Maximum Likelihood Estimation - Econometric Analysis of Panel Data - Lecture Slides, Slides of Econometrics and Mathematical Economics

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Econometric Analysis of Panel Data

6. Maximum Likelihood Estimation of

the Random Effects Linear Model

The Random Effects Model

 The random effects model

 c

i

is uncorrelated with x

it

for all t;

 E[c

i

| X

i

] = 0

 E[ε

it

| X

i

,c

i

]=

it it i it

i i i i i

i i i i i i i

N

i=1 i

1 2 N

y = +c +ε , observation for person i at time t

= +c + , T observations in group i

= + + , note (c , c ,...,c )

= + + , T observations in the sample

c=( , ,... ) ,

x β

y X β i ε

X β c ε c

y Xβ c ε

c c c

N

i=1 i

Σ T by 1 vector

Notation

2 2 2 2

u u u

2 2 2 2

u u u

i i

2 2 2 2

u u u

2 2

u i i

2 2

u

i

1

2

N

Var[ +u ]

= T T

=

=

Var[ | ]

 

 

 

=

 

 

 

 

′

• ×

′







=







i

i

T

T

ε i

I ii

I ii

Ω

Ω 0 0

0 Ω 0

w X

0 0 Ω





   







   



ε

ε

ε

ε

ε

σ σ σ σ

σ σ σ σ

σ σ σ σ

σ σ

σ σ

i

(Note these differ only

in the dimension T )











 



Maximum Likelihood

i

it i

i1 i2 iT i i

i i

2 2

u

Assuming normality of and u.

Treat T joint observations on [( , ,... ),u ] as one T

variate observation. The mean vector of u is zero

and the covariance matrix is = I.

logL=

ε

ε

ε ε ε

′ σ + σ

i

ε i

Ω ii

N

i=1 i

2 2

i u i

i

logL where

logL ( , ) = T log 2 log | | ( ) ( )

2

= T log 2 log | |

2

ε

Σ

  ′ σ σ π + +

 

 ′  π + +

 

-

i i i i i i

-

i i i i

β, Ω y - X β Ω y - X β

Ω ε Ω ε

Panel Data Algebra (3, cont.)

i

2 2 2 2 2

u

T 2

t 1

2 2

= = [ ]=

| |=( ) , = a characteristic root of

Roots are (real) solutions to =

= = + or ( ) ( 1)

Any vector whose elements sum to zero (

ε ε ε

ε

=

′ ′ σ + σ σ + ρ σ

σ λ

λ

′ ′ λ ρ ρ λ

∏

i

i

T

i t

Ω I ii I ii A

Ω A

Ac c

Ac c c ii c i i c = - c

i i

2 2

i

=0)

is a characteristic vector that corresponds to root = 1.

There are T -1 such vectors, so T - 1 of the roots are 1.

Suppose 0. Premultiply by to find

( ) ( 1) = T ( )

′

λ

′ ′ ≠

′ ′ ′ ′ ρ λ ρ

i c

i c i

i i i c = - i c i c

i

i i

2

i

T

2 T 2 T 2

t i

t 1

=( 1). Since 0,

divide by it to obtain the remaining root =1+T.

Therefore, | |=( ) ( ) (1 T )

ε ε

=

′ ′ λ ≠

λ ρ

σ λ = σ + ρ

∏ i

- i c i c

Ω

Panel Data Algebra (3, conc.)

i i

2 2

2 2 i i

i i i i i 2 2 2

i u

N

i 1 i

2 N N 2 N

i 1 i i 1 i i 1 i 2

logL T log 2 log | |

2

(T ) -1 1

T log 2 T log log(1 T )

2 T

logL logL

-1 1

[(log 2 log ) T + log(1 T )]

2 2

ε

ε

ε ε

=

ε = = =

ε

 ′ 

= π + +

 

    σ ε

 

′ = π + σ + + ρ + −

   

σ σ + σ

   

 

= Σ

′ = π + σ Σ Σ + ρ − Σ

σ

-

i i i i

Ω ε Ω ε

ε ε

ε ε

2 2

i i

i 2 2

i u

(T )

T

ε

ε

 

σ ε

−

 

σ + σ

 

Direct Maximization of LogL

2 2 2

u i i i i

2

i i i i i i i

Simpler : Take advantage of the invariance of maximum

likelihood estimators to transformations of the parameters.

Let =1/ , = / , R T 1, Q / R ,

logL (1 / 2)[ ( Q (T ) ) logR T log T l

ε ε

θ σ τ σ σ = τ + = τ

′ = − θ − ε + + θ +

i i

ε ε og 2 ]

Can be maximized using ordinary optimization methods (not

Newton, as suggested by Hsiao). Treat as a standard nonlinear

optimization problem. Solve with iterative, gradient methods.

π

Application – ML vs. FGLS

Simulated Likelihood Function

i

T 2 N 2 2 2

u i 1 t 1 it u i

i

i

The conditional log likelihood for the sample is then

logL( , , | ) ( 1 / 2)[log 2 log (y - - v ) ]

The unconditional log likelihood is obtained by integrating v out of

L ( ,

ε = = ε ε

′ σ θ = Σ Σ − π − θ + θ σ

it

β v x β

β

i

i

2 2

u i i u

T 2 2 2

t 1 it u i i i

2

v i u i

, | v ); logL ( , , )

( 1 / 2)[log 2 log (y - - v ) ] (v )dv

E logL ( , , | v )

The integral usually does not have a closed form. (For the normal distribution

ab

ε ε

∞

= ε ε

−∞

ε

σ θ σ θ =

′ Σ − π − θ + θ σ φ

= σ θ

∫

it

β

x β

β

ove, actually, it does. We used that earlier. We ignore that for now.)

Computing the Expected LogL

i

i

2

i i

T 2 2 2

t 1 it u i i i

2

v i u i

How to compute the integral: First note, (v ) exp( v / 2) / 2

( 1 / 2)[log 2 log (y - - v ) ] (v )dv

E logL ( , , | v )

(1) Numerical (Gauss-Hermite) quadratur

∞

= ε ε

−∞

ε

φ = − π

Σ − π − θ + θ σ φ

= σ θ

it

x β

β

2 H

v

h h

h 1

e for integrals of this form is

remarkably accurate;

e g(v)dv w g(a )

∞

−

=

−∞

Example: Hermite Quadrature Nodes and Weights, H=

Nodes: -2.02018,-0.95857, 0.00000, 0.95857, 2.

Weights: 1.99532,0.39362, 0.94531, 0.39362, 1.

Applications usually use many more points, up to 96 and

Much more accurate (more digits) representations.

Gauss-Hermite Quadrature

i

i

T 2 2 2

t 1 it u i i i

2

i i

i i i i i i

2 T 2 2

i t 1 it u

( 1 / 2)[log 2 log (y - - v ) ] (v )dv

(v ) exp( v / 2) / 2

Make a change of variable to a v / 2 ,v = 2 a , dv = 2 da

exp( a ) [log 2 log (y - - 2

∞

= ε ε

−∞

∞

= ε ε

−∞

Σ − π − θ + θ σ φ

φ = − π

− Σ π − θ + θ σ

π

it

it

x β

x β

i

i

2

i i

2 T 2 2 2

i t 1 it u i i

2 T 2 2 2

i t 1 it u i i

2 H

i i i h 1 h h

a ) ] 2 da

exp( a ) [log 2 log (y - - ( 2 a )) ] da

exp( a ) [log 2 log (y - - ( 2 a )) ] da

exp( a )g a da w g(a )

∞

= ε ε

−∞

∞

= ε ε

−∞

∞

=

−∞

− Σ π − θ + θ σ

π

− Σ π − θ + θ σ

π

π π

it

it

x β

x β

Simulation

i

i

2

i u

T 2 2 2

t 1 it u i i i

2

v i u i v i

The unconditional log likelihood is an expected value;

logL ( , , )

= ( 1 / 2)[log 2 log (y - - v ) ] (v )dv

E logL ( , , | v ) = E g(v )

An expected value can be

ε

∞

= ε ε

−∞

ε

σ θ

′ Σ − π − θ + θ σ φ

= σ θ

it

β

x β

β

{ }

i

i

R

T 2 2 2

v i t 1 it u i,r

r 1

T 2

t 1

'estimated' by sampling observations

and averaging them

1

ˆ

E g(v ) ( 1 / 2)[log 2 log (y - - v ) ]

R

The unconditional log likelihood function is then

1

( 1 / 2)[log 2 log

R

= ε ε

=

= ε

′ = Σ − π − θ + θ σ

Σ − π − θ

∑ it

x β

{ }

N R

2 2

it u i,r

i 1 r 1

2

u i,1 i,R

i,r

(y - - v ) ]

This is a function of ( , | , , v ,..., v ),i 1,...,N

The random draws on v become part of the data, and the function

is maximized with respect to the

ε

= =

ε

′

• θ σ

θ σ =

it

i i

x β

β, y X

unknown parameters.

MSL vs. ML - Application

Unbalanced Panel Data

t=1 t=2 t=3 t=

i=1 ● ●

i=2 ● ●

i=3 ● ● ● ●

i=4 ● ●

i=5 ● ●

i=6 ● ● ●

i

T

i t 1 it

z. z

t

N

t i 1 it

z. z