Computers Opns Res. Vol. 17, No. 2, PP. 123-132. 1990 0305-0548/90 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright 0 1990 Pcrgamon Press plc

TESTS OF EFFICIENCY IN DATA ENVELOPMENT ANALYSIS

JATI K. SENGUPTA*

Department of Economics, University of California, Santa Barbara, CA 93106, U.S.A.

(Received January 1989; revised July 1989)

!kope and Purpose-Models of data envelopment analysis (DEA) which. have been widely used in recent times in operations research for measuring productive etliciency, differ from regression models in one fundamental aspect that the latter uses a predictive criterion for choosing the best model. This paper develops a set of statistical tests for the DEA models, which provide a predictive framework. By using the canonical correlation theory in the multiple input case, these tests provide an important link between the parametric and nonparametric approaches to the estimation of productive efficiency. Some empirical applications illustrate the various tests which enhance the applicability of the DEA models by examining the appropriateness of their input-output mix.

Abstract-Canonical correlation theory is utilized here to modify the set of efficiency measures in data envelopment analysis (DEA). This involves (1) the mean square error criterion, (2) the canonical correlation test, and (3) the set of redundancy tests. An empirical application to educational production frontier data shows the usefulness of these tests and how they can be statistically tested as a two-stage procedure. This may help provide the bridge between the parametric and the nonparametric approaches to the estimation of production frontiers.

INTRODUCTION

Data envelopment analysis (DEA) has been widely applied in several fields of operations research today to measure the productive efficiency of a set of decision-making units (DMUs) or firms. This technique provides a significant generalization of the concept of the production function often employed in economic theory to measure a systematic relationship between different input bundles and the corresponding maximum levels of output. Several special problems arise when one seeks to estimate the optimal production function, also called the production frontier when there are multiple outputs and multiple inputs and the sample data contain random errors. Two types of approaches to the problem have been followed in the current literature. One is the parametric approach where a functional form is specified and then estimation performed for the stochastic frontier. A critical survey of this approach is provided by Forsund, Love11 and Schmidt Cl], Schmidt [2] and Sengupta [3]. The second is the nonparametric approach which uses mathematical programming techniques, e.g. Farrell [4], Timmer [S] and Charnes, Cooper and Rhodes [6]. The term ‘data envelopment analysis’ was originally proposed by Charnes, Cooper and Rhodes (CCR), who have made major contributions to the theory of efficiency measurement. Most recently the technique of DEA has been widely used in management science and operations research applications, e.g. in public sector organizations such as schools, hospitals and other service agencies where the data on the prices of inputs and outputs are frequently unavailable. A recent survey of the various issues related to the DEA methods may be found in Seiford [7] and Sengupta [8-lo].

Two important issues arise in the empirical application of DEA models. The first issue is that such models use linear programming (LP) formulations to specify one or more efftciency facets. An undesirable consequence is that it wastes a lot of information. Farrell [4] noted this pessimistic bias in the LP method of convex hull estimation, where only a handful of the so-called ‘efficient’ points contribute directly to the estimate, the rest being ignored. The second major issue deals

‘Jati K. Sengupta is Professor of Economics and Operations Research at the University of California, Santa Barbara. He received his doctoral degree from Iowa State University and held important managerial positions including the Directorship of the Indian Institute of Management at Calcutta. He has done extensive theoretical and empirical research on stochastic systems and models and has had consulting experience with the World Bank, Ford Foundation and other agencies. His publications include six research monographs and over 200 research papers in such journals as Management Science, Journal of Optimization Theory and Applications, Operations Research, Econometrica, international Journal of Systems Science and Journal of Mathematical Analysis and Applications. His latest research monograph: Efficiency Analysis by Production Frontiers: the Nonparametric Approach (1989) provides an up-to-date survey of research on data envelopment analysis discussed in this paper.

123

124 Jm K. SENOUPTA

with the distribution of the data points which are found to be efftcient by the DEA model. Specifically, in the multiple output case, how should various outputs be aggregated into a composite output, and furthermore how can the sensitivity of data points around the efficiency frontier be analyzed?

Our object here is to discuss these issues in terms of the following set of efficiency tests: (1) tests of predictive power, (2) canonical correlation tests, and (3) tests of redundancy. All these tests are designed to explore (a) the role of data points which are close in some sense to the efficiency frontier in some sense, and (b) the need for modifying the optimal solutions of the traditional DEA model. These tests are illustrated by an empirical application to study the comparative performance of public school districts in California, which has been utilized extensively in previous studies by Sengupta [9].

DEA MODEL AND ITS EFFICIENCY MEASURES

In the multiple-output case the DEA model tests if unit k is efficient by solving the following LP model:

Min gk = /YxL subject to (s.t.) (a, a)~ C (1)

where the constraint set C is given by

C= {~Xj2a’Yj;O!‘Y~= l,aase,/?Zse).

Here the observed data points dj = (Xj, yj) comprise input (xi) and output (Yj) vectors of dimensions nt and s respectively for each unit j = I, 2, . . . , n. The column vectors a and /I are constrained to be positive by using E as a non-Archim~ean in~nitesimal and e is a column vector of ones. Let x = a: and fi = /?z be the optimal solutions of the DEA model (1). By the CCR criterion the kth DMU for which the objective function is given in (1) is efficient if /3f’x, - a:‘~, = 0 and the slack variable s: = fi$‘x, - a,*y, = 0. If however it holds that /Qxk - af’y, > 0, then the kth DMU is not efficient. By varying k in the objective function (1) within the index set I,, = { 1,2, . . . , PI} all the efficient units can be determined. The set of eflicient points has been classified by Charnes, Cooper and Thrall [ll] in three subsets: (i) the subset E, of DEA extreme points which are unique for each k, (ii) the subset E, of nonextreme points which are DEA eflicient, and (iii) the subset E, of units which are scale efficient but not technically efficient, i.e. at least one component of the vector (a, fl) is zero. Finally, all other units belong to the remaining subset of units which are not efficient by the CCR criterion. Denote by set E the union of the subsets E,, E, and E3 and assume that it contains n, units and let N be its complement in the entire data set D containing the remaining units. It is clear that by taking suitable convex combinations of the points (xlr, yt) for kEE one could thus specify the convex hull of the efficiency frontier. Since the optimal weight vectors (a$, /It) can be interpreted as virtual multipliers for the LP model which is dual to (l), the CCR criterion can be viewed as one which reduces the multi-input multi-output situations to single input (u: = fiz’xn) and single output (~2 = a,*‘yJ ones through the use of virtual multipliers, Thus if unit k is efficient, i.e. ke E, then v[: - u:* = 0, otherwise it is ine~cient, i.e. v: > ut implying ktz N.

Several aspects of the CCR efficiency test have to be noted. First of all, if the observed data points dj contain noise elements or errors, the efficiency surface specified above would be subject to stochastic perturbations. Thus the points belonging to N which are very close to the efficiency frontier would have to be admitted. Secondly, for each k E E we have one pair (a:, /I:) of vectors which characterize one facet of the efficiency surface. Which of the different efficiency facets one would use for decision-making purposes? By adopting the criterion of minimizing the L,-norm (i.e. sum of absolute values of the errors), Timmer considered the mean LP model as follows to characterize one efficiency facet:

Min S = fi’Z s.t. (a, 8) E c (2)

where

C = ( FXj 3 Q’yj; a’j = 1, a 2 se, b > se> allj=1,2 ,..., n.

Here CT= (X, 7) are the mean input and output vectors computed on the basis of n observed data points. But if all units are not efficient by the CCR criterion, a more appropriate version of the mean LP model (2) should use the mean values (a, J?) based on those efficient units which belong

Tests of efficiency in DEA 125

to set E only. It is clear however that in many cases (e.g. if the noneficient points belonging to set N are very closely clustered around the mean vector Jcomputed on the basis of trl points only) the optimal solutions would remain the same. Finally, the DEA model (1) does not capture stochastic variations in the data set D = (dj = (xj, yj);j = 1,2, . . . , PI}. One implication of this situation is that the effkiency frontier may be sensitive to some data points and robust in case of others and the DEA model fails to distinguish between these two subsets of data points.

TESTS OF PREDICTIVE POWER

We may now propose some tests of predictive power of the DEA models. To fix ideas let us consider the mean model (2) and assume that 8= (b, &) is the optimal solution vector. The error of prediction may then be defined as

.k?j=pxj-a’yj, jc:O. (3)

Based on this estimated error of prediction, three different criteria may be defined as follows:

(i) Mean square error (MSE(a)) of prediction:

MSE(s)= i +I. (4) j= 1

(ii) Correlation criterion:

r(O) = cov(xO, y”)/[(var x’)(var y”)]rlz (5)

where

X0 =$x,9 Y0 = &‘Yj, j ED.

(iii) Mean square error over cone domains:

MSE(s), = MSE(.s) over j E Do (6)

where Do is a cone domain of the data set D, which may be specified suitably, e.g. if we consider the set E of efficient points, one particular efficiency facet may provide a more probable cone domain in terms of the data distribution.

Although we have fixed the mean model (2) to denote the estimate @= (b, a), it is clear that the method is very general and can be applied to any other procedure of estimating 0. Besides the DEA procedure, two other such procedures are: (a) to consider subsets of the data set D by deleting sample observations in a certain order, and (b) to consider additional inputs and outputs to augment the predictive power of the ekiency model. Clearly our objective should be to choose that procedure which has the lowest MSE(.s), or the highest r(@ or the lowest MSE(E)~~ among all feasible methods that are comparable. These criteria depend on the number of observations n, the number of parameters m + s and the distribution of the data points dj = (xj, yj). Clearly the degree of freedom is n - m - s, which has to be positive before any statistical significance can be attached, e.g. to the correlation measure r(0).

A direct way to incorporate the predictive power criterion into the DEA model is to transform the latter formulation as follows:

Min gR = p’zi - cr’V,$

s.t. a’VYYa = 1 = fl’V,.p

and

e=&a)EC.

(7)

Here we have used the mean model (2) and transformed it so that the estimator 8 maximizes the correlation criterion defined in (5). Note that V,, is the variance-covariance matrix of the input and output vectors and the first set of constraints of (7) is only the normalization condition. On using a linear normalization condition a’e s = 1 = /3’e,,, where e,, e, are column vectors of ones with

126 JATI K. SENCZJITA

s and m elements, we obtain the LP model:

Min 4s s.t. a’eS = 1 =/Ire,,,; BoC.

Several results can be stated on the basis of this transformed LP model of efficiency.

(8)

Theorem

If there is an optimal solution of the mean DEA model (2), then there must exist an optimal solution B of the transformed model (8). Furthermore, if the former solution is unique with all positive components, so is the latter.

Proof

If the data set Et has all positive components, then the mean DEA model (2) must have an optimal feasible solution by the Slater condition since the set c is compact. On adding the linear normalization conditions in (8), the augmented set of c remains compact. Hence the result.

Remark 1

Let g be an optimal solution of the nonlinear programming (NLP) model (7) at which the correlation r(g) attains the positive value r&j> 0). Then we must have

r&) 2 r(pi

where r(@ is the correlation associated with the standard optimal solution 8= ($, &) of the mean DEA model (2).

Remark 2

For any a preassigned so as to satisfy (S), the multiple-output model collapses to a single composite output case, where the error of prediction can be written as

where yj” = a”‘yj is the single composite output.

Remark 3

Now consider the maximum correlation model

Max J = a’V&? s.t. a‘t;p = 1 = /J’V,,fi (9)

in relation to the mean DEA model (2). Let 8’ = (PO, a’) be the optimal solution of (9) such that it satisfies a0 > se, p” 3 se where E is a non-Archimedean (positive) infinitesimal. Then it can be easily shown that

r(e”) 2 r,(B) 2 r(Q).

Furthermore, the relative improvement in predictive power due to a switch from 8 to 0 and from 8 to 8’ may be estimated by the correlation gains (r(g) - r(e)} and {r(O’) - (8)) respectively.

It is to be noted that the MSE(e), or MSE(s),, criterion is closely related to the correlation criterion but the mean square error is easier to relate to the distribution of errors (E). Since the estimates (3 or g are not unbiased, minimizing MSE(e) is more appropriate for the transformed DEA models, However this would generate nonlinear programming problems. The MSE(s) criterion can however be easily applied to empirically choose between the different effkiency facets specified by the LP model (1) when k varies over the index set 1,.

CANONICAL CORRELATION TESTS

The method of canonical correlation constructs composite inputs and composite outputs by choosing the optimal weights for combining the inputs and outputs. Since the DEA model can also be viewed as an alternative approach for selecting suitable weights for combining various inputs and outputs, it is clear that the canonical correlation method can be easily combined with the DEA model. Such an integration would enhance the potential applicability of the DEA models in situations where the input-output data are subject to stochastic errors. Besides predictive power,

Tests of efkiency in DEA 127

the tests of canonical correlation are intended to measure the mutual coherence of the input and output vectors and their partial and total correlations.

The method of canonical correlation may be introduced into the standard DEA model through a two-stage procedure. In the first stage we solve for the h efficient units (h $ n) from the sequence of LP models (I ) by varying k in the objective function (k E 2,). This may include only the subset E, of DEA extreme points which are unique for each kelh. Based on these data points dj~& we form the composite inputs and outputs as xf = @xj, Yf = O[‘Yj, where Xi, Yj are treated as random m-tupie and s-tupie vectors for each je Z,,. At this stage we can apply the canonical correlation method to determine the weight vectors 0 = (fi, a). Classically the problem of canonical correlation consists of finding vectors Q1, &, . . . , fltrnt where we assume m 6 s and express the composite input and output as x& = p;,X and Ysi, = aii,y, i = 1,2, . . . , m such that the foi~owing conditions hold:

6)

(ii)

(iii)

(iv)

x& y&, are the two linear functions of x and y respectively, with variance unity, which have correlation coefficient r(xT1,, yt,) = rcl, with maximum absolute value; for i < m, xTi, is the linear function of x with variance unity, uncorreiated with

X;1), * * * 3 $i- 1) and y&, is the linear function of y with variance unity, uncorrelated with

Yil)3+**,YTi-i> such that the pair (X;i,, vsit) has a correlation coeffkient with maximum absolute value; for 1 < i < m the pair of variables (Xfi,, Y;i~) is called the ith pair of canonical variables and the absolute value of its correlation coefficient rco is called the ith canonical correlation; and by construction the ordering condition I& 2 r&, 2 . . .a r$, is satisfied.

Clearly the first set of canonical variables zfr, = (x;,,, Y;J associated with the canonical vectors 8 (,) = (&,, atlJ is by delinition the largest and hence it represents the most substantial relationship; the second one qz, with ((zt,,, 0 & is the second largest relationship and so forth. Clearly it holds that

r:,,~r~~,~...~rr:,)~O. (10)

Two types of tests can be made about these canonical correlations. One is the test of statistical significance of the canonical correlations, when sample observations are used. The second is to apply the tests of canonical regression theory where the canonical variate as a dependent variable is regressed on the observed explanatory variables. Thus on taking the first canonical correlation we estimate the linear regression model

Y;,t=Y,+Y,x, -4-Y2X2+-*-+Ym~,+e (11)

where the standardized regression coefficients y& are proportional to the standardized canonical weights /&, i.e. yk = j&T(1). It is useful to perform a statistical assessment of the contribution of each explanatory variable xk on the composite dependent variable yiI, in terms of the size of the coefficient ylr and its standard error.

For the first case a commonly used test [12] for the statistical significance of the canonical correlations is Bartlett’s chi-square approximation of Wilk’s lambda criterion. To test the null hypothesis (HO) that the s output variables are unrelated to the m input variables

H,: V,, =o (12)

H, : vyx # 0

one could use Wilk’s lambda criterion from the likelihood ratio test:

where pPbe is the sample analog of the variance-covariance matrix of p and g, M = min(s, m) and I is the maximum eigenvaiue associated with the jth canonical correlation, i.e.

~~~~~=~~~~~~~ti~; j=1,2,...,M.

Bartlett uses a large sample approximation ofthis lambda statistic A in terms of a chi-square variate

X2 =-[(h-l)-$(m+s+l)]lnA. (14)

The null hypothesis H0 in (12) is rejected if X2 > X,’ with a = ms degrees of freedom. If the null

128 JATI K. SENGUPTA

hypothesis can be rejected, then the contribution of the first canonical variate pair can be removed from A and the statistical significance of the remaining canonical variate pairs assessed. This sequentiai testing procedure has often been criticized on the ground that the sample size required may be too high and the test may be extremely conservative. However a detailed Monte Carlo study by Mendoza, Markos and Gonter [13] involving one thousand sample correlation matrices for each of 7 population canonical correlations, e.g. three with strong correlation, i.e. 0.9, 0.8, 0.7 and three with weak correlation, i.e. 0.3, 0.2 and 0.1 and one with zero correlation showed the following result: for the 3 cases of strong population canonical correlation, a sample size as small as 50 will detect these over 90% of the time, for a moderate correlation of 0.50 a sample size of 100 is needed to detect it about two-thirds of the time, while a weak population canonical correlation of 0.30 would require a sample size of 200 to be detected about 60% of the time. Furthe~ore one has to remember that the canonical correlations only show relations between the composite variables and not between the observed dependent and independent variables. Hence if only one or a few variables have a high degree of association with the canonical variable, the total amount of variance in the y-set variables accounted for by the canonical variate can be small. In such cases there is no general relationship between the two composite canonical variates, since the canonicai correlation method indicates only a specific relationship between one or few dependent and predictor variables

c141. The canonical regression model (11) provides another test which can be used to compute the

explained variance in canonical regression where the canonical composite variable is used as the dependent variable. However one should add the caveat that the measures of standard error and the squared multipIe correlation coefficient R2 pertain to the unobserved canonical variate yti, or y&. One should however exercise some caution in interpreting a specific value of R2 as a measure of explanatory power. This is because the weights are specifically chosen here to maximize the correlation without paying attention to the response of the individual predictor variables. But this criticism can be lessened considerably by suitably rede~ning the first composite canonical variate to include only those variables which contribute to the composite very significantly. A second interpretation of the canonical regression model (11) may be given by adopting the dummy variable approach of Stone [15). In this approach we introduce several outside instrument variables 4, 4, I& say and compute the indi~dual correlation coefficient between yfi, and D,, DZ, & respectively. Finally, we choose that outside variable as the dummy variable to represent yfi, which has the maximum value of the squared correlation, provided the latter is quite large, e.g. exceeding 0.85. Thus if D, is the most important variable to represent y;,, we would have the linear regression model

Di = ye + f YiXi + e (15) i=l

which can be used for the usual regression diagnostics. Since the surrogate variable D, is an observed variable and since it is widely used in applied econometric studies, it may be more useful in providing suitable economic insight. Stone [Is] used per capita national income as the D, variable having correlation of over 0.95 with the composite variable yf,,. In our educational application the test scores in reading, writing, spelling and mathematics are the component output variables, for which a suitable D, variable may be the state-wide IQ score. Note that if the error term e in (15) satisfies the standard assumptions of the least squares model, i.e. identically and independently distributed like a unit normal variate, then the usual r-statistics can be applied to test the significance of the individual regression coefftcients pi, Also the proportion of variance of Di explained by the above linear model may be tested statistically. Thus let r be the sample correlation coefficient between D, and B1, where 6, = p0 + xpixi and let p be the population correlation coeficient. To test the null hypothesis H,: p = po, p0 specified versus the alternative Hi: p #p. we may use Fisher’s z-statistic, i.e. let

z=!*nl+r and 2 I-r

(16)

then as the sample size n + cc one gets

1 2-N (,----- c > n-l

Tests of eflicicncy in DEA 129

and under Ho:

Jn-l)(z - lo) - N(O9 1)

where N(p, q) denotes the normal distribution with mean p and variance 4. On using (16) a statistical test or confidence interval for [ can be easily specified. This approach based on the observed surrogate variable D, in (15) has two most important implications. One is the interface between the econometric and the mathematical programming approaches, as it may bring these two approaches much closer. Secondly, methods can be explored in the programming approach to improve the correlation level from p = p. to p = p1 where pr > p. > 0, and the number of predictor variables is increased from m to m + 1. This would quantify the relative cost of improving the predictive power of the DEA technique.

TESTS OF REDUNDANCY

Two most important problems in canonical correlation theory is to decide how many significant canonical correlations to take when there are M and how to capture the variance overlap between the two composite canonical variates x;,+ y&,; k = 1,2, . . . , M. In linear multiple regression the proportion of explained variance of the dependent variable is measured by the squared multiple correlation coefficient. In the theory of canonical correlation however, a squared canonical correlation coefficient only tells us the amount of variance that the two composite canonical variates y’, xc share and does not necessarily capture the variance overlap between the two sets of variables x = (Xi, x2, . . . ,x,) and I’= (yl, y2,. . . , y,). Hence Stewart and Love [ 163 introduced an index of redundancy to measure the amount of informational overlap between the two sets of variables.

The central idea behind this redundancy index is quite simple; It is the weighted average of the amount of variance in set Y that is predictable from the different canonical variates for set X. Denoting this redundancy in set Y given set X by R,,, and the squared canonical correlation between the canonical variate J+, and the corresponding canonical variate in set X by prc we obtain

M

Rrtx = 1 cc&,,, (17) k=l

where V,,r, is the variance in the Y-set extracted by the kth canonical variate y;&... Thus it is clear that the redundancy index is equal to the average squared multiple correlation for predicting the variables in the Y set from the variables in the X set. Note however that this redundancy index ignores all the intercorrelations of the variables being predicted. Thus we would obtain the same amount of variance accounted for with this redundancy index no matter whether the y variables are highly correlated or not. This is undesirable in the same sense as it would be undesirable if the multiple correlation in a regression context were unaffected by the intercorrelations of the predictor variables.

Note that the redundancy index (17) above is nothing but the average variance of the variables in the Y set that is explained by a canonical variate of the X set. Hence one can also write

R R,,,, = Cr:k&Ju,,,,~sl (18)

where R,,, is the redundancy of the dependent variable given the kth canonical variate of the predictors, RyIx is the overall redundancy, f,,9,k, = a’V,,,_ is the vector of loadings of the y-variables on their kth canonical correlation. Unlike canonical correlation, the redundancy index (18) is nonsymmetric; hence given a canonical correlation coefficient, the associated redundancy of the Y-variables will be different from that of the X-variable. Clearly the redundancy measure Rrlx in (18) can be viewed as a two-step procedure, i.e. r& occurs in the first step and explains the variance of the canonical variate of the Y-set, given its counterpart in the X-set, whereas the second part of the formula specifies the mean explained variance of the y-variables by their kth canonical component. Compared to this two-step explained variance approach the standard canonical correlation analysis only maximizes one part of the redundancy formula (18).

Another interesting way to interpret the redundancy index is to regress each variable in the Y set in turn on all the variables in the predictor set and then averaging the resulting squared multiple

130 JA,~ K. SENGIJPTA

correlation coefficients, i.e.

Yt = Pm + z PriXi + 5; R: = squared multiple correlation (19) i=l

t= 1,2,. . . ,s; mcs

&IX= (l/s) i R:. (20) r=1

Thus it is clear that the redundancy index expresses the amount of variance in the Y set that is ‘redundant’ to the variance in the predictor set X. Since it combines the individual multiple correlation coefficients Rf, one could easily assess the goodness of fit of each equation in (19) and in particular one could generalize it to a weighted average measure, i.e.

R w = i w,R:I~w,, w, 2 0 (21) 1=1

where the weights w, may reflect the importance of each y, variable in the Y-set. It is clear that the ideal situation where the redundancy index can play an effective role as a

measure of similarity of the output and the input sets in the DEA model arises when the canonical variates are meaningful in terms of external evidence (e.g. the dummy variable method of Stone) and they account for a large proportion of the variance in the original data. In such situations the redundancy index may be used to decide if new input (output) variables are to be added or the old ones to be deleted.

In this sense the redundancy index may be used to examine the appropriateness of the input-output mix of different DEA models. However it is an open question as to what is the minimum acceptable value of the redundancy index to justify a meaningful interpretation of the canonical functions. As in the theory of principal components there is no definite theory in this regard, although a priori information and simulation experiments in suitable empirical cases may provide valuable guidelines.*

EMPIRICAL APPLICATIONS

Some empirical applications are now considered using input-output data for selected public elementary schools in California for the years 1976-77 and 1977-78 over which separate regressions estimates of educational production functions were made in earlier studies [17, 181. Statistics of enrollment, average teacher salary, standardized test scores were all obtained from the published official statistics. Out of a larger set of 35 school districts, 25 were selected in the three contiguous counties of Santa Barbara, Ventura and San Luis Obispo on the basis of separate homogeneity tests based on the Goldfeld-Quandt statistic [19]. Four output variables were the standardized test scores in reading (yl), writing (y2), spelling (y3) and mathematics (yJ. Two measures of aggregate output can then be defined. One is the composite output y’ = a’y based on the optimal weights and the other is the average output jj = (yl + y, + y, + yJ4 with equal weight on each output. As input variables we had a choice of eight variables of which the following four were utilized in the LP estimates: x1 = average instructional expenditure, x2 = proportion of minority enrollment, xj = average class size and xq = a proxy variable representing the parental socio- economic background. Again one can define a composite input xc = #I’x, and an average input, though the latter may not be very meaningful due to the diverse nature of the four inputs.

On the basis of these statistical data three types of tests are made in order to test the predictive power and robustness of the LP estimates of production frontier. The first test computes the mean square error MSE(.s) of prediction defined in (4) over the nondegenerate solutions only. Three types of output are used, e.g. one is the average output (J), the second is the composite output (yi) defined by the DEA model and the third is the composite output (fR) which maximizes the

*The author is thankful to one anonymous referee for this important suggestion.

Tests of efkiency in DEA 131

correlation between inputs and outputs. The results are as follows:

MSE(&) over Nondegenerate LF solutions

LP, v YS Y’R -~

: 0.0140 0.01 OS 0.0008 0.0001 0.00021 0.00006 5 0.0101 0.0003 0.00006

E: 0.0140 0.0108 0.0004 0.0083 0.00171 0,0~6 I6 0.0101 0.0024 0.00172 18 0.0140 0.0001 o.oQOu6 25 0.0101 0.0005 0.00004

Mean LP 0.01 or 0.0003 0.00006

It is clear that the mean square error is uniformly lower for the composite output model and more so for the canonically weighted case. When the degenerate cases are also included, this pattern is upheld, although the mean square error is higher, e.g. the average of MSE(ts) for the degenerate cases altogether turns out ta be 0.1451,0.0038 and 0.0002 respectiveiy for J, y$ and fR. Thus the use of canonical variates which maximizes the correlation between inputs and outputs helps improve the predictive power of the regression model and also the DEA modei. This is more so when the data points are suitably clustered around cone domains, e.g. for points close to the mean LP modef the predictive power is observed to be the highest.

The second test computes the various statistics related to the canonical correlation theory based on the composite output which is denoted as yfk) for the kth canonical correlation k = 1,2,3,4 in the two-stage procedure as follows:

k

Method

A. Two-stage DEA (a) canonical correlation: rfkr

IW G, (c) redundancy: Rvlx f%)

8. Ordinary canonical approach ta) hkl

W f& [c) rt3dundancy (%)

1 2 3 4

0.92 0.71 0.46 0.02 0.83* 0.50* 0.20 0.71 0.20 0.09 -

0.84 0.70 0.30 0.12 0.70 0.49 0.09 0.01 0.65 0.30 0.04 0.01

DEA model improves the canonical correlation, Thus if the effidient units are first identified and screened and then the canonical correlation method applied, it tends to improve the predictive power. Secondly, the first two squared canonical correlations are found to be statistically significant at the 0.05 level and the null hypothesis of p 0 = 0.50 is strongly rejected at the 0.001 level when we apply the z-statistic defined in (16). Finally, the measure of redundancy in the Y-set, given the X-set shows that the third and fourth variate have negligible ~ont~butio~s. This raises a strong case for ignoring the third and fourth canonical correlations in our results.

Finally, we apply Bartlett’s test criterion (14) to test the null hypothesis that the third canonical correlation pf3) = 0: i.e.

&J: P(3) = 0; K:P(3)#oO.

The value of X2 = 3.08 turns out to be less than X:6,0.05 = 26.296 thus rejecting the null hypothesis. But the other null hypotheses such as pfl, = 0 and pf2) = 0 were strongly rejected at 0.001 level. This suggests that only the first two canonical output variables may need to be considered in running canonical regression estimation. Finally, we computed the correlation coefficients of the first canonical output y&, and the three dummy variables such as I), = average IQ score, DZ = CPA

132 J~rt K. SENGUPTA

score and 4 = average budget of the school districts

DI & 4

Yh 0.95* 0.72 0.48

clearly the dummy variable D, seems to represent best the composite output yT1, in terms of the highest degree of correlation which is statistically significant at the 0.001 level.

CONCLUDING REMARKS

The multivariate theory of canonical correlations and their related statistical tests provide a signi~cant generalization of the DEA method of efficiency measurement. Two new areas where this theory can be applied deal with the recent approaches of nonparametric estimation of the production frontier. First of all, different types of composite outputs based on canonical correlations of different orders may be used to estimate the distribution of input-response in a nonparametric manner 19, 10-J. Secondly, the theory of rank correlations may be exploited in the various tests of predictive power, which would generaiize the nonparametric method of efficiency measurement followed in the DEA approach. Also there is great scope of applying simulation techniques here from the input and output data generated by different dasses of nonnormal distributions.

Acknowledgements-The author is most thankful to two anonymous referees for their helpful comments and suggestions and to Dr Samuel J. Raff for expediting the review process.

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