La...

as aw
..u..‘w._ mm.”

~. 14.
www
.71... fit .3.

kw » u

.

.-
II
7.

WC! . 4. ‘ 4 . “V:
u 44 I
A M: a. 5&3? ,
k 11:3».
1 .. ~£:T..
1" . 4? :1
5 i. » 2&3.»
. L1! In . ‘ A} y
.mvwvfiia f. . <
L. in. , {51"
3...?
a;
.l‘ ,r
.
.i
«in
a? 1.2.3.

11.522.

 

.-

s 3.1}

‘1", xvsitvtil

3% 13231039.
.I\S

list.
.iIn. \1
Vi}...
I1

 

. l‘ .
:5...)
a r...

QOOq

 

URBARY
Michigan State
University

 

 

 

This is to certify that the
dissertation entitled

THREE ESSAYS IN ECONOMETRICS

presented by

PAN UTAT SATC HAC HAI

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Economics

 

m swr

 

Major Professor’s Signature

your 22, 2009

 

Date

MSU is an Affirmative ActiorVEquaI Opportunity Employer

 

.—.- -.--.-

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5/08 K IProj/Accapres/CIRCIDateDue nndd

 

THREE ESSAYS IN ECONOMETRICS
By

Panutat Satchachai

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILSOPHY
Economics

2009

ABSTRACT
THREE ESSAYS IN ECONOMETRICS
By

Panutat Satchachai

In the first chapter, we consider GMM estimation when there are more moment
conditions than observations. Due to the singularity of the estimated variance matrix of
the moment conditions, the quick solution of using the generalized inverse, although
temping, is shown to be unfruitful.

In the second and third chapters, we consider the problem of point estimation of
technical inefficiency in a simple stochastic frontier model with panel data.

In the second chapter, we wish to correct the bias of the estimates of technical
inefficiency based on fixed effects estimation that previously shown to be biased upward.
Previous work has attempted to correct this bias using the bootstrap, but in simulations

the bootstrap correct only part of the bias. The usual panel jackknife is based on the

assumption that the bias is of order T'1 and is similar to the bootstrap. We show that

"2 , not T" , and

when there is a tie or a near tie for the best firm, the bias is of order T‘
this calls for a different form of the jackknife. The generalized panel jackknife is quite
successfully in removing the bias. However, the resulting estimates have a large
variance.

In the third chapter, we focus on how we could decrease the variance and MSE of
a jackknife-type estimate of the frontier intercept found in the previous chapter. We

consider the split-sample jackknife proposed by Dhaene, Jochmans and Thuysbaert

(2006), which is simply two times the original estimate based on the whole sample minus

the average of the two half-sample estimates, and the “generalized” version proposed by
Satchachai and Schmidt (2008), which is relevant in the case of an exact tie or a near tie.
Although these estimators also successfully remove the bias, their variance is still large.
We also consider whether or not there is an “optimal” split-sample jackknife estimator
that has small variance and/or small MSE. For a special case of N = 2 , we derive the
“optimal” weights for the original estimate and the half-sample estimates. Although the
“optimal” split-sample jackknife has even smaller variance and MSE, it does not properly
remove the bias, and it appears that there is not much gain in terms of mean square error

from applying the jackknife procedure.

ACKNOWLEDGEMENTS

First and foremost, I would like to thank my advisor, Professor Peter Schmidt.
Without his guidance and encouragement, this thesis would have not appeared in its
finished form. His teaching approach, research experience and knowledge in economics
also help me developing my teaching method as well as training and shaping the direction
of my research interests and of this dissertation. His help in editing the dissertation is
also gratefully acknowledged. I would also like to thank Assistant Professor Emma
Iglesias who was always willing to spend time in clarifying some doubts encountered and
gave invaluable feedback during the process of writing this dissertation, and to Professor
Timothy Vogelsang for his invaluable comments. I am also grateful for the teaching
opportunity given by the Department of Economics through the graduate assistantship.

I am also thankful of my undergraduate friends, fellow graduate students and the
staff at the Department of Economics who also contributed to the completion of this
dissertation through their friendship and help. A special thank you also goes to my
friends at the Michigan State University badminton club and the Thai student community
at Michigan State University for their warm-hearted friendship.

Finally, I am greatly indebted to and thankful of my parents for their support, both
financially and mentally. My special thanks also go to my sisters for their love and
encouragement and to my grandmother who was always supportive of my decision. I

could only wish that she could share this joyfirl moment with me.

iv

TABLE OF CONTENTS

LIST OF TABLES ................................................................................. vii
Chapter 1
GMM with More Moment Conditions than Observations ................................. l
1.1 Introduction ............................................................................ 2
1.2 The Model .............................................................................. 2
1.3 TheCaseofn=1 ..................................................................... 4
1.4 The General Case ..................................................................... 5
1.5 Concluding Remarks .................................................................. 7
1.6 Appendix: Proof of Result 1.2 ...................................................... 7
1.7 Appendix: Proof of Result 1.3 ...................................................... 8
Chapter 2
Estimates of Technical Inefficiency in Stochastic Frontier Models with Panel Data:
Generalized Panel Jackknife Estimation ....................................................... 13
2.1 Introduction ........................................................................... 13
2.2 Fixed Effects Estimation of the Model ........................................... 15
2.3 Deriving the Order in Probability of the Bias .................................... 18
2.3.1 The Case of No Tie ...................................................... 19
2.3.2 The Case of an Exact Tie ................................................ 20
2.3.3 The Case of a Near Tie ................................................... 21
2.4 Correcting Bias with the Panel Jackknife and the Generalized Panel
Jackknife ............................................................................. 23
2.4.1 The Panel Jackknife ...................................................... 23
2.4.2 The Generalized Jackknife .............................................. 25
2.4.3 The Generalized Panel Jackknife When Bias is of Order T "V 2 ....28
2.4.4 What If The Wrong Jackknife Is Used? ......................................... 29
2.5 Design of the Monte Carlo Experiments .......................................... 30
2.6 Results of the Monte Carlo Experiments ......................................... g 35
2.7 Concluding Remarks ................................................................ 39
2.8 Output Tables ........................................................................ 42
2.9 Appendix: Proof of Lemma 2.2 ................................................... 54
2.10 Appendix: Supplementary Tables ................................................. 58
Chapter 3
Estimating Stochastic Frontier Models with Panel Data Using Split-Sample
Jackknife ............................................................................................. 94
3.1 Introduction ........................................................................... 94
3.2 The Model ............................................................................ 96
3.3 The Split-Panel Jackknife .......................................................... 97
3.3.1 No Tie ...................................................................... 98
3.3.2 An Exact Tie .............................................................. 98

3.3.3 What If The Wrong Version Is Used? ........................................... 99

3.4 The “Optimal” Split-Panel Jackknife ............................................ 101
3.4.1 Unconstrained “Optimal” Split-Panel Jackknife ................... 103

3.4.2 Constrained “Optimal” Split-Panel Jackknife ...................... 104

3.5 Simulations ......................................................................... 107
3.5.1 Design of Experiments ................................................. 108

3.5.2 Results .................................................................... 110

3.6 Conclusions ......................................................................... 112
3.7 Output Tables ....................................................................... 114
3.8 Appendix: Deriving the Expected Value and Variance of the Max. . . . . 122
3.9 Appendix: The “Optimal” Split-Sample Jackknife ............................ 126
3.9.1 § that minimizes MSE(5£) without constraint ..................... 127

3.9.2 5.7 that minimizes MSE(Ei) subject to a0 + a] +a2 =1 .......... 128

3.9.3 07 that minimizes va.r(b?) subject to a0 + a1 +a2 =1 ............ 128
BIBLIOGRAPHY ................................................................................ 129

vi

LIST OF TABLES

2.1 (Experiment I: No Tie) T =10 , Bias of the Estimates .................................. 42
2.2 (Experiment I: No Tie) T =10 , Variance of the Estimates .............................. 43
2.3 (Experiment I: No Tie) T =10 , MSE of the Estimates .................................. 44
2.4 (Experiment 11: Exact Tie) T =10 , Bias of the Estimates .............................. 45
2.5 (Experiment 11: Exact Tie) T =10 , Variance of the Estimates ......................... 46
2.6 (Experiment 11: Exact Tie) T =10 , MSE of the Estimates .............................. 47
2.7 (Experiment III: Near Tie) T =10 , Bias of the Estimates .............................. 48
2.8 (Experiment III: Near Tie) T =10 , Variance of the Estimates ......................... 49
2.9 (Experiment III: Near Tie) T =10 , MSE of the Estimates .............................. 50
2.10 (Effect of Changing T) p: = 1, N = 20 , Bias of the Estimates ........................ 51
2.11 (Effect of Changing T) ,u: =1, N = 20 , Variance of the Estimates .................. 52
2.12 (Effect of Changing T) p: =1, N = 20 , MSE of the Estimates ........................ 53
2.13 (Experiment I: No Tie) T = S , Bias of the Estimates .................................... 58
2.14 (Experiment I: No Tie) T = 5 , Variance of the Estimates .............................. 59
2.15 (Experiment I: No Tie) T =5, MSE ofthe Estimates....................................60
2.16 (Experiment 11: Exact Tie) T = 5 , Bias of the Estimates ................................ 61
2.17 (Experiment 11: Exact Tie) T = 5 , Variance of the Estimates ........................... 62
2.18 (Experiment 11: Exact Tie) T = 5 , MSE of the Estimates ............................... 63
2.19 (Experiment III: Near Tie) T = 5 , Bias of the Estimates ................................ 64
2.20 (Experiment III: Near Tie) T = 5 , Variance of the Estimates ........................... 65
2.21 (Experiment III: Near Tie) T = 5 , MSE of the Estimates ............................... 66

vii

2.22 (Experiment I: No Tie) T = 20 , Bias of the Estimates .................................. 67

2.23 (Experiment I: No Tie) T = 20 , Variance of the Estimates ............................. 68
2.24 (Experiment I: No Tie) T = 20 , MSE of the Estimates ................................. 69
2.25 (Experiment 11: Exact Tie) T = 20 , Bias of the Estimates .............................. 70
2.26 (Experiment II: Exact Tie) T = 20 , Variance of the Estimates ........................ 71
2.27 (Experiment 11: Exact Tie) T = 20 , MSE of the Estimates .............................. 72
2.28 (Experiment III: Near Tie) T = 20 , Bias of the Estimates .............................. 73
2.29 (Experiment III: Near Tie) T = 20 , Variance of the Estimates ........................ 74
2.30 (Experiment III: Near Tie) T = 20 , MSE of the Estimates .............................. 75
2.31 (Experiment I: No Tie) T = 50 , Bias of the Estimates .................................. 76
2.32 (Experiment I: No Tie) T = 50 , Variance of the Estimates ............................. 77
2.33 (Experiment I: No Tie) T = 50 , MSE of the Estimates ................................. 78
2.34 (Experiment 11: Exact Tie) T = 50 , Bias of the Estimates .............................. 79
2.35 (Experiment 11: Exact Tie) T = 50 , Variance of the Estimates ........................ 80
2.36 (Experiment 11: Exact Tie) T = 50 , MSE of the Estimates .............................. 81
2.37 (Experiment III: Near Tie) T = 50 , Bias of the Estimates .............................. 82
2.38 (Experiment III: Near Tie) T = 50 , Variance of the Estimates ........................ 83
2.39 (Experiment III: Near Tie) T = 50 , MSE of the Estimates .............................. 84
2.40 (Experiment I: No Tie) T = 100 , Bias of the Estimates ................................. 85
2.41 (Experiment I: No Tie) T = 100 , Variance of the Estimates ........................... 86
2.42 (Experiment I: No Tie) T = 100 , MSE of the Estimates ................................ 87
2.43 (Experiment 11: Exact Tie) T = 100 , Bias of the Estimates ............................ 88
2.44 (Experiment 11: Exact Tie) T = 100 , Variance of the Estimates ........................ 89

viii

2.45 (Experiment 11: Exact Tie) T = 100 , MSE of the Estimates ........................... 90

2.46 (Experiment III: Near Tie) T = 100 , Bias of the Estimates ............................ 91
2.47 (Experiment III: Near Tie) T = 100 , Variance of the Estimates ....................... 92
2.48 (Experiment III: Near Tie) T = 100 , MSE of the Estimates ........................... 93
3.1 (Experiment I: No Tie) T =10 , Bias of the Estimates ................................. 114
3.2 (Experiment I: No Tie) T = 10 , Variance of the Estimates ........................... 115
3.3 (Experiment I: No Tie) T = 10 , MSE of the Estimates ................................ 116
3.4 (Experiment 11: Exact Tie) T =10 , Bias of the Estimates ............................ 117
3.5 (Experiment II: Exact Tie) T =10 , Variance of the Estimates ........................ 118
3.6 (Experiment II: Exact Tie) T = 10 , MSE of the Estimates ............................ 119
3.7 Weight Comparisons between Estimators, N = 2 and T =10 ........................ 120

3.8 N = 2 (Restricted), Bias, Variance and MSE of the “Optimal” Split-Sample
Jackknife Estimates ........................................................................ 121

ix

CHAPTER ONE

Satchachai, Panutat and Peter Schmidt, 2008, GMM with More Moment Conditions than
Observations, Economics Letters, 99, 252-255.

Chapter 1

GMM with More Moment Conditions than Observations

1.1 Introduction

In this paper, we consider GMM estimation when there are more moment
conditions than observations. This can occur in practice, for example, in dynamic panel
data models, as in Han et a1. (2005). It is well known that the optimal weighting matrix
for GMM is the inverse of the variance matrix of the moment conditions. However,
when there are more moment conditions than observations, the usual estimate of the
variance matrix of the moment conditions is singular. In that case it is tempting to use its
generalized inverse as the weighting matrix. The point of this paper is to demonstrate
that this is not a good idea. When the continuous updating form of GMM is used, the
value of the criterion function equals one for all values of the parameter. When the two
step GMM estimate is used, the value of the criterion function is less than or equal to one,

and again the usefulness of such procedure is doubtful.

1.2 The Model
We consider GMM estimation based on the population moment conditions
E(g(.v,6’o))=0- (1-1)
Here g is a k X] vector of moment conditions, and 90 is the population value of
a p - dimensional parameter 6. We assume k 2 p so there are enough moment

conditions to identify 60.

The data y] ,..., y n are a random sample from a population that satisfies (1.1). In

the usual case, n > k , but in this paper we consider cases where n < k .

We define the sample moment conditions as
1 n
a. =;Zg(y,-,6). (1.2)
i=1

Let C = Eg(y,-,6l)g(y,- , 6), , the variance matrix of the population moment conditions.

The usual estimate of C is

A

me) = £2g<yp6>g<yp0i . (1.3)

i =1
Note that C" (6) is singular when n < k.
We now distinguish two different forms of the GMM estimator. The “continuous
updating” estimator éCUE is the value of 6 that minimizes the criterion function
EUEM)=§.(6)’én(6)“§.(6>. (1.4)
The “two step” estimator (92 STEP is the value of 6 that minimizes the criterion function
35mm)=§n(6>'én(é)“§n(6). (1.5)
where (9 is some initial (consistent) estimator.
When n < k , these estimates do not exist. However, we can replace the inverse
C" (6)"1 and C" ((9)4 by the “generalized inverse” C” (t9)+ and C" (60+ , in (1.4) and

(1.5) respectively.l For example, Han, Orea and Schmidt (2005) follow this procedure.

 

I If for any m x n matrix H with rank of r , then it is always possible to find matrices R ,
a m x r matrix, and S , each having a rank of r , such that H = RS . Then the

generalized inverse of H , H + , is

l. 3 The Case of n =1
Suppose that n =1, so §n(6) = g(y1,6).

We note that, if A is a k x k matrix ofrank 1, so that A = :g' , where 4’ is a

kxl,then A+ =(§'§)’2A.

Result 1.1 (i) QEUE (6) =1 for all 6.
(ii) Q35TEP(éstEP) $1-
Proort (i) QSUEre)=gm,6>'[g(yr.6)g(y1,6)'1+g<y1,6)

= gm ,a'gryl :9)[g(y1,9)g(y1 ,6)'1'2 gty1.6)'g(yr.6)
= 1.

(ii) Q35”? (éZSTEP) s QESTEP (é) = 1 for an arbitrary é. a
Part (i) of this result says that éCUE is not defined, because the criterion function

SUE (6) equals one for all 6. Clearly this criterion function contains no information at

all about 6.

Part (ii) of this result says that the minimized value of Q35 TEP (62 STEP) is

bounded above by one. It may or may not equal one.

 

H+ = (RS)+ = S'(SS')"1(R'R)‘1R'.
The generalized inverse (or “Moore-Pemose pseudo inverse”) H + is the unique matrix
that satisfies the following properties
(i) HH+ and H +H are symmetric;
(ii) H+HH+ = H+; and
(iii) HH+H = H.
If H is invertible, then H + = H ’1.

Example 1.1 Suppose that y is k X], and every element of y has expectation equal to
,u. Thus we have
E<y—yek)=0 (1.6)

where e k is a k x1 vector of ones. We are not restricting the variance matrix of y. We

have one observation, yl (k x1). Note that an obvious estimator is [I = -Il:e},y1, and this

would have certain optimality properties if the elements of y are uncorrelated and have

equal variance, but not otherwise.

By part (i) of Result 1.1, the continuous updating estimate of ,u is undefined. We

have (y-wk) [(y-l‘ek)(}"#ek) ]+(y-#ek) =1 for all #-
An interesting result that holds for this example is that the two step estimate based

on the initial estimate [1 (as above) is also undefined. We have

[(Y‘WkTU-fleknz.
[ty-izek) (y-izemz

 

(y—pekiKy—fiek)(y-fleh'rty—pek)=

and this equals one for all ,u , because e}, (y - jiek ) = 0 for this specific choice of [1.

1.4 The General Case

The problem is exactly as Section 1.2. We define ‘g’n (6) as in equation (1.2), and

the criterion function QSUE (6) and Q3STEP (6) as in equation (1.4) and (1.5). We have

n observations, with n < k.

One basic result is essentially the same as for the case of n = 1.

Result 1.2 (i) QSUE (9) :1 for all 0.

(ii) QriSTEP (éZSTEP) S 1 -
Proof. See Appendix. [3
So, as in the case of n = 1, the CUE criterion function contains no information

about 6. Similarly QESTEP (6) is bounded above by one, and is not of any obvious use.

Unlike the case of n =1, it does not seem generally possible to say anything

useful about circumstance in which QESTEP (62 STEP) = l . An exception is the following

example.

Example 1.2 This is the same as Example 1.1 except that now n = 2 . So we observe y]

and y2 , where E(y,-) = pek and V(y,-) = 2, an unrestricted k x k matrix, for i= 1,2.

Define y = %(y] +y2), which is k X], and define the scalar T = e}?

1

= 2—k 21:12,]; y i]- . Let [i be the two step estimator based on the initial estimator i.

That is, )7 minimizes the criterion function
+

2 I
Z<y.-—§et)(y,-—iek) (Ix—wk). (1.7)

i=1

1

ZSTEP
" 2

(#)=(T—Wk)’

From Result 1.2 we know that Q3575}, (,2?) s l. The following result (provided in the

Appendix) gives the condition for equality.

Result 1.3 In Example 1.2, we have )7 = i and QESTEPUT) =1 , if

k k
2011' ‘71)2 =ZU’2)‘ -?2)2- (13)

i=1 i=1

1.5 Concluding Remarks

When there are more moment conditions than observations, the usual estimate of
the variance matrix of the moment conditions is singular, and so the usual “optimal”
weighting matrix cannot be calculated. This paper shows that this problem cannot be
solved by using the generalized inverse of the estimated variance matrix.

In such cases, one can always just drop moment conditions. Of course, then the
question is which ones to drop, and in particular whether any rule based on the data will
be useful. In work not summarized in this paper, we have investigated the use of
principal components, as suggested by Doran and Schmidt (2006) for cases of near-
singularity. We were unable to come up with any solid results that would indicate that
principal components are useful when the estimated weighting is singular, although this

remains a topic worth exploring.

1.6 Appendix: Proof of Result 1.2

Define 6(6) = [g(y1,6),...,g(yn,6)], k x n. Let en be an n x1 vector ofones.
Then 3(6) = lame, and (3,,(9) = ~1—G(6)G(6)'. Therefore the CUE criterion
n n

fimction is

1 I r I
95(0) =;e,.G(6) [0050009) Immen. (1.9)
Now we use the fact that, for A such that AA' is singular but A'A is nonsingular,

(AA')+ = A(AA')_2A'. Therefore

SUE (6) = gegctai 0(0)[G<6>' 0(6>]'2 6060' G<6>en

=1,;1en (1.10)
n
=1.

The proof of part (ii) is exactly the same as for the case of n = 1.

1.7 Appendix: Proof of Result 1.3

Let [2 = i be the initial estimate. Let xi = (y,- —flek) for i= 1,2 and

1 2 . . .+ 1 .+
X=[xl x2]. LetB=|:32i=1(yi‘#ek)(J’i-#ek)] =(5XX) .Thenthetwo

step efficient GMM estimator is

I

‘3ka
ejcBek O

 

x7=argmin(r—izek)B(r—xiet) =
,u

We can rewrite Ii as

 

 

and we can rewrite (y—fiek) as %Xe2 +(fi—Ii)ek.

Hence,
Q(fi)=(?-flek) BG-flek)
1 A ~ ' 1 S N
= [2X62 +(p—p)ek) B(-2-X82 +(#".U)ek)
l.’ a~2r 1...... l~~"
=1?er BXez +(p—,u) ekBek +§(,u-,U)ekBXez 701-7!)er Bet

= fieeXithX'XflX' e2 +(fi-fi)zeiBe/t +(x‘2-me'2X'Ber

2
r ' BX I
=l(26'282)+ -lek'BX82 ekBek -lek' e2 e'zX Bek
4 2 ekBek 2 ekBek

 

 

=1+l(eitBXez)2 __1_(eitBX92)2
4 ejBek 2 ejBek

_1_1(e),(2X(X'X)‘2X'))(e2)2
4 ejcBek

_1__1(e;,X(X'X)“ez)2
2 ejcBek
<1.

 

We have equality when
e},X(X'X)"'e2 = 0. (1.11)
Now we show that this occurs when condition (1.8) of Result 1.3 holds.

We note that

erX=kl(rr—iz> (re—72)]. (1.12)

Adding and subtracting 719k and he k , and rewriting, we have
yr #3 = 0/1 -?19k)+(71 -fi)ek;
y2 -fi = (Y2 —?2ek)+(72 'fi)ek-

Now define A = 571 — 72. Then we can rewrite (1.12) as

ekX = é—kAfi —1]. (1.13)

Lemma 1.1 Define Sij = (y,- —y,-ek )'(yj —yjek) for i,j =1,2. Then

1 2 1 2
s +—kA s -—kA
11 4 12 4

I

X X: 1 2 1 2 .
s ——kA s +—kA
2| 4 22 4

Proof

(Yr-9190,01 —5>rer> =I(y1 —rret)+<'y‘r —iz)etll(yr ~rret>+trr —fi)er]
_(y1-?rek)+ _ (J’l-ylek)+

_ 1_ _ _ 1- _
In - 5 (yr - y2 )Jet _ (yr - -2-(y1 - yz )Jek

_ 1 ' _ 1
= (yr-yrek)+-2-Aek] (YI'ylek)+'2'Aek]

 

 

_ ' _ , _ l
=(y1—y19k) (Y1 -}’Iek)+ Aek(y1-y1ek)+ZkA2
_ ' _ 1 2
=(y1-yrek) (yr ‘J’lek)+ZkA

=5” “Fl-ICAZ
4

similar for the other terms. I:

10

Lemma 1.2

1 2 1 2
s +—kA s ——kA
11 4 12 4

S21 -:11-kA2 S22 +%kA2

1 2 1 2 1 2]
= s +—kA s +—kA — s ——kA
(H 4 )(22 4 ] ('2 4

2 1 2
= (511322 —S12)+ZkA (S11 +S22 + 2812)

det(X'X) = det

Using Lemma 1.1 and Lermna 1.2, we obtain

1 2 1 2
s +—kA —s +—kA
1 22 1 12 1

(X X)‘1 =——, 1 1 (1.14)
dCt(X X) -S21+—kA2 S“ +—kA2
4 4
Combining (1.12) and (1.14),
. _ 1 kA
ekX(X X) l=----—.-—IS22 +512 ‘512 ‘5111
2 det(X X)
and then
r - 1 kA
2 det(X X)

11

1 (kA)2

Lemma 1.3 ejcBek =—* ' [(S22 +312)2 +(S12 +5102].
[det(X X)]2

 

Proof:

413e,, = 2e;,X(X'X)‘2X'e,c
= 2[e;,X(X'X)'I ] [(X'X)‘1 X'ek]

21kA[s+s s s]1kA[s+s s s]
= ——r—_ 22 12 ‘12‘11 ——v 22 12' — 12-11
2 det(X X) 2 det(X X)

2
=% (kA) [(322 + 512)2 + ($12 + 5102]
det(X X)

 

From (1.15) and Lemma 1.3, we have

 

 

(eitX(X 20432)2 = (322 “511)2 _ (1.16)
ekBek (822 +312)2 +(822 +512)2
. k _ k _
Therefore, If s“ = $22 , or Zj=l(ylj — yr): Zj=1 ( yz j — yz) , then
(ta;X(X'X)“e2)2 _ 0 D

 

ejc Bek

12

Chapter 2
Estimates of Technical Inefficiency in Stochastic Frontier Models with

Panel Data: Generalized Panel Jackknife Estimation

2.1 Introduction

In this chapter we consider the stochastic frontier model with time-invariant
technical inefficiency in a panel data setting. This model was first considered by Pitt and
Lee (1981), who estimated the model by MLE given a distributional assumption for
technical inefficiency. Without such a distributional assumption, Schmidt and Sickles
(1984) proposed fixed effects estimation. In this approach, the frontier intercept is
estimated as the maximum of the estimated firm-specific intercepts, and a firm’s level of
inefficiency is measured by the difference between the frontier intercept and the firm’s
intercept.

It is well understood that the “max” operation causes the estimated frontier
intercept, and therefore the estimated inefficiency levels, to be biased upward. Schmidt
and Sickles (1984), Park and Simar (1994) and Kim, Kim and Schmidt (2007) discuss
this problem. Hall, Hardle and Simar (1995) show that the bootstrap is asymptotically (as
T —> 00 with N fixed) valid in this setting, provided that there is a unique best firm (no
tie for the largest population intercept), and Kim, Kim and Schmidt (2007) use the
bootstrap to construct a bias-corrected estimate of the frontier intercept (and therefore of
inefficiency levels). The bootstrap is used to estimate the bias, which is then subtracted

from the original estimate. In their simulations, Kim, Kim and Schmidt (2007) found that

13

the bias correction was partially successful. It removed some but not all of the bias.
Often it seemed to remove about half of the bias.

In this chapter we consider instead bias corrections based on the jackknife. If the

bias of the fixed effects estimate is of order T‘l , the usual delete-one panel jackknife
estimator (as in Hahn and Newey (2004)) should remove the bias. However, intuitively
we would expect the jackknife bias correction to be similar to the bootstrap bias

correction, which was only partially successful. Thus it would seem that the finite-

sample relevance of the bias being of order T ‘1 may be questionable.
In this chapter we analyze the case of an exact tie for the best firm. In this case
the bootstrap is not asymptotically valid. Furthermore, we show that the bias of the fixed

T4”, not T‘l. In this case the

effects estimate of the frontier intercept is of order
usual delete-one panel jackknife does not properly remove the bias. Indeed, we show that
it removes (approximately) half of the bias. A different form of the jackknife, which we
call the generalized panel jackknife, does remove the bias.

In the simulations of Kim, Kim and Schmidt (2007) there was not an exact tie,
and an exact tie may also be unlikely in actual data. However, if there is nearly a tie, in
the sense that there is substantial uncertainty ex post about which is the best firm, it is not
clear whether asymptotics that assume no tie are more relevant than asymptotics that
assume an exact tie. In order to further analyze a near tie, we give a specific definition

(involving a local pararneterization) of “near tie,” and we show that the bias is again of

order T"1/2

, so that the generalized panel jackknife is needed to successfully remove the
bias.

We then perform simulations to assess the finite-sample relevance of these results.

14

The plan of the chapter is as follows. In Section 2.2, we define some notation and

give a brief review of fixed effects estimation of the stochastic frontier model with panel

data. In Section 2.3 we show that the bias is of order T—1 / 2 for the case of an exact tie
or a “near tie.” Section 2.4 describes the generalized panel jackknife that is appropriate
in this circumstance. In Section 2.5 we explain the design of our Monte Carlo
experiments, and Section 2.6 gives its results. Finally, Section 2.7 contains our

concluding remarks.

2.2 Fixed Effects Estimation of the Model

Consider a single-output production function with time-invariant technical

inefficiency u ,2 0, There are N firms, indexed by i =1,...,N , over T time periods,
indexed by t = 1,...,T. We consider the linear regression model of Schmidt and Sickles
(1984):

yi, = a + xftfl + vi, — u,-,i=1,...,N;t =1,...,T, (2.1)
where yi, is the logarithm of output for firm i at time t; xi, is a vector of K inputs
(e.g., in logarithms for Cobb-Douglas production function); ,6 is a K x1 vector of
coefficients; and vi, is an i.i.d. idiosyncratic error with mean zero and finite variance.
The vi, represent uncontrollable shocks that affect level of output, e.g., luck, weather, or
machine performance. The time-invariant technical inefficiency u ,- satisfies u ,- 2 O for
all i and ui > 0 for some i. There is no distributional assumption on u,- except that it is

one-sided.

15

Defining a,- = a - u,- , we can write (2.1) as a standard panel data model:

yit = “i + xirfl + Vit- (2-2)
Obviously, a,- s or since ui 2. 0. When a,- (and u ,-) is treating as fixed, (2.2) leads to a

fixed effects estimation problem in which neither a distribution for technical inefficiency

nor the independence between technical inefficiency and x i, or v,-, (or both) is needed.
We assume strict exogeneity of the regressors xi, in the sense that (x,1,...,x,-T) is

independent of (vn ,..., ViT )'. There is no restriction on the distribution of v”.

To estimate 6 , we use the fixed effects estimate ,6 , which can be estimated as
“least squares with dummy variables,” by regressing y i, on xi, and a set of N dummy
variables, or as the “within estimator,” by regressing (y), — yi) on (x,-, — ii) . Given the
estimate ,6 , the estimates 6, can be recovered as the averages of the firm-specific
residuals, i.e., d,- = y,- —f,~',6 where y, = T'lztyi, and 2?,- =T-lztx,-, , or
equivalently as the coefficients of the firm-specific dummy variables.

The within estimator )6 is consistent as N or T —+ co , and the firm-specific

intercepts Li, are consistent as T -—) 00. To estimate a and ui , Schmidt and Sickles

(1984) suggested the following estimators:

121° =é-ai, i=1,...,N. (2.3)

Park and Simar (1994) show that these estimates are consistent as N -> oo , T —-) oo , and

T‘“2 ln(N)—>O.

16

In this chapter, to maintain the connection to the earlier literature on
bootstrapping of this model, and also the literature on the jackknife, we will consider
asymptotic arguments as. T —-> 00 with N fixed. In this case we can hope only to
measure inefficiency relative to the best of the N firms.

For ease of presentation, we follow Kim, Kim and Schmidt (2007) and rank the

intercepts ai such that a“) S “(2) S S “(111), so that (N) indexes the firm with the
largest value of or, among N firms, which we will call the best firm. Similarly, we rank
the levels of technical inefficiency u ,- in the opposite order such that
u“) 2 ”(2) 2 2 u( N) . Obviously, a“) = a — ”(1') for all i and specifically
“(M =a 'uUV)‘

Now we define the relative inefficiency measures

u; =u,- —u(N) =a(N) —a,-. (2.4)

These are the focus of this chapter since, as T —) 00 with N fixed, 6 is a consistent
estimate of am), not a , and ii; are consistent estimates of u; , not u i.

Although it is consistent for a( N ) (as T —> 00 with N fixed), it is biased upward
for finite T. This is true because (it 2 ti:( N) and E(6(N)) = “(N)' That is, the max
operator in (2.3) induces upward bias: the largest 6,- is more likely to contain positive

estimation error than negative error. The upward bias in the estimate 6 induces the

upward bias in the estimates of relative technical inefficiency. That is,

E(&) - a( N) = E(12,-) — u;. Therefore we will simply evaluate the bias of c? as an

17

estimate of “(111); there is no need to separately evaluate the bias of the estimates of

relative technical inefficiency.

The bias of 6 as an estimate of 6( N) corresponds to what Kim, Kim and

Schmidt (2007) call the “first-level bias.” To correct this first-level bias, Kim, Kim and
Schmidt (2007) consider a bootstrap bias correction for the fixed effects estimate. They

6 boot

evaluate the “second-level bias,” E ( )— 6 , and use it to correct the first-level bias.

That is, if the second-level bias equals the first-level bias, we would want to evaluate

at — [E(6b"°’ ) — a] = 26 — E(o‘t”00’). (2.5)
The feasible version of this is
eggs” = 2a — 13‘1 2117:1521”), (2.6)

where “b ” represents a single bootstrap replication and “ B ” is the total number of
bootstrap replications. In their simulations (see their Table 4), this estimate removes
some but not all of the bias in 6 . Often it seems to remove about half of the bias.

In this chapter, we will consider the jackknife as a simple alternative to the

bootstrap.

2.3 Deriving the Order in Probability of the Bias

In this section, we show that the bias of 6 is of order T ‘1 if there is no tie for the

best firm; that is, if “(111) is distinct from all the other 6,» However, if there is a tie for

the best firm, or if there is a “near tie” (in a sense defined precisely below), the bias is of

order T—l/z.

I8

For simplicity, we will discuss the simple case of no regressors:

yr, =6,- +v,-,,i=1,...,N;t =1,...,T, (2.7)
where vi, are i.i.d. with mean zero and variance 02. Thus 6,- = y). The various 6) are
independent and JT(6,- — (If) —> N (0, 0'2). However, the inclusion of regressors would
not alter our results since the within estimator of 6 is unbiased, and our results really

only depend on the vector whose i’h element is s/T(6,~ — 05,-) being normal with mean

zero and finite variance matrix. See Hall, Hardle and Simar (1995), Appendix (i),

equation (A.l) for this condition, which would still hold with regressors.

2.3.1 The Case of No Tie

Suppose first that there is no tie for the best firm. That is, there is a unique firm

6‘ -”
I

such that 6( N) = 6;.

Hall, Hardle and Simar (1995) show the equivalence of (i) there is no tie for the
best firm, and (ii) the asymptotic distribution of 6 is normal. More precisely, they show

that if there is no tie, P(6 = 6(N)) —> 1 as T —> oo , so that the asymptotic distribution of
6 is the same as the asymptotic distribution of 6( N) , the estimate of 6( N) that would be
used if the identity of the best firm was known. Since 6( N) is unbiased, it follows that

JT times the bias of 6 must go to zero as T —> 00. Thus we conclude that the bias of

6 is of an order smaller than T‘” 2. We presume that it is of order T—l .

19

2.3.2 The Case of an Exact Tie
Suppose now that there is a tie for the best firm (the largest 61,-). Specifically

suppose that the first “k ” firms are tied, so that 6( N) = 61 = 62 = = 6k for

2 S k S N . Again the discussion in Hall, Hardle and Simar (1995, Appendix (i)) applies.

With a probability that approaches one as T —-> oo , 6 will equal 6,- for some i with

1 s i _<_ k , that is, the estimated best firm will be one of the k truly best firms. Therefore

with a probability that approaches one,

fi(&_a(N)) = fimaxfli’] -a(}v),&2 -a(N)"°°’&k —a(N))
(2.8)
= max{\/T(5t] —a(N)),\/T(62 -a(N))a---afi(ék _a(N))}

and therefore JT (6 — 6(N)) —> Z where Z is the maximum of a set of k normals with
zero means. For k >1, Z is not normal, and E (Z ) > O. The bias of 6 is therefore, for

large T, T—l/2E(Z), which is oforder T'l/z.

We can give an explicit expansion for the case of N = k = 2 and the simple

model above (with no regressors). We first state the following Lemma.
Lemma 2.1 Suppose X] and X2 are i.i.d. N(,u,0'2), i.e.,
X 2
[1]~N(#} o 02 ,
X 2 .U 0 0'

EImaX(X19X2)-#]=(1/~/;)0'- (2.9)

then

20

Proof. Let

Y XI 1“ 0'2 0'2
= ~ N , .
Z X1 -X2 0 0'2 20'2
So, p=0'2/\/20'20'2 =1/x/2 and
E(XllX1>X2)=E(Y|Z>O)
= p + (1/ J2)0'/1(0), where 2(-) is the normal hazard function
= p + (1/ «(2' waif/7 ), since 4(0) = ¢(0)/(1- mm» = 72/7

= xx +(1/J'r?)o.
Hence, E(Xl 1X1 > X2 ) — ,u = (1H; )0' and
E[max(Xl,X2)]= (1/2)E(X1 1X1 > X2)+(1/2)E(X2 |X2 > Xr)aby Symmetry
= E(X1|X1> X2),sinceXl andX2 arei.i.d.
Therefore, bias = E[max(X] ,Xz )] — ,u = (1/ J; )0' .
In the present setting, “ X11” and “ X 2 ” are 61 and 62 ; “ ,u ” = 61 = 62 ; the
variance “0'2 ”is oz/T ; and the bias of 6 = max(61,62) equals (I/x/gfl-Uza.

Clearly, this is proportional to T‘“ 2.

2.3.3 The Case of a Near Tie

In the previous sections we saw that the bias of 6 is of order T-1 if there is no

tie for the best firm, while it is of order T '1/ 2 if there is an exact tie. It is not clear how

relevant either set of results will be in finite samples if there is (in some sense) nearly a

21

tie. Intuitively that will depend on how close we are to a tie, which depends not only on
how close the a,- are to each other, but also on T'” 20' , which is the standard deviation
of the 6,-.

One way to model this is by a “local to tie” parameterization. So, to keep things
T-1/2

simple, let N = 2, a] > 62 , and 62 =61 — c for c > O , where c does not depend

on T. Then in our simple (no regressors) model, ~/T(61 — 61) —> N (0,02) . Also
Jfldz —a2) —> N(0,0'2) and so 61022 -61+T_1/zc)—> N(O,0'2), or
«fl—"(62 —61)—> N(—c,0'2). Then
filmax(0?ra&2)-a1]= maxtfirér — at Hflo‘zz — a] )1
(2.10)
—> Z,
where “ Z ” is the max of a N (0, 02) random variable and a N (-c,0'2) random variable.
Clearly E(Z) 2 E(N(O,02)) = 0 and the bias of 6 is again (for large T) T'l/2E(Z) ,
which is oforder T‘l/Z.
A similar analysis applies if 62 = 0!] — T'yc where c > 0 and 7 21/2. The
value of c matters (as above) when y = 1/ 2 but it does not affect the limit distribution if

7 > 1/ 2 . So the asymptotics for the case of a “near tie” are very similar to those for an
exact tie if a tie is near enough.

Once again we can give an explicit expression for the case of N = k = 2 and the

simple model (no regressors).

22

Lemma 2.2 Let X I and X 2 be independent normals, where X1 ~ N (0,02) and
X2 ~ N(,u2,0'2).Then
ElmaX(X1 .X2 )1 = [one N5 >24 + «EMMA/510. (2.11)
where ,u. = 212/0".
Proof See Appendix.
To apply this to our model, “X1 ” and “X2 ” are 61 and 62 ; “0'2 ”is az/T;

p2 =—T'l/2c;and ,ut =—T—l/2c/T'“20'=—c/0'. So the bias is

bias = [<D(—c/\/20')(-c/0') + J2¢(—c/«/—2-0') + (c/0')]T"l / 20', (2.12)
which is indeed proportional to T'” 2 .
2.4 Correcting Bias with the Panel Jackknife and the Generalized Panel

Jackknife

2.4.1 The Panel Jackknife

Jackknife estimation is an automatic bias reduction tool under assumption of a
series expansion for the bias of an estimator. Quenouille (1956) and Tukey (195 8) show
that using the jackknife estimates based on removing data and then recalculating the
estimator removes the first order bias from an initial estimator. For a comprehensive
background on jackknife estimation, see Miller (1974).

To describe the jackknife in a general setting, let the data be indexed by

' t=1,2,...,T. Let 6 be the estimator based on all T observations, and let 6“) be the

23

“delete-observation— t ” estimator that omits observation I and uses the other T —l

observations. Then the jackknife estimator is

J(6) = T6-(T —1)T“Zté(,). (2.13)

This estimator is said to remove the bias of order T ‘1 , in the following sense. Suppose
that

E(6) =6+T_IB+T_2D+0(T—3). (2.14)
Then

E[J(6)] = o + Gail—JD + 0(T‘2) = e + 0(T‘2). (2.15)

So if the bias is of order T‘l , in the sense that (2.14) holds, the jackknife leaves only the

bias of order T‘Z.
Hahn and Kuersteiner (2004), Hahn and Newey (2004), and Femandez-Val and
Vella (2007) apply the jackknife to nonlinear panel data models and nonlinear dynamic

panel data models. In the panel data setting, even though there are really NT

observations, we treat the number of observations in (2.13) as T , and to calculate éfl)

we delete the t’h period observation for each cross-sectional unit. (This is done because,

in the models they consider, the bias is of order T"l .) We refer to this procedure as the

“panel jackknife.”

Other similar versions of the jackknife can remove bias of order T '1 . For
example, Dhaene, Jochmans and Thuysbaert (2006) propose the split-sample jackknife

estimator:

24

SSJ(6) = 26—(1/2)(6(1) +6121), (2.16)
where 6 is the fixed effects estimator based on the full sample; 6(1) and 6(2) based on
first- and second- half of panel sample, where each half-panel consists of T/ 2
consecutive observations over time for all cross-sectional units. They show that the split-
sarnple jackknife estimator, which is an extension of the panel jackknife with T = 2 , also

removes the bias of order T4 from the fixed effects estimator. However, for the case

that the bias is of order T’l , we will consider only the standard panel jackknife as
described above.
It is obvious that when there is no tie, the panel jackknife will remove the first-

level bias of the estimate of a( N) (hence, the bias of the estimates of relative technical

. . * . . . - .
1nefficrency ui ) srnce the bras rs of order T l . For the cases of an exact tre and a near

tie, however, we need a jackknife estimator that can handle bias of order T'” 2 . The

difference in the order of the bias leads us to the generalized jackknife.

2.4.2 The Generalized Jackknife

Schucany, Gray and Owen (1971) were the first to propose a jackknife estimator
that can handle a more general form of bias. It was not until later that Gray and
Schucany (1972) gave it the name “generalized jackknife.” Gray and Schucany (1972)

define the generalized jackknife as the following.

25

Definition Gray and Schucany (1972)’s Definition 2.1. Let 61 and 62 be the estimators

for 6. Then, for any real number R #1, the generalized jackknife estimator 0(61 ,62) is

defined as

. ~ 6—R6
G(6|,62)=—1——2—.

1_R (2.17)

The usual (Quenouille) jackknife corresponds to 6] = 6 , 62 = T‘] Z r 6 (r) , and

R = (T —1)/T .
If we can express the bias of the estimators in terms of the sample size T and the

true parameter 6 , we can choose R so that the generalized jackknife is unbiased.

Theorem 2.1 Gray and Schucany (1972)’s Theorem 2.1. If the bias of the estimators 61
and 62 can be expressed as

E(6k) = 9+1),c (T,6),k =1,2;

b2 (739) i 0;

and

 

R _ bl(T,6)

— at 1,
b2 (T .6)

then

E[G(61 ,6} )1 = 6.

26

Proof

[6 + b] (T, 6)] - R[6 + b2 (T, 6)]

£166. .632 )1 = 1_ R

 

 

+ bl (T, o) - sz (T, o)
l—R

=6

= 9, since sz (T, 6) = b1(T,6).

D
In general, we do not have a bias expression of the form of the previous theorem,
but we have a series expansion of the bias with leading term of known order. Then the

generalized jackknife removes the first term of the bias expansion.

Theorem 2.2 Gray and Schucany (1,972)’s Theorem 2.2. If the bias of the estimators 6]

and 62 can be expanded as infinite series:

E(6k) = e + ZZIbk,(T,6),k =1,2

 

and
R =———b“(T’6) =1
b21(T ,6)
then
00 oo
. . ._ b -(T,6)-R ._ b -(T,6)
E[G(191.6’2)]=0+ 2'” " 2'4 2’ .
1 — R

Proof Similar to proof of Theorem 2.1. :1

27

2.4.3 The Generalized Panel Jackknife When Bias is of Order T ‘1/ 2

We are specifically interested in the case that the bias of 6 is of order T—l/ 2.

Suppose that the following expansion holds:
E(é)=o+T‘1/ZB+T“D+0(T‘3/2). (2.18)

As before, we let 61 = 6 (T observations) and 62 = T4 2160) . Then the weight R in

Theorem 2.2 is equal to
R=(B/JT)/(B/JT—1)=JT—1/JT (2.19)
and the generalized jackknife is

J? . Jr: -. ~
Wig-WT 24%-

6(6) = (2.20)

It is then easy to verify that the bias of 0(6) is of order T '1 ; that is, the T‘“ 2 term in

the bias of 6 has been removed.

In the panel data case, once again we treat the number of observations as T , and
60) is calculated by deleting the t’h time period observation for each cross-sectional

unit. We will call this the generalized panel jackknife.

The generalized jackknife removes bias more aggressively than the usual
jackknife, in the sense that the weights attached to 6 and to T"1 2’ 6(,) are larger. For

example, for T =10 we have

J(6) = 106 40-12160)

G(6) =19.sé —18.5(T'12t6(,)).

28

Similarly for T = 50 we have

J(6) = 506 — ”(T—1216(0)

0(9‘) = 99.56 — 98.5(T'l 2‘ 6(0).

2.4.4 What If The Wrong Jackknife Is Used?

We have seen that the usual panel jackknife is appropriate when the bias is of
order T'] , whereas the generalized panel jackknife is appropriate when the bias is of

order T "l 2 . This raises the question of what happens if the wrong version of the

jackknife is used.

Theorem 2.3 If the bias of 6 is of order T'“ 2 , the usual panel jackknife corrects

approximately half of the bias.

Proof We have E (6) = 6 + T’1 / 2B + higher order terms. So, dropping the higher order

terms, we calculate

E[J(6)]=6+B~/T—B\/T—l=6+ (2.21)

B
W'

Comparing the bias on (2.21) to the original bias of T ’l/ 2B , we have removed about

half of the first-order bias term. Cl

Theorem 2.4 If the bias of 6 is of order T ”I , the bias of the generalized panel jackknife

is approximately the negative of the bias of the original estimate.

29

Proof Suppose E (6) = 6 + T "l / 2B + higher order terms. So, again dropping the higher

order terms,
" _ VT -I _ VT -1 42 _ -1

=VIE—l-Jf—T[w/T6+T-l/zB—x/TTl6—(T—1)‘”2B]

=e+ "“Z—Uudr“fi3 a2»

__1_[T
JT-JT-r

—9+ 1 {JTTT—JT B
- JT—JT—H\JT¢T—l

=o—cHVJT—UT43.
So the bias of G(6) , — (JT/JT -1)T_IB , is approximately the negative of the original

bias, T-IB. [:1

2.5 Design of the Monte Carlo Experiments

In this section, we conduct Monte Carlo simulations to investigate the finite

sample performance of the following estimators of 6( N) : (i) 6 , the maximum of the

' fixed effects estimates; (ii) J (6) , the panel jackknife estimate; (iii) 0(6) , the
. boot

generalized panel jackknife estimate; and (iv) a BC , the bias-corrected bootstrap point

estimate.

30

We are primarily interested in the bias of these estimators. However, we will also
report their variance and mean square error. These measures are defined precisely later
in this section.

The model is the simple panel data model with no regressors given in (2.7). Thus,
the data generating process is

yit = a+vit -ui
(2.23)
= a,- + vi,,i=1,...,N;t =1,...,T,

where a,- = a—ui; the u,- are i.i.d. half-normal: ui = iUil where U,- ~ N(0,0'3);and the

vi, are normal with mean zero and variance 0,? . These distributional assumptions are

not used in estimation. They just characterize the process that generates the data.

The set of parameters is {6, 0'3 , 0'3 , N, T} but this can be reduced somewhat. All

of the results (bias, variance, and MSE) are invariant with respect to a , so we set it equal

to one, without loss of generality. Also, only ratios of variances matter. If we multiply

both 03 and 03 by a constant q , the biases of the estimates change by J; and the
MSEs change by q. So we really only need to consider three parameters: N, T, and a
relative variance parameter. Kim, Kim and Schmidt (2007) used the relative variance
parameter 7* = (03 )e /[03 + (03 )e ] , where (03 )e = var(u) = ((72' — 2)/7r)0'3. We will
use instead the parameter to defined by

_ (021)“

y, _ , (2.24)
T—l/ 2 av

31

This is not a matter of substance. We use [1* because we find it easier to interpret. It
measures the standard deviation of the 6,- in units of the standard deviation of the 6;. So
our parameter space is {#1: , N ,T} .

We set scale by setting 03 / T = 0.1. Then, for a given ,6... , (03 )e is determined.
We consider ,ur- : 10'l,10-“2,1,10“2, and 10. With 03 /T = 0.1 , for a given value of
pr: , the values of (03 )t and 03 can be determined:

(1) n. =10‘l = 0.1: (o3)... = 0.001; of? = 0.0028;

(2) p. =10‘“2 = 0.3162: (o3). = 0.01; 03 =0.0275;

(3) 2:. =1: (o3). =01; o3 =0.2752;

(4) ,u. =10“2 =3.1623; (o3). =1; o3 =2.7519;

(5) 2:. =10: (o3). =10; of} = 27.5194.

We consider sample sizes N = 2,10, 20,50, and 100 , and we set T =10. We also
considered T = 5,20,50, and 100 , and the results for these values of T are shown in
Supplemental Table 2.13 - 2.48.

The basic outcomes that we would expect in the simulations are as follows. First,

bias will be larger when N is larger, but the effect of N on the relative performance of

the various bias-corrected methods is not obvious. Second, bias will be larger when [Jar
is smaller, since then the variability of the 6,- is smaller relative to the sampling
variability of the 6,-. We might expect the panel jackknife or the bootstrap to be better

than the generalized panel jackknife when ,ue is large (we are farther from a tie), and

32

vice-versa. Third, conditional on pt , we do not expect T to be very important. When
we change T in our experiment, holding constant p: and 03 / T , it means that 03

increases proportionally to T , and (03 )t is unchanged. Therefore neither the variability

of the 6,- nor the sampling variability of the 6,- changes. The only reason that T should

matter is that the jackknife’s weights on 6 and T—12,é(t) depend on T.

We consider three different variations of the setup we have just described.
Experiment I (No Tie). The setup of this experiment is exactly as just described.

There are no restrictions on the 6;. They just follow from the draws of the half-normal
u ,- . This setup is very similar to that of Kim, Kim and Schmidt (2007).

Experiment 11 (Exact Tie). We generate data as described above. Now we (the

data generator) know which firm is the best and the value 6( N) of its intercept. We
randomly select one of the other (N — 1) firms and set its intercept also equal to “(NT

Therefore we have created an exact two-way tie for the best firm.
Experiment III (Near Tie). We start as in Experiment 11. However, once we have

observed the best firm and 6( N) , we randomly select one of the other (N — 1) firms and
set its intercept equal to

am) - T““2[a(N) - a(N_1)]. (2.25)
So, for example, if T =10 , we have now created a new second-best firm whose intercept

is J1_0 = 3.162 times closer to a( N) than the previously second-best firm’s intercept.

33

For each configuration of {[1:1- , N,T} , we perform 1,000 replications. Within each

replication, the bias-corrected bootstrap point estimate is based on 1,000 bootstrap

replications.
For each of the estimators (6, J (6),G(6), 6229’ ) we calculate bias, variance and
mean square error. The parameter being estimated, a( N) , varies across replications

because of the random draws of the half-normal u,- that determine a,- = a — u ,-.

Therefore we will explicitly state our definition of bias, variance and MSE. First define

(i) NREP = number of replications; (ii) r = index of replication, r =1,..., NREP; (iii)
6, = value of 6(N) in replication r; (iv) 6, = estimate of 6, in replication r (for any
of the four estimators listed above); and (v) 6 = NREP‘l Zr 6, and
5 = NREP-12,6, .
The definition of bias is straightforward:
bias(6) = NREP—'1 2,053, - 6,) = 5 — 0. (2.26)
Then we define the mean squared error as

MSE(6) = NREP" Zr (6, — 6, )2
. (2.27)
= NREP-1 2,109, — 6,) - bias(6)]2 + bias(6)2

34

and the variance as

var(6) = MSE(6) — bias(6)2
(2.28)
= NREP-1 Zuni, — e, ) — bias(6)]2.

2.6 Results of the Monte Carlo Experiments

Tables 2.1, 2.2, and 2.3 give the results of Experiment I in which there is no tie.
All of these results are for T = 10. Table 2.1 gives the bias of the estimates, while Table
2.2 gives variance and Table 2.3 gives MSE. In all three tables, column (1) gives results

for 6 ; column (2) gives results for the panel jackknife J (6) ; column (3) gives results

for the generalized panel jackknife 6(6) ; and column (4) gives results for the bias-

- - .. boot
corrected bootstrap pornt estimate a BC .

Consider first Table 2.1, which gives the bias of the various estimates as an

estimate of (luv). This is equivalent to the bias of estimated relative technical

inefficiency 6; as an estimate of u;. As expected, the bias of 6 is larger when N is

larger (the “max” is taken over more firms) and when ,ut is smaller (we are closer to a

tie). The panel jackknife and the bias-corrected bootstrap are less biased than the fixed
effects estimate 6 . However, they only correct part of the bias. In most cases the
jackknife corrects more of the bias than the bias-corrected bootstrap. The generalized

panel jackknife overcorrects (so the original upward bias now becomes a downward

bias).

35

When p:- is very small, so that the variability of the 6,- is very small relative to
the sampling variability of the 6,- , we are in a sense close to a tie. In these cases the “no
tie” asymptotics appear to be relevant: the generalized panel jackknife is nearly unbiased,
and the panel jackknife (and also the bias-corrected bootstrap) corrects about half of the
bias, as predicted by Theorem 2.3. Conversely, when ,us is large we are far from a tie,
the panel jackknife and the bias-corrected bootstrap are nearly unbiased, and the
downward bias of the generalized panel jackknife is almost as large as the upward bias of
6 , as predicted by Theorem 2.4.

Table 2.2 gives the variance of the various estimates. They are easy to
summarize. The variance of the 6 is less than the variance of the bias-corrected
bootstrap point estimate, which is less than the variance of the panel jackknife, which is
less than the variance of the generalized panel jackknife. The variance of the generalized
panel jackknife is considerably larger than the variance of the other estimators. To

properly interpret these variances, remember that we are ultimately interested in

estimating the relative size of the u ,- , whose variance is (03 )s , and that in our setup

(o3)... = 0.001,0.01, 0.1, 1, and 10 for rt. = 10‘1 ,10’1/2,1,10“2, and 10, respectively.
So the variance of these estimators is large enough to be an issue, except perhaps for the
larger values of pr .

Table 2.3 gives the MSE of the estimates. In terms of MSE, the two varieties of
the jackknife are dominated by the bias-corrected bootstrap. The bias-corrected bootstrap
is also generally better than the fixed effects estimate 6 , except in those cases where the

bias of 6 is small (i.e., when N is small and ,us is large).

36

Now we turn to Experiment 11, the case of an exact tie. These results are in Table
2.4, 2.5, and 2.6.

In terms of bias, we see in Table 2.4 that the generalized panel jackknife is clearly
the best. It overcorrects the bias, but not by as much as the panel jackknife and the bias-
corrected bootstrap undercorrect. As expected from Theorem 2.3, the panel jackknife
corrects about half of the bias. The bias-corrected bootstrap, which is not valid
asymptotically in the case of an exact tie, also appears to correct about half of the bias.

In Table 2.5, the variances of the estimates are rather similar to the variances for
the case of no tie (Table 2.2). The main difference is that now the variance does not

depend as strongly on pr. , presumably because, once we have forced a tie, the similarity

of the other 6,- is not of as much importance. The ranking of the estimators, in order of

increasing variance, is still the same as in Table 2.2 (6 , bias-corrected bootstrap, panel
jackknife and generalized panel jackknife).

In terms of MSE, we see in Table 2.6 that the bias-corrected bootstrap still
dominates both varieties of the jackknife. It is also generally better than the fixed effects
estimate 6 . This favorable performance of the bias-corrected bootstrap is perhaps
surprising, given that it is not asymptotically valid in the case of an exact tie.

Our last experiment is Experiment III, the case of a near tie. The results for this
experiment are given in Table 2.7, 2.8, and 2.9. As a general statement, the results are
between those of Experiment I and Experiment 11, which is not surprising.

For small values of ,us (nearer tie), the bias results in Table 2.7 are quite similar

to those of Table 2.4 for an exact tie. In these cases the generalized panel jackknife has

little bias, while the panel jackknife and the bias-corrected bootstrap correct about half of

37

the bias. For large values of ,us (less near tie), the panel jackknife and the bias-corrected

bootstrap still correct only some of the bias, but the generalized panel jackknife
overcorrects. Still, it is generally true in Table 2.7 that the generalized panel jackknife
has the smallest bias.

In terms of variance (Table 2.8) and MSE (Table 2.9), the results are fairly similar
to those for both the case of no tie and the case of an exact tie. Once again the bias-
corrected bootstrap is generally the best, and the generalized panel jackknife is the worst.

The last issue we consider is the effect of changing T. We consider the same

three kinds of experiments as just described, with T = 5,20,50, and 100 (in addition to
T =10 , which we have just discussed). These results are given in Supplemental Table
2.13 — 2.48. In this chapter we will display the results only for [1* =1 and N = 20.

Tables 2.10, 2.11, and 2.12 give the bias, variance, and MSE of the various estimates.

As discussed in Section 2.5, we do not expect changes in T to be very important,

because we are holding constant N , ,us , and 03 / T , or equivalently we are holding

constant N , (03 )e , and 03 / T . Indeed, the motivation for adopting this

pararneterization was that we expected it to make one of the parameters (T) unimportant.
We expect changing T to be more important for the jackknife estimates than for the other
two estimates, because the value of T affects the weights that the jackknife puts on the
original estimate versus the average of the delete-one-observation estimates.

What we see in Tables 2.10 — 2.12 is not surprising. In Table 2.10, the effect of
changing T on the bias of the estimates is very minor. In Table 2.11, changing T does
not affect the variance of the fixed effects estimate or the bias-corrected bootstrap point

estimate very much, but the variance of the jackknife estimates increases noticeably as T

38

increases. Correspondingly, in Table 2.12 the MSE of the jackknife estimates increases
as T increases. However, it remains true that the value of T is much less important than

the values of N and ,us in determining the relative performance of the various estimates.

2.7 Concluding Remarks

In the stochastic frontier model with panel data, the fixed effects estimate of the
frontier intercept is biased upward. Previous work found that the bias-corrected bootstrap
corrected only part of this bias. This chapter has tried to explain that finding and to see
whether we can more successfully remove the bias using the jackknife.

The bootstrap is known to be asymptotically (as T —> 00 with N fixed) valid if
there is no tie for the best firm, and not valid if there is an exact tie. So whether there is a
tie, and how close we are to having a tie if there is not an exact tie, is a reasonable issue
to focus on.

When there is an exact tie, we show that the bias of the fixed effects estimate is of
order T’“ 2 rather than T" . Not only is the bootstrap not valid, but the usual panel

jackknife, which is based on the assumption that the bias is of order T '1 , also does not
work correctly. More specifically, we show that it removes (approximately) half of the

bias. A different form of the jackknife, which we call the generalized panel jackknife, is

needed to remove the bias of order T‘” 2 .

If there is no tie, the bootstrap is valid and the panel jackknife should also be

effective in removing bias, since now the bias is of order T" . In this case the

39

generalized panel jackknife will not work correctly, and indeed we show that its bias is
the negative of the bias of the fixed effects estimate; it reverses the bias.
We also consider the case of a near tie, which we define as the case that the

difference between the frontier intercept and the intercept of the second-best firm is

0(T'” 2 ). In this case the bias is again of order T_” 2 and so the generalized panel

jackknife should remove it.

Our simulations support the finite-sample relevance of these arguments. When
there is a tie or a near tie, the generalized panel jackknife removes the bias effectively,
whereas the panel jackknife and the bias-corrected bootstrap remove about half of the
bias. When there is not a tie, the generalized panel jackknife overcorrects the bias, and
the panel jackknife and the bias-corrected bootstrap are much better at removing the bias.

The major drawback of the jackknife is that its variance is large. This is true for
both versions of the jackknife but the variance is the largest for the generalized panel
jackknife. There does not seem to be any good reason to prefer the panel jackknife to the
bias-corrected bootstrap, since it has a larger variance and does not do a better job of
correcting bias. However, while the generalized panel jackknife is clearly dominated by
the bias-corrected bootstrap in terms of MSE, it does do a very good job of removing bias
when there is an exact tie or a near tie. Empirically, presumably that corresponds to cases
where the identity of the best firm is in substantial doubt.

The inability of the generalized panel jackknife to beat the bias-corrected
bootstrap in terms of MSE when there is an exact or a near tie is perhaps surprising, since
the bootstrap is not valid if there is a tie. However, “not valid” here has a specific

meaning, namely that we cannot claim that the distribution of the bootstrap estimate

40

around the original estimate matches the distribution of the original estimate around the
true parameter. Apparently the bias-corrected bootstrap is nevertheless a useful point

estimate.

41

2.8 Output Tables
Table 2.1: (Experiment I: No Tie) T = 10 , Bias of the Estimates

 

 

 

 

 

 

 

 

 

42

 

 

. ( 1) “(2) (3) (4)

'u‘ N E(a "a(N)) E(J(Cl)-C¥(N)) E(G(0!)"a(1v)) E(égoé” -a(N))
10"1 2 0.1671 0.0810 - -0.0099 0.1006
10'1/2 2 0.1391 0.0522 -0.0394 0.0743
1 2 0.0887 0.0230 -0.0463 0.0368
101/2 2 0.0462 0.0125 -0.0230 0.0204
10 2 0.0294 0.0199 0.0100 0.0204
10‘] 10 0.4532 0.2113 -0.0436 0.2669
10'1/2 10 0.3935 0.1556 -0.0951 0.2111
1 10 0.2809 0.0828 -0.1261 0.1218
101/2 10 0.1504 0.0293 -0.0983 0.0439
10 10 0.0577 -0.0034 -0.0678 0.0046
10-1 20 0.5566 0.2724 -0.0271 0.3326
104/2 20 0.4928 0.2093 -0.0895 0.2722
1 20 0.3750 0.1176 -0.1537 0.1767
101/ 2 20 0.2349 0.0563 -0.1321 0.0895
10 20 0.1 136 0.0074 -0.1046 0.0292
10"1 50 0.6699 0.3132 -0.0629 0.3975
10‘1 / 2 50 0.6092 0.2678 -0.0921 0.3433
1 50 0.4973 0.1903 -0.1334 0.2565
101/2 50 0.3556 0.1151 -0.1385 0.1639
10 50 0.2059 0.0379 -0.1391 0.0724
10"1 100 0.7584 0.3627 -0.0544 0.4594
10“” 100 0.6949 0.3127 -0.0901 0.4012
1 100 0.5809 0.2293 -0.1413 0.3086
101/2 100 0.4433 0.1652 -O.128O 0.2182
10 100 0.2950 0.0922 -0.1217 0.1281

—

Table 2.2: (Experiment I: No Tie) T = 10 , Variance of the Estimates

 

 

 

 

 

(1) (2) (3) (4)

,1. N 8 J0?) Ga?) 422’?”
10‘1 2 0._0666 0.1 130 0.2127 0.0805
10"“2 2 0.0696 0.1159 0.2149 0.0839
1 2 0.0803 0.1 183 0.2002 0.0936
101/ 2 2 0.0932 0.1133 0.1582 0.1004
10 2 0.0991 0.1055 0.1189 0.1018
10"1 10 0.0355 0.1440 0.3871 0.0623
104/2 10 0.0382 0.1472 0.3901 0.0662
1 10 0.0483 0.1478 0.3665 0.0764
101/2 10 0.0696 0.1378 0.2836 0.0938
10 10 0.0890 0.1282 0.2106 0.1034
10‘1 20 0.0264 0.1419 0.4089 0.0528
10"“ 2 20 0.0284 0.1467 0.4191 0.0557
1 20 0.0359 0.1534 0.4191 0.0650
101’ 2 20 0.0518 0.1388 0.3197 0.0775
10 20 0.0757 0.1329 0.2613 0.0948
10"1 50 0.0208 0.1500 0.4570 0.0469
10"“ 2 50 0.0215 0.1489 0.4518 0.0476
1 50 0.0254 0.1479 0.4356 0.0527
101/2 50 0.0359 0.1438 0.3896 0.0638
10 50 0.0569 0.1401 0.3257 0.0832
[0‘1 100 0.0191 0.1617 0.5014 0.0466
10‘“2 100 0.0196 0.1556 0.4799 0.0467
1 100 0.0231 0.1587 0.4807 0.0515
101/2 100 0.0317 0.1585 0.4490 0.0626
10 100 0.0440 0.1400 0.3532 0.0716

 

 

 

 

 

_

43

Table 2.3: (Experiment I: No Tie) T = 10 , MSE of the Estimates

 

 

 

 

 

(1) (2) (3) (4)
11. N & J(c‘v) C(51) 45‘6“
10-1 2 0.0945 0.1 196 0.2127 0.0906
10-1/ 2 2 0.0890 0.1186 0.2164 0.0894
1 2 0.0882 0.1 188 0.2024 0.0950
101/2 2 0.0953 0.1134 0.1588 0.1008
10 2 0.1000 0.1059 0.1190 0.1022
10-1 10 0.2409 0.1887 0.3891 0.1355
10-1/2 10 0.1931 0.1715 0.3991 0.1107
1 10 0.1272 0.1547 0.3824 0.0912
101/2 10 0.0922 0.1386 0.2932 0.0958
10 10 0.0923 0.1282 0.2152 0.1035
10-1 20 0.3362 0.2162 0.4096 0.1634
10-1/2 20 0.2712 0.1905 0.4271 0.1298
1 20 0.1765 0.1673 0.4427 0.0962
101/ 2 20 0.1070 0.1420 0.3472 0.0855
10 20 0.0886 0.1329 0.2722 0.0956
10-1 50 0.4696 0.2481 0.4610 0.2049
10-1/ 2 50 0.3925 0.2206 0.4603 0.1655
1 50 0.2727 0.1841 0.4534 0.1184
101/2 50 0.1623 0.1571 0.4087 0.0907
10 50 0.0993 0.1415 0.3451 0.0885
10-1 100 0.5942 0.2932 0.5044 0.2576
10-1/2 100 0.5025 0.2534 0.4880 0.2077
1 100 0.3605 0.21 12 0.5006 0.1467
101/2 100 0.2282 0.1858 0.4654 0.1102
10 100 0.1310 0.1485 0.3680 0.0880

 

 

 

44

 

 

Table 2.4: (Experiment 11: Exact Tie) T = 10 , Bias of the Estimates

 

 

 

 

 

 

 

 

 

 

(1) (2) (3) (4)

lb N 1‘3(62 ‘a(N)) E(J(é)-a'(7v)) E(G(&)-a(1v)) E(ég‘g” _a( N))
** 2 0.1828 0.0997 0.0120 0.1163
10"1 10 0.4580 0.2165 -0.03 80 0.2724
10-1/2 10 0.4077 0.1693 -0.0820 0.2256
1 10 0.3261 0.1268 -0.0832 0.1678
101/2 10 0.2499 0.1 122 -0.0330 0.1323
10 10 0.2052 0.0817 -0.0424 0.1145
10"1 20 0.5593 0.2696 -0.0357 0.3353
104/2 20 0.5020 0.2158 -0.0859 0.2827
1 20 0.3988 0.1412 -0.1302 0.2006
101, 2 20 0.2938 0.0976 -0.1093 0.1421
10 20 0.2282 0.0954 -0.0446 0.1 194
10'"1 50 0.6707 0.3104 -0.0693 0.3985
104/2 50 0.6134 0.2715 -0.0889 0.3492
1 50 0.5124 0.21 17 -0. 1052 0.2739
101/2 50 0.3926 0.1579 -0.0895 0.2038
10 50 0.2899 0.1237 -0.0514 0.1530
10"1 100 0.7615 0.3702 -0.0422 0.4642
10-1/2 100 0.7023 0.3240 -0.0747 0.4117
1 100 0.5950 0.2523 -0. 1089 0.3271
101/ 2 100 0.4665 0.1843 -0.1 132 0.2434
10 100 0.3382 0.1236 -0. 1026 0.1660

Note: ** value of ,w- is irrelevant when N = 2 and there is an exact tie.

45

Table 2.5: (Experiment 11: Exact Tie) T = 10 , Variance of the Estimates

 

 

 

 

 

(1) (2) (3) (4)
p. N 8 J(&) 0(8) 62°C“
** 2 0.0658 0.1 107 0.2063 0.0792
10“ 10 0.0346 0.1408 0.3785 0.0607
10'“ 2 10 0.0362 0.141 1 0.3785 0.0626
1 10 0.0424 0.1396 0.3598 0.0682
101/2 10 0.0528 0.1210 0.2729 0.0748
10 10 0.0620 0.1308 0.2785 0.0813
10'1 20 0.0264 0.1449 0.4207 0.0531
10-1/ 2 20 0.0282 0.1464 0.4187 0.0556 ‘
1 20 0.0351 0.1484 0.4048 0.0640
101/2 20 0.0471 0.1444 0.3596 0.0746
10 20 0.0562 0.1228 0.2694 0.0778
10'1 50 0.0208 0.1495 0.4570 0.0474
10-1/ 2 50 0.0225 0.1554 0.4721 0.0506
1 50 0.0274 0.1554 0.4506 0.0579
101/2 50 0.0369 0.1505 0.4032 0.0685
10 50 0.0497 0.1385 0.331 1 0.0765
10"1 100 0.0192 0.1614 0.5011 0.0471
10-1/2 100 0.0204 0.1594 0.4892 0.0492
1 100 0.0247 0.1606 0.4784 0.0556
101/2 100 0.0313 0.1551 0.4395 0.0620
10 100 0.0430 0.1443 0.3680 0.0715

 

 

 

 

 

#

Note: ** value of 1a is irrelevant when N = 2 and there is an exact tie.

46

Table 2.6: (Experiment 11: Exact Tie) T = 10 , MSE of the Estimates

 

 

 

 

 

(1) (2) (3) (4)
,1. N a, J(c‘z) 0(6) 582%”
H 2 0.0993 0.1207 0.2065 0.0928
10-1 10 0.2444 0.1877 0.3811 0.1350
10-1/2 10 0.2024 0.1698 0.3852 0.1135
1 10 0.1437 0.1556 0.3667 0.0964
101/2 10 0.1153 0.1336 0.2740 0.0923
10 10 0.1041 0.1379 0.2803 0.0944
10-1 20 0.3392 0.2176 0.4220 0.1655
10-1/2 20 0.2802 0.1929 0.4260 0.1355
1 20 0.1941 0.1683 0.4218 0.1043
101/2 20 0.1334 0.1539 0.3716 0.0948
10 20 0.1083 0.1319 0.2714 0.0920
10-1 50 0.4706 0.2459 0.4618 0.2063
10-1/2 50 0.3987 0.2291 0.4800 0.1726
1 50 0.2900 0.2002 0.4617 0.1340
101/2 50 0.1910 0.1754 0.4112 0.1101
10 50 0.1337 0.1538 0.3337 0.0999
10-1 100 0.5991 0.2984 0.5029 0.2626
10-1/ 2 100 0.5136 0.2644 0.4948 0.2187
1 100 0.3787 0.2243 0.4903 0.1626
101/2 100 0.2489 0.1801 0.4523 0.1212
10

Note: ** value of ,u: is irrelevant when N = 2 and there is an exact tie.

 

100

 

 

0.1573

 

 

0.1596

47

 

 

0.0991

 

Table 2.7: (Experiment III: Near Tie) T = 10 , Bias of the Estimates

 

 

 

 

 

(1) (2) (3) (4)
#9 N E(é-a'uvfl E(J(é)-a(1v)) E(G(C2)"a(N)) E(égzv‘” _a(N))
10'1 2 0.1774 0.0934 0.0049 0.1 107
10"“ 2 2 0.1661 0.0807 -0.0094 0.0994
1 2 0.1361 0.0503 -0.0402 0.0707
101/2 2 0.0792 0.0107 -0.0616 0.0252
10 2 0.0323 0.0003 -0.0334 0.0053
10‘] 10 0.4578 0.2164 -0.0381 0.2724
10-1/ 2 10 0.4067 0.1678 -0.0841 0.2244
1 10 0.3210 0.1207 -0.0904 0.1620
101/2 10 0.2273 0.0861 -0.0627 0.1084
10 10 0.1403 0.0326 -0.0809 0.0560
10"1 20 0.5592 0.2695 -0.0358 0.3353
10—1/2 20 0.5018 0.2158 -0.0857 0.2826
1 20 0.3971 0.1390 -0.1331 0.1986
101/2 20 0.2844 0.0851 -0.1249 0.1314
10 20 0.1906 0.0534 -0.0912 0.0808
10-1 50 0.6707 0.3104 -0.0693 0.3985
10"1 I2 50 0.6133 0.2713 -0.0891 0.3491
1 50 0.5119 0.2112 -0. 1059 0.2754
101/2 50 0.3898 0.1537 -0.0952 0.2004
10 50 0.2755 0.1053 -0.0741 0.1367 '
10'1 100 0.7615 0.3702 -0.0422 0.4642
10‘“ 2 100 0.7023 0.3240 -0.0748 0.41 17
1 100 0.5950 0.2522 -0.1090 0.3270
101/2 100 0.4659 0.1835 -0.1141 0.2426

10 100 0.3340 0.1190 -0.1075 0.1613

 

 

 

48

 

 

Table 2.8: (Experiment III: Near Tie) T = 10 , Variance of the Estimates

 

 

 

 

 

 

 

 

 

 

(I) (2) (3) (4)

,1. N 8 J0?) 6(6) 45%“
10‘1 2 0.0660 0.1 l 17 0.2092 0.0795
10‘“ 2 2 0.0666 0.1 136 0.2137 0.0804
1 2 0.0696 0.1 193 0.2251 0.0840
101/ 2 2 0.0820 0.1234 0.2086 0.0963
10 2 0.0956 0.1 174 0.1622 0.1030
10‘1 10 0.0347 0.1408 0.3794 0.0607
10'”2 10 0.0363 0.1412 0.3786 0.0626
1 10 0.0426 0.1396 0.3605 0.0683
101/ 2 10 0.0548 0.1250 0.2817 0.0762
10 10 0.0686 0.1243 0.2539 0.0840
10‘1 20 0.0264 0.1449 0.4206 0.0531
104/2 20 0.0282 0.1464 0.4188 0.0556
1 20 0.0351 0.1483 0.4050 0.0639
101/ 2 20 0.0475 0.1468 0.3663 0.0750
10 20 0.0597 0.1277 0.2794 0.0804
10'1 50 0.0208 0.1495 0.4570 0.0474
10‘“ 2 50 0.0225 0.1554 0.4772 0.0506
1 50 0.0274 0.1552 0.4502 0.0578
101/2 50 0.0368 0.1517 0.4075 0.0684
10 50 0.0504 0.1403 0.3369 0.0771
10"1 100 0.0192 0.1612 0.5011 0.0471
10““ 2 100 0.0204 0.1594 0.4893 0.0492
1 100 0.0247 0.1606 0.4784 0.0556
101/2 100 0.0313 0.1548 0.43 84 0.0619
10 100 0.0430 0.1440 0.3674 0.0715

*

49

Table 2.9: (Experiment 11]: Near Tie) T = 10 , MSE of the Estimates

 

 

 

 

 

 

(1) (2) (3) (4)
1.. N 6? .102) 0(4) 6:88”
10"1 2 0.0975 0.1204 0.2093 0.0918
10““ 2 2 0.0942 0.1201 0.2138 0.0903
1 2 0.0881 0.1218 0.2267 0.0890
101/2 2 0.0883 0.1235 0.2124 0.0969
10 2 0.0966 0.1174 0.1633 0.1031
10-1 10 0.2442 0.1876 0.3809 0.1349
10‘“2 10 0.2016 0.1694 0.3857 0.1130
1 10 0.1456 0.1542 0.3686 0.0945 .
101/2 10 0.1056 0.1324 0.2856 0.0880
10 10 0.0883 0.1254 0.2605 0.0871
10'1 20 0.3391 0.2175 0.4219 0.1655
10"“ 2 20 0.2800 0.1930 0.4261 0.1354
1 20 0.1928 0.1676 0.4227 0.1034
101/ 2 20 0.1284 0.1540 0.3819 0.0922
10 20 0.0960 0.1305 0.2877 0.0870
10‘1 50 0.4706 0.2459 0.4618 0.2062
10-1/ 2 50 0.3986 0.2290 0.4801 0.1725
1 50 0.2895 0.1998 0.4614 0.1336
101/2 50 0.1887 0.1753 0.4166 0.1085
10 50 0.1263 0.1514 0.3424 0.0958
10'1 100 0.5991 0.2984 0.5029 0.2626
10‘“ 2 100 0.5136 0.2644 0.4949 0.2187
1 100 0.37 86 0.2242 0.4903 0.1626
101/2 100 0.2483 0.1885 0.4514 0.1208
10 100 0.1545 0.1582 0.3789 0.0975

 

 

 

50

 

 

Table 2.10: (Effect of Changing T) ,u. =1,N = 20 , Bias of the Estimates

 

 

Experi .. (1) (2) (3) (4)

mm“ T M“ ‘a(N)) E(J(é)-a(1v)) E(G(é)-a(1v)) E(ag‘g” —a(N))

1 5 0.3842 0.1333 -0.1472 0.2098

10 0.3750 0.1176 -0.1537 0.1767

N0 20 0.3795 0.1417 -0.1023 0.1782

TIE 50 0.3821 0.1380 -0.1085 0.1750

100 0.3764 0.1292 0.1193 0.1659

11 5 0.4032 0.1566 -0.1 191 0.2272

10 0.3988 0.1412 .0.1302 0.2066

EXACT 20 0.3908 0.1407 0.1220 0.1900

TIE 50 0.4037 0.1556 -0.0950 0.1969

100 0.3979 0.1635 -0.0720 0.1877

111 5 0.4012 0.1547 0.1210 0.2252

10 0.3971 0.1390 -0.1331 0.1986

NEAR 20 0.3950 0.1389 .0.1242 0.1885

TIE 50 0.4031 0.1563 0.0930 0.1962

100 0.3973 0.1614 -0.0757 0.1871

 

 

 

51

 

 

Table 2.11: (Effect of Chaning T ) ,u. = 1, N = 20 , Variance of the Estimates

 

 

 

 

 

 

 

 

Experiment T (I) (22 (32 (4)
a J(a) 0(a) 5527‘”
1 5 0.0365 0.1208 0.3162 0.0647
10 0.0359 0.1534 0.4191 0.0650
N0 20 0.0365 0.1678 0.4717 0.0669
TIE 50 0.0363 0.2247 0.7607 0.0663
100 0.0378 0.3265 1.0631 0.0702
11 5 0.0334 0.1139 0.3029 0.0593
10 0.0351 0.1484 0.4048 0.0640
EXACT 20 0.0361 0.1867 0.5337 0.0650
TIE 50 0.0384 0.2568 0.7893 0.0714
100 0.0366 0.3046 0.9902 0.0680
111 5 0.0332 0.1137 0.3030 0.0590
10 0.0351 0.1483 0.4050 0.0639
NEAR 20 0.0360 0.1868 0.5343 0.0649
TIE 50 0.0384 0.2542 0.7810 0.0713
100 0.0366 0.3089 1.0055 0.0680

52

Table 2.12: (Effect of Changing T) ,u. = 1, N = 20 , MSE of the Estimates

 

 

 

 

Experiment T (1) (22 (32 (4)
02 J(a) 0(a) 623%”
1 5 0.1841 0.1386 0.3379 0.1087
10 0.1765 0.1673 0.4427 0.0962
N0 20 0.1805 0.1879 0.4822 0.0987
TIE 50 0.1824 0.2637 0.7725 0.0969
100 0.1795 0.3432 1.0773 0.0977
11 5 0.1959 0.1384 0.3171 0.1109
10 0.1941 0.1683 0.4218 0.1043
EXACT 20 0.1935 0.2065 0.5486 0.1011
TIE 50 0.2014 0.2810 0.7989 0.1102
100 0.1949 0.3313 0.9954 0.1033
111 5 0.1942 0.1376 0.3176 0.1097
10 0.1928 0.1676 0.4227 0.1034
NEAR 20 0.1924 0.2061 0.5497 0.1004
TIE 50 0.2009 0.2787 0.7896 0.1098
100 0.1945 0.3349 1.0113 0.1030

 

 

53

 

 

2.9 Appendix: Proof of Lemma 2.2

Lemma 2.3 Let X] and X 2 be independent bivariate normals with different mean, but

[me] [(02 :21] (2.29)

E1m3X(X1.X2)1=P(X12X2)°E(X1|X1 >X2)+1"(X2 ZX1)'1‘3(X2|X2 >X1)-

identical variance, i.e.,

Then

Proof Let m=max(X1,X2), s=1ifX12X2 and =2ifX22X1,and m=X1ifs=1
and =X21fs=2.Then

f(m)=f(m,s=1)+f(m.s=2)
=P(s=1)f(m|s=l)+P(s=2)f(m|s=2)

and

E(m) = [mf(m)dm
=P(s=1)E(m|s=l)+P(s=2)E(m|s=2)
= P(s =1)E(X1 I X, 2 X2) 4 P(s = 2)E(X2 1 X2 2 X.)
= P(Xl 2 X2)E(X] IX] 2 X2) + P(X2 2 X1)E(X2 | X2 2 Xl ).

Lemma 2.4 By Lemma 2.3,

_ 1 _
E[max(X1,X2)] = $121—$218] + 361% (m #2)]

\1 20'2 20':2

\
(fir-#1) _ _1_ _(#2-.U1)
+<I>[-————20-2 ] [#2 + (50% ——————202 ].

/

 

Let X1 and X 2 be independent normals, where X I ~ N (0,02) and X 2 ~ N (112,02).

Then

54

E1max<X1.X2>]=[d> (ff—114+f 4%)] (2.30)

where p: :52.
0'

Proot. By Lemma 2.3, Without loss of generality, let ,u] = 0 and 14. be some constant
such that ,uz = pm (since “0'” is %—, ,uz —,u1—-> OasT —) 00). Then

P(X] 2 X2) = P(X] —,1’2 2 0)
P[(X1-X2)-(#1-#2) , 0-(#1-#2)]

7202 — 7202
(#1- #2)
= 1 — <1)
1- 1—7—1
_ ¢[(#1-#2)].
20'2
Similarly,

P(X2 2 X1) = M].

V202

From the facts about truncated bivariate normal distribution,

2

(#1 #2)
E X X >X = +— 0'). ———-— ,
( 1| 1 2) #1 if; [ 20

1
E(X2IX2>X1)=#2+ (#2 42—2].

TE" 77

55

 

 

 

 

 

 

0 and ,uz = ,um' (hence

Since we assume ,ul

 

 

 

if 7012
r x a
.62 1_F2\
17 m
+ \.II/
[\l/ulL fifi amH

 

 

\ z 2 ”—5 \W/ M. 2 2
26 (Mr/1c 26. {167 .47.

{2.\ .m. {.K + M
._.f w. 8 fl

56

Lemma 2.5 By Lemma 2.4, the bias is proportional to 0' when means are unequal:

bias = [4%]... + ޢ(%] - 11.] - 0'. (2.31)

bias = E[max(X],X2)]— 112
Elmax<X1.X2 11— m

\

= _<0(%\.w + 424-25)]9-440

2)
{48149481410-

 

 

 

57

2.10 Supplementary Tables
Supplemental Table 2.13: (Experiment I: No Tie) T = 5 , Bias of the Estimates

 

 

 

 

 

 

 

. (1) “(2) S3) (4)

#* N E(a—a(1v)) E(J(a)-a(1v)) E(G(a)-a(/v)) E(ég‘é?’ -050”)
10"1 =2—'=Ii-=-0.1597 0.0737 -0.0225 0.0996
10‘“ 2 2 0.1317 0.0513 -0.03 86 0.0747
1 2 0.0829 0.0251 -0.0395 0.0397
101/2 2 0.0384 0.0060 -0.0301 0.0157
10 2 0.0185 0.0059 -0.0081 0.0099
10“1 10 0.4632 0.2342 -0.0219 0.2991
10'“2 10 0.4028 0.1771 -0.0753 0.2420
1 10 0.2868 0.0907 -0.1285 0.1455
101/2 10 0.1524 0.0220 -0.1238 0.0564
10 10 0.0623 0.0006 -0.0684 0.0167
10‘1 20 0.5551 0.2601 -0.0697 0.3519
10‘“ 2 20 0.4961 0.2125 -0.1046 0.2988
1 20 0.3842 0.1333 -0. 1472 0.2098
101/2 20 0.2477 0.0785 -0.1 107 0.1219
10 20 0.1230 0.0273 -0.0797 0.0496
10‘l 50 0.6730 0.3228 -0.0687 0.4283
10'1/2 50 0.6119 0.2734 -0.1050 0.3727
1 50 0.5012 0.1999 -0.1371 0.2855
101/2 50 0.3627 0.1319 -0.1262 0.1934
10 50 0.2179 0.0730 -0.0890 0.1044
10‘1 100 0.7429 0.3411 -0.1081 0.4638
10‘“ 2 100 0.6827 0.2915 -0.1459 0.4099
1 100 0.5769 0.2311 -0.1556 0.3288
101/2 100 0.4343 0.1492 ~0.1694 0.2263
10 100 0.2853 0.0786 -0.1525 0.1307

 

 

 

58

 

 

Supplemental Table 2.14: (Experiment I: No Tie) T = 5 , Variance of the Estimates

 

 

 

 

 

(1) (2) (3) (4)

y. N (i J(é) 0(0) 07%”
10—1 2 0.0676 0.1030 0.1813 0.0804
10'“2 2 0.0704 0.1063 0.1840 0.0836
1 2 0.0790 0.1088 0.1732 0.0909
101/2 2 0.0886 0.1056 0.1445 0.0953
10 2 0.0939 0.1033 0.1236 0.0970
10‘1 10 0.0351 0.1090 0.2776 0.0603
10‘“ 2 10 0.0373 0.1 153 0.2929 0.0636
1 10 0.0462 0.1 182 0.2769 0.0731
101/2 10 0.0629 0.1204 0.2468 0.0841
10 10 0.0772 0.1054 0.1712 0.0881
10‘1 20 0.0281 0.1202 0.3363 0.0551
104/2 20 0.0295 0.1207 0.3347 0.0568
1 20 0.0365 0.1208 0.3162 0.0647
101/2 20 0.0537 0.1 194 0.2674 0.0789
10 20 0.0767 0.1210 0.2230 0.0942
10-1 50 0.0207 0.1088 0.3244 0.0447
10-1/2 50 0.0225 0.1128 0.3314 0.0479
1 50 0.0280 0.1207 0.3420 0.0560
101/ 2 50 0.0383 0.1203 0.3082 0.0662
10 50 0.0570 0.1187 0.2561 0.0815
10'1 100 0.0180 0.1113 0.3406 0.0421
104/2 100 0.0194 0.1146 0.3466 0.0451
1 100 0.0224 0.1 144 0.3351 0.0488
101/2 100 0.0298 0.1204 0.3306 0.0567
10 100 0.0439 0.1 187 0.2867 0.0694

_

 

 

 

59

 

 

Supplemental Table 2.15: (Experiment I: No Tie) T = 5 , MSE of the Estimates

 

 

 

 

 

(1) (2) (3) (4)
,1. N .2 J02) 6050 64538”
10'1 2 0.0931 0.1084 0.1818 0.0903
10‘“ 2 2 0.0877 0.1089 0.1855 0.0892
1 2 0.0859 0.1094 0.1748 0.0925
101/2 2 0.0901 0.1056 0.1454 0.0956
10 2 0.0943 0.1033 0.1237 0.0971
10‘1 10 0.2497 0.1638 0.2781 0.1498
10'1/2 10 0.1996 0.1467 0.2986 0.1222
1 10 0.1285 0.1265 0.2934 0.0943
101/2 10 0.0862 0.1209 0.2621 0.0873
10 10 0.0811 0.1054 0.1759 0.0884
10" 20 0.3362 0.1878 0.3411 0.1789
10"“ 2 20 0.2756 0.1658 0.3457 0.1461
1 20 0.1841 0.1386 0.3379 0.1087
101/2 20 0.1150 0.1256 0.2797 0.0937
10 20 0.0919 0.1217 0.2294 0.0967
10‘1 50 0.4736 0.2130 0.3291 0.2282
10‘“ 2 50 0.3969 0.1875 0.3424 0.1868
1 50 0.2792 0.1606 0.3608 0.1375
101/2 50 0.1698 0.1377 0.3241 0.1036

10 50 0.1045 0.1241 0.2640 0.0923 -
10'1 100 0.5699 0.2276 0.3523 0.2573
10‘“2 100 0.4855 0.1996 0.3679 0.2131
1 100 0.3552 0.1678 0.3594 0.1569
101/2 100 0.2184 0.1427 0.3594 0.1079
10 100 0.1253 0.1249 0.3100 0.0865

 

 

 

60

 

 

Supplemental Table 2.16: (Experiment 11: Exact Tie) T = 5 , Bias of the Estimates

 

 

 

 

 

(1) (2) (3) (4)

#* N E(é—auvfl E(J(C2)—a(1v)) E(G(C2)-a(1v)) E(égg” -510“)
** 2 0.1760 0.0892 -0.0078 0.1154
10"1 10 0.4669 0.2387 -0.0164 0.3027
10"”2 10 0.4159 0.1941 -0.0538 0.2554
1 10 0.3292 0.1382 -0.0753 0.1884
101/ 2 10 0.2506 0.1059 -0.0560 0.1454
10 10 0.2138 0.1097 -0.0067 0.1366
10'1 20 0.5569 0.2641 -0.0633 0.3540
10'“2 20 0.5014 0.2194 -0.0959 0.3039
1 20 0.4032 0.1566 -0.1 191 0.2272
101/2 20 0.3005 0.1210 -0.0797 0.1671
10 20 0.2372 0.1130 -0.0259 0.1449
10‘1 50 0.6713 0.3183 -0.0765 0.4249
10‘“2 50 0.6100 0.2647 -0.1213 0.3678
1 50 0.5049 0.2005 -0.1400 0.2867
101/ 2 50 0.3836 0.1460 -0.1196 0.2090
10 50 0.2853 0.1159 -0.0734 0.1594
10'1 100 0.7427 0.3407 -0.1088 0.4635
10-1/2 100 0.6841 0.2961 -0.1377 0.4118
1 100 0.5831 0.2339 -0. 1565 0.3352
101/2 100 0.4643 0.1839 -0.1297 0.2605
10 100 0.3489 0.1456 -0.0817 0.1955

 

 

 

 

 

Note: ** value of p: is irrelevant when N = 2 and there is an exact tie.

61

Supplemental Table 2.17: (Experiment II: Exact Tie) T = 5 , Variance of the

 

 

 

 

 

 

 

 

 

 

 

 

Estimates
(1) (2) (3) (4)
p, N a, J(&) 0(0) 0523‘"
——i— — —i-— 111——
" 2 0.0670 0.1014 0.1776 0.0797
10-1 10 0.0350 0.1104 0.2845 0.0601
10-1/ 2 10 0.0369 0.1132 0.2893 0.0618
1 10 0.0457 0.1 197 0.2878 0.0718
101/ 2 10 0.0591 0.1234 0.2657 0.0825
10 10 0.0661 0.1 106 0.2060 0.0837
10-1 20 0.0278 0.1 197 0.3366 0.0544
10-1/ 2 20 0.0286 0.1162 0.3238 0.0548
1 20 0.0334 0.1 139 0.3029 0.0593
101/ 2 20 0.0446 0.1 131 0.2670 0.0689
10 20 0.0548 0.1070 0.2226 0.0738
10-1 50 0.0205 0.1099 0.3314 0.0444
10-1/2 50 0.0216 0.1147 0.3442 0.0465
1 50 0.0250 0.1 129 0.3246 0.0504
101/2 50 0.0330 0.1093 0.2901 0.0578
10 50 0.0469 0.1 148 0.2662 0.0708
10-1 100 0.0179 0.1116 0.3434 0.0418
10-1/2 100 0.0186 0.1095 0.3335 0.0426
1 100 0.0233 0.1 187 0.3484 0.0500
101/2 100 0.0312 0.1210 0.3304 0.0597 .
10 100 0.0414 0.1149 0.2834 0.0670

Note: ** value of y» is irrelevant when N = 2 and there is an exact tie.

62

Supplemental Table 2.18: (Experiment II: Exact Tie) T = 5 , MSE of the Estimates

 

 

 

 

 

(1) (2) (3) (4)
,4 N 0 J02) 0(0) 5.59.01
** 2 0.0980 0.1094 0.1777 0.0930
10-1 10 0.2530 0.1674 0.2848 0.1517
10-1/2 10 0.2098 0.1509 0.2922 0.1270
1 10 0.1541 0.1388 0.2935 0.1073
101/2 10 0.1219 0.1347 0.2689 0.1036
10 10 0.1118 0.1226 0.2061 0.1023
10—1 20 0.3379 0.1895 0.3406 0.1797
10-1/2 20 0.2799 0.1644 0.3330 0.1471
1 20 0.1959 0.1384 0.3171 0.1109
101/2 20 0.1349 0.1278 0.2734 0.0968
10 20 0.1110 0.1198 0.2232 0.0947
10-1 50 0.4712 0.2112 0.3373 0.2249
10-1/2 50 0.3936 0.1847 0.3590 0.1817
1 50 0.2799 0.1531 0.3441 0.1326
101/2 50 0.1802 0.1306 0.3044 0.1015
10 50 0.1283 0.1282 0.2716 0.0962
10—1 100 0.5695 0.2276 0.3552 0.2566
10-1/2 100 0.4866 0.1972 0.3524 0.2121
1 100 0.3634 0.1734 0.3729 0.1624
101/2 100 0.2468 0.1548 0.3472 0.1275
111- .10 0163.1 _- 11 - ,, 011 02-105 .

 

Note: ** value of p. is irrelevant when N = 2 and there is an exact tie.

 

 

 

 

63

 

   

 

 

Supplemental Table 2.19: (Experiment III: Near Tie) T = 5 , Bias of the Estimates

 

 

 

 

 

 

 

 

64

 

 

. (1) “(2) 53) (4)

#* N E(a_a(N)) E(J(a)-a'(1v)) E(G(a)-a(N)) E(dg‘é‘” _a(N))
10-1 2 0.1689 0.0835 0.0121 0.1088
10-1/ 2 2 0.1547 0.0712 0.0221 0.0955
1 2 0.1 193 0.0443 0.0396 0.0650
101/ 2 2 0.0652 0.0136 0.0441 0.0272
10 2 0.0262 0.0006 0.0305 0.0084
10-1 10 0.4666 0.2383 0.0169 0.3023
111-“ 2 10 0.4146 0.1925 0.0558 0.2540
1 10 0.3229 0.1318 0.0817 0.1816
101/ 2 10 0.2206 0.0744 0.0890 0.1144
10 10 0.1319 0.0439 0.0545 0.0634
10-1 20 0.5568 0.2640 0.0634 0.3540
10-1/ 2 20 0.5011 0.2192 0.0960 0.3036
1 20 0.4013 0.1547 0.1210 0.2252
101/ 2 20 0.2887 0.1071 0.0960 0.1540
10 20 0.1912 0.0664 0.0732 0.0988
10-1 50 0.6713 0.3183 -0.0764 0.4249
111-“ 2 50 0.6099 0.2646 0.1214 0.3677
1 50 0.5045 0.2000 0.1404 0.2863
101/ 2 50 0.3807 0.1425 0.1239 0.2056
10 50 0.2694 0.0980 -0.0936 0.1422
10-1 100 0.7427 0.3407 0.1088 0.4635
10-1/ 2 100 0.6840 0.2960 0.1378 0.41 17
1 100 0.5830 0.2337 0.1569 0.3350
101/ 2 100 0.4632 0.1819 0.1325 0.2590
10 100 0.3422 0.1365 0.0934 0.1877

Supplemental Table 2.20: (Experiment III: Near Tie) T = 5 , Variance of the

 

 

 

 

 

 

 

65

 

 

Estimates
(1) (2) (3) (4)
,1. N 3 Na.) 0(02) 423%”
10-1 2 0.0670 0.1008 0.1758 0.0796
10'“ 2 2 0.0675 0.1013 0.1763 0.0800
1 2 0.0716 0.1068 0.1816 0.0849
101’ 2 2 0.0821 0.1087 0.1664 0.0930
10 2 0.0909 0.1062 0.1405 0.0965
10-1 10 0.0350 0.1 104 0.2845 0.0601
111-“ 2 10 0.0368 0.1132 0.2894 0.0618
1 10 0.0457 0.1 184 0.2839 0.0716
101 / 2 10 0.0623 0.1262 0.2694 0.0857
10 10 0.0781 0.1132 0.1946 0.0918
10-1 20 0.0278 0.1 197 0.3366 0.0544
10-1/ 2 20 0.0285 0.1162 0.3235 0.0547
1 20 0.0332 0.1137 0.3030 0.0590
101/ 2 20 0.0448 0.1 139 0.2699 0.0690
10 20 0.0602 0.1 135 0.2321 0.0794
10-1 50 0.0205 0.1099 0.3314 0.0444
111-“ 2 50 0.0216 0.1147 0.3443 0.0465
1 50 0.0250 0.1129 0.3244 0.0504
101’ 2 50 0.0330 0.1088 0.2890 0.0577 ‘
10 50 0.0478 0.1158 0.2680 0.0715
10-1 100 0.0179 0.1116 0.3433 0.0418
10-1/ 2 100 0.0186 0.1095 0.3334 0.0426
1 100 0.0233 0.1186 0.3482 0.0500
101’ 2 100 0.031 1 0.1207 0.3295 0.0595
10 100 0.0416 0.1156 0.2856 0.0672

Supplemental Table 2.21: (Experiment [11: Near Tie) T = 5 , MSE of the Estimates

 

 

 

 

 

 

 

 

66

 

 

<1) <2) . <3) <4)
,1. N 3 J18) 6050 4860’

10‘1 2 0.0956 0.1078 0.1759 0.0917-
10‘“ 2 2 0.0914 0.1063 0.1768 0.0892
1 2 0.0859 0.1088 0.1831 0.0891
101’ 2 2 0.0864 0.1089 0.1684 0.0937
10 2 0.0916 0.1062 0.1414 0.0966
10-1 10 0.2527 0.1672 0.2848 0.1515
10-1/ 2 10 0.2087 0.1503 0.2925 0.1263
1 10 0.1499 0.1358 0.2906 0.1046
101/2 10 0.1109 0.1318 0.2773 0.0988
10 10 0.0954 0.1151 0.1976 0.0958
10'1 20 0.3378 0.1895 0.3406 0.1797
10-1/ 2 20 0.2796 0.1642 0.3327 0.1469
1 20 0.1942 0.1376 0.3176 0.1097
101/ 2 20 0.1281 0.1253 0.2791 0.0927
10 20 0.0968 0.1 179 0.2374 0.0892
10-1 50 0.4711 0.2112 0.3372 0.2249
113-“ 2 50 0.3936 0.1847 0.3590 0.1817
1 50 0.2795 0.1529 0.3442 0.1324
101’ 2 50 0.1779 0.1291 0.3043 0.1000
10 50 0.1204 0.1254 0.2767 0.0918
10-1 100 0.5695 0.2276 0.3552 0.2566
10'“ 2 100 0.4865 0.1971 0.3524 0.2121
1 100 0.3632 0.1732 0.3728 0.1622
101/ 2 100 0.2457 0.1538 0.3470 0.1266
10 100 0.1587 0.1343 0.2943 0.1024

Supplemental Table 2.22: (Experiment I: No Tie) T = 20 , Bias of the Estimates

 

 

 

 

 

.. (1) “(2) “(3) (4)

#3 N E(a-a(1v)) E(J(a)-a(1v)) E(G(a)"a(N)) E(C’ig‘é?‘ _a(N))
10‘] 2 0.1788 0.0873 -0.0065 0.1083
10-1/2 2 0.1503 0.0686 0.0151 0.0830
1 2 0.0942 0.0249 -0.0463 0.0396
101/2 2 0.0425 0.0089 -0.0255 0.0131
10 2 0.0258 0.0170 0.0079 0.0169
10'1 10 0.4581 0.2121 -0.0403 0.2643
10‘” 2 10 0.3993 0.1589 -0.0877 0.2099
1 10 0.2920 0.0953 -0.1064 0.1291
101/2 10 0.1630 0.0354 -0.0956 0.0524
10 10 0.0675 0.0145 -0.0398 0.0149
10'1 20 0.5581 0.2761 -0.0132 0.3253
10"“2 20 0.4964 0.2231 -0.0573 0.2695
1 20 0.3795 0.1417 -0.1023 0.1782
101/ 2 20 0.2370 0.0607 -0.1203 0.0897
10 20 0.1213 0.0333 -0.0569 0.0391
10"1 50 0.6849 0.3532 0.0130 0.4082
10'“2 50 0.6243 0.3127 -0.0070 0.3551
1 50 0.5097 0.2218 -0.0736 0.2641
101/2 50 0.3668 0.1354 -0.1020 0.1696
10 50 0.2199 0.0614 -0.1011 0.0818
10-1 100 0.7621 0.3897 0.0076 0.4544
10‘“ 2 100 0.6999 0.3367 -0.0360 0.3979
1 100 0.5852 0.2448 -0.1045 0.3040
101/ 2 100 0.4414 0.1450 0.1592 0.2038
10 100 0.2923 0.0805 -0.1367 0.1133

 

 

 

67

 

 

Supplemental Table 2.23: (Experiment I: No Tie) T = 20 , Variance of the Estimates

 

 

 

 

 

(1) (2) (3) (4)
#1 N 0 J(&) 0(0) 5,3031
10-1 2 0.0721 0.1343 0.2703 0.0876
10-1/2 2 0.0735 0.1314 0.2577 0.0885
1 2 0.0824 0.1396 0.2634 0.0971
101’ 2 2 0.0971 0.1242 0.1806 0.1069
10 2 0.1014 0.1069 0.1203 0.1034
10-1 10 0.0347 0.1811 0.5272 0.0610
111-“ 2 10 0.0381 0.1836 0.5215 0.0663
1 10 0.0484 0.1768 0.4621 0.0782
101/2 10 0.0679 0.1619 0.3663 0.0934
10 10 0.0853 0.1231 0.2109 0.0995
10-1 20 0.0287 0.1893 0.5668 0.0577
111-“ 2 20 0.0301 0.1886 0.5601 0.0598
1 20 0.0365 0.1678 0.4717 0.0669
101/2 20 0.0516 0.1688 0.4344 0.0804
10 20 0.0754 0.1396 0.2853 0.0950
10-1 50 0.0221 0.1916 0.5987 0.0504
11,-“ 2 50 0.0225 0.1879 0.5863 0.0503
1 50 0.0271 0.1906 0.5753 0.0566
101/ 2 50 0.0405 0.1829 0.5033 0.0733
10 50 0.0630 0.1773 0.4247 0.0937
10-1 100 0.0187 0.2177 0.7015 0.0474
111-“ 2 100 0.0189 0.2004 0.6434 0.0471
1 100 0.0229 0.2038 0.6401 0.0532
101/ 2 100 0.0312 0.2069 0.6190 0.0628

10 100 0.0449 0.1696 0.4499 0.0748

 

 

 

68

 

 

Supplemental Table 2.24: (Experiment I: No Tie) T = 20 , MSE of the Estimates

 

 

 

 

 

 

 

 

 

 

(1 ) (2) (3) (4)
p, N do J(a) 0(0) 67%"
10-1 2 0.1041 0.1419 0.2704 0.0993
111-“ 2 2 0.0961 0.1361 0.2579 0.0953
1 2 0.0913 0.1402 0.2656 0.0986
101/ 2 2 0.0990 0.1243 0.1813 0.1071
10 2 0.1020 0.1072 0.1203 0.1037
10-1 10 0.2446 0.2261 0.5289 0.1309
111-“ 2 10 0.1975 0.2089 0.5292 0.1 103
1 10 0.1336 0.1859 0.4734 0.0948
101/ 2 10 0.0944 0.1632 0.3754 0.0962
10 10 0.0898 0.1234 0.2125 0.0998
10-1 20 0.3402 0.2655 0.5670 0.1635
10-1/ 2 20 0.2765 0.2383 0.5634 0.1325
1 20 0.1805 0.1879 0.4822 0.0987
101/ 2 20 0.1078 0.1725 0.4489 0.0884
10 20 0.0901 0.1408 0.2885 0.0965
10-1 50 0.4912 0.3163 0.5988 0.2170
10-1/ 2 50 0.4122 0.2856 0.5864 0.1764
1 50 0.2869 0.2398 0.5807 0.1263
101/2 50 0.1750 0.2012 0.5137 0.1021
10 50 0.1113 0.1811 0.4349 0.1004
10-1 100 0.5995 0.3695 0.7016 0.2539
10-1/2 100 0.5088 0.3138 0.6447 0.2055
1 100 0.3653 0.2637 0.6510 0.1456
101/ 2 100 0.2261 0.2279 0.6444 0.1043
10 100 0.1304 0.1761 0.4686 0.0876

69

Supplemental Table 2.25: (Experiment 11: Exact Tie) T = 20 , Bias of the Estimates

 

 

 

 

 

. (1) “(2) f3) (4)

l“ N E(a-a(N)) E(J(a)-a(N)) E(G(a)-a(N)) E(ag‘g” _a(N))
** 2 0.1947 0.1005 0.0038 0.1234
10-1 10 0.4617 0.2213 0.0253 0.2681
10"“2 10 0.4098 0.1712 -0.0737 0.2197
1 10 0.3218 0.1141 -0.0991 0.1534
101/2 10 0.2408 0.0867 -0.0714 0.1131
10 10 0.1940 0.0732 -0.0508 0.0991
10"1 20 0.5589 0.2701 -0.0262 0.3248
10’1/2 20 0.5009 0.2214 -0.0653 0.2717
1 20 0.3968 0.1407 -0. 1220 0.1900
101/ 2 20 0.2949 0.1 115 -0.0767 0.1378
10 20 0.2272 0.0918 -0.0470 0.1141
10-1 50 0.6826 0.3344 0.0229 0.4029
10‘“ 2 50 0.6203 0.2693 -0.0907 0.3450
1 50 0.5139 0.2065 -0.1089 0.2649
101/2 50 0.3862 0.1405 -0.1116 0.1831
10 50 0.2824 0.1010 -0.0851 0.1343
10‘1 100 0.7606 0.3815 -0.0073 0.4514
10'“2 100 0.6957 0.3089 -0.0880 0.3892
1 100 0.5828 0.2170 -0. 1584 0.2969
101/2 100 0.4534 0.1646 -0.1316 0.2160
10 100 0.3322 0.1116 -0.1148 0.1505

 

 

 

 

 

Note: ** value of ,u» is irrelevant when N = 2 and there is an exact tie.

70

Supplemental Table 2.26: (Experiment 11: Exact Tie) T = 20 , Variance of the

 

 

 

 

 

 

 

 

 

 

 

Estimates
(1) (2) (3) (4)
p, N a, J(&) G(&) 0229‘
*‘1‘ 0.0722 0.1352 0.2766 0.0878
10‘1 10 0.0340 0.1695 0.4919 0.0591
10‘“ 2 10 0.0363 0.1750 0.4999 0.0622
1 10 0.0450 0.1812 0.4922 0.0724
101/2 10 0.0562 0.1564 0.3786 0.0808
10 10 0.0658 0.1451 0.3228 0.0856
10'1 20 0.0283 0.1925 0.5822 0.0570
10"“2 20 0.0288 0.1801 0.5408 0.0563
1 20 0.0361 0.1867 0.5337 0.0650
101/ 2 20 0.0479 0.1603 0.4130 0.0745
10 20 0.0586 0.1435 0.3375 0.0795
10-1 50 0.0231 0.2004 0.6242 0.0524
10"“ 2 50 0.0251 0.2053 0.6351 0.0564
1 50 0.0296 0.2046 0.6163 0.0612
101/ 2 50 0.0377 0.1763 0.5008 0.0670
10 50 0.0496 0.1640 0.4312 0.0764
10‘1 100 0.0188 0.2195 0.7099 0.0473
10‘“ 2 100 0.0196 0.2180 0.7032 0.0483
1 100 0.0227 0.2058 0.6520 0.0521 '
101/2 100 0.0284 0.1840 0.5518 0.0570
10 100 0.0389 0.1681 0.4652 0.0670

Note: ** value of ,m is irrelevant when N = 2 and there is an exact tie.

71

Supplemental Table 2.27: (Experiment 11: Exact Tie) T = 20 , MSE of the Estimates

 

 

 

 

 

 

 

 

 

 

(1) (2) (3) (4)
,1. N a“: J (6%) 6(2) 022%“
1 2 0.1101 0.1453 0.2766 0.1030
10-1 10 0.2472 0.2185 0.4925 0.1310
10-1/2 10 0.2043 0.2043 0.5053 0.1104
1 10 0.1486 0.1942 0.5021 0.0959
101/2 10 0.1142 0.1639 0.2837 0.0936
10 10 0.1025 0.1505 0.3254 0.0954
10-1 20 0.3407 0.2655 0.5829 0.1625
10-1/ 2 20 0.2797 0.2291 0.5451 0.1302
1 20 0.1935 0.2065 0.5486 0.101 1
101/ 2 20 0.1348 0.1727 0.4188 0.0935
10 20 0.1102 0.1520 0.3397 0.0925
10-1 50 0.4890 0.3122 0.6248 0.2148
111-“ 2 50 0.4099 0.2779 0.6433 0.1754
1 50 0.2937 0.2472 0.6282 0.1314
101/ 2 50 0.1868 0.1960 0.5132 0.1006
10 50 0.1293 0.1742 0.4384 0.0944
10-1 100 0.5973 0.3651 0.7100 0.2511
10-1/2 100 0.5036 0.3134 0.7110 0.1998
1 100 0.3624 0.2529 0.6771 0.1403
101/2 100 0.2340 0.2111 0.5691 0.1036
10 100 0.1493 0.1805 0.4784 0.0897

“

Note: *‘1‘ value of ,w is irrelevant when N = 2 and there is an exact tie.

72

Supplemental Table 2.28: (Experiment [11: Near Tie) T = 20 , Bias of the Estimates

 

 

 

 

 

 

. (1) “(2) f3) (4)

[a N E(a-a(N)) E(J(a)-a(N)) E(G(a)-a(N)) E(ggg‘g” -a(N))
10’1 -2 0.1906 0.0974 0.0017 0.1193
10"1 l 2 2 0.1820 0.0899 -0.0047 0.1 106
1 2 0.1579 0.0705 -0.0193 0.0874
101/2 2 0.1052 0.0337 -0.0397 0.0429
10 2 0.0455 0.0026 -0.0404 0.0073
10‘1 10 0.4615 0.2212 -0.0254 0.2679
10‘“ 2 10 0.4091 0.1705 -0.0744 0.2190
1 10 0.3182 0.1106 -0.1023 0.1493
101/2 10 0.2237 0.0686 -0.0905 0.0948
10 10 0.1404 0.0295 -0.0842 0.0483
10‘l 20 0.5589 0.2700 -0.0263 0.3248
10"“2 20 0.5006 0.2208 -0.0662 0.2714
1 20 0.3955 0.1389 -0.1242 0.1885
101/ 2 20 0.2875 0.1022 -0.0879 0.1294
10 20 0.1998 0.0594 -0.0847 0.0867
10"1 50 0.6826 0.3344 -0.0229 0.4029
10"“ 2 50 0.6203 0.2693 -0.0907 0.3450
1 50 0.5137 0.2057 -0.1102 0.2646
101/2 50 0.3845 0.1384 -0.1141 0.1811
10 50 0.2733 0.0933 —0.0912 0.1241
10"1 100 0.7606 0.3815 -0.0073 0.4514
104/2 100 0.6957 0.3088 -0.0881 0.3892
1 . 100 0.5827 0.2167 -0.1588 0.2968
101/2 100 0.4529 0.1640 -0.1325 0.2154
10 100 0.3292 0.1091 -0.1168 0.1471

 

 

 

73

 

 

Supplemental Table 2.29: (Experiment III: Near Tie) T = 20 , Variance of the

 

Estimates

 

 

 

 

(1 ) (2) (3) (4)

,1 N C; J(a) 0(a) 61%”
——

10-1 2 0.0722 0.1341 0.2712 0.0878
10-“ 2 2 0.0724 0.1349 0.2703 0.0882
1 2 0.0733 0.1336 0.2614 0.0894
101/ 2 2 0.0785 0.1280 0.2359 0.0936
10 2 0.0914 0.1206 0.1857 0.1021
10-1 10 0.0340 0.1694 0.4915 0.0591
10-1/ 2 10 0.0363 0.1745 0.4987 0.0621
1 10 0.0449 0.1780 0.4837 0.0720
101/ 2 10 0.0569 0.1572 0.3811 0.0811
10 10 0.0691 0.1500 0.3276 0.0890
10-1 20 0.0283 0.1925 0.5823 0.0570
10-1/ 2 20 0.0288 0.1801 0.5410 0.0563
1 20 0.0360 0.1868 0.5343 0.0649
101/ 2 20 0.0480 0.1591 0.4098 0.0744
10 20 0.0608 0.1557 0.3683 0.0819
10-1 50 0.0231 0.2004 0.6242 0.0524
111-“ 2 50 0.0251 0.2054 0.6352 0.0563
1 50 0.0296 0.2051 0.6185 0.0612
101/ 2 50 0.0377 0.1755 0.4984 0.0669
10 50 0.0499 0.1629 0.4273 0.0764
10-1 100 0.0188 0.2195 0.7100 0.0473
10-1/ 2 100 0.0196 0.2180 0.7033 0.0483
1 100 0.0227 0.2060 0.6525 0.0522
101/ 2 100 0.0284 0.1843 0.5533 0.0569
10 100 0.0388 0.1659 0.4595 0.0667

 

 

 

74

 

 

Supplemental Table 2.30: (Experiment III: Near Tie) T = 20 , MSE of the Estimates

 

 

 

 

 

(1) (2) (3) (4)
y. N g) J(d) G02?) 03%”
10-1 2 0.1085 0.1436 0.2712 ' 0.1021—
10-1/2 2 0.1055 0.1430 0.2704 0.1005
1 2 0.0983 0.1386 0.2618 0.0971
101/ 2 2 0.0895 0.1292 0.2374 0.0955
10 2 0.0933 0.1206 0.1873 0.1022
10-1 10 0.2470 0.2183 0.4922 0.1309
10-1/2 10 0.2037 0.2036 0.5042 0.1 100
1 10 0.1461 0.1902 0.4941 0.0943
101/2 10 0.1070 0.1619 0.3893 0.0901
10 10 0.0888 0.1509 0.3347 0.0913
10-1 20 0.3406 0.2655 0.5830 0.1624
10-1/ 2 20 0.2794 0.2289 0.5454 0.1300
1 20 0.1924 0.2061 0.5497 0.1004
101/ 2 20 0.1307 0.1696 0.4175 0.0912
10 20 0.1007 0.1592 0.3755 0.0894
10-1 50 0.4890 0.3122 0.6247 0.2148
10-1/ 2 50 0.4099 0.2779 0.6434 0.1754
1 50 0.2935 0.2475 0.6307 0.1312
101/ 2 50 0.1855 0.1947 0.5114 0.0997
10 50 0.1246 0.1716 0.4356 0.0918
10-1 100 0.5973 0.3651 0.7100 0.2511
10-1/2 100 0.5036 0.3134 0.7110 0.1998
1 100 0.3623 0.2529 0.6777 0.1402
101/2 100 0.2335 0.2112 0.5709 0.1033
10 100 0.1472 0.1778 0.4731 0.0883

A

 

 

 

75

 

 

Supplemental Table 2.31: (Experiment I: No Tie) T = 50 , Bias of the Estimates

 

 

 

 

 

 

 

 

 

 

. (1) “(2) S3) (4)

#1 N E(“ - a(N)) E(J(a) - a(N)) E(G(a) - a(N)) E(C‘gg‘é?’ _ “(N9

10-1 2 0.1655 0.0769 0.0127 0.0913 —
10-1/2 2 0.1368 0.0452 0.0474 0.0647
1 2 0.0865 0.0228 -0.0415 0.0301
101/2 2 0.0419 0.0036 0.0352 0.0136
10 2 0.0183 0.0080 -0.0023 0.0075
10‘1 10 0.4538 0.2058 -0.0447 0.2549
10‘“ 2 10 0.3937 0.1505 -0.0952 0.1987
1 10 0.2802 0.0742 -0.1339 0.1091
101/2 10 0.1480 0.0076 -0.l342 0.0323
10 10 0.0582 0.0051 -0.0486 0.0030
10‘1 20 0.5617 0.2849 0.0053 0.3243
10"” 2 20 0.5013 0.2284 -0.0472 0.2707
1 20 0.3821 0.1380 -0. 1085 0.1750
101/ 2 20 0.2327 0.0397 0.1552 0.0770
10 20 0.1156 0.0175 -0.0815 0.0284
10‘l 50 0.6795 0.3383 -0.0064 0.3949
104/2 50 0.6166 0.2721 -0.0759 0.3379
1 50 0.5022 0.1998 -0.1056 0.2480
101/2 50 0.3571 0.1113 0.1371 0.1519
10 50 0.2035 0.0566 -0.091 7 0.0612
10"1 100 0.7560 0.3936 0.0276 0.4371
10'“2 100 0.6934 0.3274 -0.0423 0.3800
1 100 0.5830 0.2597 -0.0668 0.2950
101/2 100 0.4393 0.1623 -0.1 175 0.1966
10 100 0.2864 0.0641 -0.l604 0.1011

76

Supplemental Table 2.32: (Experiment I: No Tie) T = 50 , Variance of the Estimates

 

 

 

 

 

(1) (2) (3) (4)
,1 N a, J(o?) 6(4) 61%"
10-1 2 0.0743 0.1576 0.3519 0.0900
10-1/ 2 2 0.0758 0.1646 0.3752 0.0917
1 2 0.0821 0.1424 0.2884 0.0961
101/ 2 2 0.0906 0.1320 0.2291 0.0992
10 2 0.0948 0.1058 0.1311 0.0981
10-1 10 0.0378 0.2504 0.7629 0.0680
10-1/ 2 10 0.0399 0.2421 0.7280 0.0701
1 10 0.0480 0.2277 0.6688 0.0773
101/ 2 10 0.0687 0.2036 0.5307 0.0953
10 10 0.0873 0.1394 0.2628 0.1023
10-1 20 0.0301 0.2505 0.7964 0.0612
11,-“ 2 20 0.0309 0.2603 0.8358 0.0613
1 20 0.0363 0.2447 0.7607 0.0663
101/ 2 20 0.0499 0.2320 0.6830 0.0776
10 20 0.0674 0.1643 0.3958 0.0870
10-1 50 0.0212 0.2923 0.9773 0.0500
10-1/ 2 50 0.0212 0.2979 0.9987 0.0494
1 50 0.0253 0.2609 0.8493 0.0554
101/ 2 50 0.0349 0.2543 0.7934 0.0652
10 50 0.0545 0.1898 0.5094 0.0818 -
10-1 100 0.0174 0.2939 1.0040 0.0440
10-1/2 100 0.0188 0.3061 1.0381 0.0466
1 100 0.0221 0.2845 0.9516 0.0510
101/2 100 0.0304 0.2590 0.8317 0.0602
10 100 0.0.474 0.2337 0.6835 0.0785

*

 

 

 

77

 

 

Supplemental Table 2.33: (Experiment I: No Tie) T = 50 , MSE of the Estimates

 

 

 

 

 

 

 

 

 

 

 

(1) (2) (3) <4)
,1. N 81 J0?) 6(4) 6218‘”
- —

10'1 2 0.1017 0.1635 0.3521 0.0984
10-1/ 2 2 0.0945 0.1667 0.3775 0.0959
1 2 0.0896 0.1429 0.2901 0.0970
101/2 2 0.0924 0.1320 0.2303 0.0994
10 2 0.0951 0.1058 0.1311 0.0982
10"1 10 0.2438 0.2928 0.7649 0.1330
10‘“ 2 10 0.1949 0.2647 0.7370 0.1096
1 10 0.1265 0.2332 0.6848 0.0892
101/2 10 0.0906 0.2037 0.5487 0.0963
10 10 0.0907 0.1394 0.2652 0.1023
10'1 20 0.3456 0.3317 0.7965 0.1664
10"“2 20 0.2822 0.3125 0.8381 0.1346
1 20 0.1824 0.2637 0.7725 0.0969
101/ 2 20 0.1040 0.2336 0.7070 0.0836
10 20 0.0807 0.1646 0.4024 0.0878
10‘1 50 0.4828 0.4068 0.9773 0.2059
10'“ 2 50 0.4015 0.3720 1.0045 0.1636
1 50 0.2775 0.3009 0.8605 0.1 169
101/2 50 0.1624 0.2666 0.8122 0.0883
10 50 0.0959 0.1930 0.5178 0.0855
10-1 100 0.5890 0.4489 1.0048 0.2351
10‘“2 100 0.4995 0.4133 1.0399 0.1910
1 100 0.3620 0.3520 0.9560 0.1380
101/ 2 100 0.2234 0.2854 0.8455 0.0989
10 100 0.1294 0.2378 0.7093 0.0887

b

78

Supplemental Table 2.34: (Experiment I]: Exact Tie) T = 50 , Bias of the Estimates

 

 

 

 

 

.. (1) 52) S3) (4)

#11 N E(a - a(N)) E(J(a) - a(N)) E(G(a) - a(N)) E(égg?’ - “(110)
** 2 0.1820 0.0943 0.0058 0.1077
10-1 10 0.4571 0.1801 -0.0997 0.2577
10‘“ 2 10 0.4066 0.1573 -0.0945 0.2124
1 10 0.3200 0.1051 -0.1 120 0.1487
101/2 10 0.2471 0.1092 -0.0301 0.1218
10 10 0.2066 0.0887 -0.0305 0.1119
10"1 20 0.5628 0.2611 -0.0438 0.3246
111-“ 2 20 0.5063 0.2244 0.0605 0.2747
1 20 0.4037 0.1556 -0.0950 0.1969
101/2 20 0.2942 0.1009 -0.0943 0.1349
10 20 0.2240 0.0993 -0.0266 0.1105
10-1 50 0.6803 0.3342 0.0155 0.3956
10"“ 2 50 0.6202 0.2906 -0.0423 0.3419
1 50 0.5131 0.2297 -0.0565 0.2606
101/ 2 50 0.3814 0.1196 0.1449 0.1730
10 50 0.2745 0.0884 -0.0995 0.1209
10"1 100 0.7569 0.3767 -0.0072 0.4385
10‘“ 2 100 0.6969 0.3278 -0.0450 0.3856
1 100 0.5889 0.2496 -0.0932 0.3017
101/2 100 0.4582 0.1768 -0.1075 0.2166
10 100 0.3321 0.1062 -0.1218 0.1450

     

Note: *‘1' value of 131 is irrelevant when N = 2 and there is an exact tie.

 

 

 

79

 

 

Supplemental Table 2.35: (Experiment H: Exact Tie) T = 50 , Variance of the

 

 

 

 

 

 

 

 

 

Estimates
(1) (2) (3) (4)

N, N 0*, J(a) 0(a) 613291
** 2 0.0742 0.1572 0.3462 0.0900
10-1 10 0.0384 0.2842 0.8880 0.0695
111-“ 2 10 0.0409 0.2571 0.7858 0.0726
1 10 0.0494 0.2452 0.7225 0.0824
101/ 2 10 0.0600 0.1887 0.4997 0.0867
10 10 0.0666 0.1809 0.4589 0.0872
10-1 20 0.0310 0.2822 0.9012 0.0634
111-“ 2 20 0.0327 0.2628 0.8248 0.0654
1 20 0.0384 0.2568 0.7898 0.0714
101/2 20 0.0496 0.2243 0.6469 0.0799
10 20 0.0579 0.1739 0.4419 0.0807
10-1 50 0.0219 0.3086 1.0319 0.0514
111-“ 2 50 0.0230 0.2952 0.9821 0.0530
1 50 0.0275 0.2623 0.8435 0.0586
101/ 2 50 0.0391 0.2774 0.8571 0.0733
10 50 0.0520 0.2252 0.6421 0.0822
10-1 100 0.0173 0.3098 1.0634 0.0438
113-“ 2 100 0.0181 0.3005 1.0319 0.0448
1 100 0.0224 0.2968 0.9978 0.0517
101/ 2 100 0.0316 0.2735 0.8756 0.0638
10 100 0.0445 0.2433 0.7281 0.0758

Note: ** value of ,w- is irrelevant when N = 2 and there is an exact tie.

80

Supplemental Table 2.36: (Experiment 11: Exact Tie) T = 50 , MSE of the Estimates

 

 

 

 

 

(1) (2) (3) (4)

#1 N 6: J (68) 6(0) 633%?!
** 2 0.1073 0.1661 0.3463 0.1016
10-1 10 0.2473 0.3167 0.8979 0.1359
10"“2 10 0.2062 0.2819 0.7947 0.1177
1 10 0.1518 0.2563 0.7351 0.1045
101/2 10 0.1210 0.2006 0.5006 0.1015
10 10 0.1093 0.1887 0.4599 0.0997
10'1 20 0.3477 0.3504 0.9031 0.1688
10'“2 20 0.2891 0.3131 0.8285 0.1409
1 20 0.2014 0.2810 0.7989 0.1102
101/2 20 0.1361 0.2345 0.6558 0.0981
10 20 0.1081 0.1838 0.4426 0.0929
10"1 50 0.4847 0.4202 1.0321 0.2079
10"” 2 50 0.4076 0.3797 0.9839 0.1699
1 50 0.2907 0.3151 0.8467 0.1265
101/ 2 50 0.1845 0.2917 0.8781 0.1033
10 50 0.1273 0.2330 0.6520 0.0968
10’1 100 0.5902 0.4517 1.0635 0.2361
111-“ 2 100 0.5037 0.4079 1.0339 0.1935
1 100 0.3693 0.3591 1.0065 0.1427
101’ 2 100 0.2415 0.3048 0.8871 0.1107
10 100 0.1548 0.2546 0.7430 0.0969

 

 

 

 

 

_

Note: *‘1' value of p11 is irrelevant when N = 2 and there is an exact tie.

81

Supplemental Table 2.37: (Experiment III: Near Tie) T = 50 , Bias of the Estimates

 

 

 

 

 

A (1) Am A(3) (4)

141 N E(a - a(N)) E(J(a) -a'(N)) E(G(a') - a(N)) E(c‘rfi?’ _ “(N9
10’1 2 0.1796 0.0926 0.0047 0.1054
10-1/ 2 2 0.1745 0.0865 0.0024 0.1002
1 2 0.1593 0.0698 0.0205 0.0852
101/ 2 2 0.1221 0.0391 0.0448 0.0519
10 2 0.0696 0.0128 0.0444 0.0214
10-1 10 0.4570 0.1802 0.0994 0.2576
10-1/ 2 10 0.4061 0.1564 0.0958 0.2118
1 10 0.3175 0.1021 0.1155 0.1458
101/ 2 10 0.2359 0.0995 0.0384 0.1096
10 10 0.1688 0.0572 0.0556 0.0740
10-1 20 0.5628 0.2610 0.0438 0.3246
111-” 2 20 0.5062 0.2242 -0.0607 0.2746
1 20 0.4031 0.1563 0.0930 0.1962
101/ 2 20 0.2900 0.0967 0.0986 0.1302
10 20 0.2060 0.0705 -0.0664 0.0915
10-1 50 0.6803 0.3342 0.0155 0.3956
10-1 / 2 50 0.6202 0.2906 0.0424 0.3419
1 50 0.5129 0.2295 0.0567 0.2604
101/2 50 0.3801 0.1180 0.1467 0.1714
10 50 0.2677 0.0800 0.1097 0.1129
10-1 100 0.7569 0.3767 0.0072 0.4385
10-1/ 2 100 0.6969 0.3278 0.0449 0.3856
1 100 0.5889 0.2493 -0.0936 0.3016
101/ 2 100 0.4579 0.1759 0.1089 0.2161
10 100 0.3298 0.1038 0.1246 0.1424

 

 

 

82

 

 

Supplemental Table 2.38: (Experiment III: Near Tie) T = 50 , Variance of the

 

 

 

 

 

 

 

83

 

 

Estimates
(1) (2) (3) (4)

,1. N 61 M) 0(2) 6:88”
10" 2 0.0743 0.1550 0.3389 0.0901
10‘“ 2 2 0.0744 0.1565 0.3452 0.0903
1 2 0.0753 0.1625 0.3635 0.0916
101/2 2 0.0798 0.1602 0.3517 0.0966
10 2 0.0913 0.1487 0.2819 0.1057
10‘1 10 0.0384 0.2839 0.8870 0.0695
10‘“ 2 10 0.0409 0.2575 0.7871 0.0726
1 10 0.0494 0.2450 0.7234 0.0823
101’ 2 10 0.0604 0.1842 0.4836 0.0868
10 10 0.0697 0.1726 0.4209 0.0898
10’1 20 0.0310 0.2823 0.9015 0.0634
10‘“ 2 20 0.0327 0.2627 0.8243 0.0654
1 20 0.0384 0.2542 0.7810 0.0713
101/ 2 20 0.0499 0.2277 0.6587 0.0802
10 20 0.0600 0.1891 0.4901 0.0829
10'" 50 0.0219 0.3086 1.0318 0.0514
10‘” 2 50 0.0230 0.2952 0.9818 0.0530
1 50 0.0274 0.2619 0.8424 0.0586
101/2 50 0.0391 0.2753 0.8490 0.0733
10 50 0.0523 0.2243 0.6357 0.0825
10‘1 100 0.0173 0.3097 1.0634 0.0438
10'1/2 100 0.0181 0.3005 1.0320 0.0448
1 100 0.0225 0.2973 0.9996 0.0517
101’ 2 100 0.0316 0.2734 0.8754 0.0638
10 100 0.0446 0.2429 0.7262 0.0759

Supplemental Table 2.39: (Experiment III: Near Tie) T = 50 , MSE of the Estimates

 

 

 

 

 

 

 

 

 

84

 

 

(1) (2) (3) <4)
,1. N 131 J(é) 0(6) 128%“
_ —

10" 2 0.1065 0.1636 0.3389 0.1012
10‘“ 2 2 0.1048 0.1640 0.3452 0.1003
1 2 0.1007 0.1674 0.3639 0.0989
101/2 2 0.0947 0.1618 0.3537 0.0993
10 2 0.0961 0.1489 0.2839 0.1062
10-1 10 0.2472 0.3164 0.8969 0.1358
10‘“ 2 10 0.2058 0.2820 0.7963 0.1175
1 10 0.1502 0.2555 0.7367 0.1035
101/2 10 0.1161 0.1941 0.4851 0.0988
10 10 0.0982 0.1758 0.4240 0.0953
10"1 20 0.3477 0.3504 0.9034 0.1688
10"“ 2 20 0.2890 0.3129 0.8280 0.1409
1 20 0.2009 0.2787 0.7896 0.1098
101/2 20 0.1340 0.23 70 0.6684 0.0971
10 20 0.1024 0.1941 0.4945 0.0913
10‘1 50 0.4847 0.4202 1.0321 0.2079
10-1/2 50 0.4076 0.3796 0.9836 0.1699
1 50 0.2905 0.3146 0.8456 0.1263
101’ 2 50 0.1836 0.2892 0.8705 0.1027
10 50 0.1239 0.2307 0.6477 0.0952
10'1 100 0.5901 0.4517 1.0634 0.2361
10‘” 2 100 0.5037 0.4079 1.0340 0.1935
1 100 0.3692 0.3595 1.0084 0.1427
101/2 100 0.2412 0.3044 0.8873 0.1105
10 100 0.1534 0.2537 0.7417 0.0962

Supplemental Table 2.40: (Experiment I: No Tie) T = 100 , Bias of the Estimates

 

 

 

 

 

 

A (1) A(2) A(3) (4)

AAA N E(a - a(N)) E(J(a) - a(N)) E(G(a) ‘- a'(N)) E(dfi‘éo’ - “(N9
-10“ 2 0.1695 0.0762 00-17-7- 0.0942
10-1/ 2 2 0.1404 0.0415 0.0580 0.0672
1 2 0.0895 0.0303 -0.0293 0.0315
101/ 2 2 0.0451 0.0175 -0.0102 0.0170
10 2 0.0248 0.0159 0.0070 0.0142
10'1 10 0.4640 0.2064 -0.0526 0.2667
10‘“ 2 10 0.4066 0.1840 -0.0397 0.2146
1 10 0.2970 0.1308 -0.0362 0.1307
101/2 10 0.1625 0.0316 -0.1000 0.0469
10 10 0.0709 0.0198 -0.0315 0.0164
10'1 20 0.5576 0.2625 -0.0342 0.3179
10‘” 2 20 0.4956 0.2123 -0.0725 0.2615
1 20 0.3764 0.1292 -0.1 193 0.1659
101/ 2 20 0.2289 0.0357 -0. 1584 0.0711
10 20 0.0967 -0.0115 -0.1203 0.0013
10-1 50 0.6700 0.3282 0.0153 0.3783
10-“ 2 50 0.6109 0.2823 0.0480 0.3277
1 50 0.4996 0.1941 -0.1 129 0.2423
101/2 50 0.3540 0.1097 0.1359 0.1441
10 50 0.2067 0.0463 -0.1 149 0.0588
10’1 100 0.7513 0.3438 -0.0656 0.4244
10"“ 2 100 0.6898 0.2877 -0.1 163 0.3693
1 100 0.5796 0.2135 -0.1545 0.2853
101/2 100 0.4396 0.1704 -0.1002 0.1957
10 100 0.2879 0.0883 -0.1123 0.1046

 

 

 

85

 

 

Supplemental Table 2.41: (Experiment I: No Tie) T = 100 , Variance of the

 

 

 

 

 

Estimates

(1) (2) (3) (4)
p. N a“: J (6%) 0(0) éboo’
10'1 2 0.0705 0.1865 0.4701 0.0859
10‘“ 2 2 0.0731 0.1921 0.4892 0.0890
1 2 0.0816 0.1518 0.3241 0.0959
101/2 2 0.0933 0.1269 0.2174 0.1005
10 2 0.0999 0.1084 0.1323 0.1031
10‘1 10 0.0358 0.3330 1.0973 0.0650
10‘“ 2 10 0.0377 0.2942 0.9353 0.0676
1 10 0.0456 0.2399 0.7306 0.0750
101/2 10 0.0643 0.2228 0.6160 0.0895
10 10 0.0833 0.1437 0.2943 0.0961
10‘1 20 0.0291 0.3623 1.2233 0.0594
10"“ 2 20 0.0307 0.3459 1.1639 0.0615
1 20 0.03 78 0.3265 1.0631 0.0702
101/2 20 0.0497 0.2791 0.8717 0.0783
10 20 0.0728 0.2062 0.5430 0.0944
10'1 50 0.0207 0.3849 1.3494 0.0483
10"“ 2 50 0.0223 0.3790 1.3195 0.0505
1 50 0.0284 0.3729 1.2720 0.0604
101/2 50 0.0400 0.3196 1.0399 0.07311
10 50 0.0587 0.2583 0.7655 0.0880
10‘1 100 0.0179 0.4360 1.5507 0.0447
10"“ 2 100 0.0188 0.4158 1.4741 0.0467
1 100 0.0219 0.4145 1.4647 0.0513
101/2 100 0.0282 0.3212 1.0869 0.0585

10 100 0.0422 0.2800 0.9065 0.0720

 

 

 

86

 

 

Supplemental Table 2.42: (Experiment I: No Tie) T = 100 , MSE of the Estimates

 

 

 

 

 

(1) (2) (3) (4)
,1 N 13 J(a) 0(a) 02‘5”
10-1 2 0.0993 0.1923 0.4704 0.0948
111-“ 2 2 0.0928 0.1938 0.4926 0.0936
1 2 0.0896 0.1528 0.3249 0.0969
101/ 2 2 0.0953 0.1272 0.2175 0.1008
10 2 0.1005 0.1086 0.1323 0.1033
10-1 10 0.2512 0.3756 1.1001 0.1362
10-1/2 10 0.2030 0.3281 0.9369 0.1137
1 10 0.1338 0.2570 0.7319 0.0921
101/ 2 10 0.0908 0.2238 0.6259 0.0917
10 10 0.0883 0.1441 0.2953 0.0964
10-1 20 0.3400 0.4312 1.2245 0.1604
10-1/ 2 20 0.2763 0.3910 1.1691 0.1299
1 20 0.1795 0.3432 1.0773 0.0977
101/ 2 20 0.1021 0.2804 0.8968 0.0834
10 20 0.0822 0.2064 0.5574 0.0944
10-1 50 0.4697 0.4927 1.3496 0.1914
10-1/2 50 0.3955 0.4587 1.3218 0.1579
1 50 0.2780 0.4106 1.2847 0.1 191
101/ 2 50 0.1653 0.3317 1.0583 0.0939
10 50 0.1014 0.2605 0.7787 0.0915
10-1 100 0.5823 0.5542 1.5550 0.2248
10-1/2 100 0.4946 0.4986 1.4876 0.1831
1 100 0.3578 0.4601 1.4885 0.1327
101/ 2 100 0.2214 0.3502 1.0970 0.0968

10 100 0.1251 0.2878 0.9191 0.0829

 

 

 

87

 

 

Supplemental Table 2.43: (Experiment II: Exact Tie) T = 100 , Bias of the Estimates

 

 

 

 

 

A (1) A(2) A(3) (4)

,u. N E((2 - a(N)) E(J(a) - a(N)) E(G(a) - 0~'(N)) H0325” _ “(NV
** 2 0.1862 0.0873 -0.0122 0.1113
10-1 10 0.4684 0.2245 0.0206 0.2717
10"“ 2 10 0.4197 0.1989 -0.0230 0.2283
1 10 0.3351 0.1362 -0.0637 0.1652
101/ 2 10 0.2604 0.1160 0.0293 0.1342
10 10 0.2188 0.1177 0.0160 0.1251
10‘1 20 0.5585 0.2624 -0.0352 0.3178
111-“ 2 20 0.5001 0.2344 0.0326 0.2650
1 20 0.3979 0.1635 -0.0720 0.1877
101/2 20 0.2927 0.1135 -0.0666 0.1310
10 20 0.2207 0.0949 -0.0315 0.1036
10-1 50 0.6716 0.3436 0.0140 0.3806
10-1/ 2 50 0.6137 0.2607 0.0941 0.3306
1 50 0.5082 0.2214 -0.0668 0.2504
101/2 50 0.3820 0.1305 -0.1223 0.1698
10 50 0.2799 0.1109 -0.0589 0.1254
10‘1 100 0.7512 0.3446 -0.0641 0.4247
104/2 100 0.6917 0.3228 -0.0480 0.3727
1 100 0.5869 0.2257 -0. 1373 0.2942
101/2 100 0.4614 0.1853 -0.0922 0.2157
10 100 0.3423 0.1154 -0.1127 0.1550

Note: *‘1' value of ,m is irrelevant when N = 2 and there is an exact tie.

 

 

 

88

 

 

Supplemental Table 2.44: (Experiment 11: Exact Tie) T = 100 , Variance of the

 

 

 

 

 

 

 

 

 

Estimates
(1) (2) (3) (4)

,1 N 0 J(a) 0(0) 0229‘
** 2 0.0700 0.2014 0.5216 0.0852
10-1 10 0.0357 0.3111 1.0118 0.0643
10-1/ 2 10 0.0371 0.2893 0.9412 0.0653
1 10 0.0447 0.2763 0.8681 0.0741
101/ 2 10 0.0564 0.2394 0.6940 0.0824
10 10 0.0638 0.181 1 0.4765 0.0845
10-1 20 0.0289 0.3698 1.2550 0.0591
10-1/ 2 20 0.0303 0.3233 1.0794 0.0608
1 20 0.0366 0.3046 0.9902 0.0680
101/ 2 20 0.0468 0.2519 0.7850 0.0741
10 20 0.0606 0.2158 0.5960 0.0833
10-1 50 0.0203 0.3741 1.3142 0.0475
10-1/2 50 0.0213 0.4284 1.5061 0.0491
1 50 0.0265 0.3465 1.1800 0.0567
101/ 2 50 0.0379 0.3123 1.0184 0.0699
10 50 0.0524 0.2648 0.8073 0.0806
10-1 100 0.0170 0.4431 1.5913 0.0426
11,-“ 2 100 0.0169 0.3995 1.4257 0.0417
1 100 0.0203 0.4190 1.4803 0.0480
101/2 100 0.0279 0.3389 1.1418 0.0578
10 100 0.0403 0.3175 1.0317 0.0713

Note: *‘1' value of ,w. is irrelevant when N = 2 and there is an exact tie.

89

Supplemental Table 2.45: (Experiment 11: Exact Tie) T = 100 , MSE of the Estimates

 

 

 

 

 

(1) (2) (3) (4)

p, N 0 J(0) 0(0) 0329’
** 2 0.1047 0.2090 0.5217 0.0976
10-1 10 0.2550 0.3615 1.0123 0.1381
10-1/2 10 0.2132 0.3288 0.9417 0.1174
1 10 0.1570 0.2949 0.8722 0.1014
101/ 2 10 0.1242 0.2528 0.6948 0.1004
10 10 0.1117 0.1949 0.4767 0.1001
10-1 20 0.3408 0.4387 1.2562 0.1601
111-“ 2 20 0.2804 0.3782 1.0805 0.1310
1 20 0.1949 0.3313 0.9954 0.1033
101/ 2 20 0.1325 0.2648 0.7894 0.0913
10 20 0.1093 0.2248 0.5970 0.0940
10-1 50 0.4714 0.4922 1.3144 0.1923
10-1/ 2 50 0.3980 0.4964 1.5149 0.1584
1 50 0.2847 0.3995 1.1845 0.1194
101/ 2 50 0.1839 0.3293 1.0334 0.0987
10 50 0.1307 0.2771 0.8108 0.0963
10-1 100 0.5814 0.5618 1.5954 0.2229
10-1/ 2 100 0.4953 0.5037 1.4280 0.1806
1 100 0.3648 0.4700 1.4992 0.1345
101/ 2 100 0.2409 0.3732 1.1503 0.1043
10 100 0.1575 0.3309 1.0444 0.0953

 

 

 

 

 

“

Note: '1‘" value of ,u. is irrelevant when N = 2 and there is an exact tie.

90

Supplemental Table 2.46: (Experiment III: Near Tie) T = 100 , Bias of the Estimates

 

 

 

 

 

 

 

 

 

 

91

 

 

A (1) A0) A0) (4)

AAA N E(a-a(N)) E(J(a)-a(N)) E(G(a)-a(N)) 5(05’3‘5” -a(N))
10'1 2 0.1845 0.0845 - -0.0161 0.1096
10"”2 2 0.1811 0.0832 -0.0151 0.1063
1 2 0.1709 0.0768 -0.0178 0.0965
101/2 2 0.1434 0.0460 0.0518 0.0714
10 2 0.0932 0.0183 -0.0570 0.0346
10‘I 10 0.4683 0.2241 -0.0212 0.2716
10'“2 10 0.4195 0.1979 -0.0248 0.2280
1 10 0.3336 0.1350 -0.0645 0.1636
101/ 2 10 0.2531 0.1097 0.0344 0.1262
10 10 0.1932 0.0883 -0.0171 0.0995
10-1 20 0.5584 0.2624 0.0352 0.3178
10'“2 20 0.5000 0.2346 -0.0321 0.2649
1 20 0.3973 0.1614 -0.0757 0.1871
101/2 20 0.2895 0.1076 -0.0752 0.1272
10 20 0.2072 0.0784 -0.0510 0.0893
10-1 50 0.6716 0.3436 0.0140 0.3806
10"“2 50 0.6137 0.2607 -0.0941 0.3305
1 50 0.5081 0.2211 -0.0673 0.2503
101/2 50 0.3813 0.1275 -0.1276 0.1689
10 50 0.2762 0.1165 -0.0440 0.1214
10‘1 100 0.7512 0.3446 -0.0641 0.4247
10‘“2 100 0.6917 0.3227 -0.0481 0.3727
1 100 0.5869 0.2257 -0.1373 0.2941
101/2 100 0.4612 0.1845 -0.0935 0.2154
10 100 0.3409 0.1194 -0.1033 0.1533

Supplemental Table 2.47: (Experiment III: Near Tie) T = 100 , Variance of the

 

 

 

 

 

 

 

 

92

 

 

Estimates
(1) (2) (3) (4)

,1. N 0 1(3) 0(0) «2121”
10‘1 2 0.0701 0.2060 0.5374 0.0853
104/2 2 0.0702 0.2000 0.5174 0.0854
1 2 0.0708 0.1934 0.4980 0.0861
101/ 2 2 0.0742 0.1957 0.5052 0.0899
10 2 0.0854 0.1884 0.4379 0.1011
10-1 10 0.0357 0.3121 1.0158 0.0643
10'“2 10 0.0370 0.2914 0.9496 0.0653
1 10 0.0446 0.2749 0.8651 0.0740
101’ 2 10 0.0570 0.2377 0.6864 0.0831
10 10 0.0665 0.1946 0.5070 0.0876
10-1 20 0.0289 0.3698 1.2548 0.0591
10'“2 20 0.0303 0.3221 1.0746 0.0608
1 20 0.0366 0.3089 1.0055 0.0680
101/ 2 20 0.0469 0.2527 0.7887 0.0743
10 20 0.0614 0.2239 0.6266 0.0836
10" 50 0.0203 0.3741 1.3142 0.0475
10'“2 50 0.0213 0.4284 1.5059 0.0491
1 50 0.0265 0.3467 1.1807 0.0567
101/2 50 0.0380 0.3167 1.0340 0.0700
10 50 0.0528 0.2451 0.7314 0.0809
10‘1 100 0.0170 0.4431 1.5913 0.0426
10'“ 2 100 0.0169 0.3996 1.4262 0.0417
1 100 0.0203 0.4189 1.4801 0.0480
101’ 2 100 0.0279 0.3404 1.1478 0.0578
10 100 0.0403 0.3046 0.9839 0.0712

Supplemental Table 2.48: (Experiment III: Near Tie) T = 100 , MSE of the

 

 

 

 

Estimates
(1) (2) (3) (4)

,1. N 0 1(3) 0(a) 0806”
10-1 2 0.1041 0.2131 0.5376 0.0973
10"“ 2 2 0.1030 0.2069 0.5176 0.0967
1 2 0.1000 0.1993 0.4984 0.0954
101’ 2 2 0.0948 0.1979 0.5079 0.0950
10 2 0.0941 0.1887 0.4412 0.1023
10-1 10 0.2549 0.3623 1.0163 0.1380
10'“2 10 0.2130 0.3306 0.9502 0.1172
1 10 0.1559 0.2932 0.8693 0.1008
101/ 2 10 0.1210 0.2497 0.6876 0.0990
10 10 0.1038 0.2024 0.5073 0.0975
10-1 20 0.3408 0.4386 1.2561 0.1601
10"“ 2 20 0.2803 0.3771 1.0756 0.1309
1 20 0.1945 0.3349 1.0113 0.1030
101/ 2 20 0.1307 0.2643 0.7943 0.0905
10 20 0.1043 0.2301 0.6292 0.0916
10‘1 50 0.4714 0.4921 1.3144 0.1923
10"“ 2 50 0.3980 0.4964 1.5148 0.1584
1 50 0.2846 0.3956 1.1852 0.1194
101/2 50 0.1834 0.3330 1.0503 0.0986
10 50 0.1291 0.2587 0.7333 0.0957
10'1 100 0.5814 0.5618 1.5954 0.2229
10‘” 2 100 0.4953 0.5037 1.4285 0.1806
1 100 0.3647 0.4699 1.4989 0.1345
101/2 100 0.2406 0.3744 1.1565 0.1042
10 100 0.1565 0.3189 0.9946 0.0947

 

 

 

93

 

 

Chapter 3
Estimating Stochastic Frontier Models with Panel Data Using the Split-

Sample Jackknife

3.1 Introduction

The aim of this chapter is to consider bias corrections based on the jackknife in
stochastic frontier models with panel data. It is well known (e.g., Kim, Kim and Schmidt
(2007)) that the usual estimates of technical inefficiency based on fixed effects are biased
because the frontier is estimated as the maximum of the firm-specific estimated frontier
intercepts, and the “max” operation induces an upward bias in the estimated frontier. If

there is no tie for the best firm (in terms of the true frontier intercepts), the bias is of order

T ‘1 and the usual panel jackknife of Hahn and Newey (2004) removes the bias of that

order. However, Satchachai and Schmidt (2008) showed that, if there is an exact tie for

the best firm, the bias is of order T"l/2 and a different jackknife (the “generalized panel
jackknife”) is needed to remove the bias of that order. In either case, if the correct
version of the jackknife is used, their simulations indicated that the jackknife did indeed
remove bias quite effectively. However, the variance (and therefore the MSE) of the
estimators was very large.

Dhaene, Jochmans and Thuybaert (2006) proposed a “split sample jackknife” that

was intended to remove bias of order T ‘1 without increasing variance as much as the
usual panel jackknife. It is simply two times the original estimate (based on the whole

sample) minus the average of the two half-sample estimates. The weights on the various

94

estimates are of much smaller magnitude than for the panel jackknife and as a result its

variance is smaller. This estimator is relevant for the case of no tie for the best firm, so

that the bias is of order T"1 . However, in the case of an exact tie, where the bias is of

order T '1/2

, a different version of the split sample jackknife is needed to remove the
leading bias term. We derive that estimator (the “generalized split sample jackknife”) in
this chapter. We also determine the effects on bias if the wrong version of the jackknife
is used; that is, if the split sample jackknife is used but there is an exact tie, or if the
generalized split sample jackknife is used but there is not a tie.

Since there appears to be a trade-off between bias and variance, we consider
whether there is a split-sample jackknife that is optimal in the sense of minimizing mean
square error or variance. For the special case of N = 2 firms, we find the optimal
weights for the original estimate and the two half-sample estimates. These weights
depend on whether there is a tie, and if not, how close we are to having a tie.

We then perform Monte Carlo simulations to evaluate the finite sample relevance
of these results, and we compare these to the simulation of Satchachai and Schmidt
(2008).

The plan of the chapter is as follows. In Section 3.2, we introduce the model
upon which the chapter is used. Section 3.3 describes the split sample jackknife and the
generalized split sample jackknife. In Section 3.4, we discuss the “optimal” split sample

jackknife for the case of two firms. Section 3.5 reports our simulations. Finally, Section

3.6 contains our conclusion.

95

3.2 The Model

As in Satchachai and Schmidt (2008), we consider the standard panel data

stochastic frontier model with time-invariant technical inefficiency u ,- :
y),=a,-+x;-,,B+v,-,,i=1,...,N,t=1,...,T, (3.1)

where a,- = a — u,- and u,- 2 0. We consider fixed effect estimation in which the technical

inefficiency u,~ (and hence a,) is treated as fixed and no assumption is made about the

distribution of u,- or the idiosyncratic errors vit.

Given the within estimate 6 , the estimates 6?,- are the average of the firm-specific
residuals, i.e., 0?,- = 7,0351? where )7,- = T‘th y), and 3?,- = T’lztxi, , or equivalently
as the coefficients of the firm-specific dummy variables.

As suggested by Schmidt and Sickles (1984), the estimates of the frontier
intercept a and technical inefficiency u, are as follow

62' = max 68 ° 12" = é—éi, i=1,...,N . (3.2)

j=1,...,N J ’

We will regard 07 as an estimate of a( N) = max aj and 17,- as an estimate of
j=1,...,N

111 . ,, . 1. . ‘ .
u,- = a( N) —a,- = u,- — mm u j. Because of the “max operator 1n (3.2), a 18 biased
j=1,...,N

upward as an estimate of “(NT That is, the largest 6%,- is more likely to contain positive

estimation error than negative error. That is the bias that we wish to remove using the
jackknife.
Following Satchachai and Schmidt (2008), we distinguish the following two

cases. The first is the case of “no tie,” in which there is a unique value of i such that

96

a( N) = ai (that is, there is a unique best firm). Hall, Hardle and Simar (1995) show that
in this case [r is asymptotically normal (as T —> 00 with N fixed) and that the bootstrap

is valid. In this case the bias of 0? is of order T"1 . The second case is the case of an

exact tie, so that there are two or more values of i such that a( N) = 68,-. In this case 02 is
not asymptotically normal. The bootstrap is not valid, and Satchachai and Schmidt

(2008) show that the bias is of order T '1/2 .

3.3 Split-Sample Jackknife

In their simulations, Satchachai and Schmidt (2008) found that the panel jackknife
and generalized panel jackknife remove most of the bias, but their variances are large.
The “split-sample jackknife” proposed by Dhaene, Jochmans and Thuysbaert (2006) is an
attempt to remove bias without such a substantial increase in variance.

To describe the split-sample jackknife in a general setting, let the data be indexed

by t =1,...,T . Let 19 be the fixed effects estimator based on all T observations; let 19(1)

be the “first-half” sample estimator that omits observations from t = T / 2 +1 through T

for all cross-section units and uses only the first T / 2 observations, and let 61(2) be the
“second-halt” sample estimator that omits observations from t =1 through T/ 2 for all
cross-sectional units and uses only the second T/ 2 observations. Then the split-sample

jackknife estimator is

881(61): zé—Jz-(ém + 63121) = 261—0501110500). (3.3)

97

3.3.1 No Tie

This is the case of a unique best firm. The bias of (9 (i.e., Er ) is of order T‘1 and
. . . . c B D -3
we can express 1t 1n the followmg expansmn: E(6) = 6’ + ~77 + -—2 + 0(T ) . The panel
T
jackknife of Hahn and Newey (2004) is

J(é) = T19+(T—1)§1;Zté(,) , (3.4)

where 6"“) is the delete-observation-t estimator that omits observation I and uses the
other T -1 observations. It is easy to see that, given the bias expansion above,
E[J(61)] = 6 + 0(T-2) so that the leading term of the bias expansion has been removed.

The split sample jackknife SSJ(19) also removes the leading term of the bias
expansion. The motivation is that it might have smaller variance than the panel
jackknife, because it uses smaller “weights,” for example, with T = 100 , the panel
jackknife multiplies the original estimate by 100 and the mean of the delete-one estimates
by 99, while the split sample jackknife multiplies the original estimate by two and the

mean of the half-sample estimates by one.

3.3.2 An Exact Tie

When there is a tie for the best firm (the largest a1), Satchachai and Schmidt

(2008) show that the bias of 6% is, for large T , of order T—l/z. We express it with the

following expansion: E (9) = 6 + J?— + 712 + 0(T’3/ 2

J?

Schmidt suggested the generalized panel jackknife:

). In this case, Satchachai and

98

77 ~ T—li 1 .
19 2,190).

0(6) = JT-JT—l -«/T—~/T-1 7

(3.5)

This is (even more so then the panel jackknife) an aggressively weighted estimator. (For
example, with T = 100 the weights in (3.5) are 199.5 and 198.5, versus 100 and 99 for
the panel jackknife.) Accordingly its variance is large.

For the case of an exact tie, a split sample jackknife can also be used. We

propose the f0110wing “generalized split sample jackknifez”

‘5 é——l——-1-(é(1) +0121) (3.6)

72-71 72—12
72 1

The weights used here are «FT—1 = 3.414 and —— = 2.414. It is easy to verify that

72—1

GSSJ((9) =

. . B . .
these are the correct welghts to remove the leadlng term 37:- from the b1as expans1on,

and therefore reduce the bias to order T'l. Again, the motivation is that we hope that the

variance of this estimator will be smaller than the variance of 0(9).

3.3.3 What If The Wrong Version Is Used?
In this section, we show what happens to the bias of the estimate when the wrong
version of the split-sample jackknife is used.

1/2

First we consider the case that the bias is actually of order T’ (there is an exact

tie), but we use the split-sample jackknife that is designed to remove bias of order T‘l.

99

Theorem 3.1 If the bias is of order T'l/2 , the split-sample jackknife corrects the bias by

41 .4%.

Proof We have E (19) = 61 + i + higher order terms . So, dropping the higher order

)7

terms, we calculate

E(SSJ((9)) = 2E(é) — #30911) + 912))

B 1 213

=2 0+— —— 26+—

1 11) 11 1121
=6+0.586—2—.

7?

Comparing this to the original bias of TBT , we have removed the bias by 41.4%. 1:1

Next we consider the case that the bias is actually of order T ’1 (there is no tie),

but we use the generalized split sample jackknife that is designed to remove bias of order

T"V2.

Theorem 3.2 If the bias is of order T“l , the generalized split-sample jackknife reverses

the sign and changes the bias by a factor of 72 .
Proof: Suppose E(é) = 6 +-T12+ higher order terms. So, again dropping the higher order

terms,

100

E(GSS/(é)) = ——2‘/Z_TE(0) - 21_1 %E(63(') + 1912))
_ 72 B 1 1 23
——A‘/§_l(6+?) —‘/§_15[26+§1/—2]

B
=0— 2—.
7'],

Now, the bias has a reverse sign and the estimate is overly-corrected by a factor of f2- . C]

3.4 The “Optimal” Split-Sample Jackknife

Both intuition and previous simulations indicate that the jackknife may decrease
bias but increase variance. Given such a trade-off, we may wish to consider versions of
the jackknife that are optimal in the sense that they minimize MSE (or just variance).

To keep things simple, we will consider estimators that are similar to the split
sample jackknife, in the sense that they are linear combinations of the original estimator

and the two half-sample estimators. That is, we consider the estimators of the form
19' = 11019 + 1110(1) + 1120(2). (3.7)
Then we seek the “optimal” weights 00,01 and 02. However, instead of focusing only on

the weights that correct the bias, we now seek weights that minimize variance or MSE.
We can consider estimators that do, or do not, satisfy the following constraint:

(10 +al+az =1. (3.8)
This is basically the condition for consistency of 61 . Note that if we do not impose this

constraint, minimization of the variance of 67 is a silly problem since 00 = a] = 02 = O is

the degenerate solution. But it is a well-posed problem to minimize the variance of 5

101

subject to this constraint, or to minimize the MSE of 19 either with or without the
constraint.

To obtain analytic expressions for 00,01 and a2 , we consider the special case that

there are only two firms, i.e., N = 2. Although the number of firms considered is
restricted, the number of time periods T is not restricted.

We define the following notation

I

0=100 01 02] ; (39A)

(3) =[é 01') (9(2)); (3.93)

E0129 =1®o 91 921'; (3.90)
A V00 V01 V02

V(®)EV= V10 V11 V12 . (3.91))
V20 V21 V22

We note that 5 = 0'69 , so that E(g) = 0'8 and var(g) = a'Va.

The optimal estimators that we will derive are infeasible in practice, because they
depend on (0 and V. Our interest in them is that we want to see how the optimal
weights compare to the weights used by the jackknife procedures of the previous Section.
Also, we want to see how much better (in terms of variance or MSE) the optimal
estimators are. For example, if the optimal estimator is only slightly better than the
original estimator, there will be little point in further exploration of split-sample jackknife
procedures.

We now consider the case of a simplified version of the panel data stochastic
frontier model. We assume N = 2 , and we also assume normality and we do not include

regressors in the model. So we have

102

YIt ~N a] A 0'2 0 . (3.10)
121 22 0 02

The object of estimation (19 above) is a = max(a1,a2) . Without loss of generality, we
take 012 = 0 and a1 = a > 0. Therefore a is what we are trying to estimate, and a/0' is

a measure of how close we are to a tie.
The expression for G) and V , for this simplified model, are derived in Appendix.
Given these expressions, we then seek to minimize either of the following quantities

var(fi) = a’Va; (3.11A)

1455(0) = var(0) + 6102(0) = a'Va + a'®®'a - 2000 + 02. (3.1 113)

3.4.1 Unconstrained “Optimal” Split-Sample Jackknife

The unconstrained minimization (with respect to a) of var(b?) is trivial, namely,
a = 0 , 07 a 0 and var(Ei) = 0. However, the unconstrained minimization of MSE(0?) is

not trivial.

Proposition 3.1 Unconstrained “Optimal” Split-Sample Jackknife. The estimator
67 = 0'6) that minimizes MSE(5Z) is
0 = a[(V + 00')'10]'é) (3.12)

Proof See Appendix. 13
In the case of no tie, the unconstrained “optimal” split-panel jackknife that

minimizes MSE is as in (3.12) and is a non-trivial result. In an exact tie case, without

103

loss of generality, we can take a = 0. However, this leads us to the trivial solution where

a = 0 (00 = a] = 02 = 0) and there is no-trivial solution.

3.4.2 Constrained “Optimal” Split-Sample Jackknife
To maintain the connection with the other versions of j ackknife, we impose the

consistency constraint a0 + a] + 02 =1 . It is worth mentioning that with the constraint,

the estimate may or may not be first-order biased. For example, (i) in the case of no tie,

if 00 = 2 and a1 = 02 = —% , i.e., weights of the split sample jackknife, or (ii) in the case

of an exact tie, if 00 = 752/271 and a] = 02 = --A/—.2_1_—A , i.e., weights of the generalized

split sample jackknife, then there is no bias of the first-order, in the sense that the leading
term in the expansion of the bias has been removed. However, these are the weights that
completely remove the first-order bias from the original estimate and are not the choices

that minimize variance or MSE.

We can rewrite the constraint as a0 =1— 0] — 02 and, in matrix notation, as

-—110

a=el+Aa4,wheree1=[l 0 0],,A=[_1 0 I] and a4=[al a2]'.

Then we can write

var(0) = eiVel + 2a1A' Vel + a1A'VAa1, (3.13A)

MSE(0) = eicel + 21151116111 + a1A'CAa. — 2aeim — 2011:1111 + 02, (3.1313)

where C = V +®®'.

104

Proposition 3.2 Constrained “Optimal” (Minimum MSE) Split-Sample Jackknife. The

estimator 67 that minimizes MSE(£§) subject to the constraint 00 + a] + a2 =1 is

 

 

0(min MSE) = 1100 + 1110(1) + 1120(2),
where
a = VII-2V01-2(®1-a)(®0 —®1)
2V00 + VII-4V01+ 2((90 —@1)2
and a1=02 = Voo-V01+(€‘>0-6¥)(('90-®1)2 (3,14)
2V00 + V” -4V0] + 2(90 - @1)
Proof See Appendix. :1

If there is no tie for the best firm, the “optimal” weights (3.14) depend on the true
variance matrix V and on a (the difference between the two firms’ means). For the case

of an exact tie, we simply take a = 0 and the weights simplify to

0(min MSE) = 1.37670 —0.1887d(1)—0.1880&(2).2 (3.15)

Corollafl 3.1 (i) If the bias is of order T'l/2 , the “optimal” split-sample jackknife
(3.15) corrects the bias by almost 15%; and (ii) if the bias is of order T-l , the “optimal”

split-sample jackknife (3.15) corrects the bias by about 38%.

Proof (i) Suppose E (d) = a + £— + higher order terms. Dropping the higher order

77’"

terms,

 

2 The difference between the weights on the half-sample estimates, 0.1887 and 0.1880, is
due to the randomness in the Monte Carlo evaluation of V .

105

 

 

.. _ i _ B _ B
E(0)_1.3767(0+fi] 0.1887[a+A/T/_2] 0.1880(a+m]

= a + 0.84391.

JT
.. . B . . .
(11) Suppose E (a) = a + T + higher order terms. Droppmg the h1gher order

terms,

E(§)=l.3767[a+—2]-0.1887 a+—-B—— —0.l88 a+—-Bi—
T T/2 772

= a + 0.62332.
T

Proposition 3.3 Constrained “Optimal” (Minimum Variance) Split-Sample Jackknife.

The estimator 5 that minimizes var(Ei) subject to the constraint 00 + a] + a2 =1 is

if (min var) = 0062 + 0162(1) + 0262(2),
where

V11-2V01

00 = V00 "V01
2V00 +V11-4V01

2V00 + V11-4V01 '

 

 

and a] =a2 = (3.16)

Proof See Appendix. (:1
Note that the “optimal” weights (3.16) only depend on the true variance matrix
V . In the case of an exact tie, i.e., a = 0 , the constrained “optimal” split-sample

jackknife simplifies to

0(min var) = 0.50 + 0.25011) + 0.25012). (3.17)

106

Corollafl 3.2 (i) If the bias is of order T’V2 , the bias of the “optimal” split-sample
jackknife (3.17) increases by 21%; and (ii) if the bias is of order T"I , the bias of the

“optimal” split-sample jackknife (3.17) increases by 50%.

Proof (i) Suppose E (o?) = a + -B— + higher order terms . Dropping the higher order

)7

terms,

 

1111.11.11»:0.5[1.%)015[1.%].115[.. [53]

=a+1.2071—B—.

J?

(ii) Suppose E (0?) = a + TB— + higher order terms . Dropping the higher order

terms,

2. . B B B
E (a (mm var)) — 0.50[a + T] + 025(6! + 372—] + 025(0 + m]

=a+1.50—B-.
T

(:1
As expected, minimizing the variance of the estimator, without regard'to bias, will

increase bias.

3.5 Simulations

In this section, we investigate the finite sample performance of four estimators: (i)
the split-sample jackknife estimate, SS/(é); (ii) the generalized split-sample jackknife,

GSSI(0?); (iii) the optimal split-sample jackknife that minimizes the MSE, 5 (min MSE ) ;

107

and (iv) the optimal split-sample jackknife that minimizes the variance, 5 (min var) . In
the last two cases, these are the estimators in (3.14) and (3.16) that minimized MSE and
variance subject to the restriction 00 + a] + 02 =1 . We also compare the results for these
estimators to the original estimator d , the panel jackknife estimator J (6?) , and the

generalized panel jackknife G(cir) that were analyzed in Satchachai and Schmidt (2008).

3.5.1 Design of the Experiments
We consider the model with no regressors, as in Satchachai and Schmidt (2008).
The inclusion of regressors would not change our results much because the coefficients

(,6) of the regressors are estimated so much more precisely than the a,- are.

Thus, the data generating process is

y), =a+v,-, —u,- fori=l,...,NandT=l,...,T (3.13)

=m-m
The u,- are i.i.d. half-normal: u,- =|U,-| where U,- ~ N(0,0'3); and the vi, ~ N(0,0'3).

These distributional assumptions are not used in estimation. They just characterize the
data generating process.

We employ the parameterization used in Satchachai and Schmidt (2008). Our
_ -l/ 2 2 _ 2
parameters are N ,T and ,u: — (0'u )1. / T 0", where (0,, )4 — ((7r—2)/7r)0'u . We use
,u: because of its convenient interpretation. It measures the standard deviation of the a,-

in units of the standard deviation of the 07,-.

108

To maintain the connection to Satchachai and Schmidt (2008), we use the same

parameter values: we fix 03/T :01 and consider 114 =10_1,10"V2,1,10V2 and 10.
Then, for a given value of 114, we can determine (0'3 )4 and 03, i.e.,

(1) ,1. =10‘I = 0.1: (03). = 0.001; 03 = 0.0028;

(2) ,1. = 10“'/2 = 0.3162: (03).. = 0.01; 03 = 0.0275;

(3) ,1. =1: (03). = 0.1; 03 = 0.2752;

(4) ,1. =101/2 =3.1623: (03). =1; 03 = 2.7519;

(5) ,1. =10: (03).. =10; 03 = 27.5194.

For the split-sample jackknife and the generalized split-sample jackknife, we set

T =10 and consider sample sizes N = 2,10,20,50 and 100.

However, for the “optimal” split-sample jackknife, we set N = 2. To calculate

the weights 00,01 and 02 , we need values of the parameter a = laa) — 03(1)| for a given

101. These numbers are shown in Table 3.7.

We consider two experiments with the setup described above:
(1) Experiment I (No Tie). The setup of this experiment is exactly as just

described. There are no restrictions on the a). They follow from the draws of the half-

normal u,- .

(2) Experiment 11 (An Exact Tie). We generate the data as described above. Now

'we (the data generator) know which firm is the best firm and the value of its intercept

109

a( N) = max aj. Then, we randomly select one of the other (N -1) firms and set its
j=l,...,N

intercept equal to a( N) . Hence, we have created an exact two-way tie for the best firm.

For each configuration of {113,N,T}, we perform 1,000 replications. Then we
report the bias, variance and MSE for each estimator. The results for 07 , J (0?) , and

(1(0) are taken from Satchachai and Schmidt (2008).

3.5.2 Results

Table 3.1 gives the bias of the various estimators for Experiment I in which there
is no tie. When ,m is small, we are in a sense close to a tie. The generalized jackknife
G(0?) and the generalized split-sample jackknife GSSJ(07) are closed to being unbiased.
The panel jackknife J (62) removes almost half of the bias and the split-sample jackknife
removes about 41% of the bias, as theory predicts (Theorem 3.1).

For larger values of ,u. , when we are farther from a tie, the panel jackknife and
the split-sample jackknife do a good job of removing bias, while the generalized panel
jackknife and the generalized split-sample jackknife overcorrect (reverse the sign of the
bias). Again, this is as theory predicts (Theorem 3.2).

Table 3.2 gives the variance of the various estimators. As expected, the split-
sample jackknife has a smaller variance than the panel jackknife, and the generalized
split-sample jackknife has a smaller variance than the generalized panel jackknife. In all
cases, the variance is larger than the original fixed effects estimator d .

Table 3.3 gives the MSE of the various estimators. The generalized panel

jackknife and the generalized split-sample jackknife have large MSEs. The MSE of the

110

split sample jackknife is generally smaller than the MSE of the panel jackknife, but
bigger than the MSE of the original fixed effects estimator. However, in some cases

(large N and/or small #111) the split sample jackknife is better in terms of MSE than the

original fixed effects estimator.

Table 3.4, 3.5 and 3.6 give the same results for the case of an exact tie. Generally

speaking, the results are similar to those in Table 3.1, 3.2 and 3.3 for #4 =10'1 (a near

tie). The generalized panel jackknife and the generalized split sample jackknife do a
good job of removing bias. The generalized split sample jackknife is better than the
generalized panel jackknife, in terms of variance and MSE, but in most cases its MSE is
larger than the MSE of the original fixed effects estimate or of the split sample jackknife.
The split sample jackknife is better than the original fixed effects estimator, in terms of
bias and MSE, for almost all of those exact-tie cases.

Table 3.7 shows the “optimal” weights for the two “optimal” split-sample
jackknife estimators, in which MSE and variance are minimized subject to the constraint

00 + a] + 02 =1 . For the estimator that minimizes MSE, for small values of ,u. (and a ),
the weight 00 is greater than one, while the weights 0] and 02 are negative. These

patterns of the “optimal” split-sample jackknife that minimizes MSE are similar to those
of other versions of jackknife. On the other hand, all of the weights of the “optimal”
split-sample jackknife that minimizes variance are less than one. Clearly smaller weights
are helpful in keeping the variance of the estimator small, and this agrees with the fact
that the estimators that use large weights to aggressively remove bias, e.g., the

generalized panel jackknife, tend to have large variance.

Ill

In Table 3.8 we compare the bias, variance and MSE of the “optimal” split-
sample jackknife to those of the original fixed effects estimate 6% . Comparisons to the
other estimators that we have considered can be made by referring to the entries in Tables
3.1-3.6.

First consider 07(min MSE). Its bias is slightly smaller than the bias of the

original fixed effects estimator, but its variance is slightly larger. Its MSE is very similar
to that of the original estimator. (In fact, in some cases it appears to be slightly larger,
which logically cannot be the case. This must be a reflection of numerical inaccuracy,
which is small but not small relative to the difference in the MSE of the estimators.) The
obvious conclusion is that, while split-sample jackknife methods can be used to remove
or reduce bias, they will not be successful in reducing the MSE of the estimate by any
meaningful amount.

For §(min var) the situation is somewhat different. Its variance is slightly

smaller than the variance of the original fixed effects estimator, while its bias and MSE
are slightly larger. All of these differences are small. So, again, while split-sample
jackknife methods can be used to remove or reduce bias, that objective is not compatible

with reduction of variance or MSE.

3.6 Conclusions

In this chapter, we have tried to find a jackknife-type estimator of the frontier
intercept that has small MSE and/or small variance. We have investigated the
performance of the split-sample jackknife estimator and the generalized split-sample

jackknife. We also consider the “optimal” split-sample jackknife, which minimizes MSE

112

or variance. In terms of the weights that define the estimators, these estimators are less
aggressive in removing bias than the panel jackknife and generalized panel jackknife.

When there is no tie for the best firm, we show that the split-sample jackknife is

also effective in removing bias of order T ’1 , but has smaller variance and smaller MSE
than the panel jackknife. Although the “optimal” split-sample jackknife has even smaller
variance and MSE, it does not properly remove the bias, i.e., the estimate is still biased
upward. Also it is not a feasible estimator outside the simulation setting.

When there is an exact tie, the generalized split-sample jackknife also correctly
removes the bias, but again its variance and MSE increase significantly comparing to the
original fixed effects estimate. In terms of variance and MSE, it is the worst estimator
among the four estimators considered. In this case, the “optimal” split-sample jackknife
successfully reduces the variance and MSE. This is not surprising since 6% corresponds

to 00 = 1,01 = 02 = 0 and these are not “optimal.”

113

3.7 Output Tables

Table 3.1: (Experiment I: No Tie) T = 10 , Bias of the Estimates

 

 

 

 

 

,1. I N l 0 l 1(0) 881(0) 0(0) | 0881(0)
10-1 2 0.1671 0.0810 0.0898 0.0099 0.0195
10-1/2 2 0.1391 0.0522 0.0639 0.0394 0.0424
1 2 0.0887 0.0230 0.0274 0.0463 0.0594
101/2 2 0.0462 0.0125 0.0130 0.0230 0.0340
10 2 0.0294 0.0199 0.0172 0.0100 0.0001
10-1 10 0.4532 0.2113 0.2480 0.0436 0.0423
(1)-“2 10 0.3935 0.1556 0.1921 0.0951 0.0927
1 10 0.2809 0.0828 0.1045 0.1261 0.1450
101/2 10 0.1504 0.0293 0.0343 0.0983 0.1300
10 10 0.0577 0.0034 0.0036 0.0678 0.0904
10-1 20 0.5566 0.2724 0.31 18 0.0271 0.0344
(1)-“2 20 0.4928 0.2093 0.2518 -0.0895 0.0890
1 20 0.3750 0.1176 0.1593 0.1537 0.1457
101/2 20 0.2349 0.0563 0.0762 0.1321 -0.1483
10 20 0.1136 0.0074 0.0230 -0.1046 0.1052
10-1 50 0.6699 0.3132 0.3765 0.0629 0.0383
10-1/2 50 0.6092 0.2678 0.3219 0.0921 0.0843
1 50 0.4973 0.1903 0.2352 0.1334 0.1354
101/2 50 0.3556 0.1151 0.1426 0.1385 0.1586
10 50 0.2059 0.0379 0.0574 0.1391 0.1527
10-1 100 0.7584 0.3627 0.4336 0.0544 0.0256
10-1/2 100 0.6949 0.3127 0.3744 0.0901 0.0790
1 100 0.5809 0.2293 0.2805 0.1413 0.1443
101/2 100 0.4433 0.1652 0.1956 -0.1280 0.1548
10 100 0.2950 0.0922 0.1118 0.1217 0.1474

_

 

 

 

 

114

 

 

Table 3.2: (Experiment I: No Tie) T = 10 , Variance of the Estimates

 

 

 

 

 

I“ N d J (63') SSJ(é) C(62) GSSJ(&)
10‘1 2 0.0666 0.1 130 0.0932 0.2127 0.1768
104/2 2 0.0696 0.1 159 0.0954 0.2149 0.1764
1 2 0.0803 0.1 183 0.1020 0.2002 0.1698
101/2 2 0.0932 0.1133 0.1059 0.1582 0.1477
10 2 0.0991 0.1055 0.1042 0.1189 0.1209
10‘1 10 0.0355 0.1440 0.0821 0.3871 0.2243
10'”2 10 0.03 82 0.1472 0.0861 ‘ 0.3901 0.2308
1 10 0.0438 0.1478, 0.0969 0.3665 0.2406
101/2 10 0.0696 0.1378 0.1098 0.2836 0.2289
10 10 0.0890 0.1282 0.1 173 0.2106 0.1997
10"1 20 0.0264 0.1419 0.0705 0.4089 0.2090
10'”2 20 0.0284 0.1467 0.0743 0.4191 0.2182
1 20 0.0359 0.1534 0.0860 0.4191 0.2406
101/2 20 0.0518 0.1388 0.0930 0.3297 0.2236
10 20 0.0757 0.1329 0.1043 0.2613 0.1979
10‘1 50 0.0208 0.1500 0.0670 0.4750 0.2107
10‘“2 50 0.0215 0.1489 0.0669 0.4518 0.2078
1 50 0.0254 0.1479 0.0697 0.4356 0.2064
101/2 50 0.0359 0.1438 0.0833 0.3 896 0.2281
10 50 0.0569 0.1401 0.1002 0.3257 0.2321
10'1 100 0.0191 0.1617 0.0643 0.5014 0.2056
10'“2 100 0.0196 0.1556 0.0655 0.4799 0.2085
1 100 0.0231 0.1587 0.0705 0.4807 0.2156
101 / 2 100 0.0317 0.1585 0.0802 0.4490 0.2245
10 100 0.0440 0.1400 0.0913 0.3532 0.2316

 

 

 

 

115

 

 

Table 3.3: (Experiment I: No Tie) T = 10 , MSE of the Estimates

 

 

 

 

10

———————.1_——

 

100

 

0.1310

 

0.1485

 

116

0.1038

 

0.3680

 

pt I N I a? I J ((2) I SSJ(&) I C(62) I GSSJ(&)
10“1 2 0.0945 0.1196 0.1012 0.2128 0.1772
10”“ 2 2 0.0890 0.1 186 0.0995 0.2164 0.1782
1 2 0.0882 0.1188 0.1028 0.2024 0.1733
101/2 2 0.0953 0.1134 0.1061 0.1588 0.1488
10 2 0.1000 0.1059 0.1045 0.1190 0.1209
10‘1 10 0.2409 0.1887 0.1436 0.3891 0.2261
10‘”2 10 0.1931 0.1715 0.1230 0.3991 0.2394
1 10 0.1272 0.1547 0.1078 0.3824 0.2617
101’ 2 10 0.0922 0.1386 0.1110 0.2932 0.2458
10 10 0.0923 0.1282 0.1173 0.2152 0.2078
10'1 20 0.3362 0.2162 0.1677 0.4096 0.2101
10'“2 20 0.2712 0.1905 0.1377 0.4271 0.2261
1 20 0.1765 0.1673 0.1 113 0.4427 0.2618
101/2 20 0.1070 0.1420 0.0988 0.3472 0.2456
10 20 0.0886 0.1329 0.1048 0.2722 0.2090
10'I 50 0.4696 0.2481 0.2088 0.4610 0.2122
10’” 2 50 0.3925 0.2206 0.1705 0.4603 0.2149
1 50 0.2727 0.1841 0.1250 0.4534 0.2247
101/2 50 0.1623 0.1571 0.1037 0.4087 0.2532
10 50 0.0993 0.1415 0.1034 0.3451 0.2554
10" 100 0.5942 0.2932 0.2524 0.5044 0.2062
10‘” 2 100 0.5025 0.2534 0.2056 0.4880 0.2148
1 100 0.3605 0.2112 0.1492 0.5006 0.2364
101’ 2 100 0.2282 0.1858 0.1185 0.4654 0.2485

0.2533

Table 3.4: (Experiment 11: Exact Tie) T = 10 , Bias of the Estimates

 

 

 

 

 

 

 

 

 

 

p: N d J (0?) SSI(0?) G(o}) GSSI(0'2)
*" 2 0.1828 0.0997 ' 0.1054 0.0120 -0.0042
10’1 10 0.4580 0.2165 0.2533 -0.03 80 -0.0361
10'1/2 10 0.4077 0.1693 0.2065 -0.0820 -0.0780
1 10 0.3261 0.1268 0.1505 -0.0832 -0.0979
101/2 10 0.2499 0.1 122 0.1 189 -0.0330 -0.0663
10 10 0.2052 0.0847 0.1044 -0.0424 -0.0382
10"1 20 0.5593 0.2696 0.3142 -0.0357 -0.0324
10'“ 2 20 0.5020 0.2158 0.2608 -0.0859 -0.0805
1 20 0.3988 0.1412 0.1771 -0.l302 -0.l363
101/2 20 0.2938 0.0976 0.1212 -0. 1093 -0.1230
10 20 0.2282 0.0954 0.1038 -0.0446 -0.0720
10'1 50 0.6707 0.3104 0.3773 -0.0693 -0.0376
10‘“2 50 0.6134 0.2715 0.3279 -0.0889 -0.0758
1 50 0.5124 0.21 17 0.2548 -0.1052 -0. 1095
101 ’ 2 50 0.3926 0.1579 0.1845 -0.0895 -0. 1097
10 50 0.2899 0.1237 0.1341 -0.0514 -0.0861
10‘] 100 0.7615 0.3702 0.4388 -0.0422 -0.0 l 76
10"” 2 100 0.7023 0.3240 0.3874 -0.0747 -0.0580
1 100 0.5950 0.2523 0.3055 -0.1089 -0. 1040
101’ 2 100 0.4665 0.1843 0.2240 -0.1 132 -0.1190
10 100 0.3382 0.1236 0.1523 -0. 1026 -0.1106

Note: "‘1‘ value of ,w is irrelevant when N = 2 and there is an exact tie.

117

Table 3.5: (Experiment 11: Exact Tie) T = 10 , Variance of the Estimates

 

 

 

 

Note: *** value of ,u. is irrelevant when N = 2 and there is an exact tie.

 

 

 

 

118

 

 

,1. I N 0 J(0) 881(0) 0(0) 0881(0)

'1'" 2 0.0658 0.1107 0.0926 0.2063 0.1767
10-1 10 0.0346 0.1408 0.0798 0.3796 0.2186
10-1/2 10 0.0362 0.1411 0.0809 0.3785 0.2180
1 10 0.0424 0.1396 0.0858 0.3598 0.2180
101/2 10 0.0528 0.1210 0.0896 0.2729 0.2028
10 10 0.0620 0.1308 0.0957 0.2785 0.1982
10-1 20 0.0264 0.1449 0.0702 0.4207 0.2069
111-“ 2 20 0.0282 0.1464 0.0736 0.4187 0.2132
1 20 0.0351 0.1484 0.0850 0.4048 0.2342
101/2 20 0.0471 0.1444 0.0957 0.3596 0.2416
10 20 0.0562 0.1228 0.0958 0.2694 0.2153
10-1 50 0.0208 0.1495 0.0684 0.4570 0.2166
10-1/ 2 50 0.0225 0.1554 0.0723 0.4721 0.2269
1 50 0.0274 0.1554 0.0769 0.4506 0.2292
101/2 50 0.0369 0.1505 0.0870 0.4032 0.2403
10 50 0.0497 0.1385 0.0934 0.331 1 0.2245
10-1 100 0.0192 0.1614 0.0643 0.5011 0.2048
10-1/ 2 100 0.0204 0.1594 0.0656 0.4892 0.2048
1 100 0.0247 0.1606 0.0736 0.4784 0.2220
101/2 100 0.0313 0.1551 0.0814 0.4395 0.2339

10 100 0.0430 0.1443 0.0924 0.3680 0.2410

Table 3.6: (Experiment 11: Exact Tie) T = 10 , MSE of the Estimates

 

 

 

 

,1. N 0 1(0) 881(0) 0(0) 0881(0)

*** 2 0.0993 0.1207 0.1037 0.2065 0.1767
10-1 10 0.2444 0.1877 0.1440 0.3811 0.2199
10-1/2 10 0.2024 0.1698 0.1236 0.3852 0.2241
1 10 0.1487 0.1556 0.1084 0.3667 0.2276
101/2 10 0.1153 0.1336 0.1038 0.2740 0.2072
10 10 0.1041 0.1379 0.1066 0.2803 0.1996
10-1 20 0.3392 0.2176 0.1689 0.4220 0.2079
10-1/2 20 0.2802 0.1929 0.1416 0.4260 0.2197
1 20 0.1941 0.1683 0.1163 0.4218 0.2528
101/2 20 0.1334 0.1539 0.1104 0.3716 0.2567
10 20 0.1083 0.1319 0.1066 0.2714 0.2205
10-1 50 0.4706 0.2459 0.2107 0.4618 0.2180
10-1/2 50 0.3987 0.2291 0.1798 0.4800 0.2327
1 50 0.2900 0.2002 0.1418 0.4617 0.2412
101/2 50 0.1910 0.1754 0.1211 0.4112 0.2523
10 50 0.1337 0.1538 0.1114 0.3337 0.2319
10-1 100 0.5991 0.2984 0.2569 0.5029 0.2051
10-1/2 100 0.5136 0.2644 0.2156 0.4948 0.2082
1 100 0.3787 0.2243 0.1669 0.4903 0.2328
101/ 2 100 0.2489 0.1891 0.1316 0.4523 0.2481

10 100 0.1573 0.1596 0.1155 0.3785 0.2532

 

 

 

 

 

 

*

Note: *** value of ,u: is irrelevant when N = 2 and there is an exact tie.

119

Table 3.7: Weight Comparisons between Estimators, N = 2 and T = 10

Panel 3.7.1: Minimizing MSE

 

#5 a 00 a1 02
0 0 1.3767 0.1887 W
10—1 0.0034 1.3737 0.1872 0.1865
10.112 0.0107 1.3671 0.1839 0.1832
1 0.0337 1.3463 0.1735 0.1729
10112 0.1066 1.2811 0.1408 0.1403
10 0.3371 1.0796 0.0400 0.0396

 

 

 

 

 

Panel 3.7.2: Minimizing Variance

 

 

 

 

 

#5 a 00 01 02
0 0 0.5000 0.2500 0.2500
10-1 0.0034 0.5039 0.2483 0.2478
10—1/2 0.0107 0.5039 0.2483 0.2478
1 0.0337 0.5036 0.2484 0.2480
101/ 2 0.1066 0.5004 0.2500 0.2496
10 0.3371 0.4656 0.2674 0.2671
NRE
Nate 1" a = NREP Z a(2),repl -a(l),repl] and a(2),repl = max(al,replaa2,repl) -
repl=1

Note 2: Recall,

 

“Optimal ”Split-Sample Jacklmife: 000? + 0162(1) + 020(2)

Split-Sample Jackknife: SSI(&) = 207 — 0.562(1) — 0.507(2)

Generalized Split-Sample Jackknife: 0881(0) = 3.4140 —1.2070(1)—1.2070(2)

 

omcod 0 ~ mmd

mcmod

.0: 888 :0 mm 82: 05 m .1. 2 5:3 Ego—ohm mm .3. mo 02$, 4...... .082

waned ommfio

wmw fl .o

 

 

5

TS» EEVM

m—mE

 

Gm: 88E

3

 

 

60 69m ”gm 651m.

I

85

 

 

 

 

   

 

 
 
 

 

826 $86 $85 :53 $85 386 826 $86 4086 S 2
ooaod G86 wmood $8.0 mmaod aomod mmmod mmaod moved 3 m \ _o_
w Sod 2.36 Na _.o 0286 .4306 good wwwod mowed Swod 2 _
good $36 $5 _ .o wwwod $36 2 _ fie oawod wooed 3m _ .o 3 m \ To—
33 .o vmcod mmomd Road 336 gm _ .o 336 wooed Eb _ .o 2 .13
$2 3> 85 mmE 00> mam mm: 5» 85 s i
=85 58K $012580 0
6 6 3

83.5.3.0 £528.. 8.052-230 1.380.. 2.. .6 mm: E... 85:5 .35 588386 N n 2 "3. es;

   

 

 

   

 

 
 
 

 

 

02. oz ”:31. ESE

 

 

      

 

 

121

3.8 Appendix: Deriving The Expected Value and Variance of The Max

Consider a bivariate normal (N = 2) with t =1,...,T

ylt ~N a] , 0'2 O . (3.19)
y2t a2 0 02

Suppose that a] = a > 0 and a2 = 0. Without loss of generality, we can assume

T = 2. All that we are interested in is, for each variable, the overall sample mean and the
two half—sample means. So, for each variable, we have effectively two observations (the
two half-sample means) and their mean value (since the overall sample mean is indeed
the average of the half-sample means).

We are interested in three estimators:

y11+y12 — =y21+y22.

(1)&=max(rl,r2),where71= and y2 2 ,

(2) 02“) = max(y11sy21) ; and

(3) 63(2) = maXO’IZaJ’ZZ)-
With an approach similar to Satchachai and Schmidt (2008), we can derive the

first and second moment of these three estimators:

E(&) = E[maX(?1J2)]

_____ _____ (3.20)
= P(y12y2)-E(y1|y12y2)+P(Y2 Zyfl'EUz |y2 2y”

and, for t=1,2 ,

1202(1)) = E(&(2))
= ElmaXO’Inyth (321)
= P(yu Z y2:)°E(ylz lyu 2 yzz) + P(th 2 y1:)-E(y2: | hr 2 Mr)-

122

From basic probability theory,

P012 J72) = ”571-72 2 0)
=P((J71-a)-(J72 *0) 2 _(a-0))
0'

=1-¢(-%) .
we).

 

Similarly,

P92 2 ?1)= <D(-%], P(ylt 2Y2t)=¢[-—;‘/1-—-2—], and P(y2t Zylt)=¢[- :2 J
0' 0'

We use the fact about the incidentally truncated bivariate normal:

 

E(ill71272)= E(Yl |?1-72 20)

= Eh I (a —a);<72 -0) 2120)]

- a +—l—-g—2.(— fl). since p - —1—-
«Ex/5 a ’ «5

= a +gl(—£}
2 0'

where /1(x) = or the “inverse Mill’s ratio.” Similarly,

I - (D(x)

_ _. _ a
E(yzlyz ZYI)=— [3]. Then,

E[maX()71J2 )1 = 0&1“ + 5211(- 3)] + ‘I’(' gig/1(3)]
=a-«»(:)+o-¢e)

and, fort=1,2,

123

 

E[maX0’1ny2;)l = ¢[—2‘/ia_—2-][a +%z[—T:_;]J + ¢[— Vii—31124 ;, H
=W[ :2]+fia.¢[f2:_z}

Now, we show the derivation of the variance of (2,020), and 62(2) . From the

 

 

definition of variance, we can write
Vark?) E var[maXO’—1J2)] = E[max(?1,72)]2 -[E[max(?1 072 )112 (322A)
and

var(&(])) = var(ém) E var[max(y],,y2,)]

(3.223)
= Etmaxom >12 - [Etmaxowzt >112.

With a similar approach to the one used above, we can also derive

E[maX(?1,J72 )12 and Etmaxom, >12:

Etmaxowz >12 = P01 2 .v2)-E(712 | r] 2 yz) + 10072 2 71) - 5&3 | 372 2 71) (323A)

and

E[max(y1,,y2,)]2 = P(J’lt 2 Mt) ' E(J’lzt U]: 2 Ya) +

2 (3.23B)
P(J’zz Z Yul ' E(y2t U2: 2 J41)-
From the fact about the variance of incidental truncated bivariate normal,
. 2 2
var x truncatzon = 0’ l- 6 a
( I ) x( p (y» (324)

= E (Jr2 | truncation) -[E (x I truncation)]2,

where 5(x) = 2.(x)[/l(x) - x]. So,

E (x2 | truncation) = var(x | truncation) + [E (x | truncation)]2

and

124

50120.2 22) = mm In 2 i2)+[E(?Ili1272)]2
0'2 1 2
=— 1——5(—3) + (Hg—263)
2 2 0' 2 0'
2 2 2
=g__2_5(_£)+a2+2.12(_£]+W{_£]
2 4 0 4 0 0'

 

and
E(iz2 | 72 254) = VWz U72 271)+[15'7(?2 W2 271)]2
2 \2
=9_1_15(£) + 9.1(9.
2 2 0' 2 0')
2 2 2 \
=§__2_5(£]+0_,12(£_
2 4 0' 4 0')
Then,

IEImaonJzn2 = P01 2 mm? I 21 2 22H P02 2 m-Hfi I 22 2 21)

=¢(%){%2--9§-6(-%J+a2+9§*2(-%JW(-%)}
+¢(-%J{%2-%35I-SJ+§*Z(-%l}

Substitute Mx) = T-if—STLS and 6(x) = A(x)[/1(x) — x], and we get

2
E[max(jil , 572 )]2 = 92— + £ch) + 0:043)

0'

=%j+a-Ia¢I;I+a¢I2II

= “7+0; -E[maX()7|J2)]°

and

125

var[maX071Jz)l = E[maX@IJ2)]2 —IEImax<yIJz >112
2
= 5’5- + a - Elmo—1.22)] {animation2

2
= 92—+ E[max(?1a72)] ' {a - Elmaxfifiizflk

2
So, var(c?) = %— + E (a?) - (—bias(6r)) . In a similar manner, the variances of (2(1) and

51(2) are var(dm) = 0'2 + E(d(l))-(—bias(&(l))) and var(é(2)) =

= 0'2 + E(&(2)) - (-bias(&(2))) , respectively.

3.9 Appendix: The “Optimal” Split-Sample Jackknife

Consider a new estimator c? that is a linear combination of the estimators based
on the full sample, C? , and the half samples, 62“) and (2(2) :
a” = God + aléa) + @590) , (3.25)
where

y21+y22 .

andiz =--—,

A _ _ _ +
(1) a = max<yI.y2>.whereyI #2522

(2) 02“) = max<yn,y21); and

(3) as”) = maxowzz).

We are interested in the estimator 5 that minimizes (i) MSE(&') , (ii) MSE(&')
subject to 00 + a] + 02 =1 , and (iii) var(fi) subject to 00 + a] + 02 =1.

First, we define the following notation

a=[ao a1 02],; (3.26A)

126

(3)402 c2“) (WW; (3.26B)

E(é)ao=[o, o. 0,]; and (3.26C)
. V00 V01 V02

V(®)‘=‘V= V10 VII V12 - (326D)
V20 V21 V22

Since we assume independence, V12 = V21 = 0. Using matrix algebra,
bias(§) = we — a , var(a) = a'Va and MSE(&’) = var(a') + biasz(&) =
= a'Ca — 2aa'® + 0:2 , where C = V + (96'. For the constrained cases, we can rewrite

ao+al+a2=l as 00:1—01—02. Inmatrixnotation,wecanwrite a=e1+Am,

r -1 1 0 v
wheree]=[l 0 0],A=[1 O 1]anda:=[al 02].

With the matrix simplification, we can rewrite

var(&') = (81 + Aa*)'V(e1 + A0.)
= eiVel + 20‘14' Ve] + 04A, VAa:

and

MSE(&) = (el + Aa..)'C(el + Aa.) — 2a(el + Aa.)'o + a2

= eiCe] + ZaLA'Cel + aLA'CAat - 2aei® — 2aa£A'® + az.

3.9.1: 5 that minimizes MSE(5£) without constraint.
F. 0. C
wig-2 = 2(V +®®’)a—20® =0
0

(V + oo')a = 09
a = a -(V + oo')"o.

127

w

.9.2: 5 that minimizes MSE(5) subject to 00 +01 +02 =1.

 

 

 

 

 

F.O.C
W = 2A'Ce1 + 2A'CAaa- — 2aA'G) = 0
da:
A'CAa: = aA'@ — A'Cel

a. = (A'CA)"(aA'o — A'Cel).

So,
_ _ V00 ‘VOI +(a' -®o)(®1-@o)
a] — (12 " 2
2V00 + VII-4V01+ 2(@1490)
and
a0 =1_al -02 = VII "2VOI -2(@1 ‘0)(90 ”912).

ZVOO + V11-4V01+ 2(90 —®1)
3.9.3: 5 that minimizes var(5) subject to 00 +01 + 02 =1 .
F.0C

“3““) = 2A'Ve1 + 2A' VAa. = o
a;
A'VAa... = —A'Vel

a... = -(A'VA)“A'Ve.

So a1=az= V00-V01 and ao=1—a1—az= V11-2V0] .
2V002LV11-4V01 2V00 +V11-4V01

128

BIBLIOGRAPHY

Aigner, D., C.A. K. Lovell and P. Schmidt, 1977, Formulation and Estimation of
Stochastic Frontier Production Function Models, Journal of Econometrics, 6, 21-
37.

Clark, C.E., 1961, The Greatest of a Finite Set of Random Variables, Operations
Research, 9, 145-161.

Coelli, T, 1995, Estimators and Hypothesis Tests for a Stochastic Frontier Function: A
Monte Carlo Analysis, Journal of Productivity Analysis, 6, 247-268.

Dhaene, G., K. Jochmans, and B. Thuysbaert, 2006, Split-Panel Jackknife Estimation of
Fixed Effects Models (Previous Title: Jackknife Bias Reduction for Nonlinear
Dynamic Panel Data Models with Fixed Effects), Working Paper.

Doran, H. and P. Schmidt, 2006, GMM Estimators with Improved Finite Sample
Properties Using Principal Components of the Weighting Matrix with Application
to the Dynamic Panel Data Model, Journal of Econometrics, 133, 387-409.

Efron, B., 1982, The Jackknife, the Bootstrap and Other Resampling Plans, Society for
Industrial Applied mathematics, Philadelphia, Pennsylvania.

Femandez-Val, I. and F. Vella, 2007, Bias Corrections for Two-Step Fixed Effects Panel
Data Estimator, IZA Discussion Paper 2690, Institute for the Study of Labor
(IZA)

Gray, H.L., and W.R. Schucany, 1972, The Generalized Jackknife Statistic, Marcel
Dekker, Inc., New York.

Gray, H.L., W.R. Schucany, and TA. Watkins, 1975, On the Generalized Jackknife and
its Relation to Statistical Differentials, Biometrika, 63, 637-642.

Hahn, J. and G. Kuersteiner, 2004, Bias Reduction for Dynamic Nonlinear Panel Models
with Fixed Effects, Unpublished Manuscript.

Hahn, J ., and W. Newey, 2004, J ackknife and Analytical Bias Reduction for Nonlinear
Panel Models, Econometrica, 72, 1295-1319.

Hall, P., H.W.K. Hardle, and L. Simar, 1995, Iterated Bootstrap with Applications to
Frontier Models, Journal of Productivity Analysis, 6, 63-76.

Han, C., L. Orea and P. Schmidt, 2005, Estimation of a Panel Data Model with

Parametric Temporal Variation in Individual Effects, Journal of Econometrics,
126, 241-267.

129

Johnson, N.L. and S. Kotz, 1972, Distributions in Statistics: Continuous Multivariate
Distributions, Johnson Wiley & Sons, Inc., New York.

Kim, M., Y. Kim, and P. Schmidt, 2007, On the Accuracy of Bootstrap Confidence
Intervals for Efficiency Levels in Stochastic Frontier Models with Panel Data,
Journal of Productivity Analysis, 28, 165-181.

Matyas, L., 1999, Generalized Methods of Moments Estimation, Cambridge University
Press, Cambridge; New York.

Miller, R.G., 1974, The Jackknife - A Review, Biometrika, 61, 1-15.

Olson, J .A., P. Schmidt and D.M. Waldman, 1980, A Monte Carlo Study of Estimators of
Stochastic Frontier Production Functions, Journal of Econometrics, 13, 67-82.

Park, B. U., and L. Simar, 1994, Efficient Semiparametric Estimation in a Stochastic
Frontier Model, Journal of the American Statistics Association, 89, 929-936.

Pitt, M.M., and LP. Lee, 1981, The Measurement and Sources of Technical Inefficiency

in the Indonesian Weaving Industry, Journal of Development Economics, 9, 43-
64.

Quenouille, M.H., 1956, Note on Bias in Estimation, Biometrika, 61, 353-360.

Satchachai, P. and P. Schmidt, 2007, GMM with More Moment Conditions than
Observations, Economics Letters, 99, 272-275.

 

, 2008, Estimates of Technical Inefficiency in Stochastic
Frontier Models with Panel Data: Generalized Panel Jackknife Estimation,
Department of Economics, Michigan State University.

Schmidt, P., and R. Sickles, 1984, Production Frontiers and Panel Data, Journal of
Business and Economic Statistics, 2, 367-3 74.

Schucany, W.R., H.L. Gray, and DB. Owen, 1971, On Bias Reduction in Estimation,
Journal of the American Statistical Association, 66, 524-533.

Tukey, J .W., 195 8, Bias and Confidence in Not Quite Large Sample, (Abstract), Annals
of Mathematical Statistics, 18, 614.

130

   

”'IIIIIIIIIIIIIIIIIIIIIIIIIIII‘IIIIIIIIIII‘III’ITs

063 0812