE

β

=

( 1)

ˆ

within k

β

×

=

1

( ) '( )

i

T N

it i it i

i i

−

= =

1

( ) '( )

i

T N

it i it i

i i

−

= =

, 1

it i i t u it

( it) 0

E u =

( it is) 0

E u u = for t≠s (no temporal auto in u it)

( it jt) 0

E u u = for i≠ j (no spatial auto)

( it js) 0

E u u = for i≠ and j t≠s

( it)

E u =σ i = 1, 2, , N (spatial heteroskedasticity but no temporal auto)

Estimation:

• Estimate (1) by OLS → get e it's

2 1 2 ˆ

T

it it t

T it t

e e e

−

=

∑ for i = 1, 2, , N (use all NT observations)

or

1

1

n i i

N

=

= ∑ if T is small

• Use ˆρi's to quasi-difference (1):

it

it i it it i it i it i it

u

≈



Estimate by OLS with N(T-1) observations → ˆu it

•

2

ˆ ˆ

1

i

T it t u

u

− −

∑

or

2

ˆ ˆ

1

i

T it t u

u T

−

∑

• From WLS:

1

Y X β α ρ u

The standard F test can be used to test whether the pooled or fixed-effect model is more appropriate:

H0: C1 =C2 ==C N =α *

i i

C =zα

( ) ( )

2

( 1)

1, 1

LSDV pooled LSDV U

N

−

−

−

− −



The coefficients on the time-invariant variables cannot be estimated by within estimators

*

it it i it

Y = X β +zα +ε

Denote: z iα* =c i

Fixed effects assume c i are correlated with X i

( i i)

E c X = h X =( i)

1 1

i

α

×

Random effects assume c i are uncorrelated with X i

( i i)

E c X = α= constant

then Y it = X itβ+(α+u i)+ εit

= +α X itβ+(u i+εit)

(c i and X i are not correlated)

The following assumptions are made:

( it ) 0

NT k

E ε X

× =

( it ( )) 0

NT k

E u X

× =

( it)

( it) u

( it js) 0

E ε ε = for i≠ or j t≠s or both

( i j) 0

( i js) 0

E uε = for all i,j,s

Let: ηit = + u i εit

( it) ( i it) u

E η =E u +ε =σ +σε

( it jt) ( i it)( j jt) 0

( it is) ( i it)( i is) u

(NT Y1) Xβ (NTη1)

× = + ×

1 2 ( 1)

( 1)

( 1)

NT

N T

NT

η

η

η

η

×

×

×

=  



'

00 ( ) ( )

( )

i j

T T

T T

E

ε

ε

ε

ηη

×

×

+







'

( ) ( )

( i j) 0

T T

T T

E ηη

×

×

=

So:

'

( i j)

NT NT

NT NT

E ηη

00 00

00

Σ







We can estimate ˆβRE

by GLS estimation:

( 1)

ˆ

RE

k

β

×

(X' − X) (− X' −Y)

(X X ) (− X Y )

(PX) '(PX) − (PX) '(PY)

=

1 '

P P= Σ− → 1/2

00

N

P=I ⊗ Σ−

1/2 00

1/2 00

1/2 00

−

−

−

=

Σ







00

1

T T T

T

ε

θ σ

1

u

T

ε ε

σ θ

= −

+

*

.

1

i i

i i i

iT i

Y Y

Y Y Y

Y Y

ε

θ θ σ

θ

−

=

 and the same for X *i

2 Feasible Estimation:

We need to estimated σ and ε2 2

u

σ

Estimation of σ : ε2

it it i it

Y = +α X β + u +ε

Estimate (*) using OLS and use the residuals to get the estimation of σ ε2

2

( ) ˆ

N T

it i

i t

ε

−

=

− −

∑∑

Estimation of σ : u2

*

( i i) i i

e

u +ε =Y − −α X β



. ( i i)

Var u +ε = 2 2

u

T

ε

σ

Estimation of Var u( i+εi.) is

'

e e

ε

σ

−

− Insert 2

ε

u

Whether c i =z iα* are correlated with (Xi) or not

If they are → RE will produce inconsistent estimates

If they are not → RE model may be preferable

2 ( 1)

within RE within RE within RE k

W− = β β ′ Var− β Var β −− β β χ −

H0: no correlation between Ci and X i

HA: correlation between Ci and X i

Under H0: Both ˆβFEand ˆβREare consistent estimators but only ˆβREis efficient

Example 1:

, 1

it i i t u it

( it) 0

E u = 2 2

( )

i

it u

( it jt) 0

( it is) 0

E ε ε = for t≠s (no temporal auto)

( it js) 0

E ε ε = for t≠s (no temporal auto)



1

( 1)

2

( 1)

( 1)

T

N

T

NT

ε

ε

ε

ε

×

×

×



( ')

E εε

Σ =

off-diagonal blocks t≠s:

( 1) ( 1)

( i j' ) 0

T T

× ×

=

diagonal blocks

( 1) ( 1)

( i i' )

T T

E ε ε

× ×

2 ( )

1 1

1

i

T

T

T T

ε

−

−

×







( )

NT NT

E εε

×

′

1

2

2 1 2 2

2

N N

ε

ε

ε

σ

σ

σ

Ω







with

2 2

2

ˆ ˆ

ˆ 1

i i

u i

ε

σ σ

ρ

=

−

Example 2:

Now assume no temporal autocorrelation

But allow spatial autocorrelation and cross-section heteroskedasticity

( it) 0

E u = 2 2

( )

i

it u

( it jt) ij 0

E ε ε =σ ≠ spatial autocorrelation

( it is) 0

E ε ε = for t≠s (no temporal auto)

( it js) 0

E ε ε = for t≠s (no temporal auto)

( ')

E εε

Σ =



1

( 1)

2

( 1)

( 1)

T

N

T

NT

ε

ε

ε

ε

×

×

×



Same country:

( )

NT NT

E εε

×

′

ii ii

ii ii

I

σ σ

σ σ

==







Different country i≠ : j

( 1) ( 1)

( i j' )

T T

× ×

ij ij

ij ij

I

σ σ

σ σ

==







( )

NT NT

E εε

×

′

N N

I I I

I I I

I I I

=





