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Chapter 4 Random effect model (REM)
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Objectives
(1) Introduce about Random Effect Model
(2) Estimates the slope paramaters in FEM by Within Estimator, Between
Estimator
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(3) Estimates FEM by Least Square Dummy Variables (LSDV) method
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Notes. There are too many parameters in the fixed effects model and the
i can be assumed
loss of degrees of freedom can be avoided if the α*
random
* + u it i = 1,N; t = 1,T (4.1)
4.1 Introduce Random effect model yit = x itb + a i
Here,
i is assumed to be random
α*
i are supposed to have non zero mean, with
If the individual effects α*
i)= α0
E (α*
i= α0 + αi
Then we can define cross section units effects α*
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Pre Eq. (4.1)
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2
( ,V a i
( ,E a ix it
) = 0,
( ) = E a i
) = sm 2 ,
( ,E a ia j
) = 0
( ,V u it
) = s u 2 ,
) = 0 for i ¹ j and t ¹ s
4.1.1 The assumptions on the components of errors About a i, ) = 0, ( E a i u it , About ( 2( ) = E u it ) = 0, ( ,E u itu js E u it The components of the error are not correlated
E (αiuit) =0
Remark. The αi are independent of the error term uit and the regressors
xit, for all i and t
2
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2 + s u
4.1.2 Mean and variance of errors
( ,V eit
) = sa
) = 0,
The mean and variance of the component errors are ( ) = V yit ( E eit
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The covariance of the composite error,
Cov (εit, εjs ) = E(εitεjs)= E(αi+ uit) (αj+ ujs)
= E (αiαj + uit αj + αi ujs + uitujs)
Or
α + σ2
u
i = j, t= s
α
i = j, t ≠ s Case 1. Cov (εit, εjs ) = σ2 Case 2. Cov (εit, εjs ) = σ2
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i ≠ j, t ≠ s Case 3. Cov (εit, εjs ) = 0
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'
... eiT
ei2
ei1
( E eiei
) = E
For cross section unit i, Eq. (4.1) can be written as
(
)
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ei1 ei2 ... eiT
ö ÷ ÷ ÷ ÷ ÷ ø
æ ç ç ç ç ç è
The variance- covariance matrix of εi (for individual i) is ö æ ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç è ø
2
...
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2
= E
ei1 ei2ei1 ... eiTei1
ei1ei2 ei2 ... eiTei2
ei1eiT ... ei2eiT ... ... 2 ... eiT
æ ç ç ç ç ç è
ö ÷ ÷ ÷ ÷ ÷ ø
2
2
2
...
s a
2
2
2
...
s a s a
=
2
2 + s u
2 + s u sa ... 2 sa
sa 2 + s u sa ... 2 sa
... ... sa
æ ç ç ç ç ç è
ö ÷ ÷ ÷ ÷ ÷ ø
ee'
ee'
1 T
æ 2 Q + èç
ö 2 1 ø÷ + Tsa T
2
(4.2)
= s u
2 + Tsa
2IT + sa = U = s u ( 2Q + s u
2 ee' = s u )P
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here P =
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e' = IT - Q
(4.3)
1 T (4.2) Þ U -1 =
)-1 )
2
(4.4)
where q =
2
)
(
P (4.5)
Therefore, U -1/2 =
Q +
2
ee' = e e'e( 1 ( 2 Q + qP s u s u 2 + Ts a s u 1 s u
1 2 + Tsa s u
(
)
(4.6)
Q + P
Or, U -1/2 =
2
1 s u
ö ÷ ÷ ø
2
) (4.7)
And U = s u
( 2 + Tsa
æ ç ç è ( 2(T-1) s u
1 2 + Tsa s u )
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By taking all cross section units in the sample, the variance - covariance
matrix of the error term (ε) will be of order NT x NT
) = E
( E ee'
æ ç ç ç ç è
ö ÷ ÷ ÷ ÷ ø
( 2 J T Ä IN
) = W (4.8)
0 U 0 ... 0 0 U ... ... ... ... ... 0 U 0 0 ) + sa ( 2 IT Ä I N = U Ä I N = s u where J T = ee'
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4.2 GLS estimation
Idea. The generalised least squares (GLS) is used in estimating a
random effects model when U is known.
Suppose that the variance – covariance matrix (U) is known
Pre multiply Eq. (4.1) by U-1/2 to get
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Homework. Proving why reason Eq. (4.10) equal with IT
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With Eq. (4.9) we can apply OLS to estimate parameter (γ)
The GLS estimators of γ are
B
B
(4.13)
W + qSXX
)-1
(
(
)
W + qSXy Eq. (4.12) = SXX SXy Homework. Expanding detail (4.12) to finding why (4.12) can similar (4.13)
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Eq. (4.12) can be written in expanded form as
68 If θ = 1, then GLS estimator is equivalent to OLS pooled estimator.
Remark.
-
If θ = 0, then GLS estimator will be equal to LSDV -
- The parameter θ measures the weight given to between-group
variation.
- If U is unknown, we can use a two-step GLS estimation known with
name is called FGLS (Feasible Generalized Least Squares)
4.3 FGLS estimator
α & σ2
u are unidentifed. We
Note. When U is unknown as means as σ2
can use two-step GLS estimation known as FGLS
Step 1. We estimate the “within” estimation and “between” estimation
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model to find out
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α & σ2
u are obtained from the ”between ” effect estimation, the
The σ2
“within” effect estimation, etc.
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Then, we have to caculate
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Step 2. We have to estimate the following model:
4.4 Testing of Hypotheses
Introduction. In a panel regression model, either fixed or random effect
is an issue of unobserved variables measuring heterogeneity across the
entities which renders the bias in pooled regression estimation.
4.4.1 Measuring of Goodness Fit
W) ,
Panel data can be utilities to calculate within-entity variation (R2
B ) and overall variation (R2).
between-entity variation (R2
Option 1. Testing for Pooled Regression
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yi =α+ Xiβ+ εi. (i = 1, .., n)
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ROAA = f(HHI, L_A, SIZE, ASSET_GRO, GDP, INF) + ε
Pre Example (3.1) with model
Option 2. Testing for Fix Effects
yi = eαi + Xiβ+ εi. (i = 1, .., n)
Method 1. Fix effects model is only valid when we could test the joint
significance of the dummies by:
H0: α1 = α2 =…= αN = 0
H1: αi ≠ 0
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The F test is calculated by the following formular:
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ROAA = f(HHI, L_A, SIZE, ASSET_GRO, GDP, INF) + ε
Pre Example (3.1) With model
Method 2. F test can alse check by xtreg with option fe in Stata
α = 0
Option 3. Testing of Random Effects
α > 0
H0: σ2 H1: σ2
To test this hypothesis, we can use Lagrange Multiplier (LM) test
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developed by Bresuch and Pagan (1980)
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ROAA = f(HHI, L_A, SIZE, ASSET_GRO, GDP, INF) + ε
Pre Example (3.1) With model
4.4.2 Fix or Random effect: Hausman Test
yi = eαi + Xiβ+ εi. (i = 1, .., n) (FE)
H0: E (εit|Xit) = 0
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H1: E (εit|Xit) ≠ 0
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Using this fact, we have
Therefore,
With
Then
The test statistic is
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Hausman test (H) = q’ (Var(q))-1q
