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__ .__._._, _ ._ ——___—.—__

MULTIPLE COMPARISONS WITH THE BEST FOR INFERENCE
IN STOCHASTIC FRONTIER MODELS

AND
SUBMODEL ESTIMATION FOR COUNTERFACTUAL POLICY ANALYSIS
By

William Clinton Horrace

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

DOCTOR OF PHILOSOPHY

Department of Economics

1996

ABSTRACT
MULTIPLE COMPARISONS WITH THE BEST FOR INFERENCE
IN STOCHASTIC FRONTIER MODELS
SUBMODEL ESTIMATION FOR CagnNTERPACTUAL POLICY ANALYSIS
By

William Clinton Horrace

This is a dissertation in three chapters. In the first chapter we examine a
statistical method for performing simultaneous inference on all distances from the "best"
called multiple comparisons with the best or simply MCB. We find that MCB can be
used on a fixed effects stochastic production frontier model for panel data to construct
simultaneous confidence intervals for technical inefficiency and to perform inference on
maximum efficiency measures, where previously no methods had been suggested.

In the second chapter we use the MCB analysis of chapter one to perform
inference and point—estimation on some previously analyzed stochastic frontier data and
compare these results to the inference and point estimates currently suggested but never
exploited in the stochastic frontier literature.

The third chapter is a complete departure from the first two. In chapter three we
construct a subset limited-information maximum likelihood estimator for a vector error
correction model under the cointegration hypothesis for use in a counterfactual policy

analysis model.

Dedicated to my children: Ava Deleon Horrace and Ian William Horrace

iii

ACKNOWLEDGEMENTS

I would like to thank my thesis advisor, Professor Peter Schmidt and the rest of my
committee: Professor Ching-Fan Chung, Professor Robert Rasche and Professor Jeffrey
Wooldridge. I would also like to thank the following individuals for their support: Ava,
Ian, Sarah, Diana, Zoe, Brian, Bobby, Vaishali, Ren, Stimpy, Homer, Marge, Maggie,
Lisa, Bart and the staff at the Peanut Barrel. Also special thanks to Professor Carl

Davidson.

iv

TABLE OF CONTENTS

LIST OF TABLES ...................................... viii
LIST OF FIGURES ....................................... x
INTRODUCTION TO DISSERTATION .......................... 1
CHAPTER 1
MULTIPLE COMPARISONS WITH THE BEST AND ................. 4
THE FIXED EFFECTS MODEL
1 INTRODUCTION .................................... 4
2 STOCHASTIC FRONTIER MODELS ....................... 7
2.1 Introduction .............................. . ...... 7
2.2 Conclusions and Extensions .......................... 11
3 MULTIPLE COMPARISONS WITH THE BEST ................. 12
3.1 Introduction .................................... 12
3.2 Historical Perspective .............................. 13
3.3 Hsu (1981) - Multiple Comparisons with the Best ............. 14
3.4 Two-Sided MCC Intervals ........................... 18
3.5 Edwards and Hsu (1983) - MCB Intervals from .............. 19
MCC Intervals
3.6 Extensions and Conclusions .......................... 26

4 APPLYING MCB METHODS TO STOCHASTIC FRONTIER MODELS . 28

4.1 Motivation and Considerations ......................... 28

4.2 Large N, Small T ................................. 30

4.3 Large T ....................................... 31

4.4 Small N and T .................................. 31

5 CONCLUSIONS AND EXTENSIONS ....................... 32
CHAPTER 2

CONFIDENCE STATEMENTS FOR EFFICIENCY ESTIMATES .......... 34

1 INTRODUCTION .................................... 34

2 STOCHASTIC FRONTIER MODELS ....................... 36

2.1 Cross-sectional Data ............................... 36

2.2 Panel Data ..................................... 37

3 TECHNIQUES FOR CONSTRUCTION OF CONFIDENCE INTERVALS . 38

3.1 Cross-sectional Data: JLMS Method ..................... 38
3.2 Panel Data: Battese-Coelli Method ...................... 40
3.3 Panel Data: Multiple Comparisons with the Best ............. 41
3.4 Comparison of Different Techniques ..................... 42
4 EMPIRICAL ANALYSES ............................... 44
4.1 Indonesian Rice Farms - Erwidodo (1990) .................. 44
4.2 Texas Utilities - Kumbhakar (1994) ...................... 49
4.3 Egyptian Tileries - Scale (1990) ........................ 54

vi

5 CONCLUSIONS ..................................... 55

CHAPTER 3

ESTIMATION OF A CONDITIONAL VECTOR ERROR CORRECTION ..... 57

SUBMODEL FOR COUNTERFACTUAL POLICY ANALYSIS
1 INTRODUCTION .................................... 57
2 ESTIMATION ...................................... 58
2.1 Under Cointegration ............................... 58
2.2.1 LIML Estimation .............................. 68
2.2.2 A Two-stage Estimator .......................... 74
2.2 Under No Cointegration ............................. 77
3 CONCLUSIONS ..................................... 78
CONCLUSIONS TO THE DISSERTATION ....................... 81
APPENDIX ............................................ 82
LIST OF REFERENCES ................................... 97

vii

TABLE 1:

TABLE 2A:

TABLE 2B:

TABLE 3:

TABLE 4A:

TABLE 4B:

TABLE 5A:

TABLE 5B:

TABLE 6A:

TABLE 6A:

TABLE 7:

TABLE 8A:

TABLE 83:

LIST OF TABLES

Rice Farms - Cross Sectional Estimation Results ............ 82
Rice Farms - Confidence Intervals Based on JLMS Method, ..... 83
COLS Estimates - Period 1

Rice Farms — Confidence Intervals Based on JLMS Method, ..... 83
MLE Estimates - Period 1

Rice Farms - Panel Data Estimation Results ............... 84
Rice Farms - Confidence Intervals Based on BC Method, ....... 85
CGLS Estimates - Panel Data

Rice Farms - Confidence Intervals Based on BC Method, ....... 85
MLE Estimates - Panel Data

Rice Farms - MCB Confidence Intervals - Panel Data ......... 86

Rice Farms - MCC Confidence Intervals - Farm 164 as Control . . . 86
- Panel Data

Rice Farms - Subset MCB Confidence Intervals, ............ 87
N = 9 - Panel Data

Rice Farms - Per Comparison Confidence Intervals, .......... 87
Texas Utilities - Panel Data Estimation Results ............. 88
Texas Utilities - Confidence Intervals Based on BC Method, ..... 89
CGLS Estimates - Panel Data

Texas Utilities - Confidence Intervals Based on BC Method, ..... 89

MLE Estimates - Panel Data

viii

TABLE 9A:

TABLE 9B:

TABLE 10:

TABLE 11:

TABLE 12:

TABLE 13:

TABLE 14:

Texas Utilities - MCB & MCC Confidence Intervals .......... 90
- Panel Data

Texas Utilities - Marginal (Per Comparison) Intervals ......... 90
- Panel Data

Tileries - Panel Data Estimation Results ................. 91
Tileries - Confidence Intervals Based on EC Method, ......... 92
CGLS Estimates - Panel Data

Tileries - MCB Confidence Intervals - Panel Data ........... 93
Tileries - MCC Confidence Intervals - Panel Data ........... 94
Tileries - Per Comparison Confidence Intervals - Panel Data ..... 95

LIST OF FIGURES

FIGURE 1: MCC and MCB INTERVALS ........................ 96

INTRODUCTION TO THE DISSERTATION

This is a dissertation in three chapters. It is an attempt to address topics in
econometric theory and applied econometrics in the cross sectional, panel data and time
series contexts. Each chapter contains its own introductory remarks, so this introduction
to the entire text is intended to briefly summarize the goals of this dissertation. The plan
of the dissertation is as follows.

Chapters one and two are concerned with performing inference on technical
efficiency estimates in stochastic frontiers models. These are cross sectional and panel
data models that predict or estimate technical efficiency for a set of productive or
decision-making units. Two different approaches dominate the efficiency measurement
literature: the aforementioned stochastic frontiers approach and a deterministic approach.
While the debate over which approach is "preferred" continues, a clear advantage of the
stochastic frontiers approach is that it allows quantification of the uncertainty associated
with the efficiency estimates while the deterministic approach does not. Therefore the
import of the first two chapters of this dissertation is that they detail procedures to
quantify this uncertainty in the stochastic frontiers model. This is accomplished through
confidence intervals construction.

The first chapter details a new technique for confidence intervals construction in
the fixed effects stochastic frontier model for panel data. To my knowledge this has
heretofore never been accomplished. This technique, called Multiple Comparisons with
the Best or MCB, allows simple confidence intervals construction that not only quantifies
the uncertainty of the individual technical efficiency estimates of the productive units but

also quantifies the uncertainty of which firm in the sample is the best in the population

1

2

and suggests point estimates for technical efficiency which have less positive bias than
those currently exploited in the literature. Moreover, the fixed effects formulation of the
stochastic frontier model requires weaker distributional assumptions than any other
stochastic formulation, so the benefits of these MCB intervals are clear.

The second chapter discusses several other formulations of the stochastic frontier
model. Unfortunately, all of these formulations require stronger distributional
assumptions than the aforementioned fixed effects model, but the ability to construct
confidence intervals for their technical efficiency estimates has been well documented.
Strangely, this ability has never been systematically exploited in an empirical setting.
Therefore, chapter two details the interval construction techniques of these formulations
and, along with the MCB techniques of chapter one, presents a comprehensive empirical
study in which confidence intervals on technical efficiency measures are constructed
using various estimation and interval construction methods on three different data sets.
In doing so, chapter two advances our understanding of the various sources of uncertainty
inherent in econometric models for efficiency estimation.

Chapter three is a complete departure from the first two. This chapter presents
a new method for estimating the parameters in a conditional submodel for counterfactual
policy analysis under the cointegration hypothesis. Counterfactual analysis attempts to
analyze an economy (system of equations) in which new (counterfactual) policy rules
have been substituted for the historical policy rules which generated the data. The idea
is to see how the economy would have behaved had a different policy regime been in

effect. However, as we shall see, current counterfactual analysis techniques are limited

3

in their scope, so chapter three suggests a technique that can be more universally
applicable in these types of analyses.
Finally, in the conclusion section of this dissertation I summarize the results of

the research and suggest areas for additional work.

CHAPTER 1
MULTIPLE COMPARISONS WITH THE BEST

AND THE FIXED EFFECTS MODEL

1 INTRODUCTION

It is often the case in empirical research that comparative studies are prescribed.
For instance, one may wish to compare the effectiveness of several drugs in the treatment
of a disease, or one may be interested in comparing the differences in crop yield for a
variety of fertilizers. Such experiments might involve collecting a sample of some
effectiveness or yield measure for each treatment or variety, calculating a summary
statistic for each treatment (such as a sample mean) to estimate some population
parameter, and then comparing these statistics using some inference techniques like an
F-test or t-tests to test hypotheses on comparisons between the population parameters.

As an example consider a controlled experiment where 3 different fertilizers (A,
B, and C) are each applied in the same quantity to 10 separate samples of the same soil,
all receiving the same md, sunlight and irrigation. At the end of the experiment crop
output is measured for each of the 30 soil samples, and mean output is calculated for
each of the 3 fertilizers. The three sample means are construed as estimates of
population means, so that comparative hypotheses on the populations can be tested. For
instance a typical comparative hypothesis would be that fertilizer A is better than
fertilizer B or, perhaps, that the difference in crop yield between fertilizer A and

fertilizer C is 20 units.

5

Very often in these studies it is advantageous to test a hypothesis about several
comparisons between populations, simultaneously (i.e. a multiple hypothesis). To
continue our crop example, one may be interested in testing the hypothesis that fertilizer
A is better then fertilizer B and fertilizer A is better than fertilizer C. In these instances
the advantages of multiple comparison procedures (or MCP) over single or per
comparison testing have been well documented in the statistical literature. Basically,
MCP precludes what has been deemed the multiplicity efiect. Hochberg and Tarnhane
(1987) give an excellent exposition on the multiplicity effect and the justification for
MCP.

This chapter deals with a specific case of MCP ealled multiple comparisons with
the best (or MCB) in which simultaneous inference is performed on all differences from
the unknown ”best” population parameter. This procedure has been extensively exploited
in various forms in the natural science and statistics literatures to allow ranking and
selection of treatments. For example, see Becker (1961), Dalal and Srinivasan (1977),
Gupta and Hsu (1977) and McDonald (1977). While generally ignored in the
econometrics literature, MCB methods can be use to perform inference in stochastic
production frontier models for panel data. These are econometric panel data models that
produce estimates ‘of technical efficiency for a set of firms or productive units. Our
focus is the fixed effects estimate which produces a distinct intercept term for each firm.
The firm with the highest intercept estimate is deemed "best", and differences between
the best firm’s estimate and those of the less efficient firms are technical inefi'iciencies.
See Schmidt and Sickles (1984). These inefficiency estimates can be thought of as

comparisons with the best. If we are interested in testing the hypothesis that the firm

6

with the highest efficiency estimate is, in fact, the true best, then we must test the
simultaneous hypothesis that all other firms have positive technical inefficiencies.
Therefore, our inference is necessarily simultaneous and constitutes multiple comparisons
with the best or MCB. Once MCB intervals are calculated their midpoints suggest point-
estimates of technical inefficiency and maximum efficiency which have less positive bias
than those currently exploited in the stochastic frontier literature.

It is important to stress the fact that this chapter deals with inference on the
maximal efficiency and on technical inefficiencies measured relative to this maximum.
Therefore, confidence intervals and inference on the technical inefficiencies are
necessarily complex due to the non-linearity associated with taking the maximum of the
individual efficiencies. This chapter uses a pre-existing theoretical methodology to
simplify construction of these intervals. It is also important to realize that determination
of the most efficient firm should not be approached as simple selection of the firm with
the largest technical efficiency estimate. This naive approach assumes that the estimation
procedure reveals with certainty the true efficiency rankings of the firms. Conversely,
simultaneous MCB inference on the technical inefficiency measures allows us to
determine a confidence level with which we can say that the firm with the highest
technical efficiency estimate is the true ”best".

Therefore, the importance of these procedures is that they allow us to make
confidence statements about the efliciency estimates relative to an W standard and,
subsequently, a statement about the maximum efliciency estimate ’s ability to serve as that
standard. For practical purposes these inefficiency estimates, their intervals and any

inference on the efficient firm are measures of performance or success, and as such can

7

aid in technological decisions and the evaluation of managerial performance.
Additionally, studying these intervals will provide insight into the sources of uncertainty
associated with productivity estimation.

This chapter is concerned with performing MCB in the fixed effects stochastic
frontier model. It is organized as follows. Section 2 briefly discusses estimation of the
stochastic frontier model and summaries the main algebraic and statistical results of a
fixed effects treatment. After a very brief historical account, section 3 explores
developments in MCB theory. Section 4 discusses theoretical considerations in applying
MCB theory to the stochastic frontier model. Finally, section 5 draws some conclusions

and introduces areas for additional research.

2 STOCHASTIC FRONTIER MODELS

2.1 W

Stochastic frontier models were originally due to Aigner, Lovell and Schmidt
(1977) and Meeusen and van den Broeck (1977). These models were based on cross
sectional data and strong distributional assumptions. Similar models have also been
developed for panel data. Pitt and Lee (1981) and Schmidt and Sickles (1984) were the
first to exploit the advantages of a panel data over cross sectional data. Since this is not
intended to be a comprehensive survey, the reader is referred to Comwell and Schmidt
(1995), Greene (1995), Lovell (1993), Lovell and Schmidt (1988) and Schmidt (1985)
for further details.

This chapter deals with these models only in the context of panel data.

Specifically, only the fixed effects formulation of the stochastic frontier model will be

8

detailed. However, we now present a more general discussion of the model to
incorporate alternative formulations to be used in chapter two. The basic model that we

will consider is as follows.

(1.1) yh=a+th +v,,-u,, ui 2 O; i=1,...,N, t= 1,...,T.

Here i indexes firms (or other productive units) and t indexes time periods. Typically
y, is the logarithm of output and x, is a vector of inputs or functions of inputs. v,, is
statistical noise and u, 2 0 represents technical inefficiency, assumed to be time invariant.
More specifically, if y, is the logarithm of output, technical efficiency of the i‘h firm is
TE, = exp(-u,) and technical inefficiency is l-TEi. We will refer to the composite error

as 6,, = v,, - u,. We will always assume the following:

(A.1) The v, are iid N(O, 02,).

(A.2) x, and v,, are independent fort, s = 1, ..., T, i,j = 1, ..., N.

We will sometimes but not always make the additional assumptions:

(A.3) The u, are independent of x and v.

(A.4) ui = | Ui | , where the Ui are iid N(O, 02,)

9

Assumption (A.4) implies that the u,- are half-normal, but this assumption could be

replaced by other specific distributional assumptions, as in Stevenson (1980) or Greene

(1990).

Now define a, = a - u,, so that a, s 01 for all i. Then we can rewrite (1.1) as
the usual panel data model
(1.2) Yit = a, + Xi‘B + V“, i=1, ..., N, t: 1, ..., T.

We regard zero as the absolute minimal value of u,, and hence a as the absolute
maximal value of (1,, over any possible sample (essentially, as N—too). This can be
distinguished from the minimal value of u, and the maximal value of ozi in a given sample
of size N, and this distinction is relevant when N is small and the u, (hence 01,) are
treated as fixed. Let am 5 “[21 s 5 arm be the population rankings of the 02,, so
am = max?=1 (1,, and am] 5 0:. Similarly, let um S um,” _<. S “m be the
population rankings of the u,-, so um = minim” and um 2 0. Then “m = a - am. In
this case the technical efficiency measures u, are defined by comparing ozi to the absolute
standard a. We can consider the alternative of comparing a, to the within sample
standard am]. Define u'i = “on - a, = ui - um, so that 0 S u'i s u,. Then equation

(1.2) can be rewritten as:

(1.3) Yit = am1+ Xi‘fi + Vi, " ll.“ i=1, ..., N, t: 1, ..., T.

\n
l

10

The difference between the two definitions of u is substantive and will be
considered further in chapter two.

Equation ( 1.2) is useful primarily as a basis for estimation that treats the a, (or
u,) as fixed. A fixed effects treatment may be useful because it relies only on
assumptions (A1) and (A.2), not (A.3) and (A.4), and because it is applicable when N
is small and T is large (as well as when N is large). Suppose we estimate (1.2) by the

usual fixed effects estimation involving the within transformation (or, equivalently,

dummy variables for firms), yielding estimates of a], ..., a”, B and 02,. Define
(1.4) a = maxfflaj
fi. = & ' a, I = 1, , N
U

Then, as '1‘“,m With N fixed, éi "’ 01-1, & —’ am and LQLaingj = am "" on, SO thgfii

 

 

measures inefficiency relative to the standard of the best firm in the sample. Now
consider what happens 3.53:9. Under the assumption (A.4) of half-normality, or in
fact under any mechanism for the generation of ui that allows u arbitrarily close to zero
with positive probability (density), fl» 0 and am —> a as N600 . _Thus, a a a and

it, -> u, as both N and Tag,“ “so that inefficiency is measured relative to its absolute (not

 

just within-sample) standard. This distinction will be important in the empirical analyses
in chapter two.

The statistical properties of the estimated u, are complicated because of the " max "
operation involved in the definition of d: and therefore of iii. Consistency as both N and

T -» on was argued heuristically (as above) by Schmidt and Sickles (1984). Park and

ll
Simar (1994) and Kneip and Simar (1995) established the rate of convergence of the

estimates. However, the asymptotic distributions of the estimates of a and the ui are
unknown, so that standard methods of construction of asymptotically-valid confidence
intervals based on these asymptotic distributions are currently not possible. Additionally,
the estimate 6: is essentially based on the presumption that we are certain that our
estimates of the 0:, reveal the true ranking of the corresponding population parameters.
Therefore any statistical inference or interval construction must be conditional on that
certainty, which seems dubious. Despite these problems, MCB methods allow
construction of confidence intervals and will be discussed in the next section. To this

end, we examine the covariance structure of the 6:, conditional on x,.

(1.5) Var(&,) = 0le + i,Var(B)3£,-’ i = 1, N

Cov(a,,aj) = ‘i,Var(B)'i,-’ i ¢j

It is important to note in equation ( 1.5) that the consistency of 3 implies that as
either N -’ on or T a co , Cov(&,,&,-) _. O, and Var(&9 —. 02/T or 0 respectively,
implying that asymptotically our estimates of a, are orthogonal. This orthogonality
greatly simplifies MCB analysis, so, as we will see, MCB methods are most readily

applied when N or T is large.

2.2 Conclusions and Extensions

The previous fixed effects analysis is for a balanced panel where the number of

periods of observation for each individual or firm is the same, T. Generalization to the

YES
UT.

Sift~

C0:
C01‘.

pro

is u~

 

12

unbalanced case is straightforward and will not be discussed here. However, for the

purposes of this chapter the generalization changes the variance of 61, to:

(1.6) Var(o‘z,) = oz/T, + E,Var(B)'i,’ i = 1, N

where: T, is the number of periods for the im firm.

Of course the Var(B) is slightly different in the unbalanced case, and all the previous
results hold with some minor algebraic modifications as well. Implications of the
unbalanced panel to MCB will be discussed in subsequent sections. The following

section is devoted to MCB procedures.

3 MULTIPLE COMPARISONS WITH THE BEST
3.1 mm
Multiple Comparisons with the Best (MCB) is a specific case of Multiple
Comparison Procedures (MCP) which is performance of simultaneous inferences or
construction of simultaneous confidence intervals in comparative analysis. MCB is a

procedure for constructing simultaneous confidence intervals of the form:

(1.7) u‘,= aim-a, i= 1, ...,N .
T"H E
opal-W; ot’. i’,‘

\J

1
l

.. T]

where the a, are unknown ”goodness” parameters for N populations, and the population
with the largest a, is considered "best". It is important to realize that the best population

is unknown. If this were not the case MCB procedures would not be required; some

CS

pr:

(M

(1.

sh

C01

13

multiple comparison or single comparison technique would probably suffice, depending
on the experimenter’s requirements. Even then they would typically be based on the
estimate am, = max}i,&, which assumes that estimation reveals the true ranking of 01,.
The MCB intervals may also suggest consistent point-estimates with nice small sample
properties.

MCB is similar to and can be adapted from Multiple Comparisons with a Control

(MCC) procedures which construct simultaneous confidence intervals of the form:

(1.8) aN—a, i= 1, (N-l)

where the N‘“ population is regarded as a control. Any population may be chosen for the
control, but the choice must be independent of the data.

This section is intended to serve as in introduction to procedures for performing
inference on equations (1.7) and (1.8). What follows is a brief historical survey of MCB
procedures. Sections 3.3 - 3.5 outline the main MCB results. Finally, section 3.6
extends sections 3.3 - 3.5 to a more general case and draws some conclusions on the

state of MCB literature as it pertains to stochastic frontier models.

3.2 Hi ' P r ’v

MCP theory evolved during the late 1940s and early 19503 primarily due to David
Duncan, S. N. Roy, Henry Scheffé and John Tukey. Harter (1980) gives a complete
historical account. Shortly thereafter a related body of literature called ranking and

selm surfaced with the work of Bechhofer (1954). Additional ranking procedures

 

of

dis:

14
followed due to Gupta (1956, 1965), Fabian (1962) and Desu (1970). MCC procedures

were primarily due to Dunnett (1955, 1964). MCB evolved in the early 1980’s with the
work of Jason Hsu. The primary justification for MCB is that it not only allows
significance testing and interval estimation for differences between populations, but it
allows the experimenter to simultaneously determine which population is "best".

Hsu (1981) constructed parametric and non-parametric simultaneous one-sided
upper confidence intervals for equation (1 .7) under a location model. The parametric
confidence intervals were stronger than those suggested by Bechhofer, Gupta, Fabian and
Desu, while the non-parametric intervals were new to the literature.‘ Later Hsu ( 1984)
constructed simultaneous two-sided MCB confidence intervals for equation (1.7) which
implied his 1981 results. Additionally, Edwards and Hsu (1983) provide a general
technique for adapting MCC intervals in equation (1.8) to MCB intervals in equation
( 1.7). Hochberg and Tamhane (1987) nicely summarize the main results of these three
papers. The next section recaps those results that are germane to this chapter and

provides a few additional insights.

3.3 Hso (1281) - Moltiolo Comm’sons with tho Bost
Let 1,, ..., 1rN be N independent populations or treatments. Let the distributions
of the N populations differ only by location, so for i = 1, ..., N, F(x - 01,) is the

distribution of X in 1r. Let X51, ..., X,T be an i.i.d. random sample of size T from

 

Hsu’s intervals are stronger in several senses. See Hsu (1981) for specific comparisons with
Bechhofer (1954), Gupta (1956, 1965), Fabian (1962) and Desu (1970).

bc'

ind

shc

 

 

 

COT

th‘

15
population 1,. Let the "best" population be that with the largest location parameter, (1,.

Let

be the ordered location parameters. Let F(x) = <I>(x/(t), a normal distribution with mean

0 and standard deviation (7, unknown. Let

be the sample means of the N populations, so that E[&,] = a, for all i. Let s be the usual
pooled estimate of a with v = N(T-1) degrees of freedom. Then var(&,) = (FIT and the
independence of the populations implies cov(&,,&,) = 0 for all i =1: j. Simple calculations
show that var(o"z,, - 6:,) = 202/T and cov((3(,, - 6:,, (31,, - 62,) = 02/T for all i =16 j =11: k, so that

corr(&,, - (5,, (3:, - (1,) = ‘6, i #= j ¢ k. To perform inference on equation (1.7) select

(1.9) ’9, = am ' a, i: 1, ...,N; i¢[N],

where (31",, is any of the 62,. Then

(3017091, 3’5) = Pij = P = 1’5 H‘j-

Notice that this correlation structure will emerge regardless of which of the N sample

means we seleCt as (im, provided that the choice does not depend on the data. That is,

16

the variance and covariances of the (3:, are M conditional on the rankings. This
equicorrelated structure facilitates calculation and tabulation of the necessary multivariate

critical values. Based on this structure, one-sided intervals are given by Theorem 3.1.

Theorem 3.1 (Hsu 1981).

Simultaneous (l - A) confidence intervals for am] - a, are given by:
an." - a, E {0, max(max,,., (it, - (3:, + d,0)}

where d = Tm», ,,.,,S(2/T)%

and TQ’N,,.,., is the solution in t for

a)

[0 I... 4’""[(Zp"’ + tS)(1-P)'”]d‘1’(2)dQ..(S) = 1 - A

where Q, is the distribution of a x,u"" random variable.

For a proof see Hsu (1981). Clearly, the critical point and hence the inference
hinges on the "y,’s being equicorrelated with correlation coefficient p = 1A. This means
that for more general correlation structures these intervals are less useful, but for the
purposes of this chapter they are instructive. Tables for the critical value 'I‘*’N,,,,.,, can
be found in Hochberg and Tamhane ( 1987), Bechhofer and Dunnett (1986) and Gupta,

Panchapakesan and Sohn (1985). T‘”N_,‘,‘p will be positive for values of A < 0.50, so

17

for reasonable confidence levels, (1 will be positive. Since the lower bound is constrained
positive, these intervals are not as informative as the two-sided intervals discussed later.
However, they are of particular interest to those concerned with ranking and selection
of the treatments. Therefore, we now discuss this theorem in terms of the inference
implied by the intervals and defer any discussion of the uncertainty associated with our
estimates until we discuss two-sided intervals in section 3.5.

The constructed intervals imply statements about the ranking of the treatment
means and subsequently about the most efficient treatment mean. Notice that the upper
bound is constrained non-negative, so for any treatment to be in contention for the "best"
it is necessary that its upper bound be zero. We can make this point clear by examining
specific cases. If the difference between the largest and second largest sample means is
large relative to (1, then the treatment with the largest sample mean will have an upper
bound of zero and can be considered ”best" at an l-A confidence level. Conversely, if
this difference is small relative to (1, then the upper bound will be greater than zero and
the treatment is not "best". In fact, it is easy to show that the upper bounds for the
remaining treatments (1 = [l],..., [N-l]) will always be greater than zero, so, in this
case, none of the treatments is "best".

If we are only concerned with the hypothesis that the treatment with the largest
sample mean is best, then the notion of the remaining N—l treatments always being less
than best implies that we only need test the treatment with the largest sample mean to
perform this inference. What distinguishes this inference from a single t-test is the

selection of the critical value, which is drawn from a multivariate-t distribution to

3!

IO

31

the

 

If 0'

bale

The

18

account for the uncertainty associated with the best treatment mean being unknown and
to account for the simultaneity associated with the ranking.

If our inference fails to produce a single "best" treatment, then we may also be
able to test additional hypotheses about the population by constructing two-sided
confidence intervals, the lower bounds of which will tell us which of the treatments
cannot be best at the 1-A level. This way if we cannot pick a single ”best” treatment,
at least we can determine the treatments that cannot be best, leaving us with a subset of
the treatments which might be best. Also, the two—sided intervals may suggest consistent
estimates with nice small sample properties. This particular inference technique is
explored in section 3.5, but first we examine construction of intervals for equation (1. 8)

using MCC procedures.

3.4 jlflgSidfl MCC Intervfls

Multiple comparisons with a control or MCC is primarily due to Dunnett (1955).
If one of the N treatments, say N, can be regarded as a control, then we can construct
intervals for equation (1.8). We use the same notation as section 3.3. We consider an

balanced layout where for each treatment 1 = 1...N the number of observations is T.

Theorem 3.2

Let I T | ‘*’N,,,,‘, be the solution in t for

f0 I... {‘PMKZO" + tS)(1-p)"‘l - 4’""[(Zo"‘ - tS)(1TOY/'1}d‘1’(l)dQ.(é‘>) = l - A

01‘.

 

 

011

the

11111

cn

Sta

19

A set of 1 - A simultaneous confidence intervals for

(1N ' (11, ..., C!" - (1N4
is given by

[ON-&,-d’,&N-&, +d’] i=1, ...,N—l
where

d, = 1 T 1 “’~-1,.,,S(2/1')"

Tables for l T | (”Mm can be found in Hochberg and Tamhane (1987), Dunnett
(1964), Hahn and Hendrickson (1971) and Dunn and Massey (1965). A few observations
on these intervals are in order. First, the critical value and hence the intervals are based
on the equicorrelated structure of the 62,, - 61,, but there are several approximation
technique for dealing with a more general correlation structure. We defer discussion of
these techniques until later. Second, if we restrict the bounds non—negative, these
intervals can be regarded as MCB intervals conditional on (1,, being the known best.
Third, the two-sided critical value I T | (”MN is necessarily larger than the one-sided
critical value Tom,” for equal values of N-l, v and p, meaning that if we want to make
statements about both upper and lower bounds, the upper bound is necessarily larger.
These intervals are important insofar as they lay the foundation for adapting MCC

intervals to MCB intervals which we address in the next section.

3.5 EWLLQS: and Hsu (1283) - MCB Intorvols from MCC Intervals

To perform two-sided inference on equation ( 1.7) we examine the 1983 paper of

Edwards and Hsu. If we can regard any of the N populations as a control and construct

 

ref

 

for

int

int:

ll).

wh

20
MCC intervals, they can be adapted to the MCB intervals of equation (1.7).

Computation of these MCB intervals only requires that the MCC intervals exist (i.e. can
be constructed); it does not require independence or equal variances for the N
populations, 1,. As mentioned earlier MCC intervals can be thought of as MCB intervals
conditional on the knowledge that the control is the best. Here Edwards and Hsu adapt
these to unconditional MCB intervals to incorporate uncertainty about the best treatment.
We use the same notation as section 3.3.

Let I represent the set of population indices, so I = {1, ..., N}. If there is any
reference group or control j E I such that a random confidence interval [L3, U9] exists

fori E I - {j} satisfying

P{L,i S aj-a, S U,"foreveryiEI-{j}}21->s,

and each joint distribution, P, is an element of some family of distributions at least
partially indexed by the a,, then MCC intervals on j exist and can be adapted to MCB

intervals using the following theorem.

Theorem (Edwards and Hsu 1983)

When MCC intervals on j E I exist as defined above,

P{[L, s u’, s U,vi€ I] n [[N]E 5'1}? l-A
where

f={j:U,520foriEI-{j}},

21

and for each i E I
Li = O 1' = {i}
Li = max(minjer Lij: 0) 1' ‘fi {i}

U, = max(max,,., U,‘, 0).

For a proof see Edwards and Hsu (1983). A few observations are in order.
First, it is the set of all treatments that have all non-negative MCC upper bounds. If we
construct a set of simultaneous MCC intervals for each j E I, then those j that conform
to the selection criteria of g- must have all non-negative MCC upper bounds. Second, if
a treatment i is not in the set f, then its lower bound is max(min,er L3, 0). If the
treatment i is contained in I, then its lower bound is 0. So, a treatment’s lower bound
is 0 if it is in 3' or if it is close to being in (1 Third, the lower bound is constrained non-
negative. In their paper Edwards and Hsu give a normal distribution example of the
above theorem. While not an explicit theorem in their paper, the following theorem is
adapted from their normal example. It assumes that the "y,’s possess the aforementioned

equicorrelated structure.

Theorem 3.3 (Edwards and Hsu 1983)
A set of 1 - A simultaneous confidence intervals for
(1,," - (1,, ..., (1m - (1,,
is given by
P{[L, s u‘, s U,viE I] n [[N] E g']}21-A

where

22
r = {j: (‘1, 2 max,!“=,(1,- d’}.
and for each i E I
L, = max(min,e,[&, - (‘1, - d’], 0)
U, = max(max,¢,[(‘1, - (1, + d’], 0)
where

d’ = IT I <*>N-...,.s(2m%

Edwards and Hsu refer to these as adaptive intervals. Again, tables for | T | (”N
,.,,, can be found in Hochberg and Tamhane (1987), Dunnett (1964), Hahn and
Hendrickson (1971) and Dunn and Massey (1965). Notice that the upper bound in this
case is the same as the one-sided upper bound derived by Hsu ( 1981), save for the
critical value, so its interpretation is the same. In terms of inference on which treatment
is best, the interpretation of these intervals is straightforward. If the lower bound for a
group of treatments is not zero (positive), these treatments can be eliminated as being
best at a l-A confidence level. If the upper bound for any single treatment is zero, then
it is best at the l-A confidence level. We now discuss the interpretation of the lower
bound.

The lower bound is based on the smallest treatment mean of those that satisfy the
selection criterion, the set t. g' is the set of all treatments in contention for the best
where selection is based on the MCC intervals for each treatment. A treatment is
eliminated from contention (excluded from n, if any one of its MCC upper bounds are

negative. Clearly, this means we need only test the interval that could result in the

largest negative upper bound for each treatment, namely the interval for (1, - mafo, 0:,

23

for each treatment i. This leads to the definition of r in Theorem 3.3 which is the basis
for determining the MCB lower bounds.

Two polar examples help to illustrate the purpose of f. If g- consists of one
element, say {m}, then (1,, is the best treatment with confidence level 1-A. In this case
the MCB intervals reduce to MCC intervals with the lower bound constrained positive,
and the MCC intervals provide us with all the information we need to determine that m
is best, so the MCB intervals are narrow. If 5' consists of all N treatments, then all the
treatments will have an MCB lower bound of zero, meaning that none of the treatments
can be eliminated from being the best. In this case the MCC intervals have provided us
with little information and the MCB intervals are wide.

The lower bound is constructed so that it equals zero, when the treatment in
question is in the set I or when it is within d’ of the lowest value in this set. This means
that even if a treatment is not an element of r, it may still have zero as a lower bound
and might not be eliminated as being in contention for the best. This is due to the
uncertainty associated with the best treatment being unknown.

Since we have two-sided intervals a discussion of the factors which determine
their width is more instructive than in the case of one-sided intervals. The following
factors affect the bounds and hence the width of the intervals. T and 5 effect both
bounds in the usual way though (1’. The critical value, I T | “’N_,.,_,, merits some further
examination. When N is large, the critical value is larger making the intervals wider to
account for the added uncertainty associated with estimating more parameters. This
widening also compensates for additional multiplicity in the probability integral lest the

confidence level be diminished.

24

The lower bound involves (and, as we have seen, is based on the smallest sample
mean with a positive MCC upper bound. As T gets large the probability that t contains
the single element [N] approaches 1. That is: as T—t on , P[§' = {[N]}] -’ 1 (provided that
there are no ties for the best in the population). Since this single element is the largest
in the population it will tend to be the largest in the sample as the accuracy of the
estimates improves; this forces the lower bound up, causing the intervals to get narrower.
So, increasing T narrows the lower bound in two ways: through d’ and through (2 The
former is due to improved accuracy of our estimates and the latter is due to improved
accuracy in the ranking of the sample means.

Figure 1 (in Tables and Figures Section) illustrates the relationship between MCC
intervals and MCB intervals. The width of the MCC interval (top) is determined solely
by the allowance, (1’. The width of the MCB interval (bottom) is determined by the
allowance and the uncertainty of the ranking, which is determined by the number of
elements contained in t“. So, when T is large enough that g' consists of a single element
then the uncertainty of the ranking vanishes, and the MCB interval is identical to the
MCC interval with the maximum population parameter known.

As previously stated the size of T has a direct effect on the number of elements
that satisfy the selection statement, 3'. As T gets large the probability that g' contains the
single element [N] approaches 1, meaning that the firm with the maximum sample mean
will ultimately be determined best by the confidence interval. If T is small, I will
probably consist of many elements, and we may not be able to say with any reasonable

degree of confidence that one treatment is significantly better than the rest.

25

One must keep in mind that these intervals are not based on the estimator max,“=1
(1, - 61,, for this would imply that we are certain that our estimation procedure reveals the
true ranking of the population parameters. If we do regard these intervals as based on the
estimator max};l (1, - (1,, then (ignoring the max(., 0) operators in Theorem 3.3) these
intervals can be rewritten as follows:
L, = max-=,(‘1, - (1, - d’ - (maxf=,(1,- mime, (1,)

U1 = maxilla ' (i, + d,

If we consider the mid-point of the interval a point estimate for max?=1 (1, - (1,, then an

estimate for inefficiency, u,, based on the intervals is:

(1.10) ii, = (maxfgfi, + mime, &,)/2 - (3:,

suggesting the point estimate for am:

(1.11) am, = (max?=,(1, + minje, a,)/2

which has been shown to be consistent and have less positive bias than (‘1 (Edwards and
Hsu, 1983, p967). The bias of the estimate it, = (1 - 6:, is positive, is of magnitude
(maxEL, (Si, - mitt-6, o‘z,)/2 and is caused by the over estimation of the largest (3:, of the
sample. Notice that this bias is a function of the critical value which depends on our
confidence level. As the confidence level rises so does the critical value, potentially

making min,E , (1, smaller and the bias larger. This bias is eliminated when the difference

 

26

between the largest (1, and the second largest (1, is large, the probability of which
approaches 1 as T gets large. When T is small the probability that there are several (1,

near the maximum is large, so the bias is large.

3.6 Extonsions goo Conglosions

Clearly, the inference of the preceding theorems hinges on a specific form for the
correlation of the (1,: the correlation matrix must have identical off—diagonal elements, so
the equicorrelated structure can emerge. If the correlation matrix is of a more general
form, one must appeal to a more robust inference technique. This section briefly
summarizes a few of these techniques.

If the data are unbalanced, things become more complicated. In general,
unbalanced data lead to a lack of the equicorrelated structure, making determination of
the critical value more cumbersome. In fact, we would have to calculate a more general
critical value, T‘”,.,_,.,,,,(,D or | T | “’N,,.,.p(,,, where p(ij) is the correlation between the '1,
and 1, in the unbalanced case and is not constant over all i 9‘: j. (This notation, p(ij), is
to distinguish this correlation from that of the balanced case, p,,). Tabulation of such a
critical value would be impractical, and even computation with a computer would be
costly, since it would require solving N-l dimensional integrals. Allis not lost however,

for if the correlation matrix of the '1, takes on a product structure where:

(1.12) p(ij) = 5,5,. i,j 1E N

where (5, = [T,/(T, + T,.,)]'[5 i =15 N

 

in‘

165'

this

 

 

27

then the N -1 dimensional probability integral reduces to a lower dimensional iterated

integral, so T"°N,,,,,, and | T | (MN-1,...) are the solutions in t of:

[0 I... 111"“‘1’[(5az + tS)(1-5;’)""ld4>(2)dQ.(S) = 1 - A

and

f0 I... nimbl’lwsz + tS)(1-5;’)"”] - PIGS-,1 - tS)(1-5;2)'“l}d‘1>(2)dQ.(S) = 1 - A

respectively. These can be solved using an iterative search technique or simulated. If

this is the case, then the respective allowances for the MCC and MCB intervals become:

(1.13) d,, = TWN_,.,,,S(l/T, + 1/T,)"’ d,,’ = | T | “’N,,,,,,S(1/T, + 1/T,)"5

If we cannot appeal to this special structure in the unbalanced case (or for that
matter, in the balanced case) than we must use a conservative approximation technique.
Matejcik (1992) suggests techniques for adaptive MCB intervals that are robust to a
generalization of the correlation matrix and compares their performance using computer
simulation. These techniques are based on several MCC methods that are themselves
robust and include: an MCC method based on Banjeree’s Inequality due to Tamhane
(1977), a procedure using a moment-based approximation to the Behrens-Fisher problem

due to Tamhane ( 1977), a method using all-pairwise procedures due to Dunnett (1980)

28
and lastly his own technique based on a heteroscedastic selection procedure. The

techniques will not be discussed here; the reader is referred to the above citations. An
obvious line for further research is to examine the applicability of these techniques to
stochastic frontier models.

The adaptive MCB intervals of Theorem 3.3 (Edwards and Hsu 1983) imply the
one-sided results of Theorem 3.1 (Hsu 1981), so the rest of this chapter will be in terms
of Theorem 3.3. To construct adaptive MCB intervals one need only be able to construct
the MCC intervals. So the questions remains, can we construct MCC intervals in the
fixed effects stochastic production frontiers context? The answer to this question is yes,

and the following section seeks to explain why.

4 APPLYING MCB METHODS TO STOCHASTIC FRONTIER MODELS

4.1 W

The motivation for MCB stems from the problems specified in section 2.2. MCB
methodology eliminates these problems. First, we can perform comparative inference
on the best population mean with only a normality assumption (no half normals,
exponentials or extreme value distributions). Second, MCB allows construction of
intervals for an unknown best treatment, so we don’t have to assume that the sample
rankings coincide with the population rankings. Additionally, the point estimates of
equations (1. 10) and (1.11) are consistent as T —. co and may have less positive bias than
those of equation ( 1.4) when T is small.

To use Theorem 3.3 we must be able to construct MCC intervals for the u,, so

we would like to identify the circumstances under which MCC is the most readily

29

applicable to constructing intervals for stochastic frontiers. First, it is important to point
out that the MCB intervals are not intervals constructed on the fr, but instead on 5",, for
we are not certain which firm is the true best. However, the MCC intervals (from which
the MCB intervals are adapted) are for (1N - 01,, where N is not necessarily the index of
the best firm in the population, but merely a control. Therefore, a priori knowledge of
the best firm is not necessary, and MCC intervals constructed on o, are perfectly
acceptable. This is the strength of the adaptive intervals: the experimenter first assumes
that the estimation reveals the true best firm and constructs MCC intervals. Then she
drops this assumption and adapts the intervals to accommodate this uncertainty.
Consequently one would expect MCC intervals with the lower bound constrained non-
negative to be narrower than the subsequent MCB intervals.

Construction of the MCC and MCB intervals is simple, however determination
of the appropriate critical value is non-trivial. When we can appeal to the equicorrelated
structure the critical value can be drawn from the aforementioned references, but even
then the existing tables are for small values of N only (N s 20, typically). For large
values of N the critical value would have to be calculated using a iterative search
technique or simulated. Failing the equicorrelated case, if the product structure of
equation (1.12) emerges, then we can again calculate a critical value using a iterative
search. When we cannot appeal to the equicorrelated or product structures,
approximation techniques can be employed as noted in section 3.6, but even then one
runs into problem of no critical value tables for large N.

To construct MCB intervals we must examine the correlation structure of the 1,,

which entails calculating the variance and the covariances of the 6:, in equation (1.5). If

30
the firm efficiencies are orthogonal (e. g when B is known and Var(B) = 0), and we have

a balanced panel, then var(&,) = 201/T and p,, = p = ‘A. The equicorrelated structure
emerges; the critical value can be looked up or calculated, and the MCB intervals can
be constructed directly from Theorem 3.3 without appealing to an approximation
technique. The problem is that the firm efficiency estimates are, in general, not
orthogonal as evidenced by equation (1.5), so the equicorrelated structure is unlikely to
emerge in small samples. All is not lost however, for we can appeal to the large sample
properties of the (1,. Another potential problem arises when the panel is unbalanced. If
the panel is unbalanced then the '1, do not possess the equicorrelated structure. In this
case we would like to appeal to product structure of equation (1.12). Again,
unfortunately, the lack of orthogonality within the firm efficiencies, as evidenced in
equation (1.6), precludes us from appealing to this simpler structure. However, again
we may be rescued by the large sample properties of (1,. What follows are summaries
of the properties of the MCB intervals under different configurations of the panel data

set in both the balanced and unbalanced contexts.

4.2 MM

For a balanced panel, when N gets large the covariance matrix of B approaches
zero, causing the off-diagonal elements of Var((1) to approach zero and the diagonal
elements to approach ale, hence var(y,) —> 202/T, cov('y,,'y,) » 0le and p,, -» p = 96:,
giving us the equicorrelated structure. In this case MCB procedures can be directly
applied using Theorem 3.3 (Edwards and Hsu). If the critical constant I T | (MN-1.7.9

eannot be found in the aforementioned references, because N is too large, then it can be

31
calculated or simulated. If the panel is unbalanced, as N gets large the product structure

of equation (1.12) will emerge, and we can apply Theorem 3.3 as modified by equation
(1.13) with critical value I T | (MN-1,1,6 calculated or simulated. In either case, balanced
or unbalanced, point-estimates based on these intervals follow directly; since T is small

the point estimates will have smaller positive bias than u, and (1.

4.3 14%ng

For a balanced panel, if T is large, the variance of our estimate of B is small, so
the of diagonal elements elements of the covariance matrix of the (1, become negligable.
Additionally, the bias of 0, and (1 will be minimal. In the unbalanced case, as T, gets
large for all i similar results arise. In either case MCB procedures can be directly
applied using Theorem 3.3 (or as modified by equation (1.13)) and should result in

narrow intervals for the u, and a strong statement about which treatment is best.

4.4 Small N and T

If N is less than large, the covariance matrix of (1 is non-spherical, and Theorem
3.3 can no longer be used to perform MCB inference on the estimates. However, if the

variance structure of the (1, takes on the special form;

(1.14) Var((1,) = al/T + C Cov((1,,(1,) = C C = constant

then again Var(r‘y,) = 202/T, Cov('y,, 1,) = ol/T, p,, = p = 1/2 and the correlation

structure of the ifs is the same as when N is large. Again Theorem 3.3 can be directly

32
applied. Since T is small the bias of a, and (1 will be large, so we will need to rely on

the point estimates suggested by the intervals. Similarly for the unbalanced case, if

then we can use Theorem 3.3 as modified by equation (1.13). If the special forms of
equations (1.14) or (1.15) cannot be assumed we must appeal to an approximation

technique as suggested in section 3.6.

5 CONCLUSIONS AND EXTENSIONS

In the broadest sense this chapter has attempted to provide some insight into the
sources of uncertainty in the estimation of inefficiency measures in the fixed effects
model. Specifically, we find that MCB methodology is a useful tool for understanding
this uncertainty in the panel data context. For the econometric theorist the brief MCB
survey should shed some light on the issues surrounding this uncertainty. We conclude
that the assumption that the sample rankings reveal the true population rankings may be
dubious, particularly when T is small.

Consideration of this ranking uncertainty has strong implications for the overall
productivity modelling approach. This uncertainty uncovers additional inaccuracy
inherent in the efficiency estimates of these models, which heretofore has gone
unnoticed. If we are interested in preserving the accuracy of our estimates and reducing

this uncertainty, then we would typically have to sacrifice model flexibility to achieve it.

33

Therefore, this MCB methodology may argue for more accurate and less flexible
productivity models. Of course this remains to be seen.

The next chapter is concerned with applying this MCB methodology to real data
sets and comparing the resulting intervals with those generated under other techniques

that are fairly well-established in the stochastic frontiers literature.

CHAPTER TWO

CONFIDENCE STATEMENTS FOR EFFICIENCY ESTIMATES

1 INTRODUCTION

This chapter is a comprehensive study of methods of inference associated with
technical efficiency estimates in the stochastic frontier model. We seek to characterize
the nature and the empirical magnitude of the uncertainty associated with the usual
estimates of efficiency levels.

From our perspective, deterministic approaches (e.g. , DEA) produce efficiency
measures, while statistical approaches (stochastic frontier models) produce efficiency
estimates. The relative strengths and weaknesses of these approaches have been
vigorously debated, and will continue to be. However, the strongest argument in favor
of a statistical approach has always been that it provides a straightforward basis for
inference, not just for point estimates. Thus, for example, one can construct standard
errors and confidence intervals for estimates of technical efficiency. A statistical
approach recognizes that uncertainty exists and is capable of quantifying it. In our view,
uncertainty also exists within the deterministic approach, but methods of characterizing
and quantifying it are still not well developed. Consistency of the DEA estimates has
been established by Banker (1993) and by Korostelev, Simar and Tsybakov (1992, 1995).
Korostelev, Simar and Tsybakov also establish the rate of convergence of the estimates,
and Banker (1995) considers certain types of hypothesis tests. These results are

important but they do not lead to confidence intervals. Confidence intervals can be

34

35
constructed by bootstrapping the DEA estimates; for example, Simar and Wilson (1995)

give some theoretical results and an empirical example. However, in our view
bootstrapping procedures are an imperfect substitute for an adequately developed
distributional theory.

Ironically, the ability to conduct inference on efficiency estimates in stochastic
frontier models has previously been noted approvingly, but has never been systematically
exploited in an empirical setting. This chapter seeks to fill this void and, in doing so,
advances our understanding of the various sources of uncertainty inherent in econometric
models for efficiency estimation.

Of course, the strength of the econometric approach comes at a cost: strong and
often arbitrary distributional assumptions are necessary to extract technical efficiency
estimates and ultimately to construct confidence intervals. Therefore, a major aim of this
chapter will be to show how to perform inference on efficiency estimates under different
sets of assumptions that range from the very strong to the relatively weak, and to see
how the degree of uncertainty associated with these estimates relates to the strength of
the assumptions made. Some of the methods we discuss require panel data. Most make
specific distributional assumptions for statistical noise and technical inefficiency.
However, we also make use of the methodology of Multiple Comparisons with the Best
(MCB), developed by Edwards and Hsu (1983) and described in chapter one, which uses
panel data to construct confidence intervals without the need for strong distributional
assumptions.

In this chapter, technical efficiency estimates and their confidence intervals are

generated for three different panel data sets with different dimensional characteristics,

36

using several formulations of the stochastic frontier model. We analyze these panel data
as complete data sets and also in some cases broken down into their component cross
sections to construct confidence intervals for technical efficiency estimates using different
interval construction techniques. The results highlight the relevant strengths and
weaknesses of the various techniques and data configurations. The chapter addresses
practieal aspects of interval construction that may present problems for the data analyst.

The plan of the chapter is as follows. Section 2 briefly reviews the stochastic
frontier model as it relates to this chapter. Section 3 reviews two interval construction
techniques: the Jondrow, Lovell, Materov and Schmidt (JLMS) (1982) method and the
Battese-Coelli (BC) (1988) method. Section 4 is an empirical analysis of three panel data
sets for which we construct confidence intervals for technical efficiency estimates.

Section 5 summarizes and concludes.

2 STOCHASTIC FRONTIER MODELS
In this section we detail various estimation techniques for the stochastic frontier
model of equation (1.1). Since chapter one detailed fixed effects estimation in the panel
data context, this section concentrates on other estimation techniques in both the cross-

sectional and panel data contexts.

2.1 W
In the case of a single cross-section, T = 1 and t is irrelevant and can be
suppressed. Under assumptions (A. 1)-(A.4), the model as given in equation ( 1.1) can

be estimated by maximum likelihood (MLE). Details of this estimation, including the

37
likelihood function, can be found in Aigner et al. (1977) and will not be addressed here.

MLE of equation (1.1) yields (1, B, (1,, and 6,, which are consistent as N—>oo.

Define p. = E(u,) 2 0. Under assumption (A.4), ,u. = (1r/2)"2(r,,. Ordinary Least
Squares (OLS) applied to equation ( 1.1) yields consistent estimates of (oz-p) and B. The
Corrected Ordinary Least Squares (COLS) method constructs a consistent estimate of (1
by adding a consistent estimate of u to the OLS intercept. This requires a consistent
estimate of u,,, say (1,,, which can be derived from the third moment of the OLS residuals.
Also a consistent estimate of or, can be derived from the second moment of the OLS
residuals. See Olson et a1. (1980) for details.

So, in summary, both COLS and MLE yield consistent estimates of a, B 0,, and
(1,,. COLS is less efficient than MLE. In either case, point estimates for u, and TE, =

exp(-u,) can be obtained, as described in section 3.1.

2.2 PaneLIAata

We now turn to the case of panel data with T > 1. Under assumptions (A. 1)-
(A.4) equation (1.1) can be estimated by MLE. See Pitt and Lee (1981) for the
likelihood function and other details. MLE yields estimates of the same parameters as
in the cross-sectional case: (1, B, (7,, and (1,,. These estimates are consistent as N—>oo;
therefore MLE is appropriate when N is large. Large T is not a substitute for large N.

Equation ( 1.1) can also be estimated by Generalized Least Squares (GLS). This
requires assumptions (A. 1)-(A.4), except that it does not rely on specific distributional
assumptions (normality of v, or half-normality of u). The standard panel data GLS

procedure yields estimates of (oz-a), B, 03,, and var(u,) that are consistent as N—eoo. Care

38
must be taken to distinguish var(u,) and 02,; the usual GLS procedure uses var(u,), not

02,. Under the half-normal distributional assumption, 02,, = var(u,)rr/(rr—2), so that the
estimate of var(u,) is easily converted to an estimate of 02,. This is required to estimate
a = E(u,) and to convert the intercept, exactly as in the discussion of COLS above. We
will refer to GLS with this intercept correction as the CGLS method. Point estimates for

u, and TE, can be obtained, as described in section 3.2.

3 TECHNIQUES FOR CONSTRUCTION OF CONFIDENCE INTERVALS

We use two different techniques to construct confidence intervals for technical
efficiency estimates in stochastic frontier models. The first technique is based on the
(conditional) distribution 11, | 6,, where e, = [e,,, (,2, ...,e,,~]. It was developed for the
cross-sectional case by Jondrow, Lovell, Materov and Schmidt (JLMS) (1982) and later
generalized to the panel data case by Battese and Coelli (BC) (1988). The second
technique is based on the MCB procedures described on chapter one. The MCB method
will be based on fixed effects estimates described in chapter one, while the J LMS and
BC methods will be applied to the results of the other estimation techniques described
above; this choice is primarily driven by the difference in distributional assumptions of

the models.

3.1 Cross-sootionol Data: JLMS Mothgi
For either cross-sectional estimation method, MLE or COLS, we use the JLMS
method for interval construction. The J LMS technique follows from the distribution of

u, conditional on e, (which is a scalar, since T=l for a cross section). JLMS show that

39

given distributional assumptions (A.1) and (A.4), the distribution of u, | 5, is that of a
N(p',, 02.) random variable truncated (from the left) at zero, where a', = (12,5,(02u + 02,)"
and 01. = 02,02,102, + 02“)". They evaluate E(u, | 5,), which is regarded as a point

estimate for u,. A point estimate for TE,, due to Battese and Coelli ( 1988), is given by:

(2.1) TB; = E(exp(-u,) I 5,] = expht'a + le-H 1 - <1’(Uo - H's/0-)}{1 - 4’(-u's/U.)}",

where <I> is the standard normal cdf. Implementing this procedure requires estimates of
p', and (12.; this in turn requires estimates of (12,, and (12,, and the use of e, = y, - (1 - x-B.

Empirical implementations of the JLMS technique have focused on the point
estimate E(u, | 5,). However, confidence intervals for u, or TE, are easily constructed
from the density of u, | 5,. Critical values can be extracted from a standard normal
density to place lower and upper bounds on u, | 5,. Because TB, is a monotonic
transformation of u,, the lower and upper bounds for u, | 5, translate directly into upper
and lower bounds on TE, | 5, = exp(-u,) | 5,. Specifically, a (l-A)100% confidence

interval (L,, U,) for TB, | 5, is given by:

(2.2) L, = eXP(-u’s - 2(a),
U1 = CXP('#.1 " 200.),
where: Pr(Z > 2,) = (A/2)[1 - <I>(-p',/a.)],

Pr(Z > 2”) = (1 - A/2)[l - Q(-u',/o.)],

40
with Z distributed as N(0, 1); so that

ZL = ‘1’"{1 -(N2)[1 - PPM/00]} and

lo = ‘1’"{1 -(1 - N2)[1 - ‘M‘MYUJU-

As a semantic point, we will refer to the implementation of equation (2.2) in the
cross-sectional context as the JLMS method, since it relies on the J LMS result for the
distribution of u, | 5,, even though equation (2.1) is due to BC. The BC method will refer
to the corresponding calculations in the panel—data case, described in the next section.
It should be noted that both the JLMS and the BC methods treat a, B, 02,, and 02,, as
known, so that the confidence intervals do not reflect uncertainty about these parameters.
For large N, this is probably unimportant, since the variability in the parameter estimates
is small compared to the variability intrinsic to the distribution of the u, | 5, (and due to

the presence of the statistical noise v,,).

32 WW

The BC method for construction of confidence intervals is a generalization of the
JLMS method and also follows from the distribution of u, | 5,. The BC technique can be
based on the MLE or CGLS estimates of or, B, 01,, and 02,. It extends the JLMS method
to accommodate the case of panel data (T > 1), so that now 5, = (5,,, ..., 5,7) = (v,, -
u,, ..., v,r - 11,). DefineE, = (1/T)22,5,,, p', = 02,5,(02u + (12,,/T)'l and oz. = (9,02,,(02, +
Tozu)“. The latter expressions are essentially the same as in J LMS, with 5, replacing 5,

and ozv/T replacing 02,. Then the distribution of u, | 5, is that of a N(;(',, 02.) random

41

variable truncated at zero, a point estimate for TB, is given by equation (2.1) above, and
confidence intervals are constructed as in equation (2.2) above.

The Battese-Coelli method can also accommodate the case of an unbalanced panel,
in which there are different numbers of time-series observations per firm. Suppose that
for firm i there are T, observations, where the notation reflects the fact that T, varies over
i. We simply have to replace T by T, in the definition of p', and 02. above, so that p',
= (12,5,(02u + 02,,IT,)'l and 02., = ozuazv(az,, + T,az,,)"; note that now 03., varies over i.
Then equations (2. 1) and (2.2) hold exactly as before, except that (12., replaces 03.. Thus

an unbalanced panel causes no real problems for the BC method.

3.3 BMW

MCB intervals are naturally based on the fixed effects estimates of chapter one
and use only assumptions (A. l) and (A.2) above; they do not require a distributional
assumption for the u,. In this study, when we cannot appeal to asymptotics, the condition
for the special covariance structure of equations (1.14) and (1. 15) is met or nearly met,
so MCB is at least approximately applicable.

We note in passing that several recent models in the frontiers literature have
featured time-varying technical inefficiency. For example, see Comwell, Schmidt and
Sickles (1990), Kumbhakar (1990), Battese and Coelli (1992), and Lee and Schmidt
(1993). These models imply intercepts (1,, that vary over i and t. For a given value of
t, it is natural to proceed as before to consider comparisons relative to the maximum
(over i) of these intercepts, so that we essentially have a separate MCB problem for each

t. However, there is no apparent reason to expect the equicorrelatedness condition to

42

hold for the estimated (1,, from any of these models, and if it does not hold the methods

surveyed above would not apply.

3.4 Comoag'son of Diffosont Tfihnigoos

A discussion of the differences between the interval construction techniques is in
order. First, it should be noted that MCB provides joint confidence intervals for u', =
(1m, - (1, of equation (1.3), whereas JLMS and BC provide marginal intervals for u, =
(1 - (1, of equation (1.2). The difference between u', and u, is u,,." = minim, which may
be non-trivial when N is small. Conversely, the difference between joint and marginal
intervals may be substantial when N is large. For example, one of our data sets has N
= 171. Although independence would be a poor assumption, it is instructive to note that
a set of 171 independent intervals, each holding with a marginal probability of 0.95,
would hold jointly with a probability of only (.95)‘71 = 0.000116. Conversely, joint
confidence intervals that hold with a probability of 0.95 would correspond to marginal
intervals with a confidence level far in excess of 0.95. Other things equal, we would
certainly expect joint confidence intervals to be wider than corresponding marginal
intervals, for a given level of confidence like 0.95.

The MCB and JLMS/BC methods also differ substantially in the way they handle
estimation error. One sense in which this is true is that, assuming that the equicorrelated
structure emerges for the (1,, the MCB intervals reflect the variability of B, which the
JLMS and the BC intervals ignore. This is probably not an important difference, since

uncertainty about B is not the only source, or in most cases the major source, of

43

uncertainty about u,. To be more specific, consider the following expression for the

fixed effects estimate of (1,:

ai=yi'ifi=a+(Vi'uJ';i(B’B)=ai+Vi “E(B‘B)

The term §,(B - B) reflects estimation error in B. As noted above, BC ignores this source
of uncertainty while MCB does not. This term disappears as oimg N or T—too , and is
probably not important empirically for most data sets. More fundamentally, (1, contains
the error '6, = (1/T)E,v,,; the fixed effects procedure separates u, from v,, by averaging
away the v,,. The significance of V, depends on T and on the relative sizes of 02,, and 02,;
it is most troublesome when T is small and/or 02,, is large relative to 02,. It is important
to realize that the fixed effects estimate of u',, namely (1,", - (1,, is generally biased
upward (inefficiency is overstated), because the larger (1,, such as (1m, will on average
contain positive estimation error 17,, while the smaller (1, will on average contain negative
estimation error. (That is, the (1, will obviously be more variable than the (1,.) MCB
recognizes this variability by including the sample equivalent of 0,,(2/T)"" in the formula
for the allowance, (1, above. Also, the MCB intervals can be thought of as removing the
bias just described; they are not centered on the value (1,", - (1,.

The BC method uses distributional assumptions to remove estimation error more
effectively. The first Step in the BC procedure is to calculate 5, = V, - u, (ignoring
estimation error in B), so that the v, are averaged away, as in the fixed effects procedure.
The second step is to construct p',, which equals 5, times the shrinkage factor 02,,(02u +

(12,,IT,)‘1 < 1. This corresponds to the "best linear predictor" in the random-effects panel

44

data literature; see Schmidt and Sickles (1984). It reflects the relative variability of u,
and 3,. Finally, the distributional assumptions are used to imply the further shrinkage
factor {1 - <I>((r. - p',/a.)}{1 - <I>(-p',/(r.)}" < 1 in the calculation of the expectation of

T131151-

4 EMPIRICAL ANALYSES

4.1 ' ' - i 1

We analyze data previously analyzed by Erwidodo (1990), Lee (1991) and
Schmidt and Lee ( 1993). For a complete discussion of the data see Erwidodo (1990).
171 rice farms in Indonesia were observed for six growing seasons. The data were
collected by the Agro Economic Survey, as part of the Rural Dynamic Study in the rice
production area of the Chimanuk River Basin, West Java and obtained from the Center
for Agro Economic Research, Ministry of Agriculture, Indonesia. The 171 farms were
located in six different villages and the six growing seasons consisted of three wet and
three dry seasons. Thus the data configuration features large N and small T.

Inputs to the production of rice included in the data set are seed (kg), urea (kg),
trisodium phosphate (TSP) (kg), labor (labor-hours) and land (hectares). Output is
measured in kilograms of rice. The data also include dummy variables. DP equals 1
if pesticides were used and 0 otherwise. DVl equals 1 if high yield varieties of rice
were planted and DV2 equals 1 if mixed varieties were planted; the omitted category

represents that traditional varieties were planted. DSS equals 1 if it was a wet season.

45
The are also 5 region dummy variables, DRl, DR2, DR3, DR4 and DR5, for the six

different villages in the survey.

COLS and MLE were performed on each of the six different periods (cross-
sections) in the panel. DSS, the dummy for wet season, had to be excluded for the cross
section models, because it was constant across farms for a single period. Results are in
Table 1. Unfortunately, periods 2, 3, 4 and 5 produced a positive third-order moment
of the residuals, causing the MLE estimate to coincide with the OLS estimate as
discussed in Waldman (1982). Additionally, this problem precludes COLS estimation
since (1,, is negative. Therefore only periods 1 and 6 are analyzed as cross-sections for
this data set. Since the results for the two periods were similar only the period 1 results
are reported. Technical efficiencies and confidence intervals were produced using the
JLMS technique; i.e. , equations (2.1) and (2.2) above. Confidence levels are 95 % , 90%
and 75%. These results are contained in Tables 2A and ZB. Due to the large number
of firms in the sample (171), only nine firms are reported here and subsequently: the
three firms with the highest (1,, the three firms with the lowest (1,, and the three firms
with the median (1,.

The choice of the estimation procedure (COLS versus MLE) made very little
difference, so we will discuss only the MLE results in Table 2B. Efficiency levels are
not estimated as precisely as one might hope. The firm with the highest estimated
efficiency level had estimated efficiency of 0.9452, but a 95 % confidence interval ranged
from 0.8322 to 0.9982. The median firm had estimated efficiency of 0.9053, with a
95% confidence interval of (0.7576, 0.9957); and the worst firm in the sample had

estimated efficiency of 0.8040 with a 95% confidence interval of (0.6415, 0.9694).

46

These are fairly wide confidence intervals. In fact the uncertainty about the inefficiency
level of a given firm is definitely not small relative to the within-sample variability of the
efficiency measures, and we would have little reason to have much faith in our efficiency
rankings. The reason for this lack of precision is straightforward - most of the variation
in 5, = v, - u, is due to v,, not u,. We have (for MLE, t=1) var(v,) = 02,, = 0.0579 and
var(u,) = 02,,(1-2)/1r = 0.00633, so the variance of v is over nine times as large as the
variance of 11. This makes it very difficult to estimate u, precisely.

Next, CGLS and MLE were performed on the entire panel. The variable DSS
could now be included. Results are in Table 3. Technical efficiencies and confidence
intervals were produced using the BC technique. These results are contained in Tables
4A and 4B. Efficiency levels based on the CGLS and MLE estimates are again similar.
Not surprisingly, the panel data confidence intervals are tighter than their cross sectional
counterparts, because var(u, I 5,) is smaller with six observations than with one.
Nevertheless, the confidence intervals do not shrink as much as one might hope -
compare a 95 % confidence interval for the median firm of (0.7638, 0.9945) in Table 48
to (0.7576, 0.9957) in Table 2B. This is partly due to having only six observations per
firm, and partly to getting a larger value of 02,, for the panel than for the t=l cross
section, which diminishes the value of the panel.

The fixed effects estimates were calculated for the panel, with time invariant
regressors excluded to preclude multicolinearity. These results are also in Table 3. The
covariance matrix for the (1, very nearly exhibited the equicorrelated structure necessary

to justify the MCB procedure:

47
Mean of i,Var(B)1,' = .04572
Standard deviation of §,Var(B)i,’ = .002211
Maximum of §,Var(B)x',’ = .05523

Minimum of §,Var(B)x',’ = .03918

95 %, 90% and 75 % MCB intervals were constructed for technical inefficiencies using
critical values of | T | (”170,345,111 = 3.42, 3.18 and 2.71, respectively, and are given in
Table 5A. The intervals are too wide to be of much use. For example, the firm with
the highest (1, (and hence with estimated efficiency of 100% by the usual calculation) has
a confidence interval ranging from 0.5613 to 1. Every firm in the sample has a
confidence interval with upper limit equal to one; that is, at the 95 % confidence level,
no firm is revealed to be inefficient. In fact, this is still true at the 75% confidence
level.

The MCB intervals are much wider than their BC counterparts based on CGLS
and MLE. We next attempt to determine the relative importance of three sources of
width: estimation error, uncertainty of the identity of the most efficient firm, and the
multiplicity of the probability statement. The easiest of these factors to investigate is
uncertainty about the identity of the most efficient firm. To do so we simply assume that
firm 164, which is the firm with the largest (1,, is most efficient in the sense of having
the largest 0:, (equivalently, smallest u,). Under this assumption we construct the MCC
intervals with firm 164 as the control. 95%, 90% and 75 % confidence intervals required
critical values of | T | (”170,846.15 = 3.42, 3.18 and 2.71, respectively. Results are in

Table SB. The MCC intervals are necessarily tighter than the MCB intervals, but not

48

tight enough to be useful. In other words, the width of the MCB intervals is not
significantly decreased by knowing which firm is best. We can conclude that the width
is primarily due to either estimation error or multiplicity or both.

To disentangle the effect of multiplicity on the interval width, we would like to
be able to construct marginal intervals for each firm. In the case where MCB reveals
a single firm as efficient, this can be accomplished with a simple application of the
Bonferroni inequality. This will be demonstrated later. In the present case, where there
is no single firm revealed as most efficient, the construction of marginal intervals is less
clear, because it is necessary to make a simultaneous statement about the firms to
determine a subset of firms that may be efficient, and then to reduce this joint statement
to a marginal statement about a single firm. However, we can get some idea of the
effect of the multiplicity of the intervals just by reducing the number of confidence
intervals created, which we can do by considering a subset of the firms. We therefore
redid MCB for only the nine firms for which we have reported results in Table 5A.
(However, the parameter estimates are still from the whole sample of 171 firms.) 95 % ,
90% and 75% confidence intervals required critical values of | T | “9,346,, = 2.56, 2.38
and 1.96, respectively. Results are in Table 6A. AS was the case in the MCC
experiment, controlling for multiplicity did not result in a significant tightening of the
intervals. For example, for the median firm, compare the new interval (0.3354, 0.9837)
with N=9 to the old interval (0.2899, 1.0000) with N=171. We conclude that the
multiplicity component of the interval width is small, leaving only estimation error to

account for the large width of the intervals.

49

Further evidence on this point is obtained by considering the smallest possible
subset of firms (N=2) and assuming that it is known that one of them is the most
efficient. Thus, as in our MCC calculations, we assert that firm 164 is most efficient
and we simply construct confidence intervals for 01,6, - (1, for a given value of i. This
is a standard calculation based on the estimate a,“ - (1, and its standard error, sew, =
[var(é,,,,) + var((1,) - 200v((1,,,,, 019]”, and using critical values from the standard normal
distribution. (Note that we have not imposed the equicorrelatedness assumption in this
calculation, so our results will be slightly different from the results for MCC with N =2,
which would impose this assumption.) These are called "per comparison” intervals; they
are given in Table 6B.

The per comparison intervals are indeed narrower than the MCB and the MCC
intervals, but they are still fairly wide. For example, for the median firm we still have
a 95% confidence interval of (0.3788, 0.8102). This confirms our conclusion that, for
this data set, the width of the confidence intervals is due primarily to the estimation
error. As noted above, estimation error is important for this data set because T is small
and 02,, is large relative to var(u,). There is Simply too much noise to get a clear picture
of the value of u,. The BC method does significantly better because it makes strong
distributional assumptions that allow a much better separation of v from u. For this data

set there does not seem to be a substitute for these strong assumptions.

4.2 fl'oxas Uo'lig'es - Komohflg (1294)
In this study we reanalyze data originally analyzed by Kumbhakar (1994).

Kumbhakar estimated a cost function, whereas we will estimate the production function.

50

The data set consists of observations on 10 major privately-owned Texas electric utilities
observed annually over 18 years from 1966 to 1985, and includes information on annual
labor, capital and fuel (inputs) for electrical power generation (output). Due to the
relatively small number of firms a cross sectional study of the data was precluded.
However, with 18 periods of observation per firm we have T larger than N, the opposite
of the case with the Erwidodo rice farm data.

The model was estimated by CGLS and MLE with results given in Table 7.
Notice that now 02,, is small relative to 01,, so our estimates of technical efficiency should
be more reliable than for the previous data set. It is instructive to point out that
numerical accuracy became a problem in calculating TE, using equation (2.1). The small
value for (I. produced extremely large values of u°,/(r. which, when evaluated in the
standard normal cdf <I>(.), produced technical efficiencies greater then 100%. This was
due to rounding error in the software package we originally selected. Fortunately,
another package was found that evaluated the normal cdf more accurately. Tables 8A
and 8B give our results for all 10 firms.

As expected, the efficiency estimates are much more precise than for the previous
data set. For example, for firm 8 (one of the two median firms) and using MLE,
efficiency is estimated as 0.8472 with a 95% confidence interval of (0.8264, 0.8683).
These are useful results in the sense that the uncertainty about a given firm’s efficiency
level is small relative to the between-firm variation in efficiencies; we can have some

faith in our rankings.

constr

equicc

Unit
the mc
meAI
hrhc
nwmi
mms:
Erwidr

(0.7801

MCBj
COmnb
mReu
MCBC

“Nn‘

51
The fixed effects estimator was also calculated (Fable 7), and MCB intervals were

constructed (Table 9A). The covariance matrix for the (1, again exhibited an almost-

equicorrelated structure, so that MCB was applicable.

Mean of §,Var(B)'x,-’ = .06765
Standard deviation of i',Var(B)i,’ = .00289
Maximum of §,Var(B)§,’ = .07255

Minimum of §,Var(B)3E,’ = .06166

The MCB intervals successfully determined at the 95 % confidence level that firm 5 was
the most efficient firm in the sample and that all others were inefficient. Consequently,
the MCC intervals coincided with the MCB intervals and are not reported separately.
In fact, firm 5 was identified as most efficient at the 99.9% confidence level, so
essentially we were certain that it was the best. The confidence intervals for the other
firms are wider than the corresponding BC intervals, but still not nearly as wide as the
Erwidodo rice farm data. For example, for firm 8 compare the MCB intervals of
(0.7809, 0.8603) to the BC interval of (0.8264, 0.8683).

It is interesting to note that there is very little overlap between the BC and the
MCB intervals, with the MCB intervals being generally lower. Two opposing sources
contribute to this difference. The difference between u', and u, when N is small should
make the BC intervals lower, since BC constructs a confidence interval for exp(-u,) while
MCB constructs confidence intervals for exp(-u',), and since u', < 11, implies exp(-u,) <

exp(-u',). However, this is apparently more than offset by the BC technique’s more

52

successful reduction of the effects of the estimation error. As noted above, the BC
technique can be viewed as a set of shrinkages of the fixed effects inefficiency measures,
leading to generally higher efficiency measures.

Marginal intervals were easily constructed for each firm using the Bonferroni
inequality. Since we knew the probability with which firm 5 could be identified as most
efficient, we simply constructed a joint probability statement with this and a per
comparison interval and selected the marginal confidence levels so that the Bonferroni
inequality produced the desired joint confidence level. Here, since we were essentially
certain that firm 5 was efficient, the joint probability statement essentially reduced to a
single per comparison probability statement. In the rice farm data the per comparison
intervals were conditional on firm 164 being efficient; we just assumed that this was the
case. However, for the current data, we knew with ”almost" certainty (99.9% certainty)
that firm 5 was efficient, so our marginal statement essentially coincides with the per
comparison statement, just as the MCB intervals coincide with the MCC intervals.

The marginal/per comparison intervals are contained in Table 93. Again the
actual standard errors were used, since we did not have to appeal to equicorrelatedness
to get our critical values. As a general statement, the marginal (per comparison)
intervals are comparable to the MCB intervals. Surprisingly, in many cases the marginal
intervals are actually wid_e_r than the MCB intervals. This must reflect failure of the
equicorrelatedness assumption underlying our MCB intervals, but it also is a reflection
of the relative sizes of N and T in the data. To be more specific, consider the following

expression for the standard error of the estimate (15 - (1,:

53

565 = [var(és) '1' var(&,) " 200V(&5, (33)]115

J

= [202,/T + Ewan}; + 1',V(ii)§,' - 21,V(B)i,’]'/=

When T is small and N is large (e.g. T = 6 as in the rice farm data), the term 201,,/T
is large relative to the other three terms, so any differences between §5Var(B)§,’,
i,Var(B)i,’ and i,Var(B)§,’ ate unimportant. For MCB we assume i,Var(B)i,’ =
i,Var(B)§,’ = §5Var(B)§,’, so these insignificant differences are ignored. When T is
large, however, the term 202,,IT is small and the aforementioned differences may become
significant. However, if we ignore this difference in MCB, then the standard error for
MCB may be smaller than the standard error for the some of the marginal (per
comparison) intervals. This is less of a problem when both N and T are large, because
large N tends to shrink the V(B) term so any differences in the i,V(B)x',’ term will
become less pronounced.

In cases where equicorrelatedness of the (1, cannot be assumed, there are some
conservative MCB approximations available. Matejcik ( 1992) suggests techniques for
adaptive MCB intervals that are robust to a generalization of the correlation matrix and
compares their performance using computer simulation. These techniques are based on
several MCC methods that are themselves robust and include: an MCC method based on
Banjeree’s Inequality due to Tamhane ( 1977), a procedure using a moment-based
approximation to the Behrens-Fisher problem due to Tamhane (1977), a method using
all-pairwise procedures due to Dunnett ( 1980), and his own technique based on a
heteroscedastic selection procedure. An obvious line for further research is to examine

the applicability of these techniques to stochastic frontier models.

54
4.3 E tianTilrie - cal 1

We analyze data previously analyzed by Scale (1990). For a complete discussion
of the data see Seale (1990). He observed 25 Egyptian small-scale floor tile
manufacturers over 3 week periods for 66 weeks, for a total of 22 separate observation
periods. The set contains some missing data points, so the number of separate
observation periods varies across firms, making this an unbalanced panel. The data were
collected by the Non-Farm Employment Project in 1982—1983. The firms were located
in Fayoum and Kalyubiya, Egypt. Inputs to the production of cement floor tiles are
labor (labor-hours) and machines (machine-hours). Output is in square meters of tile.

The model was estimated by OLS, fixed effects and CGLS. The third moment
of the OLS residuals was positive, so MLE was not attempted. Estimation results are
given in Table 10. It may be noted that 02,, and 02,, are of similar magnitude. For this
reason, and because the number of firms is similar to the number of periods per firm (for
most firms), this data set has characteristics that put it in the middle ground between the
Erwidodo rice farm data set (N much larger than T, 02, larger than 02,) and the
Kumbhakar utilities data set (T larger than N, 02,, larger than 02,). We should expect
confidence intervals wider than for the utilities but narrower than for the rice farms.

Table 11 gives the BC confidence intervals based on the CGLS estimates, for all
firms. As a general statement, these confidence intervals are considerably wider than for
the utility data. They are perhaps a little narrower than the confidence intervals for the
rice farm data, but this is not entirely clear because the general level of efficiency is

lower than it was for the rice farm data.

55
We next consider the MCB intervals. Because the panel is unbalanced, different

(1, are based on different numbers of observations, and we cannot expect the
equicorrelated structure to hold. However, we can still proceed with MCB if the product
structure of equation (1.12) holds. This structure held approximately, and so we
calculated the MCB intervals, which are given in Table 12. As was the case for the BC
results, the confidence intervals are generally narrower then those for the rice farm data
but wider then those for the utility data.

MCC and per comparison intervals for the fixed effects estimation are contained
in Tables 13 and 14, respectively. Once again, they are not very different from the

MCB intervals.

5. CONCLUSIONS

In this chapter we have shown how to construct confidence intervals for efficiency
estimates from stochastic frontier models. We have done so under a variety of
assumptions that correspond to those made to calculate the efficiency measures
themselves. For example, given distributional assumptions for statistical noise and
inefficiency, the Jondrow-Lovell-Materov-Schmidt or Battese-Coelli estimates are
typically used, and confidence intervals for these estimates are straightforward. With
panel data but without distributional assumptions, efficiency estimates are commonly
based on the fixed—effects intercepts, and confidence intervals follow from the statistical
literature on multiple comparisons with the best.

In our analysis of three panel data sets, we found confidence intervals that were

wider than we would have anticipated before this study began. The efficiency estimates

56

are more precise (and the confidence intervals are narrower) when T is large and when
0,2 is large relative to 0,2, and they are less precise when T is small and when 0,2 is
small relative to 0,2. However, frankly, in all cases that we considered the efficiency
estimates were rather imprecise. We suspect that, in many empirical analyses using
Stochastic frontier models, differences across firms in efficiency levels are statistically
insignificant, and much of what has been carefully ”explained" by empirical analysts may
be nothing more than sampling error.

This is a fairly pessimistic conclusion, though it may turn out to be overly
pessimistic when more empirical analysis is done. It is therefore important to stress that
deterministic methods like DEA are not immune from this pessimism. Efficiency
”measures” from DEA or other similar techniques are subject to the same sorts of
uncertainty as are our estimates. The only difference is that we can clearly assess the
uncertainty associated with our estimates while, at present, it is less clear how to assess
the uncertainty associated with the DEA measures. In our opinion this should continue

to be a high-priority item on the DEA research agenda.

CHAPTER THREE
ESTIMATION OF A CONDITIONAL VECTOR ERROR CORRECTION

SUBMODEL FOR COUNTERFACTUAL POLICY ANALYSIS

1. INTRODUCTION

Within the last few years there have been several analyses performed that
investigate the behavior of an economy under alternative sets of policy rules. These
exercises, called counterfactual policy analyses, typically involve estimation of a VAR
or VECM specification of an economy under a different set of monetary policy rules than
the historical rules that generated the data. See for example McCallum (1988, 1990,
1993), Judd and Motley (1992, 1993) and Bordo, Choudri and Schwartz (1994).
However, as Rasche (1995) points out "what is not acknowledged in these studies is that
the technique which is applied to the VAR (V ECM) to construct the counterfactual
analysis severely limits the admissible structure of the economic model that could have
generated the historical data".

Rasche identifies pitfalls associated with the practice of "VAR transplantation"
where one a) estimates a reduced form VAR (V ECM) in p variables, b) removes a
subset of n equations representing the policy rule(s), c) supplements the remaining p-n
equations with a new policy rule(s) and reanalyzes the system. He concludes that the
usefulness of VAR transplantation is quite limited. Specifically, "the selected policy
variable cannot enter contemporaneously into any of the equations of the economic

structure except the policy rule". This is clearly entirely too restrictive.

57

58

In a subsequent note, Rasche proposes an alternative approach to the problem that
overcomes the aforementioned "pitfall" of transplantation. He proposes structural (as
opposed to reduced form) estimation of the remaining n-p equations using a subset
limited information maximum likelihood (LIML) technique introduced by Rubin (1948)
and further developed by Hood and Koopmans (1953) and Dhrymes (1970). This chapter
is concerned with detailing this estimator which involves concentration of a multivariate
Gaussian likelihood function. Under cointegration the resulting likelihood is the product
of two generalized least-variance ratios, one of which is exactly the likelihood derived
by Johansen (1988, 1991). The resulting estimator can be thought of as a generalization
of Johansen’s (1988, 1991) reduced form estimator to the structural case, where the
analyst is only concerned with estimating a subset of the equations in the system.

This chapter is organized as follows. Section 2 derives the LIML estimator for
counterfactual policy analysis under the cointegration hypothesis and then under no

cointegration. Section 3 summarizes and concludes. WW

11 II vivr.

2. ESTIMATION

2.1 W
Consider the following error-correction representation of a structural VAR model

in p dimensions.

(3.1) AY,A + E?=,AY,_,I‘, + Y,,,B(1’ + x,e = 5, t = 1, T

59

where 5, (t = 1, ..., T) are independent p—dimensional Gaussian variables with mean zero
and variance matrix 23, and X, are s-dimensional instruments appearing only in the last
11 < p equations. The Y, are cointegrated with B, the matrix of cointegrating vectors,
and (1’, the matrix of error correction coefficients, being pxr matrices. The first q data
points Y0, ..., Y,H are considered fixed, while the parameters A, I‘,, ..., 1‘, and 2 vary
without restriction. The last n equations represent the historical policy rules, while the
first m = p - n equations are the subset of interest. What is typically done in a
transplantation analysis is to estimate the reduced form of equation (3.1) and discard the
last n equations. We use limited information maximum likelihood techniques to estimate

the first m equations in structural form.

Let Y., = [AY,_,, ..., AYM], I‘.’ = [I‘,’, ..., I‘,’], and for the moment let us

suppress B by writing Z,_, = Y,,,B. Collecting t vertically, equation (3. 1) becomes,

(3.2) AYA + Y.I‘. + z,,a’ + x9 = 5

We are interested in estimating the coefficients of the first m = p-n equations (a

submodel). To this end we partition the coefficient matrices as follows,

A = [At A11]; A1 Gum): An (Pm)

1" = [P-r Fa]; I‘°1 (mxcvi Fe (NM)

60

at, (mm), 0‘2, (rxn)

(1’ = [01, 0‘2’]

O = [0, C] 9 (SW). C (sxn)

E = [£11 £12 ] E11 (Hum), 212 ("1X“), 221 (nxm), 222 (mm)
’32. 23a:

where C is unrestricted. The analyses that follow also require the following data

matrices:

w = [Y. X]; F = [Y. x 1,]; J = [Y. z,,];
N = [AY Y. x 2,];

R" = T'r’s r, s = AY, Y., Z_,, X, F, J, N;
)3 = Rayay ' RAYi(Rii)-lRiAY i = F, J-

We use a subset LIML technique to transform equation (3.2) to allow
concentration of the coefficients of the last 11 equations, while leaving the coefficients of

the first In equations undisturbed and ensuring that the two subsystems are mutually

61

independent. This approach provides us with estimates of the structural parameters of
the m equations of interest, while purging the subsystem of the stochastic effects of the
n policy rules. This idea of subset LIML was first considered by Rubin (1948) and
Koopmans and Hood (1953). For a complete explanation of this technique, see Dhrymes
(1970). The resulting "concentrated" log-likelihood function per Dhrymes (equation

7.3.49, p335) is given by:
(3.3) (-2/T)lnL = c0 + 1n | AF | + 1n | 2,, | - In | A,’AFA, | + trL‘,‘, M’RNNM
where M’ = [A,’ T.,’ 0’ (1,]. To concentrate I‘., and u,, we examine the last term of

equation (3.3). Specifically,

M’RNNM = [A,’ I‘.,’ 0’ (1,]’(N’N/T)[A,’ I‘.,’ 0’ (1,]’

= [A1, 1"], at], [RAYAY Ravi ] [A], I"I, 0111’.

JAY R 11

Let 5’ = [r.,' 51,], then

62

Substituting equation (3.4) into (3.3) and taking the derivative w.r.t 6

alnL/ab = -(T/2)[2R,AYA, + 2R,,5]E,‘, = 0

implying

(3-5) 5 = '(RJJ)-lRJAYAl°

Substituting equation (3.5) into (3.4),

(3-6) M’RNNM = AI’IRAYAY " RAYJ(RJJ).1RJAY]AI = A1,)‘JAI

Substituting equation (3.6) into (3.3) and taking the derivative w.r.t 2,,

(3an62“ = ‘(T/2)1'2111 '1' 271111°*1’>\JAIE111] = 0,

implying

(3.7) 2,, = A,’A’A, .

63
Substituting equations (3.6) and (3.7) into (3.3),

(-2/T)lnL = c, + 1n I AF | + In | A,’A’A, | - In | A,’AFA, |

and suppressing c,

(3.3) L“(A0 = | Al,>‘JAl | I AF I

 

l At’AFAt |

If B is known a priori, then equation (3.8) represents the final likelihood function.
However, its form precludes a first derivative solution for A,. Fortunately, the function
is a generalized least variance ratio, solvable using the usual canonical techniques. We
defer its solution until latter. It is important to notice that the likelihood function in
equation (3.8) is only attainable because of the restrictions on 9; without them AI is not
identified. Now, in general, B is unknown so we make it explicit in our definition of A”.

First, let F = [W Y_,B], so,

RFF = (UT) W’W W’Ydfi
B’Y.I,w fi’Y.l,Y-lB
Let
Rial: = R2, n.3,
R211? R2525

Then

64
FR,’,',, F’ = WR,‘,,'=W’ + WR,‘,,22 B’Y_,’ + BY,,R,2,,‘, ’ + BY_,R,§,2,Y_,’B’

Defining the projection matrix:
P, = i(i’i)"i’; i = W, J, F
and the sum of squared error matrix:
S,}=(1/T)i’[l- P, '; r = W, J, F; i,j = AY, Y,,, X

and using the rules of partitioned inverse,

FR,:',. F’

= (l/Dipw + PWY-IBIB,SYYlY-l31-16,Y-1,PW

" YdBm’SY‘YlY-IBTIB’Y-NPW ' PWY-IBIB,SYYlY-IBIJB’Y-l’

+ Y-1Bm’SYYlY-IB]-lB’Y—l,}

(3-9) PREP F’ = (1/T){Pw + (I - Pw)Y-tB[B’S¥Yw-tfil"B’Y-i’(1 - Pw)}

We can write AP = AY’[I/T - FR,;,§ F’]AY. Substituting equation (3.9) into this

expression,

(3-10) ”(3) = S‘AVYAY- SXVYY-l Blfi’Sl’m 31-13 ’SYI-rav

Using results for the determinant of a partitioned matrix, equation (3.10) yields:

65

 

(3'11) 1 xF(B) I = I S‘Z'YAYI I B,{SYYlY-l - SYv-IAY [S‘AVYAY -ISXVYY-l I6 I
I B’SI‘SHB I

Substituting equation (3.11) into (3.8) and mah'ng B explicit in A’,

(3.12) L'2"(An B) = I S‘AVYAYI I At’A’(3)AtI I B’lsxtyu- SYv-lAY[S\XYA "SXVWJB I

I At’AF(B)At I I B’Si‘fm B I

 

Equation (3.12) is the product of two generalized least variance ratios: one in A,
and B and another in B only. The ratio in B only is the ratio derived by Johansen (1991).
So this problem can be thought of as a generalization of the Johansen likelihood. Indeed,
if we exclude instruments, X, from the specification, AF = A’, A, is not identified and
the problem reduces to the Johansen result. This raises the question of how many
instruments are required for identification. We defer the answer to this question until

later. Notice that

x? = (1/T)[AY’AY - AY’F(F’F)F’AY]
x" = (l/T)AY’[I - F(F’F)F’]AY

F_ F
A —SAYAY

and by similar logic A’ = S’mw

From the rules of projection matrices

66

1 - P1: = 1 - Pl - (1 - Pi)X[X’(1 - P1)X]"X’(I - PI)

Premultiplying by AY’IT and postmultiplying by AY

N7 = )‘J ’ SAIYXISkxIleAY

Let (1’ = SA’YXISXXI-ISXAY

then, making B explicit.

(3-13) 13(6) = NIB) + <b(B)

Substituting (3.13) into equation (3.12) and suppressing the | S‘XYAY | term,

(3- 14) L'2"(An 13) =

I At’IX’IB) + ¢(B)}At I I B’ISQYm- 33m [S‘Xyay ]'1 SAVYY-lIB I

I A,’AF(3)A, I I H’SYYIY-rfl I

 

= 1(An B)g(B)

We now have symmetry in the two variance ratios and would like to minimize this

function, so the problem can be thought of as two eigenvalue problems. The double

67

eigenvalue problem of equation (3.14) can be transformed to a single block diagonal

eigenvalue problem, which can generate a block diagonal solution. Define:

G =
0 B
_ >3:03) '1' 4K5) O
0 $331“ ' Syrav [S‘XYAY I'ISAVYY-r
A1203) 0
B = w
0 SY-lY-l

then equation (3.14) can be written as:

(3-15) L‘2”(An B) = I (”33 I I 6’30 I "

We now discuss solutions to equations (3.14) and (3.15) using the LIML estimator and
a 2-step estimator. LIML estimation of the parameters B and A, involves minimization
of equation (3.14). Equivalently, it involves minimization of equation (3.15) subject to
the block diagonality of G. The 2-step estimator involves i) estimating B by minimizing
g(B) into equation (3.14), then ii) using this estimate to minimize f(A,, B) with respect

to A,. Under certain conditions these estimators are equivalent.

68
2.1.1 LIML Estimation

Ideally we would solve equation (3.15) for G using the usual multivariate techniques
from the theory of partial canonical correlations and reduced rank regression (see
Anderson (1951) and Tso (1981)). This is equivalent to solving for A, and B
simultaneously in equation (3.14). Since matrices D and B are functions of B, the usual
techniques to produce a solution are not applicable; some numerical optimization method
must be utilized to obtain a solution. Another complication is there are restrictions on
G and, ultimately, on any estimate, G. Specifically, the upper-right and lower-left blocks
of the matrix G are zero. This, we shall see, causes the value of the likelihood in
equation (3.15), L°Z’T(A,, B), to be larger than if G were allowed to vary without
restrictions.

For now assume B and D are not fuctions of B. We can use Theorem 1 of
Anderson (1951) to solve equation (3.15), yielding an estimate of the space spanned by
G, G = [0,, ..., 9,, (1,,“, ..., 9,”, ], where O = [9,, ..., 02,] are the eigenvectors of the

equation

(3.16) IaB-Dl =0.

normed by O’BO = land ordered0 < (‘1, < < 6, < (1,,, < < 6,, < (1,,, S

. s “p,,, 3 6pm,, _<. s (‘12,. Upon substituting matrices for B and D, equation

(3.16) becomes

69

((7-1)AF - 0 O

O (O'DSYYlY-l + SIXIAY [SXYAY -rsszJ
Multiplying by Det [ 01, It: ] = (4)1 ¢ 0

yields

((rr-l)AF - d) O
(3.17) Det O = 0

(1'U)S)‘X1Y-1 ' SYYIAY [SYYAY -lSXYY-l

To minimize equation (3.15), subject to the restriction that G be block diagonal, we
select the two sets of eigenvalues ranked 6, < < (1, and (1,,, s S (1,,,“ and use
their corresponding eigenvectors to form G. Selecting the eigenvalues in this way is
necessary to ensure satisfaction of the block diagonal restrictions on G, while ensuring
a solution to equation (3.16) and minimization of the likelihood in equation (3.15). We
now justify the selection of these specific eigenvalues.

For any 0,, which is a solution to equation (3. 17),
(3. 18) D9, = 0,139,, or

(3.19) (0,13 - mi, = 0.

70

Partition the eigenvectors so 9,’ = [95,, I 9,1,, ], and substituting for B and D, equation

(3.19) becomes

[ta-1W - «119..
(3.20) = O

[(W'DSY‘Iw-r '1' S‘YN-lAY [S‘AVYAY -ISXIYY-l 191.11

Premultiply by l: 01,, 1,), :|

to get

[(0,-1)AP ' “01,1

[(1‘01)SYY1Y-1 ' 3311M [SYYAY -ISXVYY-l 191,11

This is solved when 0, satisfies:
(3.21) (a) I («r-1N - (P I = 0 on
(b) I (1-0)S¢Yw-t- SXtayls‘Xyayl" SAVYY-II = 0
Assume AF positive definite and 4: positive semi-definite; then from equation (3.21a) it

is true that ((1,-1) 2 0 (Dhrymes (1978), Proposition 62), implying a, 2 1. There will

be p of these eigenvalues. Also from equation (3.21b), it is clear that if

71

SIX,“ [S‘XYAY ]" Sr”, and 8X,“ are positive definite, then (1-0,) > 0, implying (t, < 1.
There will be p of these eigenvalues. So, any single 0, cannot simultaneously satisfy both
parts of equation (3.21) (i.e. a, 2 l and or, < 1). So, the p eigenvalues that solve
equation (3.21a) are distinct from those p eigenvalues that solve equation (3.21b).
Additionally, it is important to point out that the p eigenvalues that solve equation (3.21a)
will have p eigenvectors of the form 9,’ = [9,’,, I 0’ ], while the p eigenvalues that solve
equation (3.21b) will have p eigenvectors of the form 9,’ = [0’ I 9,’,, ]. This is due to
the block diagonal form of matricies B and D.

Now, we must select a, and the corresponding 9,, to ensure that the likelihood of
equation (3.15) is minimized while ensuring that the block diagonal restrictions on G are
not violated. Theorem 1 of Anderson (1951) shows that to minimize equation (3.15)
where the estimate of G is allowed to vary unrestricted, we select the m+r smallest
eigenvalues and let their corresponding eigenvectors form our estimate of G. However,
G is restricted block diagonal so some modification of this procedure is necessary.
Consider four sets of eigenvalues. Consider 6', < . .. < 6,, the smallest eigenvalues,
which are constrained between 0 and 1 by construction and which solve equation (3.21b).

To maximize the likelihood we select these smallest eigenvalues and their eigenvectors

which are of the form 9,’ = [0’ I 95,], for all i = 1, ..., 1. These r eigenvectors form
the basis for the B estimate. Now, consider (1,,, < < (1,,, the second smallest

eigenvalues, which are also constrained between 0 and l. Ideally, we would select these
next largest eigenvalues and their eigenvectors to minimize equation (3.15), but these

eigenvectors will not form a basis for an estimate of A,, since they are of the form

72

9,’ = [0’ I 9,’_,,], for all i = r+1, ..., p. The form precludes an estimate of A,, because
of the position of the 0. Hence, none of these eigenvalues suffice. Consider (1,,“ s
5 (rpm, the next m largest eigenvalues. We select these eigenvalues and associated
eigenvectors which are of the form 9,’ = [9,1, I 0’ ], for all i = p+l, ..., p+m, forming

a basis for A,. The last set of eigenvalues (5pm,, 5 s (‘72,) is irrelevant.

This procedure produces a likelihood function

(3-22) [4-2/T0", m, P) = IIi=r1 at 11,13,321 55-

Equation (3.22) illustrates the effect of the block diagonal restriction. We can think of
the ordering of the eigenvalues as four ranked sets. The set of lowest ordered
eigenvalues (it, to (1,) has I members. The set of second lowest eigenvalues ((1,,, to (1,,)
has p—r members. The set of third lowest eigenvalues ((1,,, to (1,,“) has In members.
The set of largest eigenvalues (&,+m+, to (1,,) has 11 members. Selecting restricted G is
equivalent to selecting the set of smallest eigenvalues and the set of third smallest
eigenvalues. Examining the likelihood of equation (3.22) we see that its value is the
product of the eigenvalues of these selected sets. The first set of smallest eigenvalues
is optimal in the sense that they minimize L'2”(r, m, p), but the set of third smallest
eigenvalues is a deviation from the usual cononical techniques and reflects the effect of
the block diagonal restriction on G.

The assumption of positive semi-defmiteness for the matrix (b will be made clear.

Consider the rank of 0. When rank(¢) = 0, there are p eigenvalues equal to 1, and they

73

represent the p largest eigenvalues. This causes the selection of the set of third smallest
eigenvalues (and their corresponding eigenvectors) to be ambiguous, so G is not uniquely
defined. When 0 < rank(¢) < 11, there are more than m eigenvalues equal to 1, and
again G is not uniquely defined. When rank(¢) = 11, there are exactly 111 eigenvalues
equal to l and G is unique. When rank(d>) > 11, there are less than m eigenvalues equal
to 1, and G is unique.

Now relax the assumption that B and D are not functions of B. Consider the case
where, rank(¢(B)) = 11. Not only does this produce a unique G, but all the eigenvalues
comprising the A, portion of G will be unity and, hence, independent of B. The
corresponding vectors of the A, portion of G will, in general, not be independent of B.
However, a two-stage estimate consisting of first estimating B in equation (3.21b), then
using this estimate in equation (3.21a) to solve for an estimate of A, is equivalent to
LIML when rank(4>(B)) = n. This can be shown by examining equation (3.20). To
minimize equation (3.15) while observing the restrictions on G, we select the set of first
smallest eigenvalues and the set of third smallest eigenvalues. As we have seen, the two
sets of eigenvalues have no common elements. In this sense their selections are
independent. This independence causes the problem to be separable, insofar as equation
(3.20) can be solved by independently solving the two characteristic equations (3.21).
The only complication is the fact that AF and (b are functions of B. So, finding the
eigenvectors of equation (3.20) reduces to finding the eigenvectors of equation (3.21),
namely 9,,, i = p+1, ..., p+m and 9,.,,, i = 1, ..., r. Suppose we are interested in the
estimate of B only and solve equation (3.21b) to get the set of smallest eigenvalues and

9,,“, i = 1, ..., r. This selection is optimal in the sense that these eigenvalues ensure that

74

the likelihood is maximized. Now suppose we use this B estimate to solve equation
(3.21a), and based on this estimate, rank(d>) = n. Then the set of third smallest
eigenvalues will be optimal in the sense that these eigenvalues ensure that the likelihood
is maximized, because they all equal their lower bound of 1. Thus, both sets of
eigenvalues are optimal. The estimate of A, consists of the same 9,,, i = p+1, ...,
p+m, as before. Thus, when rank(¢(B)) = n, this two stage procedure produces a
maximized likelihood function, which is equal to the likelihood of the LIML estimate.
The equality of likelihoods of this two-stage estimate and the LIML estimate also
occurs when rank(¢>(B)) = 0 and when rank(4>(B)) < 11, but in these cases G is not
uniquely defined in the sense that the selection of the likelihood-maximizing eigenvalues
is ambiguous. When rank(¢(B)) > n, G is unique, but the selection of the estimate of
A, may not be optimal. Therefore, the case when rank(¢(B)) = n is of particular
interest. Specifically, when rank(¢(B)) = n, the LIML estimator of equation (3.15) is

equivalent to a two-stage estimator described in the following section.

2.1.2 A ngtstago Estimator

We investigate an estimator based on a likelihood function conditional on r in two
steps when rank(4>(B)) = n, as follows. First minimize the variance ratio, g(B), using
the usual Johansen (1991) technique. This yields 8, = [9,, 9,], where v = [0,,

1),] are the eigenvectors of the equation

I 983(qu ' 8311M [S‘XYAY 1-1 SAVYY-l I = 0,

75
normed by V’sijt = land ordered 1 > p, > > a, > 0. Then the likelihood

is maximized by the usual eigenvalue product, and equation (3.14) becomes

(3.23) L-“(Anrlfin = IAI’{>\F(3r) + «Brim 11;, (1 to

1 Al,xP(Br)Al I

 

= f(AI’ Br)g(8r)

This procedure is that of J ohansen, and (under additional restrictions) produces consistent
estimates of the eigenvalues p,, the dimension of the cointegrating space, sp(B), and the
estimates of (1 and 22. As proved in Johansen (1988), the restrictions necessary for
consistency are: 1) Y, is integrated of order 1, and 2) a and B are full column rank.

That is, under these restrictions,

plim(,0, , ..., (3,) = p,, ..., p, and plim(sp(B,)) = sp(B).

Additionally, Hargreaves ( 1994) points out that in the probability limit all the r

eigenvalues of the cointegrating space are distinct.

In a fashion analogous to Johansen (1991) we can find A, = [1,2 ..., 1;], where
U = [’r,', ..., if] are the eigenvectors of the equation

I (”XIXBI) - ¢(Br) I = 0

76

normed by U’AF(B,)U = land ordered 0;, 2 ...2 a,,; > a; = u,,: =, a; = 0.
Notice that the eigenvalues and eigenvectors are functions of r, the number of
cointegrating vectors. This reflects the fact that the likelihood value is conditional on B,.
Also the order statistic for (it; is the reverse of that for B, (at; is the largest eigenvalue
while 5,, is the smallest), because AF(B,) and ¢(B,) are summed in f(A,, 9,), while the

analogous matrices are differenced in g(B). The maximized likelihood becomes

(3.24) EMU“, 1') = 531(1 ‘1' 111;) 11:1(1 ' 99

When rank((l>(B,)) = n, the m smallest eigenvalues selected, ‘,,', , , (21;, all equal zero
for all values of I, because ¢(B,) is not full-rank. This particular rank condition reduces
the problem to that of Johansen insofar as the likelihood function of equation (3.24) has

11,13, (1 + (21;) = 1 and becomes:

L'2"(m, 1') = 111;“ ' bi)

Since this two—stage procedure is equivalent to the LIML estimator when
rank(4>(B)) = n, we would like to derive the asymptotic properties of the estimates when
this condition holds. As previously stated, under certain conditions sp(¢(B,)) is
consistent. We claim that the space spanned by A, is also consistent. In lieu of a proof,
we present the following argument for why this may be. Johansen (1988) shows the T-
consistency of sp((b(B,)) in the presence of short-run dynamics (Y.I‘.). It is reasonable

to believe that if the instruments, X,, are stationary, then the same result of T-

77

consistency of sp(¢(B,)) will hold in this analysis. Let us take this fact as given. Since
Johansen Shows this in the presence of short-run dynamics it is reasonable to believe that
an estimate of a short-run dynamic parameter, A,, would also be consistent when it is
based on a T-consistent estimate of an unknown long-run dynamic parameter, B. This
claim is supported by the fact that the usual LIML estimate of a structural parameter, say
A,, is T”—consistent (Dhrymes (1970), p335), while sp(¢(B,)) (the estimate upon which
A, estimate is based) is T-consistent. This claim is further supported by the fact that in
Johansen ( 1988) the T-consistent estimate of sp(d>(B,)) yields consistent estimates of
reduced form parameters in that particular model. Formal proof of this claim will be

undertaken in future research.

2.2 MW

If there are no cointegrating relationships, then the problem reduces to subset
LIML, and the likelihood function is that of equation (3.3). If the same restrictions on
the 9 hold then the likelihood can be concentrated to the generalized least variance ratio
of equation (3.8) but with B suppressed. Suppressing the | AF | term, this ratio

becomes

(3.25) L-“(Ao = IAr’w + PM, I

 

1 Al, XI:AI 1

which can be solved as described above. The generalized least variance ratio of equation

(3.25) was first considered by Rubin (1948) and Koopmans and Hood (1953), but to our

78

knowledge has never been solved using canonical correlations. If we have enough

instruments then, the maximized value of the likelihood is

(3.26) L'2”(m) = 11,3, (1 + 5,).

Notice that the (3:, are no longer a function of r. In the case where m = 1, this
problem reduces to single equation LIML which is identical to two-stage least squares
(ZSLS) under exact identification. So, an equally valid approach to this estimation when
m > 1 would be to estimate each of m single equations separately using 2815. Then
a meaningful question would be whether there any efficiency gains to the methods
detailed herein over the single equation approach. We plan to explore this question in

future research.

3. CONCLUSIONS

The structural estimates derived in the preceding section for B and A, can be used
to derive estimates for the other structural parameters of the submodel of interest by
substituting them back through the derivation. The resulting estimates are unique in that
they permit counterfactual policy analysis without the structural limitations presented by
Rasche (1995). However, this has come at a cost. Since the estimates are based on
limited information (i.e. we have concentrated out the parameters of the last n equations),
they are inefficient, but this is the price that is paid to ensure the mutual independence

of the two subsystems.

79

We present a subset LIML estimator which is amenable to a numerical
optimization technique such as Newton—Raphson. An obvious starting value for B in such
an approach would be the estimate produced from the J ohansen analysis of the reduced
form of equation (3.1). If rank(¢(B)) equals 11, the number of policy equations, then the
LIML solution is equal to the two-Stage estimator of section 2.1.2. If rank(¢(B)) > n,
the two estimators are, in general, not equal.

The two-stage estimate involves performance of a modified Johansen (1988)
technique involving the addition of stationary instruments in the first stage. This
produces a consistent estimate of the matrix of cointegrating vectors, B. In the second
stage A, is estimated using similar canonical methods. Another way to perform this
analysis would be to perform single equation LIML on each of the m equations in the
subset, given a first stage estimate of B. However we suspect that this would be less
efficient than the subset LIML estimate derived here.

The preceding analysis represents an initial attempt to solve the problems inherent
in counterfactual policy analysis. It does not profess to be comprehensive, but provides
the fodder for additional research. While there are many unanswered questions, a few
merit brief consideration here. First, equation (3.22) suggests likelihood ratio testing of
the number of cointegrating vectors that would be similar in structure to that of Johansen.
We would expect that such tests in this context to yield results consistent with the
reduced form tests of Johansen. If these tests are not consistent, what does this imply
about the tests and for the overall modelling approach? This, of course, remains to be

seen. Second, will the estimator of B produced in the numerical analysis of the LIML

 

80

estimator be similar in magnitude and dimension to that of Johansen’s reduced form

estimate? If they are not, what can be inferred? Clearly there is much left to be done.

on let
mode
with
natur

best

diffe
quar
we 1
prec
rela‘
wer
fror
and
mo

tol

wit
esti

int:

CONCLUSIONS TO THE DISSERTATION

The primary goal of this dissertation has been to develop a method for inference
on technical efficiency estimates in the fixed effect formulation of the stochastic frontier
model. We find that under certain conditions the methodology of multiple comparisons
with the best is a useful tool in performing this inference. Additionally, the simultaneous
nature of the resulting intervals provides us with the added benefit of inference on the
best firm in the sample; a benefit not found in existing interval construction techniques.

Secondly, our comprehensive empirical analysis attempted to disentangle the
different sources of uncertainty in efficiency estimates and to provide us with a
quantification of this uncertainty. We found confidence intervals that were wider than
we would have anticipated before this study began. The efficiency estimates are more
precise (and the confidence intervals are narrower) when T is large and when 0,,2 is large
relative to (1,2. However, frankly, in all cases that we considered the efficiency estimates
were rather imprecise. We suspect that, in many empirical analyses using stochastic
frontier models, differences across firms in efficiency levels are statistically insignificant,
and much of what has been carefully "explained" by empirical analysts may be nothing
more than sampling error. This is a fairly pessimistic conclusion, though it may turn out
to be overly pessimistic when more empirical analysis is done.

Lastly, we provide an initial attempt at solving the counterfactual policy "pitfall"
with a subset LIML estimator. What comes out of the analysis are a seemingly useful
estimator and several questions about the characteristics of this estimator. It is our

intention to address these questions in subsequent research.

81

 

APPENDIX

APPENDIX

TABLE 1

Rice Farms - Cross Sectional Estimation Results

 

Variable

Period 1 COLS Period 1 MLE

5.8483

5.9540

Period 6 COLS

5.3327

Period 6 MLE

 

0.0572

0.0583

0.0983

 

0. 1036

0. 1028

0.2073

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[ TSP 0.0033 0.0034 0.0674 0.0693
lbbor 0.1970 0.1970 0.2122 0.2007
Land 0.6372 0.6374 0.5194 0.5252
II DP 0.0143 0.0138 -0. 1480 -0. 1453
DVl -0.0857 -0.0861 0.0639 0.0568
DV2 0.0853 0.0853 0.1192 0.1175
DRl 0.2192 0.2173 -0.2642 -0.2695
DR2 -0.0325 -0.0330 -0.3744 -0.3712
DR3 0.1388 0.1385 -0.3815 -0.3712
DR4 0.0814 0.0817 -0.0613 -0.0591
DR5 0.1810 0.1829 -0.2864 -0.2668
02,, 0.0185 0.0174 0.0253 0.0462
62,. 0.0632 0.0579 0.0705 0.0564
E(u,) 0.1084 0.1053 0.1268 0.1714
0. 0.1195 0.1157 0.1364 0.1593

 

The constant term reported for COLS is before correction by E(u,).

APPENDIX

TABLE 2A

Rice Farms - Confidence Intervals Based on JLMS Method, COLS Estimates - Period 1

 

 

1m Efficiency I 95% Lbnd 95 % Ubnd 90% Lbnd 90% Ubnd I 75 % Lbnd 75 96 Ubnd
. ———4——— ——+———:——,—————iI

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

164 0.9421 ' 0.8242 0.9981 I 0.8482 0.9962 - 0.8843 0.9902
1 118 0.9390 " 0.8173 0.9980 , 0.8417 0.9959 0.8786 0.9895 II
I 108 0.9380 I 0.8151 0.9979 0.8396 0.9958 0.8768 0.9892 II
92 0.9029 l 0.7514 0.9956 0.7777 0.9912 0.8196 0.9782
87 0.9027 0.7511 0.9955 I 0.7774 0.9911 0.8193 0.9781
I 61 0.9026 ' 0.7511 0.9955 : 0.7774 0.9911 0.8193 0.9781
I 79 0.8479 I 0.6811 0.9873 I 0.7067 0.9760 0.7484 0.9479
: 16 0.8441 I 0.6769 0.9863 ' 0.7024 0.9744 [I 0.7440 0.9450
'Ll 0.8066 ’ 0 _ 0.9727 I 0.6642 _ 0.9531 , 0.7042 0.9129
TABLE 28

Rice Farms - Confidence Intervals Based on JLMS Method, MLE Estimates - Period 1

 

Firm Efficiency

    
 

I

95% Lbnd 95% Ubnd

   

90% Lbnd 90% Ubnd

75% Lbnd 75% Ubnd

 

 

 

 

 

 

 

 

 

 

 

 

 

83

 

 

0.9452 0.8322 0.9982 ! 0.8553 0.9964 0.8901 0.9908
118 0.9421 ’ 0.8253 0.9981 0.8489 0.9961 0.8845 0.9901
108 0.9411 0.8232 0.9980 0.8469 0.9960 0.8827 0.9899
19 0.9054 0.7578 0.9957 0.7835 0.9914 0.8243 0.9788
61 0.9053 0.7576 0.9957 0.7832 0.9914 I 0.8241 0.9787
87 0.9052 0.7575 0.9957 0.7832 0.9914 0.8240 0.9787
79 0.8481 0.6849 0.9866 0.7099 0.9750 II 0.7506 0.9465
0.8437 II 0.6803 0.9855 0.7051 0.9730 II 0.7457 0.9431
0. 8040 II 0.6415 0.9694 I 0.6652 0.9487 II 0.7041 0.9078

 

APPENDIX

TABLE 3

Rice Palms - Panel Data Estimation Results

 

Fixed Effects

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I Seed 0.1358 0.1208 0.1327 0.1332
Urea 0.1196 0.0918 0.1133 0.1127 II
TSP 0.0718 0.0892 0.0761 0.0769 I
Labor 0.2167 0.2431 0.2230 0.2194
Land 0.4819 0.4521 0.4771 0.4810
DP 0.0077 0.0338 0.0140 0.0093 II
DV1 0.1755 0.1788 0.1772 0.1767 II
DV2 0.1356 0.1754 0.1444 0.1410 ll
DSS 0.0489 0.0533 0.0492 0.0492
DRl 0.0500 0.0511 0.0594
DR2 0.0393 0.0441 0.0480
DR3 0.0623 0.0723 0.0799
DR4 0.0248 0.0119 0.0150
DR5 0.0818 0.0751 0.0826
01.. 0.0214 0.0215
02. 0.1076 0.1076 0.1070
E(uo 0.1166 0.1170
6. 0.0987 0.0987

 

 

 

 

 

 

 

 

84

APPENDIX

TABLE 4A

RiceFarms-ConfidencelntervalsBasedonBCMethod, CGLSEstimates-PanelData

 

  

75% Ubnd

 

 

 
   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IMMI 95% Lbnd 95% Ubnd 90% Lbnd 90% Ubnd I 75% Lbnd

164

118 0.9642

5 0.9608 0.8744 0.9988 0.8930 0.9976 0.9204 0.9938

51 0.9002 0.7628 0.9944 1 0.7857 0.9890 I 0.8224 0.9738

38 0.9002 0.7628 0.9944 0.7857 0.9890 “ 0.8224 0.9738

88 0.8999 0.7624 0.9943 0.7852 0.9889 0.8219 0.9737

142 0.76601 0.6287 0.9215 0.6485 0.8951 0.6810 0.8537

145 0.7615 “ 0.6249 0.9167 0.6447 0.8901 0.6769 0.8488

143 0.74224 0.6089 0.8951 1 0.6281 0.8683 ! 0.6595 0.8274 I
TABLE 48

Rice Farms - Confidence Intervals Based on BC Method, MLE Estimates - Panel Data

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
75% Ubnd

164 0.9643 0.8837 0.9990 0.9013 0.9979 0.9270 0.9945 I
118 0.9639 0.8825 0.9989 0.9005 0.9978 0.9263 0.9944
5 0.9609 0.8746 0.9988 0.8931 0.9976 0.9206 0.9938
88 0.9010 0.7638 0.9945 0.7867 0.9892 0.8234 0.9742
51 0.9010 0.7638 0.9945 0.7867 0.9892 0.8234 0.9742
102 0.9001 0.7626 0.9944 0.7855 0.9890 0.8222 0.9738
142 0.7602 0.6238 0.9153 0.6435 0.8887 0.6757 0.8473
145 0.7564 0.6207 0.9111 0.6403 0.8845 0.6723 0.8432
143 0.7370 0.6045 0.8891 0.6236 0.8624 0.6549 0.8217

 

 

85

APPENDIX

TABLE 5A

Rice Farms - MCB Confidence Intervals - Panel Data

 

In%wm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

75% Ubnd

163 5.4838 0.4868 1.0000 0.5094 1.0000 0.5568 1.0000

166 4.9667 0.2902 1.0000 0.3037 1.0000 0.3320 1.0000

15 4.9656 0.2899 1.0000 0.3034 1.0000 0.3316 1.0000

40 4.9646 0.2896 1.0000 0.3031 1.0000 0.3313 1.0000

143 4.5982 0.2008 1.0000 0.2101 1.0000 0.2297 1.0000

117 4.5859 0.1983 1.0000 I 0.2075 1.0000 0.2269 1.0000

45 4.5496 0.1913 1.0000 J 0.2001 1.0000 10.2188 1.0000

TABLE 53
Rice Fanns - MCC Confidence Intervals - Farm 164 as Control - Panel Data

Firm 6,. 95% Lbnd 95% Ubnd 90% Lbnd 90% Ubnd 75% Lbnd 75% Ubnd
164 5.5561 I
118 5.4860 0.4878 1.0000 0.5105 1.0000 0.5580 1.0000 II
163 5.4838 0.4868 1.0000 0.5094 1.0000 0.5568 1.0000

166 4.9667 0.2902 1.0000 0.3037 1.0000 I. 0.3320 0.9266

15 4.9656 0.2899 1.0000 0.3034 1.0000 0.3316 0.9256

40 4.9646 I 0.2896 1.0000 0.3031 1.0000 0.3313 0.9247

143 4.5982 I 0.2008 0.7332 0.2101 0.7007 0.2297 0.6410

117 4.5859 I 0.1983 0.7243 0.2075 0.6921 0.2269 0.6332

45 4M% I0wm 0w” L.02001 0w” L.02188 0am I

 

 

 

 

 

 

86

IUTENDD(

TABLE 6A

Rice Fanns - Subset MCB Confidence Intervals, N = 9 - Panel Data

95 96 Lbnd

 

95% Ubnd

  

 

     

90% Ubnd

   

75% Lbnd

 

75 96 Ubnd

 

 

 

 

    

   
  
  

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rice Farms - Per Comparison Confidence Intervals with a,“ - Panel Data

0.6494 1.0000 0.6835
118 5.4860 0.5644 1.0000 0.5940
163 5 .4838 0.5632 1.0000 0.5927 1.0000 0.6418 1.0000
166 4.9667 0.3358 0.9849 0.3534 0.9358 0.3827 0.8642
15 4.9656 0.3354 0.9837 0.3530 0.9347 ILO.3822 0.8632
40 4.9646 0.3351 0.9828 0.3527 0.9338 0.3819 0.8624
143 4.5982 0.2323 0.6813 0.2445 0.6473 0.2647 0.5978
117 4.5859 0.2296 0.6730 0.2415 0.6394 0.2615 0.5905 II
45 4.5496 0.2213 0.6490 I 0.2329 0.6166 1. 0.2522 0.5695 II
TABLE 6B

 

  

95 % Lbnd

 

95% Ubnd

  

90% Ubnd

 

75% Lbnd

  

  

75 % Ubnd

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

118 0.1912 0.6409 1.0000 0.6807 1.0000 II 0.7483 1.0000
163 0.1898 0.6413 1.0000 0.6808 1.0000 II 0.7479 1.0000
166 0.1914 0.3812 1.0000 0.4049 0.7599 0.4451 0.6912
15 0.1940 0.3788 0.8102 0.4027 0.7622 0.4432 0.6924
40 0.1971 0.3761 0.8144 0.4002 0.7654 0.4412 0.6943
143 0.1915 0.2636 0.5584 0.2800 0.5257 0.3079 0.4782
117 0.1960 0.2581 0.5565 0.2745 0.5232 0.3025 0.4748
45 0.1941 0.2499 0.5347 0.2656 0.5030 0.2924 0.4569

 

 

87

Texas Utilities - Panel Data Estimation Results

   

 

APPENDIX

TABLE 7

 

    

 

   

 

 

 

 

 

 

 

 

 

 

88

 

 

I__
I -5.0532
0.0775
0.3462 0.6275 0.5882 0.5856

0.6406 0.5652 0.5807 0.5838 II

61., 0.0079 0.0266 I
02‘, 0.0029 0.0029 0.0029
E(ui) 0.0709 0.1301
a. 0.0126 0.0126

 

APPENDIX

TABLE 8A

Texas Utilities - Confidence Intervals Based on BC Method, CGLS Estimates - Panel Data

 

i
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I Firm TE I 95% Lbnd 95% Ubnd 90% Lbnd 90% Ubnd I 75% Lbnd 75% Ubnd I
_____._. __ _.__ I ___-__. . .__,_I
5 0.9982 I 0.9938 1.0000 0.9949 0.9999 I 0.9964 0.9998 I
3 0.9960 . 0.9866 0.9999 I 0.9887 0.9999 I 0.9919 0.9994 I
10 0.9649 0.9413 0.9885 0.9451 0.9848 I 0.9510 0.9788 I
[ 1 0.9325 0.9097 0.9557 0.9133 0.9519 0.9190 0.9460 I
8 0.9167 . 0.8943 0.9396 3 0.8979 0.9358 0.9035 0.9300 I
9 0.8997 I 0.8777 0.9221 I 0.8812 0.9184 I 0.8867 0.9127 I
2 0.8973 '. 0.8754 0.9197 I 0.8788 0.9160 I 0.8843 0.9103 I
I 6 0.8835 I 0.8619 0.9055 0.8653 0.9019 I 0.8707 0.8963 I
I 7 0.8788 0.8573 0.9006 0.8607 0.8971 I 0.8660 0.8915 I
. _ °-8_ - _ __°-___ -_ 8881- __ _88 II
TABLE 8B

Texas Utilities - Confidence Intervals Based on BC Method, MLE Estimates - Panel Data

 

 

 

 

 

 

 

 

 

Mflt' --I
I 5 0. 9880 I 0.9685 0. 9994 I 0.9721 0.9989 I 0.9776 0.9973 ”I
I 3 0.9793 I 0.9566 0.9979 1 0.9604 0.9962 0.9663 0.9923
I 10 0.9095 I 0.8872 0.9322 0.8907 0.9285 0.8963 0.9227
I 1 0.8649 I 0.8437 0.8865 0.8471 0.8830 II 0.8524 0.8775
I 8 0.8472 I 0.8264 0.8683 | 0.8297 0.8649 II 0.8349 0.8595
I 2 0.8322 I 0.8118 0.8530 0.8151 0.8496 0.8202 0.8443
I 9 0.8269 I 0.8066 0.8475 II 0.8098 0.8442 0.8149 0.8389
II 6 0.8214 0.8013 0.8419 0.8045 0.8386 0.8095 0.8334
7 0.8181 0.7981 0.8385 0.8012 0.8352 0.8062 0.8300
II4 0.7873 II 0.7680 0.8069 II 0.7710 0.8037 n 0.7759 0.7987

 

 

 

 

 

 

89

APPENDIX

TABLE 9A

Texas Utilities - MCB & MCC Confidence Intervals - Pand Data

_7

 

T

v

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.. _fi __ _9_ __ UU’___ _9°_5__.I.__~“____5%___d
5 49952 I 1.0000 1.0000 ; 1.0000 1.0000 I 1.0000 1.0000
3 -5.0826 I 0.3730 0.9617 0.3772 0.9571 {I 0.3337 0.9500

. 10 -s.1451 'I 0.3201 0.9035 0.3241 0.3991 I 0.3302 0.3925
1 -5.1760 0.7951 0.3759 0.7990 0.3717 'I 0.3049 0.8653
3 51940 I 0.7309 0.3603 0.7347 0.3562 0.7905 0.3499
9 52105 I 0.7631 0.3462 0.7719 0.3421 I 0.7776 0.3359

I 2 -5.2176 I 0.7627 0.3402 0.7664 0.3362 0.7721 0.3300
7 -5.2362 I 0.7437 0.3243 0.7523 0.3203 I 0.7579 0.3143
6 -5.2366 I 0.7434 0.3245 0.7520 0.3205 'I 0.7576 0.3144
4 52669 I 0.7261 0.7999 07296 0.7960 I 0.7350 0.7901

TABLE 93

Texas Utilities - Marginal (Per Comparison) Intervals - Panel Data

 

F

90% Ubnd 1 75% Lbnd

 

 

 

 

 

 

 

 

 

imam: ...,...
I—fi—fl—WrwI—w—
I 5 I I
I 3 0.0653 0.8063 1.0000 0.8230 1.0000 I 0.8501 0.9877
I 10 0.0549 0.7729 0.9586 0.7864 0.9421 0.8081 0.9169
I 1 0.0380 0.7747 0.8991 0.7840 0.8884 0.7989 0.8718
I 8 0.0356 0.7644 0.8789 0.7730 0.8691 0.7868 0.8539
9 0.0316 0.7593 0.8561 0.7666 0.8479 0.7783 0.8351
2 0.0401 0.7400 0.8660 0.7494 0.8551 0.7645 0.8383
7 0.0431 0.7222 0.8551 0.7320 0.8435 0.7478 0.8257
6 0.0442 0.7203 0.8566 0.7304 0.8447 0.7466 0.8464
4 0.0352 |[ 0.7113 0.8164 .. 0.7293 0.8074 0.7319 0.7935

 

 

 

 

 

 

90

APPENDIX

TABLE 10

Tileries - Panel Data Estimation Results

 

 

 

 

 

 

 

0.1147 0.1147

 

 

 

 

 

91

APPENDIX

TABLE 11

Tileries - Confidence Intervals Based on BC Method, CGLS Estimates - Panel Data

 

  

75% Ubnd

   

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.9499 I 0.9034 0.9903

[14 0.9433 I 0.8482 0.9976 0.8656 0.9952 n 0.8928 0.9881
I 25 0.9425 0.8457 0.9976 0.8635 0.9952 0.8911 0.9880
| 19 0.9365 0.8359 0.9971 0.8538 0.9942 0.8850 0.9858
3 0.9341 0.83085 0.9969 0.8491 0.9939 1 0.8779 0.9851
22 0.9323 0.8280 0.9968 0.8463 0.9936 0.8753 0.9843
18 0.9296 0.8266 0.9963 0.8444 0.9928 0.8765 0.9826
16 0.9151 0.7885 0.9958 0.8103 0.9916 0.8449 0.9797

5 0.9025 0.7791 0.9935 0.7993 0.9875 0.8315 0.9713
21 0.9017 0.7889 0.9919 0.8070 0.9848 0.8359 0.9670
2 0.8992 0.7841 0.9917 0.8024 0.9834 n 0.8319 0.9660

I 23 0.8866 0.7742 0.9871 0.7917 0.9769 0.8198 0.9544
4 0.8854 0.7688 0.9878 0.7870 0.9779 0.8162 0.9555
17 0.8813 0.7686 0.9850 0.7860 0.9738 0.8140 0.9498
15 0.8405 0.7296 0.9587 0.7462 0.9404 0.7731 0.9098
11 0.7228 0.6285 0.8271 0.6425 0.8091 0.6652 0.7815

I 13 0.6988 0.6077 0.7997 0.6212 0.7823 0.6431 0.7556
I 12 0.6872 0.5976 0.7864 0.6109 0.7693 0.6325 0.7431
6 0.6748 0.5868 0.7723 0.5999 0.7554 l 0.6211 0.7297

10 0.6620 0.5534 0.7855 0.5692 0.7637 ll 0.5950 0.7307
9 0.6437 0.5541 0.7436 0.5674 0.7262 0.5888 0.6998

1 0.6283 0.5314 0.7378 0.5456 0.7186 0.5686 0.6894
20 0.5973 0.5100 0.6952 II 0.5229 0.6781 JI 0.5437 0.6521
7 0.5818 0.5059 0.6657 JI 0.5172 0.6512 0.5354 0.6290

8 0.5399 n 0.4664 0.6216 ! 0.4773 0.6074 0.4949 0.5858

 

 

 

 

92

 

,

APPENDIX

TABLE 12

Tileries - MCB Confidence Intervals - Panel Data

..._ _v. ___.__ _

 

95% Lbnd 95% Ubnd 90% Lbnd 90% Ubnd 75 % Lbnd_ 75% Ubnd

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-,,_ __ 1
14 0.9881 % 0.7373 0.7566 0.7892
24 0.9825 0.7291 1.0000 . 0.7484 1.0000 1 0.7803 1.0000
x 25 0.9770 1 0.7223 1.0000 0.7417 1.0000 1 0.7737 1.0000
3 0.9691 ‘ 0.7136 1.0000 0.7330 1.0000 0.7651 1.0000
19 0.9687 0.7164 1.0000 0.7356 1.0000 0.7673 1.0000
‘ 22 0.9685 ‘ 0.7132 1.0000 ' 0.7326 1.0000 ‘ 0.7647 1.0000
18 0.9399 : 0.6987 1.0000 0.7172 1.0000 0.7478 1.0000
, 5 0.9061 : 0.6557 1.0000 . 0.6748 1.0000 0.7064 1.0000
21 0.9043 3 0.6716 1.0000 0.6896 1.0000 0.7194 1.0000
16 0.9036 , 0.6386 1.0000 0.6585 1.0000 ' 0.6916 1.0000
2 0.8834 l 0.6550 1.0000 0.6728 1.0000 g 0.7023 1.0000
L 23 0.8811 0.6588 1.0000 0.6763 1.0000 . 0.7051 1.0000
4 0.8769 0.6508 1.0000 0.6684 1.0000 0.6977 1.0000
17 0.8497 0.6384 1.0000 0.6553 10000106833 1.0000
15 0.8121 0.6149 1.0000 0.6311 1.0000 0.6581 1.0000
11 0.6240 0.5113 1.0000 0.5246 1.0000 0.5468 1.0000
1 13 0.6071 0.5027 1.0000 6 0.5158 1.0000 : 0.5376 1.0000
12 0.5796 I 0.4891 1.0000 0.5018 1.0000 0.5230 1.0000
1 6 0.5494 I 0.4745 1.0000 x 0.4870 1.0000 1 0.5074 0.9772
: 10 0.5190 0.4387 1.0000 0.45207 1.0000 f 0.4741 0.9843
‘ 9 0.4604 0.4291 0.9673 I 0.4408 0.9417 0.4601 0.9022
1 0.4475 0.4146 0.9757 ; 0.4266 0.9482 9 0.4466 0.9058
20 0.4256 0.4104 0.9433 ; 0.4219 0.9176 i 0.4409 0.8779
7 0.3987 0.4081 0.8990 1 0.4188 0.8760 ' 0.4365 0.8406
8 0.2818 F3604 0.8057 1 0.3701 0.7848 { 0.3861 0.7522

 

 

 

93

APPENDIX

TABLE 14

Tileries - Per Comparison Confidence Intervals - Panel Data

 

- r -

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

  

 

  

 

   

 

I Firm Std Err 95% Lbnd 95% Ubnd 1 90% Lbnd 90% Ubnd 1 75% Lbnd 75% Ubnd
14 1 1 ‘ l
24 0.1045 1 0.7958 1.0000 1 0.8248 1.0000
25 0.1137 1 0.7940 1.0000 1 0.8225 1.0000
3 0.1120 1 0.7950 1.0000 1 0.8223 1.0000

I 19 0.1100 1 0.7907 1.0000 0.8185 1.0000

[22 0.1075 1 0.7943 1.0000 1 0.8216 1.0000
18 0.1225 1 0.7496 1.0000 0.7791 1.0000

11 5 0.1277 1 0.7173 1.0000 0.7468 1.0000

'21 1.0000 0.7725 1.0000 11
16 1.0000 0.7198 1.0000

l 2 1.0000 0.7543 1.0000

r23 1.0000 0.7556 1.0000
4 1.0000 0.7408 1.0000
17 1.0000 0.7151 1.0000 0.7587 0.9994
15 1.0000 0.7059 0.9962 0.7435 0.9459
11 0.8603 0.5808 0.8313 0.6130 0.7876
13 0.1034 0.5578 0.8367 0.5763 0.8098 0.6066 0.7694
12 0.1118 0.5339 0.8275 0.5530 0.7989 0.5845 0.7558
6 0.1090 0.5208 0.7984 0.5390 0.7715 0.7310
10 0.1223 0.4922 0.7950 0.5116 0.7650 0.7200
9 0.1194 0.4669 0.7456 0.4848 0.7181 0.6768

1 0.1258 0.4552 0.7453 0.4736 0.7163 0.6731
20 0.1115 0.4579 0.7089 0.4728 0.6844 0.6477
7 0.1056 0.4510 0.7822 0.4662 0.6599 0.6263
8 0.1173 110.3921 0.6210 11 0.4069 0.5985 0.5648

 

 

 

 

 

 

 

 

95

APPENDIX
MCC INTERVAL

(IN = GIN]
1 =11: N
No uncertainty of population making

(1N1 known)

 

-1-

MCB INTERV
i 4: [N]
Uncertainty of population ranking

( [N] unknown )

 

4 d’ b 4 uncertaintyofranking . <

minis: aj " 0'

FIGURE ]. MCC and MCC Intervals

96

V

d9

 

l l
f j I

J-

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._. I4 ‘.

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