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use communes-p.14

A STUDY IN ESTIMATION AND INFERENCE 0N FIRM EFFICIENCY
By

Yangseon Kim

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Economics

1999

ABSTRACT
A STUDY IN ESTIMATION AND INFERENCE ON FIRM EFFICIENCY
By

Yangseon Kim

This thesis considers the problem of interval estimation of technical efficiency
levels in stochastic frontier models with panel data. First we consider a large number of
classical and Bayesian procedures to estimate technical efficiency levels of firms and to
construct confidence intervals for these efficiency levels. We then apply these methods to
three data sets with different characteristics. The fixed effects models generally perform
poorly: there is a large payoff to distributional assumptions for efficiencies. We do not
find much difference between Bayesian and classical methods if we match methods that
depend on comparable assumptions.

This thesis also provides simulation evidence on the accuracy of inferences on
technical efficiency levels. The simulation evidence suggests that MCB and MargCB are
very conservative. Inference based on bootstrapping is not very reliable in general, but it
is reasonably reliable when T is large relative to N and when the variance of the
inefficiency term is large relative to the variance of the noise. Using the bias-adjusted
bootstrap helps but does not solve the problem completely. The method which uses a
distributional assumption in estimating inefficiencies, performs surprisingly poorly when
N is small. However, there appears to be a large gain in precision and reliability of

inference from making a distributional assumption when N is large relative to T.

ACKNOWLEDGMENTS

The completion of this study owes much to the support and encouragement of
many people who deserve recognition for their efforts. I am especially indebted to my
dissertation adviser, Peter Schmidt. I am deeply appreciative of the guidance and
encouragement which I received from him over the period of this study. This work would
not have appeared in its present form without him. I would also like to thank other
members of the thesis committee, Jeff Wooldridge, Robert de Jong and Richard Baillie
for their time and effort and for their insightful comments.

Many of my fellow graduate students have also contributed to the completion of
this study through their encouragement and friendship. My special thanks go out to
Heather Bednarek, Hailong Qian, and Yi-Yi Chen. A very warm thank you is extended
to Ann Feldman and Zeynep Altinsel who have stayed with me through some tough
times. Their kind advise and warm friendship will always be appreciated.

Finally, I reserve special thank to my family. The person who deserves the most
recognition is my husband, Sang Hyop Lee. His love and support are greatly appreciated.
I would also like to thank my lovely two sons, Jihoon and Sehoon, for their patience and
love. They tried to understand me even though they frequently missed my attention. I
would never have arrived at this point without their smile. They deserve more than half

the credit that I may have achieved.

iii

TABLE OF CONTENTS

LIST OF TABLES

vi

 

CHAPTER 1
INTRODUCTION

 

 

1. The Basic Model

p—i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2. Outline of the Thesis 2
CHAPTER 2
AN EMPIRICAL COMPARISON OF BAYESIAN AND CLASSICAL
APPROACHES TO INFERENCE ON EFFICIENCY LEVELS IN
STOCHASTIC FRONTIER MODELS WITH PANEL DATA 6
1. Introduction 6
2. Classical Statistical Procedures 7
2.1 Efficiency Measurement with a Distributional Assumption for
Inefficiency 8
2.2 Estimation without a Distributional Assumption for lnefficiency
: Fixed Effects 10
2.3 Multiple and Marginal Comparisons with the Best 12
2.4 Bootstrapping l7
3. Bayesian Precedures 22
3.1 General Discussion 22
3.2 The Bayesian Fixed Effects Model 24
3.3 Bayesian Random Effects Models 26
4. Empirical Results 29
4.1 Indonesian Rice Farms 30
4.2 Texas Utilities 33
4.3 Egyptian Tileries 35
5. Concluding Remarks 36
CHAPTER 3
SIMULATION EVIDENCE ON THE ACCURACY OF INFERENCES
IN STOCHASTIC FRONTIER MODELS 47
1. Introduction 47
2. The Model 49
3. Design of the Experiments 51
4. Results 54
4.1 Estimation of Frontier Parameters 54
4.2 Estimation of Efficiencies : MLE & BC method 57
4.3 Estimation of Efficiencies : FE and MargCB or MCB method 59

 

iv

4.4 Estimation of Efficiencies : FE and Bootstrap method

 

 

5. Concluding Remarks

CHAPTER 4
MARGINAL COMPARISONS WITH THE BEST AND
THE EFFICIENCY MEASUREMENT PROBLEM

 

Introduction

 

 

Marginal Comparisons under Standard Assumptions
Marginal Comparisons with General Covariance Structure

 

 

Application to the Efficiency Measurement Problem

 

.V‘PP’NI"

Empirical Applications
5.1 Indonesian Rice Farms

 

5.2 Texas Utilities

 

 

5.3 Egyptian Tileries

 

6. Concluding Remarks
Appendix

 

 

Bibliography

61
63

74

74
75
78
80
82
82
83
84
85
87

93

LIST OF TABLES

CHAPTER 2

Table 1: Estimates of Parameters : Indonesian Rice Farm

 

38

Table 2: Estimates of Efficiencies (Fixed Effect Model) : Indonesian Rice Farm ----- 39

Table 3: Estimates of Efficiencies (Random Effect Model) : Indonesian Rice Farm - 4O

 

Table 4: Estimates of Parameters : Texas Utilities

Table 5: Estimates of Efficiencies (Fixed Effect Model) : Texas Utilities ----------

Table 6: Estimates of Efficiencies (Random Effect Model) : Texas Utilities -------

 

Table 7: Estimates of Parameters : Egyptian Tileries

Table 8: Estimates of Efficiencies (Fixed Effect Model) : Egyptian Tileries -------

41

42

43

44

---45

Table 9: Estimates of Efficiencies (Random Effect Model) : Egyptian Tileries ------- 46

CHAPTER 3

Table 1: Bias and Standard Error of Estimated Constant Term, of: and var(u)
(N Changes with T Held Constant)

 

Table 2: Bias and Standard Error of Estimated Constant Term, of, and var(u)
(T Changes with N Held Constant)

 

Table 3: Estimation of Efficiencies : MLE & BC method

 

Table 4: Effect of Wrong Distribution on Estimation of Efficiencies

 

: MLE & BC method

Table 5: Average Sample Minimum for Different Distribution

 

Table 6: Estimation of Efficiencies : FE & MargCB or MCB
(N Changes with T Held Constant)

 

vi

65

66

67

68

69

7O

Table 7: Estimation of Efficiencies : FE & MargCB or MCB

 

 

 

(T Changes with N Held Constant) 71
Table 8: Confidence Intervals for Efficiency Estimates : FE & Bootstrap ------------- 72
Table 9: Bias Correction in the Bootstrap Intervals 4 73
CHAPTER 4
Table 1: 90% Confidence Intervals for Technical Efficiency

: Indonesian Rice Farms 90
Table 2: 90% Confidence Intervals for Technical Efficiency : Texas Utilities -------- 91

Table 3: 90% Confidence Intervals for Technical Efficiency : Egyptian Tileries ----- 92

Chapter 1

INTRODUCTION

This thesis consists of three essays on the problem of estimating the technical
efficiency of firms, using panel data. This chapter will define some basic notation and

give an overview of the thesis.

1. The Basic Model
We begin with the basic panel data stochastic frontier model of Pitt and Lee

(1981) and Schmidt and Sickles (1984):
(1.1) Ya =Ot+xztl3+vit —u,, '=1, ...... N, t=1, ....... ,T

Here i indexes firms or productive units and t indexes time periods. y“ is the scalar
dependent variable representing the logarithm of output for the i‘h firm in period t, or is a
scalar intercept, xi. is a le' vector of functions of inputs (e.g., in logarithms for the
Cobb-Douglas specification), [3 is a le vector of coefficients and v“ is an i.i.d. error
term with zero mean and finite variance. The ui satisfy 11, 20, and ui >0 is an indication of
technical inefficiency. Note that ui is time-invariant. For a logarithmic specification such
as this the technical efficiency of the i'h firm is defined as r; = exp(-ui ), so technical
inefficiency is l-ri , For small values of u; U. is approximately equal to 1- exp(-ui) = 1-ri,
so that ui itself is sometimes used as a measure of technical inefficiency.

Now define 0ti = a—ui. With this definition, (1.1) becomes the standard panel

data model with time-invariant individual effects :

(12) yit :al +xitB+vlt

Obviously we have on. _<_ or and u, = a—al. As before, technical efficiency is r. =
exp(-ui).

The previous discussion regards zero as the minimal possible value of ui, and or as
the maximal possible value of at, over any possible sample; that is, essentially, as N—)oo.
For some purposes, and especially when N is not large, it is also useful to consider the

following representation. We write the intercepts on, in ranked order as :
(1.3) (1(1)st ”Sam

so that in particular (N) is the index of the firm with the largest value of on, among firms
i=1,...,N. It is convenient to write the values of ul in the opposite ranked order, as
um) S s "(2) s “(1) , so that am: a - um for all i. Then obviously am) = 0t - um), and
firm (N) has the largest value of a, or equivalently the smallest value of ui among firms
i=1,. ..,N. We will call this firm the best firm in the sample. In some methods we measure

inefficiency relative to the best firm in the sample, and this corresponds to considering

the relative efficiency measures:

(1-4) “i: “a _u(N)= am) " an r: = exp(—u:).

2. Outline of the Thesis
Chapter 2 is primarily empirical. It first provides a survey of a large number of

classical and Bayesian methods that have been proposed to estimate technical efficiency

levels and to perform inference on these levels. Classical procedures include multiple
comparisons with the best (MCB), based on the fixed effects estimates; marginal
comparisons with the best (MargCB); bootstrapping of the fixed effects estimates; and
maximum likelihood given a distributional assumption for the inefficiency terms ui.
Bayesian procedures include a Bayesian version of the fixed effects model, and various
Bayesian models with informative priors for efficiencies.

It then applies these techniques to the three previously-analyzed data sets, on
Indonesian rice farms, Texas electric utilities, and Egyptian tileries, and compares the
point estimates and confidence intervals for technical efficiency levels. The fixed effects
models generally perform poorly; there is a large payoff to distributional assumptions for
efficiencies. There is not much difference between Bayesian and classical procedures, in
the sense that classical MLE based on a distributional assumption for efficiencies gives
results that are rather similar to a Bayesian analysis with the corresponding prior.

Chapter 3 provides simulation evidence on the accuracy of inferences on technical
efficiency levels. This is useful because the finite sample properties of the efficiency
measurement methods discussed in Chapter 2 are basically unknown. It is known that
MCB and MargCB confidence intervals are considerably wider than intervals based on
bootstrapping the fixed effects estimates or based on a distributional assumption, but it is
not known how conservative MCB and MargCB are, or whether the bootstrapping or
distribution—based estimates are reliable. The simulation evidence suggests that MCB and
MargCB are very conservative. Inference based on bootstrapping is not very reliable in
general, but it is reasonably reliable when T is large relative to N and when the variance

of the inefficiency term ui is large relative to variance of the noise v“. The main problem

with bootstrapping is finite sample bias due to the "max" operation in defining the
frontier, and using the bias-adjusted bootstrap helps but does not solve the problem
completely. There appears to be a large gain in precision and reliability of inference from
making a distributional assumption. This is so especially when N is large relative to T.
Chapter 4 provides a rigorous derivation of Marginal Comparisons with the Best
(or MargCB) which was introduced in Chapter 2. MargCB addresses the following

problem. Given estimates of the basic model, we wish to say which populations might be

best, and to construct confidence intervals for the differences u: 28”,) -—91, which

measure the amount by which a given population differs from the best. An existing
technique called Multiple Comparisons with the Best (or MCB) also addresses this
problem. MCB constructs a set S of possibly best populations, and a set of intervals

(Li ,Ui), such that:

(1.5) P[(N)ES and Li Su: S. U! forall i]2(l-c),

where l-c is a chosen confidence level (e.g., 0.90). Thus with a given confidence level we
have a set of populations that includes the best, and joint confidence intervals for all
differences from the best. These intervals are often rather wide, in part because of the
joint nature of the statement (1.5). One way to make the intervals narrower is to make a

marginal rather than joint statement, of the form :

(1.6) P[(N)eS and LI _<.u: _<.Ul ]2(l-c),

where statement (1.6) holds for a single given value of i. The MargCB technique provides

the set S and interval (L;, U,) such that (1.6) is true. This technique is then applied to the

same three data sets as were used in Chapter 2. and evidence on the width of the MCB

and MargCB intervals is provided.

Chapter 2
AN EMPIRICAL COMPARISON OF BAYESIAN AND CLASSICAL

APPROACHES TO INFERENCE ON EFFICIENCY LEVELS
IN STOCHASTIC FRONTIER MODELS WITH PANEL DATA

1. Introduction

This chapter considers the problem of interval estimation of technical efficiency
levels in stochastic frontier models with panel data. The phrase "interval estimation"
indicates that we are interested not only in point estimates of the efficiency levels of the
individual firms, but also in confidence intervals for the efficiency levels.

A number of different techniques have been proposed in the literature to address
this problem. Given a distributional assumption for technical inefficiency, maximum
likelihood estimation was proposed by Pitt and Lee (1981). Battese and Coelli (1988)
showed how to construct point estimates of technical efficiency for each firm, and
Horrace and Schmidt (1996) showed how to construct confidence intervals for these
efficiency levels. Without a distributional assumption for technical inefficiency, Schmidt
and Sickles (1984) proposed fixed effects estimation, and the point estimation problem
for efficiency levels was discussed by Schmidt and Sickles (1984) and Park and Simar
(1994). Simar (1992) and Hall, H'ardle and Simar (1993) suggested using bootstrapping
to conduct inference on the efficiency levels. Horrace and Schmidt (1996, 1999)
constructed confidence intervals using the theory of multiple comparisons with the best,
and we extend it to a univariate version. Bayesian methods have been suggested by Koop,
Osiewalski and Steel (1997) and Osiewalski and Steel (1998). They propose a model with

an uninformative prior for firm-specific intercepts that is intended to be similar to the

classical fixed effects model, and also models with informative priors, which are
comparable to classical models that assume a distribution for inefficiency.

These models have been applied to various data sets, but there has been no
systematic attempt to compare them all on a common data set.In this chapter, we apply
these models to three previously-analyzed data sets and compare the results. The major
emphasis is to try to understand the relationship between the assumptions underlying the
various models and the empirical results. More specifically, we are interested in two
types of questions. First, we wish to see how much is gained, in terms of tightness of the
confidence intervals, by being willing to make a distributional assumption for technical
inefficiency. This is phrased as a classical question, but from a Bayesian perspective the
question is simply rephrased as seeing how much is gained, in terms of tightness of the
posterior distribution, by imposing an informative prior. Second, we wish to compare the
results from Bayesian and classical analyses, where we match as far as possible the
strength of the assumptions underlying the analyses. We find large gains from
distributional assumptions, and we do not find much difference between classical and

Bayesian analyses that rely on assumptions of comparable strength.

2. Classical Statistical Procedures

In this section we will discuss classical statistical procedures for the estimation of
the model presented in section 1 of the chapter 1. Bayesian procedures will be discussed
in the next section. We will distinguish procedures that make an assumption about the

distribution of ui from those that do not.

2.1 Efficiency Measurement with a Distributional Assumption for Inefficiency

In this subsection we consider estimation under strong assumptions that are
similar to those made in the cross-sectional case (Aigner, Lovell and Schmidt (1977)).
We assume independence across firms (values of i). We assume that the explanatory

variables xit are strictly exogenous: (x,,,...,x,T) is independent of (v ..,vIT,ui). We

11 "
assume that the v“ are i.i.d. as N(0,o§) and are independent of u,. Finally, we assume a

specific distribution for u,. The distributions most commonly considered in the literature
have been the half-normal and the exponential. Other suggestions include the truncated
normal (Stevenson (1980)) and gamma (Greene (1990)) distributions. In this chapter we
will use the exponential distribution, primarily because it is easiest to handle in the
Bayesian fi’amework, and we want to be able to make comparisons across Bayesian and
classical approaches based on the same distribution for u. Thus we assume the following
density for u: f(u) = ¢"exp(—u/¢), with E(u) = ¢.

The preferred method of estimation is maximum likelihood estimation (MLE).
The likelihood fianction has been derived by Aigner, Lovell and Schmidt (1977) for the
cross-sectional case (T=1), and is easily extended to the case of panel data as follows:
N 'r 2

ln(21to2)-—lnT— ZNIii—(E +6”)]
V 2 203,1:1 z=1 T ‘ T‘b

N(T-l)
2

N fi_ 0v

+§m[1—¢(:ei+—fi¢)]

(2.1) 1n L = -ln(¢>)-

 

 

where (D denotes the cdf of the standard normal distribution, 8,, = vit — u, = y,, — a — xnfi

T
, = T423“ . The likelihood function is maximized numerically to obtain the MLE

(=1

ands

of the parameters (a, B, (b, 03 ). These estimates are consistent as N—>oo for fixed T. The

implication of this is that N needs to be large for MLE to be appropriate, whereas large T
is not required.

The model can also be regarded as the random effects model from the panel data
literature, so that estimation by generalized least squares (GLS) is possible. GLS is
consistent as N—>oo without the assumption of normality of the v“ and without an
assumption of a specific distribution for the u,. However, the model above differs from
the standard random effects model because u; does not have a mean of zero. Thus the
GLS estimated intercept will be a consistent estimate of a—E(u) rather than of a. It is
possible to adjust the intercept upward, by adding a consistent estimate of E(u), but this
requires an assumption about the distribution of u. As a result we will consider only the
MLE.

Estimation of the model yields residuals 8, = y1, —dt — xf,[3, which are naturally

regarded as estimates of 8,, = v" — ui , whereas we are interested in estimating ui itself.
The usual solution to this problem, following Jondrow et al. (1982) and Battese and
Coelli (1988), is to consider the distribution of ui conditional on 8‘ = (8”, ...... ,8,T)
evaluated at 8,. Jondrow et 81. give this distribution for the cross-sectional case (T=1),
for the cases that u is half-normal or exponential, and suggest the point estimate E(uil 8i).
Battese and Coelli (1988) give both E(ui | 8,) and E(ri | 8,) for the half-normal case with
panel data (T21). Actually, when the vat are i.i.d. normal, the distribution of ui conditional
on 8i = (8n , ...... ,8”) is the same as the distribution of ui conditional on El , regardless of

the distribution of ui. Therefore the results of Jondrow et al. can be extended to the panel

data case just by replacing 8, by 8‘ and o: by 0: /T. From their Theorem 2, we find that
the distribution of u. conditional on 83 is N(-ui ,65 /T) truncated at zero, where u. = 8 +

63 /(Td)). We can therefore calculate confidence intervals using this distribution. This

idea was suggested by Horrace and Schmidt (1996), who implemented it assuming the
half-normal distribution for u, and the results in this chapter differ only because a
different distribution (exponential) is assumed.

Conducting inference on u. using the distribution of u. conditional on Si is not
suggested as an authentic Bayesian procedure, but it obviously has a Bayesian flavor. The
main difference between this distribution and a Bayesian posterior distribution is that it
relies on asymptotics to ignore the effects of parameter estimation, whereas the
uncertainty due to parameter estimation .will figure into the Bayesian posterior. We might

expect this difference not to matter very much when N is large, however.

2.2 Estimation without a Distributional Assumption for Inefficiency: Fixed Effects

We now discuss estimation without a distributional assumption for the u;. This
subsection will give a brief review of the point eStimation problem based on the fixed-
effects estimates, and the next two subsections will consider different ways of
constructing confidence intervals for inefficiency based on these estimates.

Fixed effects estimation refers to the estimation of the panel data regression
model (1.2), treating the on as fixed parameters. Because the or. are treated as parameters,
we do not need to make any distributional assumption about the inefficiencies; nor do we
need to assume that they are uncorrelated with the xi. or the vit. We still assume strict

exogeneity of the regressors xi. in the sense that (xn,...,X.T) is independent of

10

(V..,-~,V.r)- We also assume that the V. have zero mean and constant variance 03, and
are not autocorrelated. We do not need to assume a distribution for the v...

The fixed effects estimate [3, also called the within estimate, may calculated by
regressing (y, —— 1) on (xu - 35,), or equivalently by regressing ya on xi. and a set of N
dummy variables for firms. We then obtain dc = y, - Fifi ; or equivalently the (it, are the

estimated coefficients of the dummy variables. This leads to the following expression for

A

(1,:

(2.2) (iti 0tI +Vi -§,'(l§ ’13)

A

to which we will make reference later. The fixed effects estimate [3 is consistent as
NT—>oo (i.e., as either N or T approaches infinity), and its variance is of order [N(T-1)]".

For a given firm (i), the estimated intercept (it, is a consistent estimate of 0ti as T—->oo.

Large T is needed for the term V, in (2.2) to become negligible.

Schmidt and Sickles (1984) suggested the following estimates of technical

inefficiency, based on the fixed effects estimates:

(2.3) (it = maxdt; u. =&-& , i=1,...,N.
i=1,...N ’ '

Since these estimates clearly measure inefficiency relative to the firm estimated to be the

best in the sample, they are naturally viewed as estimates of am) and u:, that is, of
relative rather than absolute inefficiency. For fixed N, (St is a consistent estimate of am)

and fit is a consistent estimate of u: as T—)oo. However, it is important to note that in

11

finite samples (for small T) d is likely to be biased upward, since (it 2 (im) and E(dqm)
= am), where rim is the estimated intercept for the unknown best firm. That is, the

"max" operator in (2.3) induces upward bias, since the largest (St‘ is more likely to
contain positive estimation error than negative error. This bias is larger when N is larger
and when the dilare estimated less precisely. The upward bias in (it induces an upward
bias in the ii: and a downward bias in if: exp(-fi,.) ; we underestimate efficiency
because we overestimate the level of the frontier.

Schmidt and Sickles argued that (it and the a: are consistent estimates of at and
the ui if both N and T approach infinity; that is, if both N and T are large, we can regard

the It: as estimates of absolute and not just relative inefficiency. The argument is simple.

As T—>00, 6t and the CI: are consistent estimates of am) and the u,, as noted above. As

N—>oo, um) should converge to zero, so that am) should converge to a and the u:
should converge to the corresponding ui. A more rigorous treatment of the asymptotics
for this model is given by Park and Simar (1994), who show that, in addition to N—->oo
and T—>oo, we need to require TV‘ln N —) 0 in order to ensure the consistency of (it as an

estimate of a. This latter requirement limits the rate at which N can grow relative to T, in

order to ensure that the upward bias induced by the "max" operation disappears

asymptotically.

2.3 Multiple and Marginal Comparisons with the Best
Multiple comparisons with the best (MCB) is a statistical technique that yields

confidence intervals for differences in parameter values between all populations and the

12

:1

 

best population. Horrace and Schmidt (1996, 1998) have suggested its use to construct

confidence intervals for the relative technical inefficiencies u: = am) —0t =

r: = exp(—u:), which are indeed differences from the best.
Let A = (0Ll ,...,0.N) be the vector of intercepts for the N firms in the panel data

regression model (1.2). (It would be natural to refer to this vector as at, but that symbol

has already been used for the intercept in model (1.1).) As before we denote jrgriax'orJ by

am). Then MCB constructs a set S of possibly best populations, and a set of intervals

(LgUg), such that:

(2.4) P[ (N) e S and Li S am) -(1i S U for all i]21-c,

where l-c is a chosen confidence level (e.g., 0.90). (Again, it would be natural to use or
in place of c for the tail probability, but the symbol a has already been used.) Thus with
a given confidence level we have a set of populations that includes the best, and joint
confidence intervals for all differences from the best. MCB was deveIOped by Hsu
(1981, 1984) and Edwards and Hsu (1983). A general exposition can be found in
Hochberg and Tamhane (1987), Hsu (1996) and Horrace and Schmidt (1999).

To perform MCB, we need an estimate A , distributed as N(A.,O'2C) with C
known, and where either cr2 is known, or we have an estimate 62, independent of A,

such that 62/02 is distributed as xi / v. In typical MCB applications to the efficiency

measurement problem, A will come from the fixed-effects estimation of the panel data

regression model (1.2), as discussed above, and there will be enough degrees of freedom

13

that we can effectively take (52 as known. Normality of A requires either that the errors
vi. are normal or that T is large enough for a central limit theorem to hold for the
expressions in (2.2) above.

Standard MCB proceeds under the further assumption that C = kIN with k known.
This assumption is usually motivated by discussion of the "balanced one way model"
(e.g., Hsu (1996), p. 43) in which we have independent observations ya (i = 1,...,N, t =
1,...,T) distributed as N(ai, 62). In this case k = l/T. This is equivalent to the panel data
regression model (1.2) if B were known, since then we have (ya - x’itB) = a. + v“, which is
the balanced one-way model. Since standard MCB would be applicable if B were known,
it is reasonable to presume that standard MCB is a good approximation if B is estimated

sufficiently precisely. Recall from the previous subsection that the variance of (i. is of

order T", while the variance of B is of order [N(T-1)]'l. Thus it may generally be the

case that standard MCB is approximately applicable when N is large. This point is
discussed in more detail in Horrace and Schmidt (1999).

Now define the following notation. Let E(‘/2) be the (N-l)x(N-1) correlation
matrix with all correlations equal to ‘/2 (i.e., diagonal elements equal one, off-diagonal

elements equal ‘/2). Let 2 be a multivariate random variable distributed as student-t with

dimension N-l, degrees of freedom v, and correlation matrix E(‘/2). Define d'(c) as the

c-level critical value of Imax [2, l; i.e., P[ max |z| |s d°(c)] = l-c. Tabulations of
=1,...N-1 I=1,...N-1

d'(c) can be found in Hsu (1996) or Horrace (1998). Define h(c) =d‘(c) (2k62)v‘, and

define the set S(c) = {i |<3ti 2 max (it J - h(c)}. Define Li and Ui as follows:
j=t,..,N

l4

(2.5) L. = max[0, IIgisi(n)citJ —dt, -h(c)] , U; = max[0, Iqa'xdtJ —dt' + h(c)]

Then MCB provides the statement (2.4) above, with S = S(c).

As noted above, standard MCB requires that the variance matrix of
A=(dt,,..,dtN) be proportional to an identity matrix; that is, the various (it, are

uncorrelated and have equal variance. General MCB allows this variance matrix to be of
an arbitrary form. A discussion of general MCB can be found in Horrace and Schmidt
(1999, section 3), and is too lengthy to include here. The empirical results in this paper
are all obtained from general MCB, but standard MCB would have yielded similar
results.

The MCB statement (2.4) is a multiple statement in the sense that the confidence
intervals all hold jointly with (at least) the specified probability. We can also consider
marginal (one at a time) confidence intervals, which are more directly comparable to the
intervals provided by the other techniques we consider. We call - this marginal
comparisons with the best, which we will abbreviate as MargCB. There are standard and

general versions of MargCB, where the standard version makes the same assumption
about the variance matrix of A as does standard MCB. We will discuss standard
MargCB, but the empirical results of this chapter use the general version. Let t'(c) be

the two-sided c-level critical value of the (univariate) student-t distribution with v degrees

of freedom; i.e., if z is distributed as student-t with v degrees of freedom, then
P[[ z ls. t‘(e)] = 1— c. Define g(c) = t'(c) (21(62)”. Define the set S(c) as above. Define

LT and U?" as follows:

15

(2.6) LT: max[0, 53313654“ -g(c/2)], Uf“ = max[0, rquéj -&. +g(c/2) ].

Then the following is true:
(2.7) P[ (N) e S(c) and LTS amr (14$ Uf“ ] 2 l-c.

A proof of this result in a more general setting will be given in Chapter 4.

Both MCB and MargCB are conservative procedures. The events given in
statements (2.4) and (2.7) hold with a probability of at least l-c, and the inequality
occurs because of uncertainty about which firm is best. There will be considerable
uncertainty about which firm is best when one or more of the a. are nearly as large as
am) and when the (itl have large sampling variance. In such cases the set S(c) will be
large and the confidence intervals will be wide. The other techniques discussed in this
chapter are not conservative and may be expected to yield narrower confidence intervals.
However, those techniques rely on stronger assumptions and/or asymptotic theory and
correspondingly may be more likely to yield inferences that are erroneous.

MCB is not designed as an asymptotic procedure. Indeed, the problem of
comparing N populations is hard to conceptualize unless N is fixed. However, since
econometricians often think in terms of asymptotics, the following comments may be
helpful. First, as just noted, standard MCB may be approximately valid when N is large.
Second, MCB assumes that the (it, are normally distributed. This should be so if the
errors vi. are normal or if T is large. Third, MCB establishes confidence intervals for the
relative inefficiencies u: or r: , but if N is large these can also be regarded as confidence

intervals for the absolute inefficiencies u; or rt.

16

2.4 Bootstrapping

We can use bootstrapping to construct confidence intervals for firnctions of the

fixed effects estimates. The inefficiency measures It: (as in equation (2.3)) and the

efficiency measures r: = exp(—fr:) are fimctions of the fixed effects estimates and so
bootstrapping can be used for inference on these measures.

Consider the general setting in which we have a parameter 9, and there is an

A

estimate 8 based on a sample 21,...,zn of i.i.d. random variables. The estimator 9 is

assumed to be regular enough so that nl""((3 -G) is asymptotically normal. The following

bootstrap procedure will be repeated many times, say for b = 1,...,B where B is large. For

iteration b, construct pseudo data 21m,...,zna’) by sampling randomly with replacement

from the original data 21,...,z,,. From the pseudo data, construct the estimate 8“). The

basic result of the bootstrap is that, under fairly general circumstances, the asymptotic
(large n) distribution of n%( B‘b)-é) conditional on the sample is the same as the
(unconditional) asymptotic distribution of n%(0 -0). Thus for large n the distribution of (3

around the unknown 9 is the same as the bootstrap distribution of 9‘”) around 0, which is
revealed by a large number (B) of draws.
We now consider the application of the bootstrap to the specific case of the fixed

effects estimates. Our discussion follows Simar (1992). Let the fixed effects estimates be

[3 and (31,, from which we calculate Grand i,'(i=1,...,N). Let the residuals be Ou= y“ -

dti- x'..[3 (i=1,...,N, t=l,...,T). The bootstrap samples will be drawn by resampling these

residuals, because the vi, are the quantities analogous to the 2's in the previous paragraph,

17

in the sense that they are assumed to be i.i.d., and the o, are the observable versions of
the v... (The sample size n above corresponds to NT.) So, for bootstrap iteration

b(= l,..,B) we calculate the bootstrap sample (If?) and the pseudo data yff’) = (it, + x'rté f

(1?). From these data we get the bootstrap estimates B‘b’, dtfb) , fir“), and if” , and the

bootstrap distribution of these estimates is used to make inferences about the parameters.

We note that the estimates II, and fi depend on the quantity max 6t 1. Since "max"
1:1,...N

is not a smooth function, it is not immediately apparent that this quantity is
asymptotically normal, and if it were not the validity of the bootstrap would be in doubt.
A rigorous proof of the validity of the bootstrap for this problem is given by Hall,

H'alrdle and Simar (1995). They prove the equivalence of the following three statements:

(i) mafia, is asymptotically normal. (ii) The bootstrap is valid as T—)oo with N fixed.

(iii) There are no ties for max or]; that is, there is a unique index i such that on = max on].
I=1.«N i=1...N
There are two important implications of this result. First, the bootstrap will not be reliable

unless T is large. Second, this is especially true if there are near ties for max a j, in other
i=1...N

words, when there is substantial uncertainty about which firm is best.
We now turn to specific bootstrapping procedures, which differ in the way they

draw inferences based on the bootstrap estimates. In each case, suppose that we are trying

to construct a confidence interval for u,'= max or J- on. That is, for a given confidence
I:1...N

level 8, we seek lower and upper bounds Li, Ug such that P[Li .<_ u'

I

_<_ Ui] = I-C. This
statement should hold exactly for large T, and for small T it will be inaccurate to an

unknown extent.

l8

The simplest version of the bootstrap for the construction of confidence intervals

is the percentile bootstrap. Here we simply take L. and U. to be the upper and lower c/2

fractiles of the bootstrap distribution of the 0:“). More formally, let F be the bootstrap

A.

cumulative distribution fiinction for fir, so that F(s) = P(u,(b’ _<. s) = the fraction of the B

bootstrap replications in which u“) s s. Then we take L = F" (8/2) and U. = F "' (1-c/2).
The percentile bootstrap intervals are accurate for large T but may be inaccurate
for small to moderate T. This is a general statement, but in the present context there is a

specific reason to be worried, which is the finite sample upward bias in max (it I as an
1:1,...N

estimate of max a1. This will be reflected in improper centering of the intervals and
1:1...N

therefore inaccurate coverage probabilities. Simulation evidence on the severity of this
problem is given in the following chapter and also given by Hall, Hardle and Simar
(1993). Several more SOphisticated (or at least more complicated) versions of the
bootstrap have been suggested to construct more accurate confidence intervals. Hall,
H'a'rdle and Simar (1993, 1995) suggested the iterated bootstrap, also called the double
bootstrap, which consists of two stages. The first stage is the usual percentile bootstrap,
which constructs, for any given c, a confidence interval that is intended to hold with
probability l-c. We will call these "nominal" l-c confidence intervals. The second stage
of the bootstrap is used to estimate the true coverage probability of the nominal l-c
confidence intervals, as a firnction of c. That is, if we define the function 1t(c) = true
coverage probability level of the nominal l-c level confidence interval fi'om the

percentile bootstrap, then we attempt to evaluate the function 1t(c). When we have done

so, we find c° , say, such that 112(6): l-c, and then we use as our confidence interval the

19

nominal 1- c' level interval from the first stage percentile bootstrap, which we "expect" to
have a true coverage probability of l-c.

The mechanics of the iterated bootstrap are uncomplicated but time-consuming.
For each of the original (first stage) bootstrap iterations B, the second stage involves a set
of B2 draws from the bootstrap residuals, construction of pseudo data, and construction of

percentile confidence intervals, which then either do or do not cover the bootstrap

estimate 9“”. The coverage probability function 7t(c) is the fraction of times coverage
occurs. Generally we take B2 = B, so that the total number of draws has increased from B
to B2 by going to the iterated bootstrap. Theoretically, the error in the percentile bootstrap
is of order n'V' while the error in the iterated bootstrap is of order n‘l. There is no clear
connection between this statement and the question of how well finite sample bias is
handled.

An objection to the iterated bootstrap is that it does not explicitly handle bias. If
the nominal 90% confidence intervals only cover 75% of the time, it simply insists on a
higher nominal confidence level, like 98%, so as to get 90% coverage. That is, it just
makes the intervals wider, when bias might more reasonably be handled by recentering

the intervals. A technique that does recenter the intervals is the bias-adjusted bootstrap of

Efron (1982, 1985). As above, let 0 be the parameter of interest, 0 the sample estimate

and 0‘”) the bootstrap estimate (for b=l,...,B), and let F be the bootstrap cdf. For n large

enough that the bootstrap is accurate, we should expect F(B) = 0.5, and failure of this to

occur is a suggestion of bias. Now define 20 = (D'1[F(B)] where (I) is the standard normal

cdf, and where E(B) = 0.5 would imply 20 = 0. Let 26:2 be the usual normal critical value;

20

e.g. for c = 0.05, 2.72 = z 025 = 1.96. Then the bias-adjusted bootstrap confidence interval

is [L.,U.] with:
(2.8) L. = i" Id><2z.- 2.2)] , U. = i" [<I><2z. + 2.2)].

For example, suppose that there is upward bias, reflected by the fact that 60% of
the bootstrap draws are larger than 9, so that E(é) = 0.4. Then 20 = -0.253, and for c =
0.05 we have <D(2 20- 2,72) = <D(-2.466) = 0.0068 and <D(220+ zen) = 0.937. Thus our

confidence interval comes from the lower tail 0.0068 fractile and the upper tail 0.063
fractile, and we have compensated for upward bias by moving the interval left. This
seems intuitively reasonable.

The assumption that justifies the bias-adjusted bootstrap is that, for some

monotone increasing fiinction g, [g(é)-g(0)] is distributed as N(-zoo,oz) and

[ g(é(b)) - g(é)] is also distributed as N(-z 0' ,oz), for some 2 , oz. (The first distribution
0 0

is from the probability law of the sample and the second is the bootstrap probability
distribution induced by resampling from the given sample.) Thus we have normality, and
also equal biases and variances, for some transformation of 0. The transformation
function g need not be known. This is an advantage in implementation, but a
disadvantage in trying to decide whether the assumption holds. It is not known whether
the bias-adjusted bootstrap is valid for our specific problem, but it performs relatively
well in the simulations reported in Chapter 3.

The final version of the bootstrap that we will consider is the bias-adjusted and

accelerated bootstrap of Efron and Tibshirani (1993). This is intended to allow for the

21

possibility that the variance of 0 depends on 0, so that a bias-adjustment also requires a

change in variance. This correction depends on some quantities defined in terms of the

so-called jackknife values of B . For i=1,...,n, let 60) be the value of the estimate based

on all observations other than observation i; and let em: n’l E Lem be the average of

these values. Then the "acceleration" factor a is defined by:

(2.9) a = s. :1, (8“,.ém)3 / 6[: IL. (fig-8,921”

With 20 and zen defined as above, define

(2.10) bl = z0 + (20 +zaz)/[l—a( z0 +Zc/2)] , b2 = 20 + (20 +z,_d2 )/[l-a( z0 +z,_d;. )]

Then the confidence interval is [L, , U, l, with Li = F" [<I>(b1)] and Ui =F" [<D(b2)]. More

discussion can be found in Efron and Tibshirani (1993, chapter 14).

3. Bayesian Procedures

In this section we discuss the Bayesian analysis of the stochastic frontier model.
Bayesian analyses have been proposed and described in a series of papers by Koop,
Osiewalski and Steel (hereafter KOS), especially KOS (1997) but also including Broeck,
Koop, Osiewalski and Steel (1994), Koop, Steel and Osiewalski (1995), and Osiewalski

and Steel (1998).

3.1 General Discussion

22

The basic Bayesian principles are straightforward. We have a set of observable
data Y = (y1,...,yn) and a vector 0 (say of dimension K) of unobservable parameters. Let
p(G) be the prior density of 9 and p(Y l 8) be the likelihood, where the prior is specified
by the data analyst and the likelihood follows from the assumed model. Then Bayes Law

says that:

(2-11) 13(9lY)0C P(9)P(Y|9)

where p(GlY) is the posterior density of 9 and "oc" indicates proportionality. The
traditional interpretation is that both the prior and the posterior reflect subjective
probability distributions of 8, one (the prior) prior to the observation of Y and the other
(the posterior) after the observation of Y. Bayes Law shows how the subjective
probability distribution of 0 is modified by the observation of Y. The concept of
subjective probability is controversial but Bayes Law itself is not, since it is just the
usual rulefor conditional probability.

Inference on the parameters is performed using the posterior distribution. Since 0
is usually multidimensional, one must face the often considerable problem of obtaining
the marginal posterior distribution for a single given parameter such as 0, (for some
specific value 1s i s K). The marginal posterior density of 0. is in principle defined by
integrating the joint posterior density of 0 with respect to all elements of 0 other than 9;,
but this integral may not be analytically tractable. An alternative is to make Monte Carlo
draws from the posterior distribution p(GIY) and to use these to reveal whatever features
of the distribution of Bi are interesting. Some numerical problems related to such Monte

Carlo procedures will be discussed below.

23

Bayesian methods treat the parameters as random and condition on the data,
which is more or less exactly the opposite of what classical methods do. However, in the

present context of the stochastic frontier model with panel data, these distinctions can

become a bit blurred. The parameters will typically included, B, 03, ul,...,uN, and

possibly some additional nuisance parameters. Existing classical treatments of this model

have always treated on, B and of, as fixed, but the inefficiency terms ul ,...,uN have often

been treated as random and sometimes assigned a distribution. As discussed in section
2.1 above, inference on u. is then performed using the distribution of ui conditional on
(8”,....,8,T ), which is certainly similar to a posterior distribution. In fact this can be
regarded as Bayesian inference in classical clothing. It differs from the Bayesian
posterior in that it treats 0t, B and the nuisance parameters in the distributions of vn and Us
as known, whereas the Bayesian posterior conditions only on the data. This difference is
not likely to be substantial in practice because the parameters being taken as known are

estimated based on NT observations (rather than just T observations for u.) and should

not contribute much variability to the Bayesian posterior.

3.2 The Bayesian Fixed Effects Model

In this section we discuss a model that KOS call the standard individual eflects
model (or SIE model). They regard it as one possible variant of the Bayesian fixed eflects
model, whereas we will just refer to it as the Bayesian fixed effects model, but this is

only a semantic point.

This model postulates an "uninformative" prior for the basic parameters

a,,...,0tN, B, 63: p((ll,...,(1N, B, 03) 0c of. (We do not regard this prior as

24

uninformative, but again this is just a semantic point.) Note that, in contrast to random

effects models to be discussed later, we do not attempt to identify a and u,,...,uN

separately. Rather we simply measure relative inefficiency, by considering u,'= martiniJ
1:1...

- on and r,'= exp(-u,°) as functions of the firm-specific intercepts. This is similar in spirit

to the classical fixed-effects treatment.

The likelihood p(Yla,,...,aN, B, 0:) is the usual (classical) normal likelihood

that would follow from treating the xi. as fixed and the V“ as i.i.d. normal. Specification of
the prior and the likelihood defines the problem and implies the form of the posterior.

The marginal posterior of or, ,...,aN, B can be calculated analytically to be (N+K)-variate

student t with N(T-l)-K degrees of freedom. For any reasonable problem the number of

degrees of freedom is large enough to treat the posterior as multivariate normal. The

A

posterior mean of B is the classical fixed-effects estimate B as in section 2.2, and

similarly the posterior mean of on is the fixed-effects estimate (it, = T, — FIB. It is also the

case that the posterior variance matrix for B and or,,...,orN is the same as the classical

result for the variance matrix of B and dtl,...,dtN. For all of these reasons the name
Bayesian fixed effects model seems appropriate.

The posterior distribution of the inefficiency estimate u: or r: is potentially
complicated, but is easily revealed by Monte Carlo draws from the multivariate normal
posterior distribution of or,,...,orN. For example, confidence intervals are easily

constmcted from the percentiles of these draws. These are the same confidence intervals

that would be constructed by a classical econometrician via a simulation from the

25

estimated distribution of (itl ,...,dtN . We suspect that they will also often be similar to the

confidence intervals constructed by bootstrapping the fixed effects estimates. They will

differ only to the extent that the empirical distribution of the residuals 9,, is not similar to

the distribution of i.i.d. normals (which would reflect a failure of the assumed model).
An important point, stressed by KOS (1997), is that the fixed effects model favors

low efficiency. An uninformative (flat) prior for or, ,...,DIN implies an uninformative (flat)

prior for u:, but an informative prior for r: = exp(-u,‘). More specifically, if u has

constant density on [0,30), then r = exp(-u) has density proportional to r'1 on (0, 1]. This is
an improper (prior) density that loosely speaking puts infinitely more weight on low
values of r than high ones; for any constant c in (0,1), no matter how small, there is
infinite weight on r in (0,c) but finite weight on c in (c,1]. One could argue about whether
this reflects a problem with this specific prior, or with improper priors in general, but in
any case it implies that we should expect the Bayesian fixed effects [model to yield
smaller poSterior efficiencies than a model with more or less any (proper) informative
prior. In a sense this fact is the Bayesian counterpart to the finite sample bias problem
discussed in section 2.2. Whether treated in a classical or Bayesian way, the fixed effects
model will tend to yield smaller efficiency values than models that assert a distribution

for inefficiency.

3.3 Bayesian Random Effects Models
In this section we consider models that have an informative prior for ui. This

allows us to distinguish u; from the overall intercept 0t, and so now we can estimate

absolute inefficiency (ui) instead of just relative inefficiency (ur). Thus the parameters of

26

the problem are or, B, 65, u,,...,uN and (l), where d) (which is present only in some

models) represents parameters in the distribution of u. In all cases we take the likelihood

p(YlB) to be the same normal likelihood as in the fixed effects case. In all cases we use

the uninformative prior for (1, B and 63: p(or, B, 03) cc of. When (b does not exist,
u,,...,uN is prior independent ofa, B, of, . When (it does exist, (1) and u,,...,uN Id) are prior
independent of or, B, o: .

The models we consider all assert in one way or another that u follows an
exponential distribution with mean 2., so that p(uilk) = A“exp(-ui/A). They differ in how it
is treated.

The first model we consider postulates an uninformative prior for r. More
precisely, the r, (i=1,...,N) are i.i.d. as uniform on (0,1]. This uninformative prior for r
implies an informative prior for u = -ln r; the u. are i.i.d. with density proportional to
exp(-u;), so that ui is exponential with A = 1. For this model, because the value of A is
specified, there are no nuisance parameters (lb) in the distribution of u.

Our second model differs from the first only because it uses a different value of A.
The first model implied prior median efficiency of 0.5, which seems low for at least some
applications. The second model chooses A to imply prior median efficiency of 0.8. This is
achieved by picking A = -ln(0.8)/ln(2) = 0.322.

Our last two models differ from the first two in that A is now treated as a
parameter. That is, we specify a hierarchical prior in which conditional on A the u. are
i.i.d. as exponential with mean parameter A, and then we specify a prior for A. Thus " " in

the notation above now corresponds to A, the nuisance parameter in the distribution of ui.

27

Whereas the u. are mutually prior independent conditional on it, their dependence on a
common value of 2. implies that unconditionally they are not prior independent. KOS
(1997, p. 86) refer to this as the "common efficiency distribution" or CED model.

For our third model, we specify an "uninformative" prior for A: p(A) 0c A'l. For
our fourth model, we follow KOS (1997) and assume that A" is exponential with mean
-1/ln(rmed), where rmcd is the specified prior median efficiency. We take rmed = 0.8 as
above (whereas they used 0.85). As noted by KOS, this hierarchical prior implies that the
prior distribution of u; is three-parameter inverted beta, but this model differs from the
model (which they call the MIED model) in which the u, are i.i.d. as three-parameter
inverted beta, because in the present model the u. are dependent due to the common value
of A.

For each of the above models, the specification of the prior and of the form of the

likelihood implies the form of the posterior, p(a,B,o§,ul,..,uN,¢|Y). In principle,

inference on u; would be conducted based on its marginal posterior, p(ule), but the
integrals needed to construct the marginal posterior analytically are intractable.
Numerical integration techniques such as those used by Broeck et a1. (1994) in the cross-
sectional case are likely to be impractical in the present setting due to the dimensionality
of the integral. We will follow Koop, Steel and Osiewalski (1995), KOS (1997) and
Osiewalski and Steel (1998) in using Gibbs Sampling to make Monte Carlo draws from
the joint posterior. These draws then reveal the posterior distribution of the parameters
such as u.; in particular, confidence intervals for u; are easily constructed from the

percentiles of the Monte Carlo draws. This is the same principle as was followed for the

28

Bayesian fixed effects model, except that there Gibbs Sampling was unnecessary because
of the simple form of the joint posterior.

Gibbs sampling is a general procedure that makes draws from a joint distribution
by making iterated sequential draws from the conditional distributions. It is useful in
cases like the present one in which the conditional distributions are much simpler than the

joint distribution, so that we know how to make draws from them. We split the set of

parameters into three subsets: (a,B,o§ ), (u,,..,uN) and 4). Starting from some arbitrary

starting values, say (or,B,03 )(O), (u,,..,u_.,)(°) and 6“”, we generate random draws in

sequence from the conditional distributions, and then we iterate. Thus, at step j, make the

following draws:
«1.13.63 )0) from p(am: Mu. ,...u.)“'”,¢“"’)
(u,,..,uN)(j) from p(u,,..,uN|Y,(0t,B, oi )U'”,¢0'l))
i“) from p(<l>lY,(a,B, oi >0").(u.....u.)“‘”) .
For large enough s that convergence has occurred, (or,B,oi )(S), (uI ,..,uN)‘s) and (1)“) can

be treated as draws from the joint posterior. The reader is referred to Dorfrnan (1997) for
more discussion of Gibbs Sampling, and to KOS (1997, Appendix) for the forms of the

conditional distributions needed in the present case.

4. Empirical Results
We now proceed to apply the classical and Bayesian procedures described above
to three previously-analyzed data sets. These data sets were chosen to have rather

different characteristics. The first data set consists of N=171 Indonesian rice farms

29

observed for T=6 growing seasons. We have 0: = 0.108 and of, = 0.007 (these values
being the exponential MLE's). Inference on inefficiencies will be very imprecise because

T is small and because 03 is large relative to of. The second data set consists of N=10
Texas utilities observed for T=18 years, with of = 0.003 and o; = 0.020. For this data set
we can estimate inefficiencies much more precisely because T is larger and of, is smaller
relative to 03. The third data set consists of N=25 Egyptian tileries observed for a

maximum of T=22 production periods, with of, = 0.113 and of, = 0.057. This is a case

that is intermediate between the other two. We will see that the precision of estimation of
the efficiency levels will indeed differ strikingly across these data sets, and that choice of

technique will matter more where precision is low.

4.1 Indonesian Rice Farms

These data are due to Erwidodo (1990) and have been analyzed subsequently by
Lee (1991), Lee and Schmidt (1993), Schmidt and Horrace (1996, 1999) and others.
There are N=17l rice farms and T=6 six-month growing seasons. Output is rice in
kilograms and inputs are land, labor, seed and two types of fertilizer. The functional form
is Cobb-Douglas with some dummy variables added for region, seasonality, and some
types of farming practices. For a complete discussion of the data see Erwidodo (1990).

Table 1 gives point estimates for the regression parameters for the fixed effects
model (within estimates); for the classical MLE's based on the half-normal and
exponential distributions for u; and for our four Bayesian models. For the Bayesian

models the point estimates are the posterior means and the standard errors are the square

30

roots of the posterior variances. Here and subsequently the heading "uninformative"
refers to our third Bayesian model, with an uninformative prior for the exponential
parameter; "hierarchical" refers to our fourth model, with an exponential prior for the
inverse of the exponential parameter; and "7.=.322“ and "1:1." refer to our second and
first models, with exponential priors with specified values of A, so as to imply median
prior efficiencies of 0.8 and 0.5 respectively. The results in Table l are quite similar
across techniques. If we were primarily interested in the regression parameters, as
opposed to the firm-specific efficiency levels, it would really not make much difference
which technique we picked.

Table 2 gives point estimates and 90% confidence intervals for the relative
efficiency measures rl', for the classical and Bayesian fixed effects models. For the

classical fixed effects model, we give the usual point estimate based on the within
estimates; the MCB and MargCB confidence intervals; and confidence intervals based
on three versions of the bootstrap. We used the percentile method and the iterated (two
stage) percentile method, and also the bias-adjusted and accelerated bootstrap (labelled
BC.) The bias-adjusted bootstrap (without acceleration) gave very similar results to the
bias-adjusted and accelerated bootstrap so we do not report them separately. We used
B=1000 bootstrap replications, except that for the iterated bootstrap we used only
B=B2=200 replications, to shorten the computational time. (For the other two data sets,
which are smaller, we used B=B2=1000 replications for the iterated bootstrap.) For the
Bayesian fixed-effects model, we report the posterior mean and the 90% confidence

intervals based on the appropriate percentiles of the simulated posterior distribution.

31

There are 171 firms and so we report results only for a few of them. We report
results for the three firms (164, 118 and 163) that are most efficient; for the firms at the
75th percentile (31), 50th percentile (15) and 25th percentile (16) of the efficiency
distribution; and for the two worst firms (117, 45). All of these rankings are according to
the classical fixed effects estimates.

In terms of the point estimates of efficiency levels, the classical and Bayesian
fixed effects estimates are relatively similar. The Bayesian estimates are a little lower,
especially for the most efficient firms. The efficiency estimates overall are rather low,
with median efficiency only a little over 0.5. This is as expected.

We next discuss the confidence intervals for efficiency levels. The MCB and
MargCB intervals are quite wide, especially for the less efficient firms. In fact, they
really are too wide to be of much use. The bootstrap and Bayesian intervals are narrower,
but still disappointingly wide. The percentile method of the bootstrap and the Bayesian
fixed effects model give intervals that are quite similar. As noted above, this is as
expected. With only T=6 observations per firm, the accuracy of the percentile bootstrap is
suspect. No such statement applies to the Bayesian method, but with only T=6
observations per firm, the prior is certainly not dominated by the data, and since the prior
is arguably unreasonable, so are the posterior results. As expected, the iterated bootstrap
intervals are wider than those from the percentile method, and the bias-adjusted and
accelerated bootstrap confidence intervals are shifted to the right (in the direction of
higher efficiency levels). Simulations reported in Chapter 3 suggest that the bias-
adjustment is helpful, but with so few observations per firm all of the bootstrap methods

are probably unreliable.

32

Table 3 gives the estimated efficiencies and the associated 90% confidence
intervals for our random effects models, including the classical MLE's based on the half
normal and exponential distributions and our four Bayesian models with informative
priors. We will not discuss the half normal results other than to note that they are not too
different from the exponential. All of the confidence intervals are disappointingly wide,
as they were for the fixed effects models of Table 2. However, comparing Tables 2 and 3,
it is apparent that the efficiency levels from the random effects models of Table 3 are
considerably higher than those from the fixed effects models. As noted above, from the
classical point of view this is a reflection of the bias in the fixed effects efficiency
estimates, while from a Bayesian point of view it reflects the influence of the underlying
prior which is very heavily weighted toward low efficiency levels.

The exponential MLE and the Bayesian model with an uninformative prior for the
exponential parameter give strikingly similar results. It is not surprising that the results
are similar but it is perhaps unexpected that they are so similar. The Bayesian model with
2:1 has a lower prior median efficiency (equal to 0.5) than the hierarchical model or the
model with 7t=0.322 (both of which have prior median efficiency of 0.8) and
correspondingly has lower posterior efficiencies. The choice of prior matters a fair

amount, which is a reflection of the small amount of data (T=6) per firm.

4.2 Texas Utilities
We next consider the Texas utility data of Kumbhakar (1996), which was also
analyzed by Horrace and Schmidt (1996, 1999). There are N=10 privately owned Texas

electric utilities observed for T=18 years. Kumbhakar estimated a cost function, whereas

33

we will estimate a Cobb-Douglas production function. Output is electric power generated
and the inputs are measures of labor, capital and fiiel. For more details on the data see
Kumbhakar (1996).

Table 4 gives the regression parameter estimates. We will not comment on these
except to note that the variance of the one-sided error is large relative to the variance of
noise (8. g. 0.020 vs. 0.003 for the exponential MLE). This is the opposite of the case for
the Indonesian rice farms. For this reason, and because T is larger here (18 vs. 6), we
expect more precise estimates of efficiency levels and less sensitivity of the results to the
choice of method for this data set than for the previous one.

Table 5 gives the estimated efficiencies and 90% confidence intervals for the
fixed effects models, while Table 6 gives the same results for the random effects models.
The format is the same as for Tables 2 and 3, except that we can display results for all
N=10 firms. We can see the same patterns here as we did for the previous data set
(though less distinctly since differences across techniques are smaller). The MCB and
MargCB intervals are wider than the other intervals. The fixed effects models give lower
estimates of efficiency levels than the random effects models. Comparable classical and
Bayesian models given comparable results: the classical fixed efi‘ects results with the
percentile bootstrap are quite similar to the Bayesian fixed effects results, and the results
from the classical exponential MLE are quite similar to those from the Bayesian model
with an uninformative prior for the exponential parameter. Bayesian models with higher
prior efficiency levels yield higher posterior efficiency levels.

However, we repeat that the main result of interest is the general comparison of

the results for this data set with those from the previous data set. For this data set we can

34

estimate efficiency levels precisely enough to make reasonable statements about them,
and the choice of technique is not critically important. The most important aspect of the
choice of technique is the choice of a fixed versus random effects model. The fixed
effects models give lower efficiencies, and are suspect forthe reasons discussed in
sections 3.2 and 4.2 above. The gain from being willing to assert a distribution for
inefficiency is large, and at least for this data set there is reasonable robustness to the

choice of distribution.

4.3 Egyptian Tileries

The last data set we consider is for Egyptian tileries. It was collected by Scale
(1990) and has been analyzed by Horrace and Schmidt (1996, 1999). There are N=25
small-scale Egyptian manufacturers of ceramic floor tiles, with observations for a
maximum of T=22 three-week production periods. There are some missing data points
(production did not occur in some periods) and so this is an unbalanced panel. Output is
square meters of tile, while inputs are labor and machine hours.

The results are given in Tables 7, 8 and 9. As with the Indonesian rice farms, we
present results only for a subset of the firms, choosing firms at the same percentiles of the
efficiency distribution as in section 4.1. We will not discuss these results in detail, but the
same comparisons across techniques that held for the other two data sets hold here as
well. Comparing results across data sets, the results here are more precise, and less
dependent on choice of technique, than for the Indonesian rice farms; they are less
precise, and more dependent on choice of technique, than for the Texas utilities. This is

predictable because, both in terms of the sizes of N and T, and also in terms of the

35

relative variances of noise and inefficiency, this data set has characteristics that are

intermediate between those of the previous two data sets.

5. Concluding Remarks

In this chapter we considered a large number of classical and Bayesian procedures
to estimate technical efficiency levels of firms and to construct confidence intervals for
these efficiency levels. We then applied these methods to three data sets with different
characteristics that determined the difficulty of the estimation problem. Comparing
results across data sets and across methods within a data set leads to some clear and
important conclusions.

First, the estimation problem is easier when T is large and when the variance of
noise is small relative to the variance of inefficiency, and harder when T is small and
when the variance of noise is large relative to the variance of inefficiency. In easier
problems we can estimate efficiency levels more precisely than in harder problems, and
there is less sensitivity of the results to the choice of technique.

Second, we do not find much difference between classical and Bayesian methods
if we match methods that depend on comparable assumptions. For example, the Bayesian
fixed effects model gives similar results to those obtained by the percentile bootstrap
applied to the fixed effects (within) estimates. As another example, the classical MLE
based on the exponential distribution gives similar results to the Bayesian model in which
the prior distribution for inefficiency is exponential, and there is an uninformative prior
for the exponential parameter. Furthermore, the two approaches face a similar problem,

in that the results may be "unreliable" when T is small. In the classical framework,

36

"unreliable" means that asymptotically valid inference may not be valid in small samples,
while in the Bayesian framework "unreliable" means that the prior will not be dominated
by the data and so there is a lack of robustness to the choice of the prior. We do not mean
to allege that these are the same problem, but simply that in either approach small T
causes problems.

Third, the multiple comparisons with the best (MCB) and marginal comparisons
with the best (MargCB) intervals are wider than any of the others we consider. This is not
a desirable feature in a confidence intervals, but on the other hand these intervals are
valid for small T, and a conservative, valid interval at least can provide the correct
message that in some cases we don't know very much.

Finally, the main difference in results is between fixed effects and random effects
models. Fixed effects models (either classical or Bayesian) yield much lower efficiency
levels than random effects models, and there are good reasons to be skeptical of the fixed
effects results. From a classical point of view, the fixed effects estimates of efficiency
levels are biased downward; from the Bayesian point of view, the fixed effects model
embodies a prior that is unreasonably heavily weighted toward low efficiency levels.
Random effects models require a distributional assumption for inefficiency, which may
be unattractive. However, making such an assumption yields large dividends in terms of

precision of estimation and in terms of more reasonable average levels of efficiency.

37

Table 1

Estimates of Parameters : Indonesian Rice Farm

 

 

RE model
Variables FE MLE Bayesian
Model ...................................................................................
Half-nonnal Exponential Uninformative Hierarchical A=.322 =1
Constant 5.199 5.181 5.187 5.187 5.319 5.419
(.194) (.193) (.193) (.195) (.206) (.220)
Seed .121 .134 .135 .135 .135 .129 .125
(.030) (.027) (.027) (.026) (.027) (.028) (.029)
Urea .092 .113 .113 .113 .113 .102 .096
(.021) (.018) (.018) (.018) (.018) (.019) (.020)
TSP .089 .076 .076 .076 .077 .084 .088
(.013) (.012) (.011) (.012) (.011) (.012) (.012)
Labor .243 .219 .217 .217 .217 .224 .234
(.032) (.029) (.029) (.029) ( .029) (.031) (.031)
Land .452 .481 .483 .483 .483 .477 .466
(.035) (.031) (.031) (.030) (.031) (.033) (.035)
DP .034 .009 .008 .008 .008 .013 .020
(.03 2) (.029) (.028) (.029) (.029) (.029) (.031)
DVl .179 .176 .176 .176 .176 .177 .177
(.041) (.038) (.038) (.038) (.039) (.038) (.040)
DV2 .175 .140 .136 .138 .138 .149 .161
(.057) (.052) (.052) (.052) (.052) (.054) (.055)
DSS .053 .049 .049 .050 .050 .050 .052
(.022) (.021) (.021) (.022) (.022) (.021) (.021)
DRl - .058 -.059 -.060 -.060 -.093 -.125
(.049) (.048) (.049) (.050) (.072) (.109)
DR2 - .047 -.045 -.046 -.046 -.075 -.112
(.058) (.057) (.057) (.059) (.076) (.105)
DR3 - .078 -.077 -.078 -.078 -.126 -.176
(.062) (.060) (.062) (.062) (.080) (. 110)
DR4 .016 .021 .018 .020 -.004 -.025
(.058) (.056) (.057) (.059) (.082) (.117)
DRS .082 .089 .088 .090 .092 .085
(.060) (.059) (.059) (.061) (.079) (. 110)
of .108 .108 .108 .110 .110 .106 .106
(.005) (.005) (.005) (.005) (.005) (.005)
of .007 .007 .007 .006
(.003)
x .081 .084 .089
(.019) (.019) (.018)

 

38

 

 

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46

Chapter 3

SIMULATION EVIDENCE ON THE ACCURACY OF
INFERENCES IN STOCHASTIC FRONTIER MODELS

1. Introduction

In the previous chapter, we considered a large number of different models, and we
discussed the relationship between the assumptions underlying the various models and
their empirical results. We found that making a distributional assumption for inefficiency
led to much tighter confidence intervals. However, it is important to know whether these
confidence intervals are accurate. For example, it is clear empirically that MCB and
MargCB confidence intervals are considerably wider than intervals based on
bootstrapping the fixed effects estimates or based on a distributional assumption, but it is
not known how conservative MCB and MargCB are, whether the bootstrapping or
distribution-based estimates are reliable. In this chapter, we conduct Monte Carlo
simulations to investigate the accuracy of the confidence intervals based on the methods
of the previous chapter

Monte-Carlo analysis of the stochastic frontier model can be found in Olson,
Schmidt and Waldman (1980), Coelli (1995), and Gong and Sickles (1989,1992). The
first two papers mainly focus on the comparison between corrected OLS and MLE in the
estimation of the regression parameters in a stochastic frontier model with cross-section
data. They find that the performance of each method in estimating the constant term and

the variance parameters depends on the relative size of the variance of inefficiency and

47

the sample size. Coelli extended the experiment to investigate the finite sample prOperties
of several tests of the existence of inefficiency.

Gong and Sickles conducted a Monte Carlo analysis to examine the relative
strengths of the stochastic frontier model and data envelopment analysis using panel data.
They considered three alternative techniques in estimating the stochastic frontier model;
MLE, corrected GLS, and the fixed effect estimator. The main focus was on investigating
the robustness of the efficiency estimates over different functional forms and different
underlying assumptions.

Our Monte-Carlo experiment is designed to investigate the relative performances
of different methods in performing inference on efficiencies in the stochastic frontier
model using panel data. The overall design of the experiment is similar to that of Olson et
a1. and Coelli. However, our study is differentiated from those two papers in that our
study considers a panel data model and more importantly our main interest is in the
efficiency estimates instead of the estimates of regression parameters. Our study also
differs from Gong and Sickles’s paper in two aspects. First, they evaluated the
performances of the different estimation methods based on the correlation coefficients
between true efficiencies and their estimates, and between true ranks and their estimates.
These criteria are useful only when our interest is in the comparison of firms in a certain
group. We focus on the evaluation of the each method in terms of the bias and accuracy
of interval estimates (confidence intervals). This allows us to make a useful statement on
the performance of each method in describing the efficiency level of a specific firm. The
other difference lies in the estimation techniques of main interest. Even though Gong and

Sickle considered three estimation techniques, inefficiency is estimated in the same way

48

in all three cases. A distribution of inefficiency was assumed to estimate frontier
parameters in MLE and to correct the constant term in corrected GLS, but the
information on the distribution of inefficiency was not used at all to estimate the firm
specific inefficiencies. The firm specific inefficiency was just defined relative to the
sample best from the mean over time of the estimated residuals, after the frontier
parameters were estimated by the different techniques. Then, the difference of three
techniques in estimating the inefficiency only comes from their relative performance in
estimating the fi'ontier parameters. Our interest here is in the comparison of techniques
that estimate inefficiencies in fundamentally different ways. In particular, we want to
compare estimation of inefficiencies using a distributional assumption to estimation of
inefficiencies in terms of the comparison of a given firm to the sample best.

Section 2 briefly introduces the model on which our study is based. Section 3
describes the design of the experiments. The experimental results and their implications

are discussed in section 4.

2. The Model
As in the previous chapter, we consider the basic panel data stochastic frontier

model as follows.

(3.1) Y. =0t+x:,B+eit where 8,, = v, —u and u. 2 0

We want to focus our attention on two estimation methods, MLE and the fixed effect
method (FE method hereafter). For ML estimation, the first thing to do is to assume a

specific distribution of u; in which case the U. can be estimated in the absolute sense. The

49

most frequently used specifications for U5, in both the theoretical and empirical literatures,
are the half-normal distribution and the exponential distribution. The truncated normal
distribution has also been frequently used as a generalization of the half normal
distribution. The gamma distribution was suggested as a generalization of the exponential
distribution, but its empirical application has been restricted by the complexity in its
estimation procedure. Here, we choose a half- normal distribution due to its popularity as
well as the simplicity of the estimation procedure. We have no evidence for other
distributions, but there is no reason to think that our results depend critically on special
features of the half normal distribution.

The point estimates and confidence intervals for inefficiencies can be calculated

from the distribution of ui conditional on a; following Battese and Coelli. If the
distribution of ui is N(ll. of) truncated from the left at zero, the distribution of u;,

conditional on 8;, is N(5, 0.2) truncated from the left at zero, where 5 and 0.2 are defined

as follows.
2 2— 2 2
(3 2) 5: no, —Toue 02 = ovau
' 2 2 ’ " 2 2
o, +Tcru <5v + Tou

The half normal distribution corresponds to the case of p.=0. To summarize, the
estimation of u; proceeds in two steps. First, we estimate the frontier parameters by MLE.
Next, the point estimate and confidence intervals for L1. are calculated from f(u,lsi)
evaluated at é,. The point estimate is the mean of the conditional distribution, E(uilei),

and the confidence intervals are based on the percentiles of the conditional distribution.

We call this the MLE & BC method.

50

In the fixed effect model, technical inefficiency is regarded as a nonnegative fixed
effect. The basic stochastic frontier model (3.1) is the standard panel data model with

time-invariant individual effects.
(3'3) Yul :al +X:IB+VII Where ul :a—al

a, can be estimated without a distributional assumption as the mean residual in the ith

A

group, i.e., a, = y, 4:5 where B is within estimator of B. a, consistently estimates

0tl as T—)oo. Inefficiencies can be estimated relative to the sample best, i.e.,

A. A

ui = max on J. —dtl. When we define technical inefficiencies in this way, the sample best
1:1,..."

firm is treated as 100% efficient. The justification for this comes from the fact that the
sample maximum approaches the population maximum as N—-)or>, so that for large N
inefficiencies can be measured relative to an absolute standard. For the confidence
statements in the FE model, two methods are employed. One is the marginal MCB
(MargCB) technique and the other is bootstrapping. The details of marginal MCB and
bootstrapping can be found in Chapter 2 and Chapter 4. These two methods will be called

FE & MargCB method, and FE & Bootstrap method.

3. Design of the Experiments
We consider the model with no regressors so that we can concentrate our interest

on the estimation of efficiencies without having to be concerned about the nature of the
regressors. In practical cases the regression parameters [3 are likely to be estimated so

much more efficiently than the other parameters that treating them as known is not likely

51

to make much difference. Then, the parameter space reduces to (on,03,aj, N, T).

Without loss of generality, we can fix the constant term or to any number, since a change
in the constant term only shifis the estimated constant term by the same amount, without
any effect on the bias and variance of any of the estimates; For simplicity, we fix the
constant term equal to one.

We need two parameters to characterize the variance structure of model. It is
natural to think in terms of oi and 03 . Alternatively, recognizing that 03 is the variance

of the untruncated normal from which u is derived, not the variance of u, we can think

. . a TC ‘ 2 1 .

instead in terms of o; and var(u), where var(u)= (————)o;. However, we obtain more
7:

readily interpretable results if we think instead in terms of the size of total variance and

the relative allocation of total variance between v and u. The total variance is defined as

of =03 +var(u). Olson et al. used 7L= 0“

 

to represent the relative variance structure, so
a

V

that their parameterization was in terms of a: and A. Coelli used a: and either 7 =

a: . _ var(u)

or — 2 . The choice among 2., y, and y'is a matter of convenience.
o, + var(u)

0'2 +0:

We decided to use 7' due to its ease of interpretation, so that we will use the parameters
of and y'. The reason this is a convenient parameterization (compared to the "obvious"
choice of of, and 03) is that, following Olsen et al., one can show that comparisons

among the various estimators are not affected by the value of total variance 0:. The

effect of multiplying of by a factor of "k" holding y' constant is as follows.

52

a) constant term: bias changes by a factor of Jk and variance changes by a factor of k

b) 03 and o: : bias changes by a factor of k and variance changes by a factor of k2
c) y'(or A or y): bias and variance are unaffected.
We set a: at 0.25, arbitrarily. Thus the only parameters lefi to consider in the

experiment are N, T, and y'. We consider three values of y' to include a case in which
the variance of v dominates, a case in which the variance of u dominates, and an

intermediate case. We take 7': O. 1, 0.5, and 0.9 to represent the above three cases. With
of =O.25, of , a: and var(u) are determined as follows for each value of y'.

a) 7‘: 0.1 : 0: =0.25, of =o.225, var(u) =0.025, of =0.069

b) y‘= 0.5 : of =o.25, 0,2 =0.125, var(u) =0.125, of =0.344

c) y'= 0.9 : of =O.25 of =0.025, var(u) =O.225, 03:0.619

Four values of N and T are considered. In order to investigate the effect of
different size of N, we allow N to vary among 10, 20, 50,and 100, while T is fixed at 10.
Similarly, we allow T to vary among 10, 20, SO, and 100, while N is fixed at 10 to

investigate the effect of different size of T. This is done for each of the three different
values of 7'.

For a given set of parameter values, in each replication we generate the (NT x1)
vector v from N(0,03) and generate the (le) vector u from u=|U|, U~N(O, a“: ). Then,
the NTxl vector y can be calculated from the model: yit =0t+vit -u,, and the point
estimates and confidence intervals are calculated from the data. We consider two

estimation techniques: MLE and the FE method. MLE calculates the estimates of or, of

53

and a: by maximizing the half- normal likelihood function, and the point estimates and
confidence intervals for ui are calculated conditional on the estimated value of or, of and

of. MLE is based on a correct distributional assumption and might be expected to
perform better than other techniques.

In the fixed effect method, the N individual effects a1, ...... a N and of, are
estimated first, and the overall constant term and firm specific inefficiencies are

estimated by a: maxdtj and 0:: axdtj- (1,. For the bootstrapping intervals, the
j=1...,N 1...,N

.-
ll

percentile intervals and bias corrected and accelerated (BCa) intervals are calculated. The
iterated bootstrap is not considered due to its computational complexity.

By repeating this process a large number (say, m) of times, we obtain the
distribution of the estimates. We calculate the means and standard errors of the point
estimates, and the coverage probability of the confidence intervals is estimated by the
fraction of times coverage occurs. For the FE & MargCB method, we use m=500 due to
the relatively simple computations required, while we use m=300 for the MLE & BC
method, and the FE & Bootstrap method. While we obtain point estimates and confidence
intervals for each of the N cross-sectional units, we just report our results averaged over

these N units.

4. Results

4.1 Estimation of Frontier Parameters

54

We begin our discussion of results by looking at the performance of the two

estimation techniques in estimating the frontier parameters oi, 03, and of: . Tables 1 and

2 provide the bias and standard error of the estimates.

Consider first the MLE estimates. The asymptotic theory for the MLE holds as
N—>oo, and our results confirm the fact that the MLE performs better for large N. In
particular the biases of the estimates disappear as N gets large, not as T gets large. It is

noteworthy that (it is biased downward, and so is 6:. This result was also found by

Olson et al. (1980), but they had no explanation for it. However, based on the analysis of
specific replications within our experiments, it appears that the bias occurs because the
exact truncation point in the distribution of u; is hard to estimate with small N. The
estimation process can be regarded as allocating ya among constant term, V“, and u, In
fact, it is not hard to distinguish the inefficiency term from the normal error even with
small N, because we assume a half normal distribution for u; that has a very different
shape from a normal error. But a problem exists in that the constant term can not be

distinguished easily from the inefficiency terms with small N. With small N, especially
when a: is big, the minimal value of ui is not close to zero. In this case, some portion of

u; is implicitly allocated to the constant term. As a result, u; tend to be underestimated and
the constant term tends to be underestimated as well (due to the negative sign in front of
ui). This explains why substantial biases in (it are found for the same parameter values as
substantial biases in the estimate of var(u), since the same logic explains the downward

bias and its decrease with N and increase with y' in the estimation of var(u).

55

The influence of y'on the estimation of the constant term depends on the size of
N. With small N, the minimal value of u, is not close to zero and it tends to be large with

bigger of. This explains why the bias of (it increases as 7' increases, at least when N is

small. With N big enough to accurately identify the location. of the distribution of u, 7°
does not have a significant influence on the bias of (it. The effect of T on the bias of d is

also closely related to the size of N and 7'. When N is big enough, the bias of or
decreases with T. Otherwise, the effect of T on the bias of (2 depends on the size of y’.
The bias decreases as T increases for small values of 7' while the opposite happens for

large values of 7'. Meanwhile, the standard error of or decreases as either N or T
increases, but the effect of N is much stronger.

Table 1 and 2 also provide the bias and standard error of the FE estimators. The
bias in the FE estimation of the constant term depends on two factors. First, the bias

comes from the fact that the sample minimum of ui is not zero. Clearly max on I:
H...”

or—“minM u j. We implicitly regard min u J. as zero by defining d=maxdtj. So, when
- i=1,...n 131...):

n‘liIL u 1. is greater than zero, it results in a downward bias in (it. This bias goes away as

i=

N——>oo and as 311“: u J. —) O. For small N, it also depends on the value of a: , since min u 1.
,,,, j=1,...N

tends to be larger for the larger values of of .

The second and more important source of bias is the fact that maxdtj is a biased
)=1....N

estimator of max 0t 1. in finite samples. For fixed N, maxétj is a consistent estimate of
i=1...” 1.1...»

mafia]. as T —>oo. However, for finite values of T, max 6t 1 is biased upward as an
i=1... i=1,...N

56

estimator of max or J. This bias is large when several of the largest 0LJ ’s are close each
1:1,...N

other (i.e., there are near ties for max on J) and the estimation error of (3tl is large.
l=t.«-N

The first source of bias decreases as N increases while the second source of bias is

large when T is small and 63 is large. The second bias also tends to be large with large

N, because near ties for max or J. become more likely, and because the bias induced by the
#1....N

"max" operator is larger when the maximization is over a larger set of a,. The final size
and sign of the bias depend on the relative size of each source of bias. For example, (it
has a negative bias at N=10, T=lO, and y°=0.5, due to the dominance of the first factor,

but it changes to a positive bias as N increases and the bias due to the first factor

disappears.

4.2 Estimation of Efficiencies: MLE & BC method

We now turn our attention to the estimation of efficiencies. Table 3 provides the
bias of the estimated inefficiencies, and the coverage rate of the confidence intervals.
These are both averages over values i=1,...,N. As is easily noticed, the bias of estimated
inefficiency is just a reflection of the bias of (it, as explained in the previous section. The
90% confidence intervals are calculated according to the BC method. Column (3) in
Table 3 shows the average width of the confidence intervals. These confidence intervals
are for r = exp(-u) and are just a transformation of the confidence intervals for u. The
coverage rates of the intervals are the same whether they are expressed in terms of u or r,
but since 05 r 51 the width of the confidence interval for r is easier to understand than the

width of the confidence interval for u. Columns (4) and (5) show the fiequency that the

57

true efficiency falls below the lower bound and above the upper bound of the confidence
interval. For valid intervals each of these should equal 0.05. Column (6) shows the
empirical coverage probability of the 90% confidence interval. The sum of the numbers
in columns (4), (5) and (6) is always one.

The sample distribution of the efficiency estimates is not centered on the true
value due to the downward bias in the estimation of inefficiency (or the upward bias in
the estimation of efficiency). Thus, the probability of the true efficiency level to be
smaller than the 90% lower bound is much greater than 0.05. However, it is also true that
the probability that the true value is greater than the upper bound is a little larger than
0.05. It appears that the confidence intervals are too narrow in general (even not
accounting for the bias problem). This occurs because the confidence intervals according
to the BC method fail to incorporate the uncertainty in the estimated frontier parameters.

As a result of both of these problems, the coverage rate is less than 0.9, and is as low as
0.29 in the extreme case of N=10, T=100, and y°=0.9.

The main determinant of the accuracy of the confidence intervals is the size of N.

When N is large (e.g., N=100) the coverage rate of the intervals is reasonably close to
0.90 (e.g., 0.884 for N=100, T=10, y'=0.5). This is so because both of the problems

discussed in the previous paragraph go away as N increases.

At this point, we need to recall that we are assuming the true distribution is
known. Given this fact, the general performance of MLE is perhaps not as good as might
be hoped. A natural question is what happens if we do not use the correct distribution of
ui. Table 4 gives some results for the case that MLE is not based on the true distribution

of the data-generating process. The assumed distribution is half-normal, but the true

58

distribution of u is N(u, of; ) truncated at zero, with p.=0.5 or u=~2. A glance at the results

reveals that the performance is very bad for u=0.5. 6t and the estimated inefficiency
have much bigger bias than in the case with correct information on the distribution of u,.
Consequently, the coverage probability is very low.

For the case of u=-2, it seems that the incorrect assumption does not distort the
results much. Especially for small N, (St is less biased and the coverage rate is actually
higher than in the case assuming the correct distribution. This is presumably because the
values of u; are more concentrated near zero than when the sample is from a half-normal
distribution, so that the inferred truncation point is much closer to zero. Table 5 might
help to understand this mechanism. It shows the average sample minimum of u, for

samples from the different distributions. For u=-2, the average sample minimum is much
smaller than when p=0 or u=0.5. This makes it less likely that a is underestimated.

These results are certainly distribution-specific, but they indicate the potential for serious
bias and incorrect coverage probabilities if MLE is based on an incorrect distributional

assumption.

4.3 Estimation of Efficiencies : FE & MargCB or MCB method
Table 6 and Table 7 summarize the performance of the FE estimates of

efficiencies. Table 6 changes N holding T constant, while Table 7 changes T holding N

constant. In both cases different values of y' are considered. We first consider the bias of
the point estimates iiI. The bias in the estimation of the inefficiencies can be defined in

two ways, depending on whether fiI is viewed as an estimate of uI or ui. Column (2) in

59

Table 6 and Table 7 shows the bias of the estimated inefficiency as an estimate of u;. It
reflects the two sources of bias discussed above and it is just the mirror image of the bias

of (it. Column (3) shows the bias of the estimated inefficiency as an estimate of the

relative inefficiency defined by uI = ul - min u I. When we determine the bias in this
1:1,...N

way, it removes the effect of min uJ :0 and only reflects the bias from the second
j=1,...N

source, which is the "max" operation over i=1,...,N. Our focus should really be on the
second type of bias since GI is most naturally thought of as an estimate of uI. Thus, the
estimated inefficiency can not be directly compared with MLE estimates because MLE
estimates the inefficiency ui defined in the absolute (not relative) sense.

As discussed in the previous section, the upward bias induced by the "max"
operation disappears as T->00. Accordingly, the effect of T is most important in the
estimation of inefficiencies in the Fixed Effect model. The bias of estimated inefficiency
decreases as N decreases, since the ui’s tend to be less tied together with small N, and the

"max" operation is over less values of i. The bias also decreases as 7' increases, since

the or; are more distinct and are estimated more precisely. Columns (4)-(11) give the
width and coverage rates of the MargCB and MCB confidence intervals. The intervals are
very conservative-they are very wide and they have coverage rates in excess of 0.90.

In fact, the coverage probability is close to one for all cases. In other to
understand the reason, it would help to see how the best set is determined. Column (12)
in Table 6 and Table 7 shows the average percentage of firms included in the best set and
column (13) shows the probability that the best set includes the true best firm. A high

percentage of firms are included in the best set, which implies a high degree of

60

uncertainty about which is the most efficient firm. The results clearly show how
conservative the best set is. It includes the most efficient firm with a probability of almost
one, even though it is designed to include the most efficient firm with a probability of at
least 0.95. The MCB and MargCB coverage probabilities would be correct if the identity
of the best firm were known, and uncertainty about which is the best firm is reflected in

the size of the best set and in the excess coverage rates of the confidence intervals.

4.4 Estimation of Efficiencies : FE and Bootstrap method

MCB and MargCB provide very reliable but excessively wide confidence
intervals in general. The confidence statements would not be useful if they are too wide,
even if they are very reliable. We consider the bootstrap method as a possible alternative.
Table 8 reports the results for confidence statements from the bootstrap method. The first
four columns show the accuracy of the confidence intervals by the percentile method and
the next four columns show the accuracy of the confidence intervals bythe bias-corrected
and accelerated (BCa) percentile method. The bootstrap intervals are much narrower than
the marginal MCB intervals, but the question is whether they are reliable.

It was discussed in Chapter 2 that the bootstrap estimator of the distribution of

[maidtj-Etaxaj] is consistent in T if there are no ties for max or I. This implies that the
1... 1 ..JJ #1...”

validity of the percentile intervals depends on large T. For small values of T, the

percentile intervals are not centered on the true values due to the bias problem previously
discussed. Since the bias is small when we have large T and y' and small N, the coverage
probability reaches almost 0.9 for these cases, but it falls in cases where the bias is big.

The width of the intervals decreases as T or y' increases just as in MargCB or MCB. But,

61

the influence of N on the width of intervals is the opposite of that in MargCB or MCB.
The intervals get narrower with larger N, while the bias increases as N increases. This
explains why the coverage probabilities of the percentile intervals falls rapidly as N
increases.

The results in Table 8 indicate that the BCa percentile intervals are better than the

uncorrected percentile intervals. Like the uncorrected percentile intervals, they are more
accurate when T and y' are large and when N is small. When T and y' are small or N is

large, they are a very considerable improvement over the percentile intervals, even
though they do not succeed entirely in yielding correct coverage rates.
The bias corrected confidence intervals are obtained by shifting the bootstrap

distribution by approximately twice the estimated bias in the bootstrap stage. If on

(1))

average [max (it I - max 6t J.] were the same as [max 6t j-maxa j], we would expect a
3:1,..." l=1

121...)! ....N 3:1,...»
properly centered interval with a coverage rate of 0.9 after the bias is corrected. In our

model, however, only some part of the bias gets be corrected. Some evidence on this

point is given in Table 9, which shows the average of maxa , max 6t I, and max (if)

121...." 1 1:1,...» i=1...,u
over different values of N, T, and y'. The fourth column in the table shows the average

bias in the FE estimates of max or I and the last column shows the average bias in the
}=1....N

bootstrap estimates. We see that [max (in I‘m-max 6t J.] is always smaller than [max 6: 1'
j=1....N l=t..-.N i=1,...u

maxaj] and the difference is substantial when y’ is small and N is large. As a result, the

bias correction is incomplete especially when 7' is small and N is large.

62

5. Concluding Remarks

In this chapter, we have analyzed the finite sample properties of two techniques
for the estimation of efficiencies, where the main difference lies in whether the technique
uses a distributional assumption in estimating inefficiencies. The MLE & BC method,
which uses a distributional assumption in estimating inefficiencies, does well when N is
large but surprisingly poorly when N is small. When N is small, there is a large
downward bias in the estimated inefficiencies, and this results in improperly centered

confidence intervals with low coverage rates. This problem is especially severe when T

and y. are large because the confidence intervals tend to be narrow with large T and large

y’, while the bias is determined mainly by the size of N. However, there appears to be a
large gain in precision and reliability of inference fi'om making a distributional
assumption when N is large relative to T.

For the Fixed Effects model things are rather different. The FE model estimates
inefliciency relative to the sample maximum. The use of the "max" operation results in an
upward bias in the estimation of the frontier, and therefore in an upward bias in the
estimation of inefficiency (or a downward bias in the estimation of efficiency), which is a
bias in the opposite direction as for the MLE & BC method. The bias in the FE estimated
inefficiencies decreases as T or 7' increases and as N decreases, but it can be very
substantial when N is large and T is small.

MCB and MargCB based on the FE estimates yield very conservative confidence
intervals. They tend to be very wide and to have coverage probabilities that are much

higher than the nominal coverage probability. While MargCB and MCB do not suffer

63

from a bias problem, they are very wide in the same circumstances that the FE point
estimates are seriously biased.

Confidence intervals based on bootstrapping of the fixed effectes tend to be
improperly centered due to the bias due to the "max" operation. As a result the
bootstrapping confidence intervals are not very reliable in general, though they are
relatively reliable when T and y. are large and N is small. Using the bias-corrected
bootstrap helps in terms of the accuracy of the intervals. It performs relatively well
overall even though the bias-correction is not complete. Like the MargCB and MCB
intervals, it performs better with small N and large T and 7', but it provides much
narrower intervals than MargCB or MCB. The bootstrap method may be a reasonable
alternative to the MLE when N is small.

The most important limitation of these results is that they are for a simple model
without regressors. With regressors included, there are a number of additional factors to

consider, including the nature of the regressors and their correlation or noncorrelation

with the inefficiency term.

64

Table 1

Bias and Standard Error of Estimated Constant Term, of, and var(u)
(N Changes with T Held Constant)

 

 

MLE FE
Constant o: var(u) Constant of,
T=10 ..........................................................................................................
bias bias bias bias bias
(std.error) (std.error) (std.error) (std.error) (std.error)
. <1) (2) (3) (4) (5)
(7 =0.1)
N=10 -.039 -.003 -.003 .109 -.002
(.111) (.030) (.022) (.111) (.032)
N=20 -.016 -.002 -.001 .170 -.001
(.080) (.022) (.016) (.094) (.022)
N=50 -.010 -.001 -.001 .228 -.001
(.048) (.014) (.010) (.081) (.015)
N=100 -.002 -.001 .000 .271 -.001
(.033) (.010) (.007) (.070) (.010)
(1' =05)
N=10 -.042 -.002 - .010 -.016 -.001
(.112) (.017) (.068) (.105) (.018)
N=20 -.019 -.001 -.006 .050 ’ -.001
(.074) (.012) (.050) (.086) (.012)
N=50 -.010 -.001 .000 .107 -.001
(.047) (.008) (.030) (.070) (.008)
N=100 -.005 -.001 .000 .148 .000
(.032) (.005) (.021) (.064) (.006)
(1' =03)
N=10 -.049 .000 -.028 -.074 .000
(.095) (.003) (.097) (.087) (.004)
N=20 -.020 .000 -.013 -.029 .000
(.059) (.002) (.070) (.063) (.002)
N=50 -.008 .000 -.006 .015 .000
(.034) (.002) (.048) (.040) (.002)
N: 100 -.003 .000 -.003 .038 .000
(.024) (.001) (.033) (.034) (.001)

 

Table 2

Bias and Standard Error of Estimated Constant Term, of and var(u)
(T Changes with N Held Constant)

 

 

MLE FE
Constant oi var(u) Constant of,
N=10 ...........................................................................
bias bias bias bias bias
(std.error) (std.error) (std.error) (std.error) (std.error)
(1) (2) (3) (4) (5)
(1' =0-l)
T=10 -.039 -.003 -.003 .109 -.002
(.111) (.030) (.022) (.111) (.032)
T=20 -.028 -.002 -.004 .050 -.001
(.086) (.021) (.016) (.080) (.022)
T=50 -.023 -.001 -.004 .008 -.001
(.055) (.013) (.013) (.056) (.014)
T=100 -.014 -.001 -.003 -.005 ' -.001
(.047) (.009) (.013) (.048) (.010)
(1' =05)
T=10 -.042 -.002 -.010 -.016 -.001
(.112) (.017) (.068) (.105) (.018)
T=20 -.039 -.001 -.010 -.037 -.001
(.095) (.012) (.060) (.089) (.012)
T=50 -.039 -.001 -.010 -.055 .000
(.078) (.008) (.057) (.076) (.008)
T=100 -.036 -.001 -.009 -.056 .000
(.077) (.005) (.057) (.073) (.005)
(7' =03)
T=10 -.049 .000 -.028 -.074 .000
(.095) (.003) (.097) (.087) (.004)
T=20 -.055 .000 -.031 -.077 .000
(.089) (.002) (.094) (.083) (.002)
T=50 -.063 .000 -.034 -.083 .000
(.084) (.002) (.092) (.079) (.002)
T=100 -.067 .000 -.036 -.084 .000
(.082) (.001) (.091) (.077) (.001)

 

Table 3

Estimation of Efficiencies : MLE & BC method

 

 

Bias Confidence Interval (90%) for Efficiency (r)
constantmefficrency width prob(<LB) prob(>UB) coverage
(11) ' rate
(1) (2) (3) (4) (5) (6)

(7' =0.1, T=10)
N=10 -.039 -.O38 .203 .280 .056 .664
N=20 -.016 -.017 .235 .159 .052 .789
N=50 -.010 -.009 .254 .086 .049 .865
N=100 -.002 -.001 .260 .064 .051 .885
(1‘ =0.5, T=10)
N=10 -.042 -.042 .209 .190 .066 .744
N=20 -.019 -.020 .210 .114 .059 .827
N=50 -.010 -.009 .210 .076 .052 .872
N=100 -.005 -.005 .210 .064 .051 .885
(7' =0.9, T=10)
N=10 -.049 -.049 .095 .334 .077 .589
N=20 -.020 -.020 .093 .204 .079 .717
N=50 -.008 -.008 .092 . 109 .067 .824
N=100 -.003 -.003 .092 .077 .062 .861
(y' =o.1, N=10)
T=10 -.039 -.038 .203 .280 .056 .664
T=20 -.028 -.028 .186 .218 .060 .722
T=50 -.023 -.022 .147 .166 .053 .781
T=100 -.014 -.014 .112 .174 .070 .756
(1' =0.5, N=10)
T=10 -.O42 -.042 .209 .190 .066 .744
T=20 -.039 -.039 .156 .226 .065 .709
T=50 -.039 -.038 .104 .280 .068 .652
T=100 -.036 -.036 .075 .344 .106 .550
(7' =0.9, N=10)
T=10 -.049 -.049 .095 .334 .077 .589
T=20 -.055 -.055 .069 .422 .081 .497
T=50 -.063 -.062 .045 .548 .066 .386
T=100 -.067 -.067 .032 .629 .077 .294

 

67

Table 4

Effect of Wrong Distribution on Estimation of Efficiencies: MLE & BC method
(Assumed Distribution is Half- Normal, True Distribution is N(u, o: ) truncated at zero)

 

 

Bias Confidencelnt.(90%) for Efficiency (r)
constant 0': var(u) u width prob(<LB) prob(>UB) coverage
rate
(1) (2) (3) (4) (5) (6) (7) (8)
(pt=0.5)
1 =0.1
N=10, T=10 -.335 -.002 -.003 -.334 .201 .831 .000 .169
N=50, T=10 -.307 .001 -.002 -.306 .254 .841 .000 .159
N=10, T=50 -.286 -.001 .002 -.285 .147 .955 .000 .045
7' =0.5
N=10, T=10 -.142 -.001 -.002 -.141 .205 .417 .021 .562
N=50, T=1O -.101 .000 .012 -.100 .205 .278 .012 .710
N=10, T=50 -.107 -.001 .010 -.106 .098 .542 .034 .424
7' =0.9
N=10, T=10 -.084 .000 .003 -.084 .090 .438 .054 .508
=50, T=10 -.O38 .000 .029 -.038 .087 .243 .031 .728
N=10, T=50 -.094 .000 -.002 -.094 .042 .636 . .056 .308
(kW-2)
1 =0.1
N=10, T=10 -.012 -.004 -.004 -.011 .187 .279 .082 .639
N=50, T=10 .028 -.002 -.001 .029 .253 .047 .098 .855
N=10, T=50 .002 -.001 -.006 .003 .144 .105 .088 .807
1' =0.5
N=10, T=10 -.002 -.002 -.029 -.002 .211 .122 .099 .779
N=50, T=10 .032 -.001 -.014 .032 .216 .033 .104 .863
N=10, T=50 -.009 -.001 -.032 -.009 .108 .153 .109 .738
7' =0.9
N=10, T=10 -.020 .000 -.045 -.020 .099 .230 .110 .660
N=50, T=10 .014 .000 -.035 .014 .098 .049 .117 .834
N=10, T=50 -.036 .000 -.052 -.035 .047 .412 .108 .480

 

68

Table 5

Average Sample Minimum for Different Distribution

 

 

N=10 N=50
yo=Ol yt=0 5 .Y :09 Y :01 yozo 5 7.: 9
p = 0 .029 .067 .085 .006 .015 .020
11 = 0.5 .266 .137 .120 .154 .035 .031
11 = -2 .019 .043 .065 .004 .009 .012

 

69

 

 

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71

Table 8

Confidence Intervals for Efficiency Estimates : FE & Bootstrap

 

Percentile Intervals (90%) for r' BC. Percentile Int. (90%) for r'

ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

width prob(<LB) prob(>UB) coverage width prob(<LB) prob(>UB) coverage

 

rate rate

. <1) (2) (3) (4) (5) <6) (7) (8)
(7 =0.1, T=10)
N=10 .355 .003 .233 .709 .330 .014 .133 .353
N=20 .345 .001 .449 .551 .326 .012 .185 .803
N=50 .324 .000 .661 .339 .311 .011 .264 .725
N= 100 .304 .000 .793 .207 .294 .006 .375 .619
(7" =0.5, T=10)
N=10 .250 .016 .136 .347 .249 .050 .031 .869
N=20 .245 .003 .247 .750 .245 .036 .1 16 .848
N=50 .230 .001 .421 .573 .230 .031 .155 .314
N=100 .218 .000 .585 .415 .217 .019 .224 .757
(7' =0.9, T=10)
N= 10 .112 .029 .089 .882 .112 .060 .066 .874

=20 .112 .017 .135 .848 .112 .063 .084 .853
N=50 .103 .004 .230 .766 .109 .033 .100 .862
N=100 .104 .001 .351 .648 .104 .034 .124 .342
(7‘ =0.1, N=10)
T=10 .355 .003 .288 .709 .330 .014 .133 .853
T=20 .280 .001 .221 .777 .260 .029 .082 .889
T=50 .195 .006 .152 .342 .186 .033 .067 .900
T=100 .144 .014 .156 .830 .140 .046 .079 .875
(y‘ =05, N=10)
T=10 .250 .016 .136 .347 .249 .050 .031 .339
T=20 .191 .014 .1 O7 .879 .190 .050 .063 ‘ .887
T=50 .123 .022 .074 .903 .127 .054 .043 .393
T=100 .093 .025 .094 .880 .093 .041 .075 .884
(7‘ =0.9, N=10)
T=10 .112 .029 .089 .882 .112 .060 .066 .874
T=20 .084 .038 .074 .889 .084 .055 .062 .883
T=50 .055 .027 .057 .915 .055 .047 .047 .906
T: 100 .039 .041 .058 .901 .039 .048 .052 .900

 

72

Table 9

Bias Correction in the Bootstrap Intervals

 

.- (b)

 

max 01 maxdt max 01
)=1....N ’ 1:1,...N ’ 1:1,...u ’
(1) (2) (3) (2)-(1) (3)-(2)

N=10, T=10 .971 1.109 1.179 .138 .070
N=10, T=50 .971 1.005 1.033 .034 .028
N=50, T=10 .993 1.223 1.336 .230 .113
N=10, T=10 .931 .981 1.018 .050 .037
N=10, T=50 .931 .942 .953 .011 .011
N=50, T=10 .985 1.107 1.173 .122 .067
N=10, T=10 .910 .921 .930 .011 .009
N=10, T=50 .910 .912 .914 .002 .002
N=50, T=10 .980 1.016 1.036 .036 .021

 

73

Chapter 4

MARGINAL COMPARISONS WITH THE BEST
AND THE EFFICIENCY MEASUREMENT PROBLEM

I. Introduction

Suppose that we have data on each of a set of N populations, indexed by a
parameter 0,, i = 1, 2, N. The parameterization is such that a larger value of 01 is
"better" than a smaller value. For example, 0; could be the mean survival time afier
medical treatment i; or, in the efficiency measurement example, more efficient firms have

larger 01. Suppose that we order the 01 as follows:
(4.1) 9(1) S 9(2) S S 901) ,

so that population (N) is best. The identity of the best population is not assumed to be
known, which is the challenging aspect of the problem. Let 0 E (01, 02, , 0N) be the

(unordered) vector of parameters. We presume that we have data on each of the

populations and correspondingly there is an estimate 0 of 0. Based on this estimate we
wish to say which populations might be best, and to construct confidence intervals for the
differences 9(N) - 01, which measure the amount by which a given population differs from
the best.

One solution to this problem is given by the technique of multiple comparisons
with the best, or MCB. MCB constructs a set S of possibly best populations, and a set of

intervals (L1,Ui), such that:

74

(4.2) P[ (N) e S and L1 30m) -01 S U. for all i ] 2 l-a,

where 1-01 is a chosen confidence level (e. g. 0.90). Thus with a given confidence level we
have a set of populations that includes the best, and joint confidence intervals for all
differences from the best.

An alternative to the multiple confidence intervals in (4.2) is a marginal (i.e.,
univariate) confidence interval for 9(N) - 01, for a single given value of i. In the efficiency
measurement example, this would amount to a confidence interval for the technical
inefficiency of a given firm, which is a natural and useful object of interest. Perhaps
surprisingly, the construction of a marginal confidence interval for 9m) - 0. is a previously
unsolved problem. In this chapter we show how to construct these marginal confidence
intervals. More precisely, for a given value of i, we provide a set S and and interval
(L':n ,UI“) such that P[(N)eS and L? s 004) - 01 5 UI“] 2 1-01. The point is that marginal
confidence intervals are natural to consider, and also that we would expect marginal

confidence intervals to be narrower than joint ones.

2. Marginal Comparisons under Standard Assumptions

Throughout this chapter we will maintain the following two assumptions. First,
we have an estimate 0, distributed as N(9,O'2C) with C known. Second, either (32 is
known, or we have an estimate 62, independent of 0, such that 62/02 is distributed as

x3/v. In any applications we envision, there will be enough degrees of freedom (v will be

large enough) that we can effectively take 02 as known.

75

Standard MCB proceeds under the fithher assumption, which we will maintain in
this section, that C = kIN with k known. This assumption is usually motivated by
discussion of the "balanced one way model" (e.g., Hsu (1996), p. 43) in which we have
independent observations ya (i = 1,...,N, t = 1,...,T) distributed as N(Gi, 02). In this case k
= 1/T. This is also the case of the panel data regression model with fixed individual
effects, if we treat the slope coefficients as known, as will be discussed below.

We define the following notation. E(‘/2) is the (N-l) x (N-l) correlation matrix
with all correlations equal to ‘/2 (i.e., diagonal elements equal one, off-diagonal elements

equal ‘/2). Let 2 be a multivariate random variable distributed as student-t with dimension

N-1, degrees of freedom v, and correlation matrix E(‘/2). Define d'(01) as the a-level

critical value of max I2. I; i.e., P[ max |zi lsd‘(01)] = 1-01. Tabulations of d°(01) can
1=1,...N-1 l=1,...N-1

be found in Hsu (1996) or Horrace (1998). Define h(a) = d'(01)(2k62)'6, and define the

set S(a) = {i 19. 2 {3197501. - 11(3)}. Define L. and U. as follows:

(4.3) L. = max[0, min 61. - 9.- 11(3)] , U. = max[0, max 0]. - é.+ 11(6)]
168(9) H

Then MCB provides the statement (4.2) above, with S = S(a).

Our marginal comparison with the best is given by the following theorem.

THEOREM 1: Let t'(01) be the two-sided a-level critical value of the
(U nivariate) student-t distribution with v degrees of freedom; i.e., if z is distributed as

student-t with v degrees of freedom, then P[|z| S t'(or)] = l-OL. Define 8(a) =

t'(a)(2k62)"/’. Define the set S(a) as above. Define LT and UI“ as follows:

76

 

(4.4) U? = max[0, min 0] - 0.- g(6/2)], up = max[0, max 0] - é,+ g(a/2)].

)‘ES(0.)

Then
(4.5) P[ (N) e 5(a) and LTS 0m - 01$ UT] 2 1-01.

The proof is given in the Appendix. It is a relatively straightforward application of the

Bonferroni inequality.

We can note that the marginal comparison (4.5) uses the 01/2-level critical value
of a univariate student-t while the multiple comparison (4.2) uses the a-level critical
value of an N—l dimensional student t. It is not clear whether there is a general inequality
between these, but for commonly chosen values of 0L (e.g. 0.05 or 0.10) the marginal
intervals are narrower than the multiple intervals. For example, for on = 0.05 and v=oo, the
univariate odZ-level critical value is 2.11, while the N-l variate a-level critical value is
2.34 for N=3, 3.16 for N=10, and so forth.

We may also wish to consider one-sided confidence intervals. One of the possible
motivations for doing so is the following. In many applications, the lower bound for
9m) -0i turns out to be zero for many observations, because the set S of possibly best

populations is large. We might choose to forgo the calculation of a lower bound, in which

case a tighter upper bound is possible. This result is given in the following theorem.

THEOREM 2: Let g(01) be defined as in the statement of Theorem 1. Then the

following are true:

77

(4.6) P[ (N) e 8(a) and 0m, - e. s max[0, max 0] - 3. + g(01)] 1 2 1-(1

(4.7) new - 0. s max[0, mjgx é) - é,+ g(2a)]] 2 1-6.

The proof is given in the Appendix, but we can note the following. Comparing
(4.6) to (4.5), the fact that we make only one statement instead of two allows us to use the
01/2-level one-sided univariate student-t critical value, which is the same as the a-level
two-sided critical value, instead of the 01/2—leve1 two-sided critical value. (For example,
for 01 = 0.05 and v=oo, we use 1.96 instead of 2.24.) Considering (4.7), we note that the
upper bound does not require the definition of the possibly best set S(a). If we do not
wish to consider S(a), we can devote the full confidence level 1-01 to the upper bound,
and we can use the a-level one-sided critical value of student-t, which is the same as the
201-level two-sided critical value. (Thus, for example, with 01 = 0.05 and v=oo, we can

now use the critical value 1.64 instead of 1.96 or 2.24.) As a result we get more a precise

upper bound.

3. Marginal Comparison with General Covariance Structure

The previous section considered the commonly-assumed special case that the
covariance matrix of 0 was proportional to an identity matrix. In this section we consider

the general case that 0 is distributed as N(0,02C) with C known but unrestricted. This

case arises in, among other cases, the panel data regression model with nontrivial

regressors.

78

We first need to define a little notation. For a given value of j, define Q as the

(N-l)xl vector whose typical element is of the form 01-01, for i = 1,...,N, itj. Formally

61=D10 where Dj is an (N-l)xN differencing matrix. The covariance matrix of 6 j is osz,
where B,- = DjCDj’. Let R; be the corresponding correlation matrix. In the special case that

C is proportional to identity, Rj = E('/2), as discussed in the previous section. In the

general case, R,- will depend on j and has no special structure, but it is easily calculated.

Define dI(or) as the two-sided a-level critical value of the multivariate student-t

distribution with dimension N-l, degrees of freedom v, and correlation matrix Rj. This
critical value will typically depend on j and will generally need to be calculated

numerically (e. g. by a simulation), since tabulation is impossible except in special cases.
NOW define 6' ij2 = 6'112 = [O 2(C11+ij-2Cij)], 1111((1) = dI(or) 6' ij, Lfi =éj'éi - 11(0),
U3 =éj-é i + hji(a), the possibly best set S(or) = {i | Uji 2 0 V j¢i } = {i | 01 2 éj - hij((1)

V j¢i}, and the lower and upper bounds L. = max[0, min)L’I] and U. = max[0, max U3].
165(0- :1

Then MCB provides the statement (4.2) above, with S = S(or).

We now provide the corresponding marginal comparison result.

THEOREM 3: Define the set S(0L) as above, and let t'(oz) be the two-sided 01-
level critical value of the univariate student-t distribution. Define gij(01)=t'(01) 61,-.

Define LT and U“ as follows:

(4.3) L'f' =max[0, min) (3,- - gum/2)) -3 1], 11:“ =max[0, rqax (é ,- + g.,-(6/2))-é .1.
:65 a. x

Then

79

 

(4.9) P[ (N) e S(a) and LTS 90:) - 0. S UI“ ] 2 1-01.

The proof is similar to the proof of Theorem 1 and is therefore omitted.
As in the standard case, we may also consider one-sided confidence intervals.
The following Theorem (also presented without proof) is the result corresponding to

Theorem 2.

THEOREM 4: The following are true:

(4.10) P[ (N) e 8(a) and 0m) - 01 S max[0, n}a'x (éj + gij(01)) - 0.] ] 2 1-01
(4.11) P[ 9(N) - 91 S max[0, max (éj + gij(201)) - 01] ] 2 l-a.

As in the previous section, one possible motivation for one-sided confidence
intervals is that they yield more precise upper bounds. However, in the case of general
covariance structure the one-sided intervals given in equation (411) also offer

considerable computational advantages, because they do not require the calculation of the

possibly best set S(or). The calculation of S(or) requires the N critical values dI(or),

j=1,...,N, each of which is from an N-l dimensional student-t distribution and is
generally calculable only numerically (via simulation). Especially when N is large, this is

a very complicated and time-consuming set of calculations.

4. Application to the Efficiency Measurement Problem

80

In this section we consider some empirical applications of our marginal
comparisons with the best to the efficiency measurement problem. Here we will give a
very brief discussion of the problem. See Chapter 2 for more detail on the problem.

We begin with the fixed-effect panel data regression model:
(4.12) Yit=ai+Xitrfi+Vit, 1:1,...,N, t=1,...T

Here i indexes firms and t indexes time periods. For purposes of exposition we assume a

"balanced" panel (T is the same for all i). We assume that the v;- are iid N(0, 0,2,) and we

treat the explanatory variables x“ as fixed. The parameters of interest are the intercepts 01,,

which correspond to the 01 in the previous sections. In our applications the estimator of [3,
say, [3 is the fixed-effect or "within" estimator obtained by least squares of (y, — a) on
(in —i), where y, and Z are the means of the T observations for firm 1. Then we
obtain an estimate of or. as (31i = _- — 2:0.

Here the specific context is the efficiency measurement problem, in which y is the
logarithm of output and x is a vector of functions of inputs into the productive process.

Larger a1 is better because it corresponds to more output for the same inputs. We define

uI= 0‘(N)‘ 01;, where am is the largest of the N ot's, and the technical ejficiency of firm i
is typically defined as rI = exp(-uI). Since uIz 0, 0 S rIS 1. MCB and our marginal
comparisons with the best procedures will provide confidence intervals for uI , and these

are easily converted into confidence intervals for rI. In particular, if (with a given

81

probability) L S uIS U, then exp(-U) S rI S exp(-L), so that lower bounds for uI convert

to upper bounds for rI and conversely.

Let or = (01,,012,...,01N). Under our assumptions the covariance matrix of 61 is
V(dt) = (03 /T)IN +2 V03) 2', where V(0) is the covariance matrix of B and i is the
matrix whose ith row is XI. In general this is not proportional to an identity matrix and so

we should allow for a general covariance structure. The methods of section 2 ("standard"

MCB or marginal comparisons) would apply if B were known, and can be viewed as

applying approximately if the proportion of V(dt) due to the variance of [3 is small. This

may generally be so when N is large relative to T, as discussed in Horrace and Schmidt
(1998). However, we present only the results that allow for the general covariance

structure.

5. Empirical Applications
5.1 Indonesian Rice Farms

These techniques are applied to the same three data sets as were used in Chapter
2. We first analyze the data of Erwidodo (1990), which contain information on N = 171
rice farms for T = 6 growing seasons. Our results are given in Table 1. We choose 01 =
0.10 (hence 90% confidence intervals). We give results for the three most efficient (best)
firms, the firms at the 75th percentile, 50th percentile, 25th percentile, and the two least
efficient (worst) firms. For each firm, we present the value of 611; the point estimate of
technical efficiency; the confidence intervals corresponding to the marginal and multiple

comparisons with the best; and the corresponding one-sided marginal and one-sided

82

multiple comparisons with the best. The one-sided marginal comparisons correspond to
equation (4.11) of section 3.

In these data we do not estimate the intercepts 011 very precisely. This occurs
because we have only six observations per firm and because the value of of turns out to

be large relative to the variation in the 01.. Correspondingly, our confidence intervals are
rather wide. In fact, they are wide enough to suggest that the efficiency measurement
exercise has more or less failed to distinguish efficient and inefficient firms. The possibly

best set S contains 98 of the 171 firms, and the other 73 firms are sufficiently close to
being in the possibly best set that the MCB upper bounds for rI equal one for all firms,

even for the two-sided intervals. (For the one-sided intervals, the upper bound is
automatically one, but this is an identity, not a data-detennined outcome.)

The marginal comparison intervals are considerably shorter than the MCB
intervals, as they should be. They use the 5% critical value of the univariate student-t
distribution, 1.96, while the multiple intervals use 10% critical values of the 170-

dimensional student-t distribution, which vary a little over comparison populations (i.e.

dI above depends on j) but equal 3.18 on average. The greater precision of the marginal

as opposed to multiple confidence intervals is most noticeable for the more efficient
firms. For example, for the most efficient firm compare the marginal interval of [074,1]
to the multiple interval of [058,1]. The extra width of the MCB intervals is the price one

has to pay for making a multiple statement, of course (the so-called "multiplicity effect").

5.2 Texas Utilities

83

We next analyze the data of Kumbhakar (1994). The results are given in Table 2.
The confidence intervals (both marginal and multiple) are much narrower for this data set
than for the previous one. We are now able to make statements about efficiencies that are
precise enough to be meaningful. For example, only two observations are in the possibly

best set, and the confidence interval for the efficiency of the (apparently) most efficient

firm is [095,1]. This occurs primarily because 03 is smaller and T is larger here, so that

the or. are estimated more precisely. In the case of MCB, the intervals will also tend to be
narrower because the multiplicity effect is not as strong with N = 10 as it is with N = 171.
The marginal intervals are narrower than the multiple intervals but the difference in width
is not as large as it was in the previous data set, again because the multiplicity effect is
weaker here. Numerically, the marginal intervals use the univariate student-t 5% critical
value of 1.96, while the multiple intervals use 10% nine-variate student-t critical values,

which are on average equal to 2.38.

5.3 Egyptian Tileries

Lastly, we analyze the data of Scale (1990). The results are given in Table 3. In
terms of the width of the confidence intervals, the results are somewhere between those
for the Indonesian rice farms and those for the Texas utilities. This is true also in terms of
the extent to which the marginal intervals are narrower than the multiple intervals. The
marginal intervals use the critical value 1.96, while the multiple intervals use critical
values that are on average equal to 2.70. As was the case in the Indonesian rice farm data,
the difference between the marginal and multiple intervals is considerable for the more

efficient firms but is not very large for the less efficient firms.

84

6. Concluding Remarks
In this chapter we have considered the general problem of creating confidence
intervals for measures of the difference between a given population and the best

population. More precisely, population 1 is characterized by a parameter 0., and we wish

to construct a confidence interval for the difference 00.) - 0., where 9(N) = mars 0J . This is
j: ...-

a challenging problem because we do not know which population is best. One solution is
given by MCB, which provides the complete set of N such confidence intervals, all of
which hold simultaneously with at least a specified confidence level. Perhaps
surprisingly, the seemingly simpler problem of providing a confidence interval for a
single difference 60.) - 6. had not previously been solved. We provide these confidence
intervals, and refer to them as marginal comparisons with the best.

Whether one prefers multiple or marginal comparisons will no doubt depend on
the context. For an example of the arguments in favor of multiple comparisons, see Hsu
(1996, p. 7). However, in some cases a marginal comparison may be natural. It seems
reasonable to be able to perform either type of inference, just as one wishes to be able to
test a set of hypotheses either individually or jointly.

In the context of the efficiency measurement problem, marginal comparisons
correspond to the construction of the confidence interval for a given firm's technical
efficiency level, and this is indeed a natural thing to consider. For example, models that
assume a distribution for u. yield marginal confidence intervals, constructed in somewhat
more straightforward ways than here. See Horrace and Schmidt (1996) or Koop et al.

(1997) for some examples. A marginal comparison with the best is directly comparable,

85

 

and provides evidence on the gain in precision from assuming a distribution for u.. These
comparisons are harder when MCB is used because the multiplicity effect and the effect

of assuming or not assuming a distribution become confounded.

86

 

APPENDIX
Proof of Theorem 1
As in the text, suppose that z is a multivariate random variable distributed as

student-t with dimension N-l, degrees of freedom v, and correlation matrix E(‘/2). Define

dI(01) as the a-level critical value of .mahx1z.; i.e., P[max. 2. S dI(0t)]= 1-01. Note that

dI(01) is the one-sided critical value corresponding to the two-sided critical value d‘(01)
used in MCB, and that d','(6t/2) = d'(et). Similarly define h.(6t) = d,°(6t)(2k6?-)"= and
note that 111((1/2) = h(et), with h(a) = d’(6t) (2k62 )V: as used in MCB.

Consider the event E. (01)={0 (N) - 0. S 004) - 0 j +h1(01) Vj¢(N)}. This is the one-
sided multiple comparisons with a control (MCC) event, with (N) as control, and is

constructed so that P[E.(or)] = 1-01. See Dunnett (1955, 1964) or the discussion in Hsu

(1996, chapter 3). The event E.(ot) implies the event {(N) e S1(01)}, where S.(a) is the

set of indices 81(01) = {i 10. 2 grails 6,- -h1(01)}. Note that 81(01/2) = S(or) c_: S(or/2).

Therefore
(A1) P[(N) e S1(01/2)] = P[(N) e S(01)] 2 1-01/2,

a standard result of the "ranking and selection" literature; e.g., see Gupta (1965).
Now pick a value of i (= 1,...,N), and consider the event A.(01) = {004) - 0. - g(or)

S 0 (N) - 0. S 00..) - 0. + g(or)}, where g(0t) was defined in the statement of the Theorem.
Note that g(or) was constructed so that P[A.(01)] = 1-01. By the Bonferroni inequality, it

follows from (A1) and P[A.(or/2)] = 1-01/2 that

87

 

(A2) P[ (N) e 8(a) and A.(01/2) ] 2 1-01 .

This inequality is not immediately useful because it is not in terms of observable
quantities, since (N) is unknown. So, we need to show that the event {(N)eS(0L) and
A.(01/2)} implies the marginal comparison event given in (4.5) of the main text. Consider

first the lower bound. The event whose probability is given in (A2) implies that

(A3) min 0.-0.-g(or/2)S0m)-0.-g(01/2)S6(N)-0..

19301)

Also 0 S 9m) - 0.. Thus the event whose probability is given in (A2) implies the lower

bound

(A4) max[0, “sll“)éi - 0.- g(or/2)] gem-0..
)5 01

The treatment of the upper bound is similar. If (N) = i, then 00..) - 0. = 0. If (N) #1,

the event in (A2) implies

(A5) 900 - 913 max 91 - 61+ 901/2)-
1:1

Therefore we have the upper bound

(A6) 9(N) - 0. S max[0, max 0 J- - 0. + g(or/2)].
jxr

88

 

Finally, since the event {(N) E 5(a)} and the bounds (A4) and (A6) are implied
by the event in (A2), they hold with at least the probability of that event; that is, with a

probability no smaller than 1-01.

Proof of Theorem 2

As in the proof of Theorem 6, we have P[(N) e 5(a)] 2 1-01/2. Now we also have
P[Gm) - 0. S 004) - 0. + g(01)] = 1-01/2, since g(01) is based on the a-level two-sided
student—t critical value, or equivalently the 01/2-1eve1 one-sided critical value. Thus the
Bonferroni inequality implies that P[(N) e S(or) and 0m) - 0. S 0nd) -0. + g(or)] 2 1-01.
Then the same logic as was used in the discussion leading up to (A6) yields the result in
(4.6).

To establish equation (4.7), we note that the upper bound does not require the
definition of the possibly best set S(a). We simply start with the statement: P[Om) - 0. S
00..) - 0. + g(201)] = 1-o1, which follows from the fact that the two-isided 201-level critical
value in g(201) is the same as the or-level one-sided critical value. Then we again apply

the same logic as was used in the discussion leading up to (A6) to obtain (4.7).

89

 

 

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92

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