Estimation of Random Components and Prediction in

One- and Two-Way Error Component Regression Models

Subhash C. SharmaDepartment of Economics

Southern Illinois University Carbondale

And

Anil K. BeraDepartment 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:

are randomly distributed across m cross-

sectional units

One problem:

,it it i ity X u

i

i

1,2, ,i m

1,2, ,t n

• Idea borrowed from Bera and Sharma (1999), “Estimating Production Uncertainty in Stochastic Frontier Function Models,” Journal of Productivity Analysis, 12, pp. 187-210.

• Model:

: represents technical inefficiency

Estimation of ?

,( , )i i i iy f x u 0iu

( , )i i i if x SF u

iu

iu

• We can recover , which is a “sufficient statistic” for

• Use “Rao-Blackwellization” and obtain

Then,

i

iu

( | ).i iE u

[ ( | )] ( )i i iE E u E u

[ ( | )] ( )i i iVar E u Var u

It is easy to derive the conditional density

Using that we can obtain conditional moments of any order. gives a point estimate of (inefficiency)

Jondrow, Lovell, Materov and Schmidt (1982), JE.

can be viewed as the production uncertainty due to technical inefficiency.

( | )i if u

( | )i iE u iu

( | )i iVar u

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 given ) to obtain conditional skewness and kurtosis measures.

iu

i

For the panel data model

Can construct confidence interval using

it it i ity X u

1 ,i i i iy X u 1

1

1

i iX

( | ) ( | )1 ( | )i i i i i i iE y X E E u

( | ).i iVar

Introduction

We have the panel regression model (for m cross sectional units over n time periods)

or

ori = 1,2…m, t = 1,2…n (1)

yit is an observation on the dependent variable, is a unit specific term, is the i-th observation on (k-1) non-stochastic explanatory variables at time t, is a (k-1)x1 vector of unknown parameters

itkkit33it22itiituβxβxβxαy

1,2...nt1,2...m,i u

β

β

β

...xxxαit

k

3

2

kit3it2iti

it

*

itiituβxαy

itx*β

iα

When is fixed it yields the fixed effect model.

In other situations it might be more appropriate to consider

where α is the mean intercept and is randomly distributed across cross-sectional units.

Thus, model (1) can be written as

(2)

Model (2) is known as the one-way random effect or the one-way error component model.

iα

iδααi

iti

*

itit uδβxαy 1,2...n t1,2...m,i

iδ

Instead of just cross sectional effect one should also capture the

time effect, i.e.,

where α is the overall effect, is an individual cross sectional

effect, and is the time effect,

So, for this case the model becomes

i = 1,2…m , t = 1,2…n (3)

• When and are fixed, (3) is called a two-way fixed effect

model. • However, when and are random, (3) is called a two-way

error component model, or two-way random effect model.

tiitλδαα

itu

tλ

iδβ

itxα

ity

iδ

tλ

iδ tλ

iδ tλ

2. One-Way Error Component Model

Consider the random effect model given by equation (2)

(2)

where for all t and j; if if and are assumed to be distributed as normal.

Let or (4)

1,2...n, t 1,2...m,i ,uδβxy itiitit

2 2 2 2i it i δ it u it jE(δ ) = E(u ) = 0,E(δ ) = σ ,E(u ) = σ ,E(u δ ) = 0,

E it js(u u ) = 0 ,jior st

0)δE(δ ji ji iti u and δ

itiituδε inii

uiδε

The joint density of δi and ui = (ui1 ui2…uin)´ is given by

(5)

From (5), the joint density of δi and εi can be easily obtained as

(6)

where

2 'i i i

i i n+1 2 2n δ u2

δ u

1 δ u uf(δ ,u ) = exp - -

2σ 2σσ σ (2π)

i i

2 2 'i δ δ i i

i i n+1 *2 *2 2n δ δ u2

δ u

(δ - μ * ) μ *1 ε εf(δ ,ε ) = exp - + -

2σ 2σ 2σσ σ (2π)

i

*2 2 22* 2 2 2δ i• δ u

δ δ 1 u δ2 2u 1

nσ ε σ σμ * = , σ = , and σ = σ + nσ ,

σ σ

with

From (6), the marginal density of ε i is

(7)

n

1titi

.n/

i

2* 'δδ i i

i n *2 2n δ u2

δ u

μ *σ ε εf(ε ) = exp - .

2σ 2σσ σ (2π)

Finally, the conditional density function of δ i given εi is

f(δi | εi) = (8)

Thus, E(δi | εi) = and Var (δi | εi) =

i

* 2i δ

* *2δ δ

(δ - μ )1 1exp - .

σ 2σ2π

2u

i2*

δ*δ

σ

εnσμ

i

,2*δ

σ

2 2

2* 2 2 2δ u

δ 1 u δ2

1

σ σσ = , and σ = σ + nσ .

σ

Thus, we propose that the random coefficient, δ i, be estimated by an estimate of E(δi | εi), i.e.

(9)

where, and

are the corresponding GLS estimates of

respectively.

2

1

i

2

δ

2

u

i

*

δ*

δi σ

εσn

σ

εσnμδ

2

i

n

1t

'ititi ),βx(y

n

1ε ˆˆ

21

2δ σ and σ,β ˆˆˆ

21

2δ σ and σβ,

Confidence Interval:

We can estimate Var (δi|εi) as

Using these estimate we can easily construct the (1-α) 100%

confidence interval for δi, as follows.

(10)

We can also test the hypothesis Ho: δi = 0, using the

approximate t-statistic given by

.ˆˆˆˆ 21

2u

2δ

*δ σ/σσσ

2

ˆ ˆˆ ˆ* *

i α/2 δ i α/2 δ(δ - Z σ , δ + Z σ )

.ˆˆ *δi σ/δ

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 + (11)

One can easily obtain that

f(ui | εi) (12)

where, = E(ui | εi) (13)

and (14)

i i

* *δ n uμ i μ .

exp)(2σ

1n/2n*

u ,

2σ

)μ(u)μ(u2*

u

*ui

*ui ii

i2δ

2u

2u

i2δ

2u

*

εnσ

εσ

σ

*

iuμ

.nσσ

σσσ

2δ

2u

2δ

2u*

u

2

Thus, following (11) the prediction for the i-th unit is

(15)

where are the consistent estimates of

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

(16)

where

ˆˆ ˆ ˆi i

* *ni i GLS δ uy = X β + μ i + μ

ˆ ˆi i

*δ uμ and μ * μ * and μ * .uδ ii

,εσ

nσβ Xy GLS,i2

1

2δ

GLSii

n

1tGLSiiiGLSit,GLS,i .β Xyε and ,ε

n

1ε

3. Two-Way Error Component Model

,(3)

where δi’s are iid, as normal with mean zero and variance,

λt’s are iid normal with mean zero and variance

and uit are iid normal with mean zero and variance

Moreover, δi, λt and uit are assumed to be independent of

each other.

Let (19)

or, (20)

it it i t ity x β δ λ u 1,2...n t1,2...m,i

;2δσ

;σ2λ

.2uσ

it i t itε δ λ u

i i n iε δ i λ u

1 2 n i i1 i2 in

i i1 i2 in

λ = (λ λ ...λ ) ,u = (u u ...u ) , and

ε = ε ε ...ε '.

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

(21)

From (21), one can easily obtain the joint density of f(δ i, λ, εi)

by substituting i.e.,

(22)

.

2πσσσ

2σ

uu

2σ

λλ

2σ

δexp

)uλ,,f(δ2

12nn

uλδ

2

u

i

'

i

2

λ

'

2

δ

2

i

ii

,ii uu

i

2n ni i i i i i i

+2 +2 2 2 2 2u u u uλ δ

i i 2n+1n2

uδ λ

λ λ δ ε ε λ ε i λδ i ε δexp - - - + - +

2σ 2σ 2σ σ σ σf δ ,λ,ε =

σ σ σ 2π

where

(23)

and

(24)

From (22), one can obtain

(25)

2 2+2 u λλ 2 2

u λ

σ σσ = ,

σ + σ

i

2 2+2 u δδ 2 2

u δ

σ σσ = .

σ + nσ

i

2 +2+n ni i i λ i i

n nλ i i i i+2 2 4 2u u uδ

i i n+1n2

uδ λ

.

δ ε ε σ i ε δσ exp - - + ε - i δ ' ε - i δ +

2σ 2σ 2σ σf δ ,ε =

σ σ σ 2π

Further, from (25)

(26)

(27)

(28)

and

(29)

2

i2 2i

i i

*δ+n * i i

λ δ * *ε δ

i n n/2uδ λ

με εσ σ exp - +

2σ 2σf ε = ,

σ σ σ 2π

2

i2i

i

* 2δ* δ

δ i• i•2 2 2*u λ δε

σ σμ = nε = nε ,

σ + σ + nσσ

2

i

2 2 2δ u λ*

δ 2 2 2u λ δ

σ σ + σσ = ,

σ + σ + nσ

2

i

* 2 2ε u λσ = σ + σ .

Finally, from (25) and (26), we get

f(δi | εi) = (30)

Thus, we propose that δi be estimated by

E(δi | εi) = (31)

And

Var (δi | εi) = (32)

By using one can also obtain the (1-α) 100% confidence

interval for δi.

,*δ i

σ

.

2δ

2λ

2u

2λ

2u

2δ*

δnσσσ

σσσσ

2

i

.

iεn

nσσσ

σμ

2δ

2λ

2u

2δ*

δi

i2

i

i

2*i δ

*δ

*δ

δ - μexp -

2σ.

σ 2π

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

(33)

where εt = (ε1t ε2t ….εmt)O, δ = (δ1 δ2 ….δm)O,

ut = (u1t u2t….umt)O, and im = (111…11)´.

The joint density of δ, λt and ut is given by

f(δ, λt, ut) = (34)

t t m tε = δ + λ i + u

.

2πσσσ

2σ

uu

2σ

λ

2σ

δδ- exp

2

12mm

uδλ

2

u

tt

2

λ

2

t

2

δ

From (34), one can obtain

f(δ, λt, εt) = (35)

and

f(λt,εt) = (36)

where

(37)

and

(38)

t

2t t t t t m t m t

+2 +2 2 2 2 2u u u uδ λ

2m+1m2

uλ δ

λ ε ε δ ε λ i δ λ i εδ δexp - - - + - +

2σ 2σ 2σ σ σ σ,

σ σ σ 2π

t

2 +2+m t t t δ

t m t t m tδ +2 2 4u uλ

m+1m 2

uλ δ

λ ε ε σσ exp - - + (ε - i λ ) (ε - i λ )

2σ 2σ 2σ

σ (σ σ ) (2π)

2 2+2 u δδ 2 2

u δ

σ σσ = ,

σ + σ

t

2 2+2 u λλ 2 2

u λ

σ σσ = .

σ + mσ

Further, from (36) we get

(39)

where

(40)

and

(41)

with

2

mt

2 2t

t t

*λ* + t t

λ δ * *ε λ

t m m/2uλ δ

με εσ σ exp - +

2σ 2σf ε = ,

σ σ σ 2π

t

2* •tλλ 2 2 2

u δ λ

σ mεμ = ,

σ + σ + mσ

2 2 2σ σ + σu2 λ δ*σ = ,λ 2 2 2t σ + σ + mσu δ λ

2* 2 2σ = σ + σ ,ε u δt

m

i

itt m1

From (36) and (39),

f(λt | εt) = (42)

Thus, we propose that λt be estimated by E(λt | εt), i.e.

E(λt | εt) = (43)

and

Var (λt | εt) = (44)

By using we can obtain the (1 – α) 100% confidence interval for λt and test the hypothesis H0: λt = 0.

.2πσ

2σ

2μλ- exp

*

λ

*

λ

*

λt

t

2

t

t

t

2* λλ •t2 2 2

u δ λ

σμ = mε ,

σ + σ + mσ

2

t

2 2 2λ u δ*

λ 2 2 2u δ λ

σ σ + σσ = .

σ + σ + mσ

,σ*λ t

3.3 Prediction

In model

we propose prediction of yi by an estimate of E(yi | εi). Thus

(45)

where,

are estimates of δi and λt and

= (ui | εi) is an estimate of ui.

y = X β + δ i + λ + u ,ni i i i

iy ˆ ˆ ˆ ˆnGLS* * *X β + μ i + μ + μui δ λ ii

ˆ ˆ ˆ ˆ

'* * * *μ = μ ,μ , .. μλ λ λ λn1 2

*λ

*δ ti

μ andμ*u i

μ E

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

(66)

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.

MG GDPln (Gas/Car) = α + β ln (Y/N) + β ln (P /P ) + β ln (Car/N) + ε ,1 2 3 it

Table: 1

One-way error component model estimates __________________________________________________________________ Methods of Estimation Parameter WAHU AM SWAR FUBA α 1.9058 2.1844 1.9967 2.0203

(0.1661) (0.2151) (0.1843) (0.1882)

1β 0.5434 0.6009 0.5550 0.5599 (0.0544) (0.0656) (0.0591) (0.0600)

2β -0.4711 -0.3664 -0.4204 -0.4118 (0.0389) (0.0415) (0.0399) (0.0402)

3β -0.6061 -0.6204 -0.6068 -0.6081

(0.0243) (0.0272) (0.0255) (0.0257)

21ˆ 0.030071 0.11420 0.038238 0.044041

22ˆ 0.062094 0.090217 0.072238 0.074485

23ˆ 0.071708 0.099850 0.081794 0.084041

2

1ˆ u 0.013509 0.008446 0.008524 0.008525 2

2ˆ u 0.008360 0.008064 0.008192 0.008167 2

3ˆ u 0.008909 0.008594 0.008729 0.008704

: denotes first stage estimates.

: are ANOVA type estimates based on GLS residuals.

: are ANOVA type estimates adjusted for degrees of

freedom, based on GLS residuals.

21

22

23

Table: 2 One-Way Error Component Model

Point and 95 % Confidence Interval Estimates for Cross Country Effect ( i )

Based on Fuller-Battese Method

Country LB 1_ˆi

UB LB 2_ˆi

UB LB 3_ˆi

UB Austria -0.15782 -0.11651 -0.07520 -0.15860 -0.11703 -0.07545 -0.15884 -0.11718 -0.07552 Belgium -0.24255 -0.20124 -0.15994 -0.24370 -0.20213 -0.16056 -0.24407 -0.20240 -0.16074 Canada 0.57070 0.61201 0.65332 0.57315 0.61472 0.65629 0.57388 0.61554 0.65720 Denmark -0.01556 0.02575 0.06705 -0.01571 0.02586 0.06743 -0.01577 0.02590 0.06756 France -0.20431 -0.16300 -0.12170 -0.20529 -0.16372 -0.12215 -0.20561 -0.16394 -0.12228 Germany -0.27841 -0.23710 -0.19580 -0.27972 -0.23815 -0.19658 -0.28013 -0.23847 -0.19681 Greece -0.05409 -0.01279 0.02852 -0.05441 -0.01284 0.02873 -0.05452 -0.01286 0.02880 Ireland 0.13131 0.17261 0.21392 0.13181 0.17338 0.21495 0.13195 0.17361 0.21527 Italy -0.20488 -0.16357 -0.12226 -0.20587 -0.16429 -0.12272 -0.20618 -0.16451 -0.12285 Japan -0.02880 0.01250 0.05381 -0.02901 0.01256 0.05413 -0.02909 0.01258 0.05424 Netherlands -0.17882 -0.13751 -0.09620 -0.17969 -0.13812 -0.09655 -0.17997 -0.13830 -0.09664 Norway -0.17329 -0.13198 -0.09068 -0.17414 -0.13257 -0.09100 -0.17441 -0.13274 -0.09108 Spain -0.56317 -0.52186 -0.48055 -0.56574 -0.52417 -0.48260 -0.56653 -0.52487 -0.48321 Sweden 0.18602 0.22732 0.26863 0.18676 0.22833 0.26990 0.18697 0.22864 0.27030 Switzerland -0.06980 -0.02849 0.01282 -0.07019 -0.02862 0.01295 -0.07032 -0.02866 0.01301 Turkey 0.06224 0.10355 0.14485 0.06244 0.10401 0.14558 0.06248 0.10415 0.14581 U.K. -0.09551 -0.05421 -0.01290 -0.09602 -0.05445 -0.01287 -0.09618 -0.05452 -0.01285 U.S.A. 0.57321 0.61452 0.65583 0.57567 0.61724 0.65881 0.57640 0.61806 0.65973

Table: 3 One-Way Error Component Model Absolute Prediction Error in Percentage

In-Sample n =18, m = 19 N = 342

Out of-Sample n =18, m = 4 N = 72 Method

Pred. Type

Mean Std. Dev. Min. Max. Mean

Std. Dev. Min. Max.

Wallace T/LG_1 1.49023 1.38969 0.00042 8.02895 4.46491 4.33748 0.00471 15.64474 and SB_1 1.45088 1.35907 0.00824 7.84881 4.46747 4.33841 0.00010 15.64853 Hussein T/LG_2 1.48399 1.39239 0.00248 8.03274 4.14864 4.20766 0.07061 15.12592 SB_2 1.47480 1.38147 0.00838 7.97819 4.14945 4.20806 0.07250 15.12743 T/LG_3 1.48404 1.39226 0.00039 8.03312 4.15007 4.20837 0.07393 15.12859 SB_3 1.47722 1.38374 0.00839 7.99128 4.46570 4.33776 0.00330 15.64590 Amemiya T/LG_1 1.46628 1.37290 0.00138 7.69096 4.72054 4.59062 0.11957 16.42695 SB_1 1.46233 1.36589 0.00089 7.65828 4.72180 4.59114 0.11700 16.42903 T/LG_2 1.46605 1.37317 0.00174 7.69154 4.41823 4.44542 0.12902 15.86412 SB_2 1.46119 1.36483 0.00089 7.65230 4.41935 4.44605 0.13199 15.86652 T/LG_3 1.46638 1.37278 0.00063 7.69074 4.41930 4.44602 0.13186 15.86641 SB_3 1.46278 1.36631 0.00089 7.66064 4.72166 4.59108 0.11730 16.42880 Swamy T/LG_1 1.47356 1.38062 0.00457 7.87822 4.53436 4.39655 0.01311 15.93431 and Arora SB_1 1.45785 1.36252 0.00330 7.78484 4.53666 4.39735 0.01736 15.93772 T/LG_2 1.47316 1.38030 0.00041 7.87722 4.21162 4.26563 0.01249 15.39819 SB_2 1.46626 1.37038 0.00332 7.82974 4.21251 4.26613 0.01028 15.39996 T/LG_3 1.47344 1.37995 0.00640 7.87698 4.21290 4.26635 0.00930 15.40076 SB_3 1.46828 1.37227 0.00332 7.84056 4.53528 4.39687 0.01481 15.93567 Fuller T/LG_1 1.47184 1.37904 0.00494 7.85012 4.55079 4.41156 0.00425 15.99399 And Battese SB_1 1.45860 1.36306 0.00125 7.76839 4.55290 4.41243 0.00013 15.99730 T/LG_2 1.47158 1.37881 0.00055 7.84900 4.22779 4.28000 0.03595 15.45442 SB_2 1.46505 1.36909 0.00126 7.80277 4.22872 4.28051 0.03367 15.45626 T/LG_3 1.47184 1.37848 0.00245 7.84865 4.22907 4.28071 0.03279 15.45696 SB_3 1.46701 1.37092 0.00126 7.81320 4.55169 4.41193 0.00249 15.99540

Table 4Two-Way Error Component Model Estimates

Methods of Estimation Parameter WAHU AM SWAR FUBA α 1.9101 -0.2391 2.0408 1.0308

(0.1672) (0.3501) (0.1915) (0.2660)

1β 0.5433 0.1682 0.5645 0.3947 (0.0547) (0.0804) (0.0608) (0.0683)

2β -0.4672 -0.2322 -0.4049 -0.3390 (0.0390) (0.0411) (0.0404) (0.0414)

3β -0.6058 -0.6024 -0.6094 -0.6096

(0.0244) (0.0258) (0.0259) (0.0256)

21ˆ 0.031875 0.18261 0.038340 0.046706

22ˆ 0.066418 0.15203 0.080906 0.13047

23ˆ 0.086992 0.19891 0.105930 0.13047

2

1ˆ 0 0.017412 0 0.002038 2

2ˆ 0 0.011753 0 0.002989 2

3ˆ 0 0.015211 0 0.003956

2

1ˆ u 0.013653 0.006526 0.006591 0.006591 2

2ˆ u 0.008914 0.006608 0.008661 0.007430 2

3ˆ u 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 1_

ˆi

UB LB 2_ˆi

UB LB 3_ˆi

UB Austria -0.15730 -0.11573 -0.07417 -0.16199 -0.11622 -0.07045 -0.16440 -0.11632 -0.06824 Belgium -0.20954 -0.16798 -0.12641 -0.21445 -0.16868 -0.12291 -0.21690 -0.16883 -0.12075 Canada 0.70206 0.74363 0.78520 0.70098 0.74675 0.79252 0.69932 0.74740 0.79547 Denmark 0.03555 0.07712 0.11869 0.03167 0.07745 0.12322 0.02944 0.07751 0.12559 France -0.17856 -0.13699 -0.09543 -0.18334 -0.13757 -0.09180 -0.18576 -0.13769 -0.08961 Germany -0.23031 -0.18874 -0.14717 -0.23530 -0.18953 -0.14376 -0.23777 -0.18970 -0.14162 Greece -0.16852 -0.12695 -0.08539 -0.17326 -0.12749 -0.08171 -0.17567 -0.12760 -0.07952 Ireland 0.06903 0.11060 0.15217 0.06529 0.11107 0.15684 0.06309 0.11116 0.15924 Italy -0.26615 -0.22459 -0.18302 -0.27130 -0.22553 -0.17976 -0.27380 -0.22573 -0.17765 Japan -0.06526 -0.02369 0.01787 -0.06957 -0.02379 0.02198 -0.07189 -0.02381 0.02426 Netherlands -0.15398 -0.11241 -0.07085 -0.15866 -0.11288 -0.06711 -0.16106 -0.11298 -0.06491 Norway -0.12771 -0.08614 -0.04458 -0.13228 -0.08651 -0.04073 -0.13466 -0.08658 -0.03850 Spain -0.57212 -0.53056 -0.48899 -0.57856 -0.53279 -0.48701 -0.58132 -0.53325 -0.48517 Sweden 0.07010 0.11167 0.15324 0.06636 0.11214 0.15791 0.06416 0.11223 0.16031 Switzerland -0.00736 0.03420 0.07577 -0.01143 0.03435 0.08012 -0.01370 0.03438 0.08245 Turkey -0.14617 -0.10460 -0.06304 -0.15082 -0.10504 -0.05927 -0.15321 -0.10513 -0.05706 U.K. -0.07939 -0.03782 0.00375 -0.08375 -0.03798 0.00779 -0.08609 -0.03801 0.01006 U.S.A. 0.73743 0.77900 0.82056 0.73649 0.78227 0.82804 0.73487 0.78294 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 1_ˆi UB LB 2_

ˆi UB LB 3_

ˆi UB

1960 -0.10644 -0.03834 0.02975 -0.11888 -0.03145 0.05598 -0.13213 -0.03201 0.06811 1961 -0.09808 -0.02998 0.03812 -0.11202 -0.02459 0.06284 -0.12515 -0.02503 0.07509 1962 -0.10192 -0.03382 0.03428 -0.11517 -0.02774 0.05969 -0.12836 -0.02824 0.07188 1963 -0.10134 -0.03324 0.03485 -0.11469 -0.02726 0.06016 -0.12787 -0.02775 0.07237 1964 -0.09254 -0.02444 0.04366 -0.10747 -0.02004 0.06738 -0.12052 -0.02040 0.07972 1965 -0.08960 -0.02151 0.04659 -0.10507 -0.01764 0.06979 -0.11807 -0.01795 0.08217 1966 -0.07768 -0.00958 0.05852 -0.09528 -0.00786 0.07957 -0.10812 -0.00800 0.09212 1967 -0.07010 -0.00201 0.06609 -0.08907 -0.00165 0.08578 -0.10179 -0.00167 0.09845 1968 -0.06280 0.00530 0.07340 -0.08308 0.00435 0.09177 -0.09570 0.00442 0.10454 1969 -0.06683 0.00127 0.06937 -0.08639 0.00104 0.08847 -0.09906 0.00106 0.10118 1970 -0.06132 0.00678 0.07488 -0.08187 0.00556 0.09299 -0.09446 0.00566 0.10578 1971 -0.05421 0.01389 0.08198 -0.07604 0.01139 0.09882 -0.08853 0.01159 0.11171 1972 -0.05001 0.01809 0.08619 -0.07259 0.01484 0.10227 -0.08502 0.01510 0.11522 1973 -0.04376 0.02434 0.09244 -0.06746 0.01996 0.10739 -0.07980 0.02032 0.12044 1974 -0.05522 0.01288 0.08098 -0.07686 0.01057 0.09799 -0.08937 0.01075 0.11087 1975 -0.04018 0.02792 0.09602 -0.06453 0.02290 0.11033 -0.07681 0.02331 0.12343 1976 -0.04229 0.02581 0.09390 -0.06626 0.02116 0.10859 -0.07858 0.02154 0.12166 1977 -0.03981 0.02829 0.09639 -0.06423 0.02320 0.11063 -0.07650 0.02362 0.12374 1978 -0.03974 0.02836 0.09645 -0.06417 0.02326 0.11068 -0.07645 0.02367 0.12379

Table: 7 Two-Way Error Component Model

Absolute Prediction Error in Percentage

Method

In-Sample n =18, m = 19 N = 342

Out of -Sample n =18, m = 4 N = 72

Pred. Type

Mean Std. Dev. Min. Max. Mean

Std. Dev. Min. Max.

Wallace BK_1 1.48853 1.38900 0.00265 8.01828 4.44417 4.32244 0.08768 15.54077 and SB_1 1.45149 1.35948 0.00041 7.84597 4.44912 4.32399 0.09620 15.54778 Hussein BK_2 1.48300 1.39134 0.00153 8.02139 4.46674 4.32958 0.12650 15.57273 SB_2 1.47380 1.38037 0.00042 7.96658 4.46770 4.32989 0.12815 15.57409 BK_3 1.48306 1.39119 0.00198 8.02172 4.47036 4.33074 0.13076 15.57785 SB_3 1.47615 1.38257 0.00042 7.97925 4.47182 4.33121 0.12819 15.57993 Amemiya BK_1 2.47954 1.92787 0.00468 11.86577 7.71453 7.09830 0.11524 27.04638 SB_1 1.48311 1.36907 0.01103 9.53317 6.66773 6.73220 0.06527 25.64423 BK_2 2.47941 1.92789 0.00551 11.86797 7.70709 7.09640 0.10471 27.03769 SB_2 1.54926 1.40013 0.01632 9.71797 6.76351 6.76448 0.09864 25.78135 BK_3 2.47957 1.92786 0.00446 11.86518 7.70978 7.09709 0.10852 27.04083 SB_3 1.55094 1.40124 0.01588 9.72511 6.72575 6.75241 0.03522 25.72830 Swamy BK_1 1.47052 1.37798 0.00146 7.82731 4.64396 4.50365 0.05252 16.26869 and Arora SB_1 1.45915 1.36347 0.00223 7.75425 4.64564 4.50429 0.05571 16.27134 BK_2 1.47045 1.37770 0.00125 7.82622 4.64394 4.50364 0.05249 16.26867 SB_2 1.46410 1.36810 0.00224 7.78057 4.64563 4.50428 0.05569 16.27132 BK_3 1.47071 1.37738 0.00242 7.82581 4.64681 4.50473 0.05791 16.27316 SB_3 1.46539 1.36974 0.00111 7.78897 4.65163 4.50656 0.06704 16.28072 Fuller BK_1 1.78864 1.48110 0.00254 9.66659 5.63341 5.32737 0.16250 20.39134 And Battese SB_1 1.49509 1.33861 0.00374 8.82149 5.30760 5.30888 0.00167 20.04742 BK_2 1.78980 1.47994 0.00228 9.65778 5.64040 5.32788 0.15296 20.39870 SB_2 1.54527 1.36379 0.00345 8.98992 5.39850 5.31337 0.06515 20.14384 BK_3 1.79016 1.47962 0.00065 9.65554 5.64373 5.32812 0.14841 20.40222 SB_3 1.54303 1.36343 0.00408 8.98725 5.38002 5.31251 0.04787 20.12437

5. SummaryWe consider the one-way error component model, i.e.

and propose an estimate of δi by the estimate of E(δi | εi),where, 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 cross-sectional unit is provided, i.e.,

where = E(ui | εi) , is also included besides E(δi | εi).

*

it it i ity = α + x β + δ + u i = 1,2...m, t = 1,2...n

,uiδε inii

iniGLSiiuiδβXy

iu