3.2.2 Random E ff ect Model (

(1)

3.2.2 Random E ff ect Model (

ランダム効果モデル

) Model:

y

_it

= X

_it

β + v

_i

+ u

_it

, i = 1 , 2 , · · · , n , t = 1 , 2 , · · · , T where i indicates individual and t denotes time.

The assumptions on the error terms v

_i

and u

_it

are:

E(v

_i

| X) = E(u

_it

| X) = 0 for all i ,

V(v

_i

| X) = σ

²v

for all i , V(u

_it

| X) = σ

²u

for all i and t ,

Cov(v

_i

, v

_j

| X) = 0 for i , j , Cov(u

_it

, u

_js

| X) = 0 for i , j and t , s , Cov(v

_i

, u

_jt

| X) = 0 for all i, j and t .

Note that X includes X

_it

for i = 1 , 2 , · · · , n and t = 1 , 2 , · · · , T .

(2)

In a matrix form with respect to t = 1 , 2 , · · · , T , we have the following:

y

_i

= X

_i

β + v

_i

1

_T

+ u

_i

, i = 1 , 2 , · · · , n ,

where y

_i

=

 





y

_i1

y

_i2

...

y

_iT

 



 , X

_i

=

 





X

_i1

X

_i2

...

X

_iT

 



 and u

_i

=

 





u

_i1

u

_i2

...

u

_iT

 



 are T × 1, T × k and T × 1, respectively.

u

_i

∼ N(0 , σ

²_u

I

_T

) and v

_i

1

_T

∼ N(0 , σ

²_v

) = ⇒ v

_i

1

_T

+ u

_i

∼ N(0 , σ

²_v

1

_T

1

⁰_T

+ σ

²_u

I

_T

) . Again, in a matrix form with respect to i, we have the following:

y = X β + v + u ,

where y =

 





y

₁

y

2

...

 



 , X =

 





X

₁

X

2

...

 



 , v =

 





v

₁

1

_T

v

2

1

T

...

 



 and u =

 





u

₁

u

2

...

 



 are nT × 1, nT × k, nT × 1 and

(3)

nT × 1, respectively.

The distribution of u + v is given by:

v + u ∼ N (

0 , I

_n

⊗ ( σ

²_v

1

_T

1

⁰_T

+ σ

²_u

I

_T

) ) The likelihood function is given by:

L( β, σ

²v

, σ

²u

) = (2 π )

^−nT/2

I

_n

⊗ ( σ

²v

1

_T

1

⁰_T

+ σ

²u

I

_T

)

⁻¹^/²

× exp (

− 1

2 (y − X β )

⁰

(

I

_n

⊗ ( σ

²v

1

_T

1

⁰_T

+ σ

²u

I

_T

) )

₋₁

(y − X β ) ) . Remember that f (x) = (2 π )

⁻^k^/²

|Σ|

⁻¹^/²

exp (

−

¹₂

(x − µ )

⁰

Σ

⁻¹

(x − µ ) )

when X ∼ N( µ, Σ ),

where X denotes a k-variate random variable.

(4)

The estimators of β , σ

²_v

and σ

²_u

are given by maximizing the following log-likelihood function:

log L( β, σ

²_v

, σ

²_u

) = − nT

2 log(2 π ) − 1

2 log I

_n

⊗ ( σ

²_v

1

_T

1

⁰_T

+ σ

²_u

I

_T

)

− 1

2 (y − X β )

⁰

(

I

n

⊗ ( σ

²v

1

T

1

⁰_T

+ σ

²u

I

T

) )

₋₁

(y − X β ) . MLE of β , denoted by ˜ β , is given by:

β ˜ = ( X

⁰

(

I

_n

⊗ ( σ

²v

1

_T

1

⁰_T

+ σ

²u

I

_T

) )

₋₁

X )

₋₁

X

⁰

(

I

_n

⊗ ( σ

²v

1

_T

1

⁰_T

+ σ

²u

I

_T

) )

₋₁

y

= ( ∑

ⁿ

i=1

X

_i⁰

( σ

²v

1

_T

1

⁰_T

+ σ

²u

I

_T

)

⁻¹

X

_i

)

₋1

( ∑

ⁿ

i=1

X

_i⁰

( σ

²v

1

_T

1

⁰_T

+ σ

²u

I

_T

)

⁻¹

y

_i

) , which is equivalent to GLS.

Note that ˜ β is not operational, because ˆ β depends on σ

²_v

and σ

²_u

.

(5)

3.3 Hausman’s Specification Error (

^{特定化誤差}

) Test

Regression model:

y = X β + u , y : n × 1 , X : n × k , β : k × 1 , u : n × 1 . Suppose that X is stochastic.

If E(u | X) = 0, OLSE ˆ β is unbiased because of ˆ β = (X

⁰

X)

⁻¹

X

⁰

y = β + (X

⁰

X)

⁻¹

X

⁰

u and E((X

⁰

X)

⁻¹

X

⁰

u) = 0.

However, If E(u | X) , 0, OLSE ˆ β is biased and inconsistent.

Therefore, we need to check if X is correlated with u or not.

= ⇒ Hausman’s Specification Error Test

(6)

The null and alternative hypotheses are:

• H

₀

: X and u are independent, i.e., Cov(X , u) = 0,

• H

1

: X and u are not independent.

Suppose that we have two estimators ˆ β

0

and ˆ β

1

, which have the following properties:

• β ˆ

0

is consistent and e ffi cient under H

₀

, but is not consistent under H

₁

,

• β ˆ

1

is consistent under both H

₀

and H

₁

, but is not e ffi cient under H

₀

. Under the conditions above, we have the following test statistic:

( ˆ β

1

− β ˆ

0

)

⁰

(

V( ˆ β

1

) − V( ˆ β

0

) )

₋1

( ˆ β

1

− β ˆ

0

) −→ χ

²

(k) . Example: β ˆ

0

is OLS, while ˆ β

1

is IV such as 2SLS.

Hausman, J.A. (1978) “Specification Tests in Econometrics,” Econometrica, Vol.46,

(7)

3.4 Choice of Fixed E ff ect Model or Random E ff ect Model

3.4.1 The Case where X is Correlated with u — Review

The standard regression model is given by:

y = X β + u , u ∼ N(0 , σ

²

I

_n

) OLS is:

β ˆ = (X

⁰

X)

⁻¹

X

⁰

y = β + (X

⁰

X)

⁻¹

X

⁰

u .

If X is not correlated with u, i.e., E(X

⁰

u) = 0, we have the result: E( ˆ β ) = β . However, if X is correlated with u, i.e., E(X

⁰

u) , 0, we have the result: E( ˆ β ) , β .

= ⇒ β ˆ is biased.

(8)

Assume that in the limit we have the followings:

( 1

n X

⁰

X)

⁻¹

−→ M

⁻_xx¹

, 1

n X

⁰

u −→ M

_xu

, 0 when X is correlated with u.

Therefore, even in the limit,

plim ˆ β = β + M

⁻_xx¹

M

_xu

, β, which implies that ˆ β is not a consistent estimator of β .

Thus, in the case where X is correlated with u, OLSE ˆ β is neither unbiased nor con-

sistent.

(9)

3.4.2 Fixed E ff ect Model or Random E ff ect Model

Usually, in the random e ff ect model, we can consider that v

_i

is correlated with X

_it

. [Reason:]

v

_i

includes the unobserved variables in the ith individual, i.e., ability, intelligence, and so on.

X

_it

represents the observed variables in the ith individual, i.e., income, assets, and so on.

The unobserved variables v

_i

are related to the observed variables X

_it

. Therefore, we consider that v

_i

is correlated with X

_it

.

Thus, in the case of the random e ff ect model, usually we cannot use OLS or GLS.

In order to use the random e ff ect model, we need to test whether v

_i

is uncorrelated

with X

_it

.

(10)

Apply Hausman’s test.

• H

0

: X

it

and e

it

are independent ( −→ Use the random e ff ect model),

• H

₁

: X

_it

and e

_it

are not independent ( −→ Use the fixed e ff ect model), where e

it

= v

i

+ u

it

.

Note that:

• We can use the random e ff ect model under H

₀

, but not under H

₁

.

• We can use the fixed e ff ect model under both H

₀

and H

₁

.

• The random e ff ect model is more e ffi cient than the fixed e ff ect model under H