Transcript Estimation of Random Components and Prediction in One

Estimation of Random Components
and Prediction in
One- and Two-Way Error Component
Regression Models
Subhash C. Sharma
Department of Economics
Southern Illinois University Carbondale
And
Anil K. Bera
Department of Economics
University of Illinois at Urbana-Champaign
July 2007
Outline
•
•
•
•
•
Problem
One-Way Error Component Model
Two-Way Error Component Model
Empirical Illustration
Summary
Problem
• Prediction in Panel Data Models.
Consider the one-way error component model:


yit    X it    i  uit ,
i  1, 2,
,m
t  1, 2,
,n
 i are randomly distributed across m cross-
sectional units
One problem:  i
• Idea borrowed from Bera and Sharma (1999), “Estimating
Production Uncertainty in Stochastic Frontier Function
Models,” Journal of Productivity Analysis, 12, pp. 187-210.
• Model:
yi  f ( xi ,  )   i  ui , ui  0
 f ( xi ,  )   i  SFi  ui
ui :
represents technical inefficiency
Estimation of ui ?
• We can recover  i , which is a “sufficient
statistic” for ui
• Use “Rao-Blackwellization” and obtain
E (ui |  i ). Then,
E[ E (ui |  i )]  E (ui )
Var[ E (ui |  i )]  Var (ui )
It is easy to derive the conditional density f (ui |  i )
Using that we can obtain conditional moments of
any order. E (ui |  i ) gives a point estimate of ui
(inefficiency)
Jondrow, Lovell, Materov and Schmidt (1982),
JE.
Var (ui |  i ) can be viewed as the production
uncertainty due to technical inefficiency.
Using expression for the conditional mean and
variance, we can construct confidence intervals
for firm specific inefficiency. Can also use
higher-order conditional moments (of ui
given  i ) to obtain conditional skewness and
kurtosis measures.
For the panel data model


yit    X it    i  uit
yi  X i    i 1  ui ,
 X i   i
1
 
1  
1
 
E ( yi |  i )  X i   E ( i |  i )1  E (ui |  i )
Can construct confidence interval using
Var ( i |  i ).
Introduction
We have the panel regression model (for m cross sectional
units over n time periods)
y it  α i  x 2itβ 2  x 3itβ 3     x kitβ k  u it
or
 β2 
 
 α i  x 2it x 3it ...x kit  β 3   u it
β 
 k
i  1,2...m, t  1,2...n
or
y it  α i  xitβ*  uit
i = 1,2…m, t = 1,2…n
(1)
yit is an observation on the dependent variable, α i is a unit
specific term, xit is the i-th observation on (k-1) non-stochastic
explanatory variables at time t, β is a (k-1)x1 vector of
unknown parameters
*
When α i is fixed it yields the fixed effect model.
In other situations it might be more appropriate to consider
α i  α  δi
where α is the mean intercept and δi is randomly distributed
across cross-sectional units.
Thus, model (1) can be written as
y it  α  xitβ*  δi  u it
i  1,2...m, t  1,2...n
(2)
Model (2) is known as the one-way random effect or the oneway error component model.
Instead of just cross sectional effect one should also capture the
time effect, i.e.,
α it  α  δ i  λ t
where α is the overall effect, δi is an individual cross sectional
effect, and λ t is the time effect,
So, for this case the model becomes
y  α  x β  δ  λ  u
it
it
i
t
it i = 1,2…m , t = 1,2…n
(3)
• When δi and λ t are fixed, (3) is called a two-way fixed effect
model.
• However, when δi and λ t are random, (3) is called a two-way
error component model, or two-way random effect model.
2.
One-Way Error Component Model
Consider the random effect model given by equation (2)
y it  x it β  δ i  u it , i  1,2...m, t  1,2...n,
(2)
where E(δ ) = E(u ) = 0,E(δ ) = σ ,E(u ) = σ ,E(u δ ) = 0,
for all t and j; E(uit u js ) = 0 if t  s or i  j,
i
2
i
it
2
δ
2
it
2
u
it
j
E(δiδ j )  0 if i  j and δ i and u it are assumed to be
distributed as normal.
Let
ε it  δ i  u it
or
ε i  δi i n  ui
(4)
The joint density of δi and ui = (ui1 ui2…uin)´ is given by
 δi2 u'iu i 
f(δi ,u i ) =
exp - 2 - 2 
n+1
2σ δ 2σ u 
n

2
σ δσ u (2π)
1
(5)
From (5), the joint density of δi and εi can be easily obtained
as
 (δi - μ *δ )2 μ *δ2 ε'i ε i 
f(δ i ,ε i ) =
exp + *2 - 2 
n+1
*2
2σ δ
2σ δ 2σ u 

σ δσ nu (2π) 2
1
i
i
where
μ *δ =
i
nσ*2
δ εi•
σ u2
*2
δ
,σ =
σ δ2σ u2
σ12
, and σ12 = σ u2 + nσ δ2 ,
(6)
n
with  i  t1  it / n.
From (6), the marginal density of εi is
 μ *δ2 ε'iε i 
f(ε i ) =
exp  *2 - 2  .
n
 2σ δ 2σ u 
σ δσ un (2π) 2
σ*δ
i
(7)
Finally, the conditional density function of δi given εi is
1
f(δi | εi) =
σ*δ
Thus, E(δi | εi) =
σδ =
μ*δ i
σ δσ u
2
*2
 (δi - μ*δ )2 
exp .
*2
2π
 2σδ 
1

i
nσ*δ2 εi 
σ u2
and Var (δi | εi) = σ*δ2,
2
σ1
2
, and σ1 = σ u + nσ δ .
2
2
2
(8)
Thus, we propose that the random coefficient, δi, be
estimated by an estimate of E(δi | εi), i.e.
2
nσ̂ *δ ε̂i nσ̂ δ2 ε̂i
*
δ̂ i  μ̂ δi 

2
σ̂ u
σ̂ 12
where,
n
ˆεi   1  (y it  x'it βˆ ),
n t 1
and
βˆ , σˆ δ2 and σˆ 12 are the corresponding GLS estimates of
β, σδ2 and σ12
respectively.
(9)
Confidence Interval:
We can estimate Var (δi|εi) as
*2
σˆ δ
 σˆ δ2σˆ u2 /σˆ 12 .
Using these estimate we can easily construct the (1-α) 100%
confidence interval for δi, as follows.
(δˆ i - Zα/2σˆ *δ , δˆ i + Zα/2σˆ *δ )
We can also test the hypothesis Ho: δi = 0, using the
approximate t-statistic given by δˆ i /σˆ *δ .
(10)
Prediction
In the one way random effect model, we propose the
predicted value of yi by the expected value of yi conditional on
the composed error term, εi, i.e,. E(yi|εi), where
E(yi | εi) = Xi β + E(δi | εi) in + E(ui | εi),
*
*
E(yi | εi) = Xi β + μδi in  μ ui .
(11)
One can easily obtain that
1
exp
f(ui | εi)  *n
n/2
σ u (2 )
where, μ*u i = E(ui |
σ *u 
2
and
 σ* 2 
  u2

u


ε

ε

εi)  2  i  2
2 i
σ
  u  nσ δ 
 δ 
σ 2u σ δ2
σ  nσ
2
u
 (u i  μ *u i ) (u i  μ *u i ) 

,
*2
2σ


u
2
δ
.
(12)
(13)
(14)
Thus, following (11) the prediction for the i-th unit is
yˆ i = Xi βˆ GLS + μˆ *δ in + μˆ *u
i
where
μˆ *δ and μˆ *u
i
i
(15)
i
are the consistent estimates of
μ *δ and μ *u .
i
i
For this model, the best linear unbiased prediction of the
i-th unit, is also given by Lee and Griffiths (1979),
Taub (1979) and Baltagi (2001, p. 22), which is
 nσ δ2 
ŷ i  X i β̂ GLS   2 ε̂ i ,GLS ,
 σ1 
where
1 n
ε̂i ,GLS   ε̂ it,GLS, and ε̂ i  yi  X i β̂ GLS.
n t 1
(16)
3.
Two-Way Error Component Model

yit  xit β  δi  λ t  uit , i  1,2...m, t  1,2...n
(3)
where δi’s are iid, as normal with mean zero and variance, σ δ2 ;
2
λt’s are iid normal with mean zero and variance σ λ ;
2
σ
and uit are iid normal with mean zero and variance u .
Moreover, δi, λt and uit are assumed to be independent of
each other.
Let
(19)
ε it  δi  λ t  u it
or,
(20)
ε i  δi i n  λ  u i
λ = (λ 1 λ 2 ...λ n ),u i = (u i1 u i2 ...u in ), and
ε i =  ε i1ε i2 ...ε in  '.
3.1 Estimation of Cross Sectional Effect, δi
We propose an estimate of δi by E(δi|εi) using the conditional
density f(δi|εi).
The joint density of δi, λ and ui is given by
 δ i2
λ ' λ u 'i u i 
exp  2  2 

2σ δ 2σ λ 2σ u2 

f(δ i , λ, u i ) 
.
2n  1
n
σ δ σ λ σ u  2π  2
(21)
From (21), one can easily obtain the joint density of f(δi, λ, εi)
by substituting uiu i , i.e.,
 λλ
δi2 εiε i λε i in λδi inε iδi 
exp - +2 - +2 - 2 + 2 - 2 + 2 
σu
σ u 
 2σ λ 2σ δi 2σ u σ u
f  δi ,λ,ε i  =
2n+1
σ δ  σ λ σ u   2π 
n
2
(22)
where
σu2σ 2λ
= 2
,
σu + σ 2λ
(23)
σu2σδ2
σ = 2
.
2
σu + nσδ
(24)
σ +2
λ
and
+2
δi
From (22), one can obtain

f  δi ,ε i  =
δi2 εiε i σ +2
inε iδi 
λ

+2 2 +
4  ε i - i nδ i  '  ε i - i nδ i  +
2σ
2σ
2σ
σ u2 
u
u
δi



σ +n
λ exp -
σδ  σ λσ u   2π 
n
n+1
2
.
(25)
Further, from (25)
*2 
 εε
μ
δi 
*
i i
σ +n
2 +
λ σ δi exp *
*2 
2σ
2σ

εi
δi 
,
f  εi  =
n
n/2
σδ  σ λσ u   2π 
(26)
2
σ*δi
σδ2
μ = *2 nεi• = 2
nεi• ,
2
2
σ
+
σ
+
nσ
σ εi
u
λ
δ
*
δi
2
σ*δi =
σ δ2  σ u2 + σ 2λ 
σ + σ + nσ
2
u
2
λ
2
δ
(27)
,
(28)
and
*2
εi
σ = σ u2 + σ 2λ .
(29)
Finally, from (25) and (26), we get

f(δi | εi) =
 δ - μ*
 i δi
exp *2
2σ

δi

σ*δi 2π

2




.
(30)
Thus, we propose that δi be estimated by
E(δi | εi) =
μ*δ i
2


σ
δ
n ε .
  2
2
2  i
 σ u  σ λ  nσ δ 
(31)
And
Var (δi | εi) =
*2
σδ i


σδ2 σu2  σ 2λ
 2
.
2
2
σu  σ λ  nσδ
(32)
By using σ *δ i , one can also obtain the (1-α) 100% confidence
interval for δi.
3.2 Estimation of the time effect, λt.
We propose an estimate of λt by E(λt | εt), which is the mean
of f(λt | εt). To obtain f(λt | εt) we rearrange the observations
in (19), as
εt = δ + λ t im + ut
(33)
where εt = (ε1t ε2t ….εmt)́, δ = (δ1 δ2 ….δm)́,
ut = (u1t u2t….umt)́, and im = (111…11)´.
The joint density of δ, λt and ut is given by
f(δ, λt, ut) =
 δδ λ t2 ut u t 
exp - 2  2 
2 
 2σ δ 2σ λ 2σ u  .
2m  1
m
σ λ σ δ σ u  2π  2
(34)
From (34), one can obtain
f(δ, λt, εt) =
 δδ
λ t2 εt ε t δε t λ t imδ λ t imε t 
exp - +2 - +2 - 2 + 2 - 2 +

σu
σ u2 
 2σ δ 2σ λ t 2σ u σ u
σ λ  σ δσ u 
m
 2π 
2m+1
2
,
(35)
and

f(λt,εt) =
where

λ t2 εtε t σ δ+2

+
(ε
i
λ
)
(ε
i
λ
)
t
m t
t
m t 
+2
2
4
 2σ λ t 2σ u 2σ u

σ δ+mexp -
σ λ (σ uσ δ ) (2π)
m
m+1
2
σu2σδ2
σ = 2
,
σu + σδ2
+2
δ
and
σu2σ λ2
σ = 2
.
2
σu + mσ λ
+2
λt
(36)
(37)
(38)
Further, from (36) we get
*2 
 εε
μ

σ*λt σδ+ exp - t *t2 + λ*t 2 
 2σ εt 2σ λ t  ,
f  εt  =
m
m/2
m
σ λ  σδσ u 
 2π 
(39)
where
σ 2λmε•t
μ = 2
,
σu + σδ2 + mσ 2λ
*
λt


2 + σ2
σ2
σu
2
λ
δ
σ*λ =
,
2
2
2
t σu + σ + mσ
δ
λ
(40)
and
2
2 + σ2 ,
σ*ε = σu
δ
t
with
 t    it m
i 1
m
(41)
From (36) and (39),


 λ  μ* 2 


t
λ
exp 
*
2σ
λ


.
σ *λ 2π
t
2
f(λt | εt) =
t
(42)
t
Thus, we propose that λt be estimated by E(λt | εt), i.e.

σ 2λ
E(λt | εt) = μ =  σ 2 + σ 2 + mσ 2
δ
λ
 u
*
λt

 mε•t ,

(43)
and
Var (λt | εt) =
2
σ*λ t =
σ 2λ  σ u2 + σ δ2 
σ + σ + mσ
2
u
2
δ
2
λ
.
(44)
By using σ*λ t , we can obtain the (1 – α) 100% confidence
interval for λt and test the hypothesis H0: λt = 0.
3.3
Prediction
In model
y i = Xi β + δi i n + λ + u i ,
we propose prediction of yi by an estimate of E(yi | εi). Thus
*
*
*
ˆ
ˆ
ˆ
ˆ
X
β
+
μ
i
n
+
μ
+
μ
ŷ i  i GLS
ui
δi
λ
where,
'


μˆ * =  μˆ * ,μˆ * , .. μˆ * 
λ  λ λ
λn 
 1
2

μ̂ *δi and μ̂ *λ t are estimates of δi and λt and
μ̂*u i = Ê (ui
| εi) is an estimate of ui.
(45)
4.
Empirical Illustration
We consider an example, first used by Baltagi and
Griffin (1983) and later by Baltagi (2001), the demand for
gasoline in a panel of 18 OECD countries covering the period
1960-1978.
The gasoline demand equation considered by Baltagi and
Griffin (1983) and Baltagi (2001, p. 21) is
ln (Gas/Car) = α + β ln (Y/N) + β ln (PMG /PGDP ) + β ln (Car/N) + ε ,
1
2
3
it
Gas/Car is motor gasoline consumption per auto,
Y/N is real per capita income,
PMG/PGDP is real motor gasoline price and
(Car/N) denotes the stock of cars per capita.
(66)
Table: 1
One-way error component model estimates
__________________________________________________________________
Methods of Estimation
Parameter
WAHU
AM
SWAR
FUBA
α̂
1.9058
(0.1661)
2.1844
(0.2151)
1.9967
(0.1843)
2.0203
(0.1882)
β̂ 1
0.5434
(0.0544)
0.6009
(0.0656)
0.5550
(0.0591)
0.5599
(0.0600)
β̂ 2
-0.4711
(0.0389)
-0.3664
(0.0415)
-0.4204
(0.0399)
-0.4118
(0.0402)
β̂ 3
-0.6061
(0.0243)
-0.6204
(0.0272)
-0.6068
(0.0255)
-0.6081
(0.0257)
ˆ 21
0.030071
0.11420
0.038238
0.044041
ˆ 22
0.062094
0.090217
0.072238
0.074485
ˆ 23
0.071708
0.099850
0.081794
0.084041
ˆ u21
0.013509
0.008446
0.008524
0.008525
ˆ u22
0.008360
0.008064
0.008192
0.008167
ˆ u23
0.008909
0.008594
0.008729
0.008704
 21 :
denotes first stage estimates.
 22 :
are ANOVA type estimates based on GLS residuals.
 23 :
are ANOVA type estimates adjusted for degrees of
freedom, based on GLS residuals.
Table: 2
One-Way Error Component Model
Point and 95 % Confidence Interval Estimates for Cross Country Effect (  i )
Based on Fuller-Battese Method
Country
Austria
Belgium
Canada
Denmark
France
Germany
Greece
Ireland
Italy
Japan
Netherlands
Norway
Spain
Sweden
Switzerland
Turkey
U.K.
U.S.A.
LB
ˆi _ 1
UB
LB
ˆi _ 2
UB
LB
ˆi _ 3
UB
-0.15782
-0.24255
0.57070
-0.01556
-0.20431
-0.27841
-0.05409
0.13131
-0.20488
-0.02880
-0.17882
-0.17329
-0.56317
0.18602
-0.06980
0.06224
-0.09551
0.57321
-0.11651
-0.20124
0.61201
0.02575
-0.16300
-0.23710
-0.01279
0.17261
-0.16357
0.01250
-0.13751
-0.13198
-0.52186
0.22732
-0.02849
0.10355
-0.05421
0.61452
-0.07520
-0.15994
0.65332
0.06705
-0.12170
-0.19580
0.02852
0.21392
-0.12226
0.05381
-0.09620
-0.09068
-0.48055
0.26863
0.01282
0.14485
-0.01290
0.65583
-0.15860
-0.24370
0.57315
-0.01571
-0.20529
-0.27972
-0.05441
0.13181
-0.20587
-0.02901
-0.17969
-0.17414
-0.56574
0.18676
-0.07019
0.06244
-0.09602
0.57567
-0.11703
-0.20213
0.61472
0.02586
-0.16372
-0.23815
-0.01284
0.17338
-0.16429
0.01256
-0.13812
-0.13257
-0.52417
0.22833
-0.02862
0.10401
-0.05445
0.61724
-0.07545
-0.16056
0.65629
0.06743
-0.12215
-0.19658
0.02873
0.21495
-0.12272
0.05413
-0.09655
-0.09100
-0.48260
0.26990
0.01295
0.14558
-0.01287
0.65881
-0.15884
-0.24407
0.57388
-0.01577
-0.20561
-0.28013
-0.05452
0.13195
-0.20618
-0.02909
-0.17997
-0.17441
-0.56653
0.18697
-0.07032
0.06248
-0.09618
0.57640
-0.11718
-0.20240
0.61554
0.02590
-0.16394
-0.23847
-0.01286
0.17361
-0.16451
0.01258
-0.13830
-0.13274
-0.52487
0.22864
-0.02866
0.10415
-0.05452
0.61806
-0.07552
-0.16074
0.65720
0.06756
-0.12228
-0.19681
0.02880
0.21527
-0.12285
0.05424
-0.09664
-0.09108
-0.48321
0.27030
0.01301
0.14581
-0.01285
0.65973
Method
Table: 3
One-Way Error Component Model Absolute Prediction Error in Percentage
In-Sample n =18, m = 19
Out of-Sample n =18, m = 4
N = 342
N = 72
Pred.
Type
Std.
Std.
Mean
Dev.
Min.
Max.
Mean
Dev.
Min.
Max.
Wallace
and
Hussein
T/LG_1
SB_1
T/LG_2
SB_2
T/LG_3
SB_3
1.49023
1.45088
1.48399
1.47480
1.48404
1.47722
1.38969
1.35907
1.39239
1.38147
1.39226
1.38374
0.00042
0.00824
0.00248
0.00838
0.00039
0.00839
8.02895
7.84881
8.03274
7.97819
8.03312
7.99128
4.46491
4.46747
4.14864
4.14945
4.15007
4.46570
4.33748
4.33841
4.20766
4.20806
4.20837
4.33776
0.00471
0.00010
0.07061
0.07250
0.07393
0.00330
15.64474
15.64853
15.12592
15.12743
15.12859
15.64590
Amemiya
T/LG_1
SB_1
T/LG_2
SB_2
T/LG_3
SB_3
1.46628
1.46233
1.46605
1.46119
1.46638
1.46278
1.37290
1.36589
1.37317
1.36483
1.37278
1.36631
0.00138
0.00089
0.00174
0.00089
0.00063
0.00089
7.69096
7.65828
7.69154
7.65230
7.69074
7.66064
4.72054
4.72180
4.41823
4.41935
4.41930
4.72166
4.59062
4.59114
4.44542
4.44605
4.44602
4.59108
0.11957
0.11700
0.12902
0.13199
0.13186
0.11730
16.42695
16.42903
15.86412
15.86652
15.86641
16.42880
Swamy
and Arora
T/LG_1
SB_1
T/LG_2
SB_2
T/LG_3
SB_3
1.47356
1.45785
1.47316
1.46626
1.47344
1.46828
1.38062
1.36252
1.38030
1.37038
1.37995
1.37227
0.00457
0.00330
0.00041
0.00332
0.00640
0.00332
7.87822
7.78484
7.87722
7.82974
7.87698
7.84056
4.53436
4.53666
4.21162
4.21251
4.21290
4.53528
4.39655
4.39735
4.26563
4.26613
4.26635
4.39687
0.01311
0.01736
0.01249
0.01028
0.00930
0.01481
15.93431
15.93772
15.39819
15.39996
15.40076
15.93567
Fuller
And
Battese
T/LG_1
1.47184
1.37904
0.00494
7.85012
4.55079
4.41156
0.00425
15.99399
SB_1
T/LG_2
SB_2
T/LG_3
SB_3
1.45860
1.47158
1.46505
1.47184
1.46701
1.36306
1.37881
1.36909
1.37848
1.37092
0.00125
0.00055
0.00126
0.00245
0.00126
7.76839
7.84900
7.80277
7.84865
7.81320
4.55290
4.22779
4.22872
4.22907
4.55169
4.41243
4.28000
4.28051
4.28071
4.41193
0.00013
0.03595
0.03367
0.03279
0.00249
15.99730
15.45442
15.45626
15.45696
15.99540
Table 4
Parameter
Two-Way Error Component Model Estimates
Methods of Estimation
WAHU
AM
SWAR
FUBA
α̂
1.9101
(0.1672)
-0.2391
(0.3501)
2.0408
(0.1915)
1.0308
(0.2660)
β̂ 1
0.5433
(0.0547)
0.1682
(0.0804)
0.5645
(0.0608)
0.3947
(0.0683)
-0.4672
(0.0390)
-0.2322
(0.0411)
-0.4049
(0.0404)
-0.3390
(0.0414)
-0.6058
(0.0244)
-0.6024
(0.0258)
-0.6094
(0.0259)
-0.6096
(0.0256)
β̂ 2
β̂ 3
ˆ 21
ˆ 22
ˆ 23
ˆ 21
ˆ 22
ˆ 23
ˆ u21
ˆ u22
ˆ u23
0.031875
0.18261
0.038340
0.046706
0.066418
0.15203
0.080906
0.13047
0.086992
0.19891
0.105930
0.13047
0
0.017412
0
0.002038
0
0.011753
0
0.002989
0
0.015211
0
0.003956
0.013653
0.006526
0.006591
0.006591
0.008914
0.006608
0.008661
0.007430
0.009032
0.006695
0.008776
0.007528
Table: 5
Two- Way Error Component Model
Point and 95 % Confidence Interval Estimates for Cross Country Effect (  i )
Based on Fuller-Battese Method
Country
LB
Austria
Belgium
Canada
Denmark
France
Germany
Greece
Ireland
Italy
Japan
Netherlands
Norway
Spain
Sweden
Switzerland
Turkey
U.K.
U.S.A.
-0.15730
-0.20954
0.70206
0.03555
-0.17856
-0.23031
-0.16852
0.06903
-0.26615
-0.06526
-0.15398
-0.12771
-0.57212
0.07010
-0.00736
-0.14617
-0.07939
0.73743
ˆi _ 1
-0.11573
-0.16798
0.74363
0.07712
-0.13699
-0.18874
-0.12695
0.11060
-0.22459
-0.02369
-0.11241
-0.08614
-0.53056
0.11167
0.03420
-0.10460
-0.03782
0.77900
UB
LB
ˆi _ 2
UB
LB
ˆi _ 3
UB
-0.07417
-0.12641
0.78520
0.11869
-0.09543
-0.14717
-0.08539
0.15217
-0.18302
0.01787
-0.07085
-0.04458
-0.48899
0.15324
0.07577
-0.06304
0.00375
0.82056
-0.16199
-0.21445
0.70098
0.03167
-0.18334
-0.23530
-0.17326
0.06529
-0.27130
-0.06957
-0.15866
-0.13228
-0.57856
0.06636
-0.01143
-0.15082
-0.08375
0.73649
-0.11622
-0.16868
0.74675
0.07745
-0.13757
-0.18953
-0.12749
0.11107
-0.22553
-0.02379
-0.11288
-0.08651
-0.53279
0.11214
0.03435
-0.10504
-0.03798
0.78227
-0.07045
-0.12291
0.79252
0.12322
-0.09180
-0.14376
-0.08171
0.15684
-0.17976
0.02198
-0.06711
-0.04073
-0.48701
0.15791
0.08012
-0.05927
0.00779
0.82804
-0.16440
-0.21690
0.69932
0.02944
-0.18576
-0.23777
-0.17567
0.06309
-0.27380
-0.07189
-0.16106
-0.13466
-0.58132
0.06416
-0.01370
-0.15321
-0.08609
0.73487
-0.11632
-0.16883
0.74740
0.07751
-0.13769
-0.18970
-0.12760
0.11116
-0.22573
-0.02381
-0.11298
-0.08658
-0.53325
0.11223
0.03438
-0.10513
-0.03801
0.78294
-0.06824
-0.12075
0.79547
0.12559
-0.08961
-0.14162
-0.07952
0.15924
-0.17765
0.02426
-0.06491
-0.03850
-0.48517
0.16031
0.08245
-0.05706
0.01006
0.83102
Table: 6
Two Way Error Component Model
Point and 95 % Confidence Interval Estimates for the Time Effect (  t )
Based on Fuller-Battese Method
Year
LB
ˆi _ 1
UB
LB
ˆi _ 2
UB
LB
ˆi _ 3
UB
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
-0.10644
-0.09808
-0.10192
-0.10134
-0.09254
-0.08960
-0.07768
-0.07010
-0.06280
-0.06683
-0.06132
-0.05421
-0.05001
-0.04376
-0.05522
-0.04018
-0.04229
-0.03981
-0.03974
-0.03834
-0.02998
-0.03382
-0.03324
-0.02444
-0.02151
-0.00958
-0.00201
0.00530
0.00127
0.00678
0.01389
0.01809
0.02434
0.01288
0.02792
0.02581
0.02829
0.02836
0.02975
0.03812
0.03428
0.03485
0.04366
0.04659
0.05852
0.06609
0.07340
0.06937
0.07488
0.08198
0.08619
0.09244
0.08098
0.09602
0.09390
0.09639
0.09645
-0.11888
-0.11202
-0.11517
-0.11469
-0.10747
-0.10507
-0.09528
-0.08907
-0.08308
-0.08639
-0.08187
-0.07604
-0.07259
-0.06746
-0.07686
-0.06453
-0.06626
-0.06423
-0.06417
-0.03145
-0.02459
-0.02774
-0.02726
-0.02004
-0.01764
-0.00786
-0.00165
0.00435
0.00104
0.00556
0.01139
0.01484
0.01996
0.01057
0.02290
0.02116
0.02320
0.02326
0.05598
0.06284
0.05969
0.06016
0.06738
0.06979
0.07957
0.08578
0.09177
0.08847
0.09299
0.09882
0.10227
0.10739
0.09799
0.11033
0.10859
0.11063
0.11068
-0.13213
-0.12515
-0.12836
-0.12787
-0.12052
-0.11807
-0.10812
-0.10179
-0.09570
-0.09906
-0.09446
-0.08853
-0.08502
-0.07980
-0.08937
-0.07681
-0.07858
-0.07650
-0.07645
-0.03201
-0.02503
-0.02824
-0.02775
-0.02040
-0.01795
-0.00800
-0.00167
0.00442
0.00106
0.00566
0.01159
0.01510
0.02032
0.01075
0.02331
0.02154
0.02362
0.02367
0.06811
0.07509
0.07188
0.07237
0.07972
0.08217
0.09212
0.09845
0.10454
0.10118
0.10578
0.11171
0.11522
0.12044
0.11087
0.12343
0.12166
0.12374
0.12379
Table: 7
Two-Way Error Component Model
Absolute Prediction Error in Percentage
Method
Pred.
Type
In-Sample n =18, m = 19
N = 342
Std.
Mean
Dev.
Min.
Max.
Out of -Sample n =18, m = 4
N = 72
Std.
Mean
Dev.
Min.
Max.
Wallace
and
Hussein
BK_1
SB_1
BK_2
SB_2
BK_3
SB_3
1.48853
1.45149
1.48300
1.47380
1.48306
1.47615
1.38900
1.35948
1.39134
1.38037
1.39119
1.38257
0.00265
0.00041
0.00153
0.00042
0.00198
0.00042
8.01828
7.84597
8.02139
7.96658
8.02172
7.97925
4.44417
4.44912
4.46674
4.46770
4.47036
4.47182
4.32244
4.32399
4.32958
4.32989
4.33074
4.33121
0.08768
0.09620
0.12650
0.12815
0.13076
0.12819
15.54077
15.54778
15.57273
15.57409
15.57785
15.57993
Amemiya
BK_1
SB_1
BK_2
SB_2
BK_3
SB_3
2.47954
1.48311
2.47941
1.54926
2.47957
1.55094
1.92787
1.36907
1.92789
1.40013
1.92786
1.40124
0.00468
0.01103
0.00551
0.01632
0.00446
0.01588
11.86577
9.53317
11.86797
9.71797
11.86518
9.72511
7.71453
6.66773
7.70709
6.76351
7.70978
6.72575
7.09830
6.73220
7.09640
6.76448
7.09709
6.75241
0.11524
0.06527
0.10471
0.09864
0.10852
0.03522
27.04638
25.64423
27.03769
25.78135
27.04083
25.72830
Swamy
and
Arora
BK_1
1.47052
1.37798
0.00146
7.82731
4.64396
4.50365
0.05252
16.26869
SB_1
BK_2
SB_2
BK_3
SB_3
1.45915
1.47045
1.46410
1.47071
1.46539
1.36347
1.37770
1.36810
1.37738
1.36974
0.00223
0.00125
0.00224
0.00242
0.00111
7.75425
7.82622
7.78057
7.82581
7.78897
4.64564
4.64394
4.64563
4.64681
4.65163
4.50429
4.50364
4.50428
4.50473
4.50656
0.05571
0.05249
0.05569
0.05791
0.06704
16.27134
16.26867
16.27132
16.27316
16.28072
Fuller
And
Battese
BK_1
1.78864
1.48110
0.00254
9.66659
5.63341
5.32737
0.16250
20.39134
SB_1
BK_2
SB_2
BK_3
SB_3
1.49509
1.78980
1.54527
1.79016
1.54303
1.33861
1.47994
1.36379
1.47962
1.36343
0.00374
0.00228
0.00345
0.00065
0.00408
8.82149
9.65778
8.98992
9.65554
8.98725
5.30760
5.64040
5.39850
5.64373
5.38002
5.30888
5.32788
5.31337
5.32812
5.31251
0.00167
0.15296
0.06515
0.14841
0.04787
20.04742
20.39870
20.14384
20.40222
20.12437
5. Summary
We consider the one-way error component model, i.e.
yit = α + xitβ* + δi + u it
i = 1, 2...m, t = 1, 2...n
and propose an estimate of δi by the estimate of E(δi | εi),
where, ε i  δi i n  ui , is like a composite error term. We also
provide expression for Var (δi | εi).
Using E(δi | εi) and Var (δi | εi) one can obtain the confidence
interval for the random component.
Next, an expression for the prediction of the i-th crosssectional unit is provided, i.e.,
ŷ i  X i β̂ GLS  δ̂ i i n  û i
where û i = E(ui | εi) , is also included besides E(δi | εi).