Between 1960 and 1990, public school enrollment in the United States grew only
19 percent but real spending in public education increased over 200 percent. Despite
rising real expenditures per student, scores on the Scholastic Achievement Test (SAT)
declined from 980 to 900 during the period. At the state level, on average, Utah’s public school students scored at the 53rd percentile, meaning that they performed 3 percentile points above the national average. However, because of the relatively low number of Utah students who are below the poverty level, Utah students would be expected to score at the 57th percentile (given the usual effects of poverty on national averages). It should be noted, that even ranking above average in the U.S. would not be as great a distinction as it might seem, given that American eighth graders only scored 17th in science and 28th in math in a recent comparison among 41 industrialized nations. {Utah Schools: A Consumer’s Guide, 1997, p. 3).
Efficiency in the public education system is a significant issue in the United
States. There may be productive or technical writer inefficiency and/or allocative or price
inefficiency (i.e., given the relative prices of inputs, the cost minimizing input
combination is not used). The magnitude of resources devoted to public education in the
United States is such that the efficient operation of school districts and individual
schools is essential. Inefficiency might be tolerated in some public programs where the
resources involved were quite small, but with billions at stake, small improvements in
school efficiency could save significant resources. Clearly, the issue of returns to scale is
important because there may be substantial savings achieved by consolidation of schools
and/or school districts.
The majority of the pro-consolidation literature rationalizes consolidation of
school districts on the basis of saving costs due to economies of scale arising out of
2 bigger size. The basic model that will be used for estimating the school district cost
function follows Downes and Pouge (1994), but an extension of the model will be made
testing the robustness of the results assuming groupwise heteroscedasticity, correlation,
and autocorrelation using panel data.
Next, the technical writer efficiency of individual school districts is measured within the
framework of an educational production function using stochastic frontier methodology
following Jondrow et al. (1982). Robustness of these efficiency estimates based on
restrictive assumptions of the parametric approach as proposed by Jondrow has been a
source of debate for the proponents of a nonparametric approach. The deterministic
nonparametric approach that developed out of mathematical programming is commonly
known as the DEA. This approach has been extensively used in measuring efficiency in
the public sector where market prices for output are not available. For example, Levin
(1974), Bessent and Bessent (1980), Bessent et al. (1982), and Fare et al. (1989b) use this
method to estimate efficiency in public education. In this context, this dissertation
measures technical writer efficiency and total factor productivity in educational production units
by utilizing a multi-output production technology using DEA. An empirical application
to Utah school districts reveals that filtering out scale, congestion, and socioeconomic
components from technical writer efficiency measures provide superior pine technical writer efficiency
estimates.
Recent work in the stochastic production function approach to measuring
technical writer efficiency has used panel data. Based on the treatment of technical writer inefficiency,
stochastic frontier panel data models fall into two classes. Pitt and Lee (1981), Battese
and Coelli (1988), Kumbhakar (1987, 1988), and Schmidt and Sickles (1984) treat
3 technical writer inefficiency as invariant over time while Cornwell, Schmidt, and Sickles (1990)
and Kumbhakar (1990) assume technical writer inefficiency as time varying. This dissertation
estimates rime-varying technical writer efficiency of Utah school districts using panel data. The
comparison of efficiency estimates from both the parametric and nonparametric
approaches using panel data reveals the convergence between these two approaches.
This dissertation is organized as follows: Chapter 1 discusses objectives and
statement of problems for each essay, Chapters 2, 3, and 4 include three different essays,
and Chapter 5 provides a summary and conclusion. Appendix B contains an extension of
the theory and estimates of Chapter 3.
1. First Essay Objectives
The main purpose of this essay is to investigate two basic questions relating to the
measure of technical writer inefficiency, which are:
1. whether economies of scale exist in the production of education, and
2. if they do, then at what level, individual school or school district?
This first essay examines the relationship between expenditure per student on
operations and capital (including construction) and the size of school districts and size of
schools. The hypothesis that district specific effects are fixed is tested against the random
effect The results indicate significant economies of scale in the size of schools, and it is
found that consolidation of schools, rather than school districts, will lead to reduction of
costs. This relationship is reexamined under conditions of heteroscedasricity, correlation,
and autocorrelation.
4
2. Second Essay Objectives
The purpose of the second essay is to:
1. define and measure technical writer inefficiency at the individual school district level using a
stochastic production frontier where efficiency is explained by variables that are within
and other variables that are beyond the control of the school districts, and
2. test if the measure of technical writer inefficiency is sensitive to the specific distributional
assumptions about the inefficiency component of the error term.
This essay measures the individual specific technical writer inefficiencies based on the
formulation provided by Jondrow et al. (1982). In the stochastic frontier model a non
negative error term representing technical writer inefficiency is added to the classical linear
model. The general formulation of the model is: yt = Pi + P2*i2 + $3xi3 + +Vk*ik + ei (!) where is output and the x/s are inputs. It is postulated that s,- = v( — where v,
~//(0,o^) and u, ~ | N(0,o*) |, i.e., u{ >0, and the «, and v, are assumed to be independent.
The error term (ej is the difference between the standard white noise disturbance (v^) and
the one-sided component (u*). The term v, allows for randomness across firms and
captures the effect of measurement error, other statistical noise, and random shocks
outside the firm’s control. The one-sided component u, captures the effect of inefficiency
(Forsund, Lovell, and Schmidt, 1980). Most of the earlier stochastic production frontier
studies could only calculate mean technical writer inefficiency of firms in the industry because
they could not decompose the residual for individual observations into the two
components. Jondrow et al. (1982) solved the problem by defining the functional form of
5 the distribution of the one-sided inefficiency component and derived the conditional
distribution of [u, | vruf] for two popular distribution cases (i.e., the half normal and
exponential) to estimate firm-specific technical writer inefficiency. technical writer inefficiencies
based on both half-normal and exponential distributions are estimated using cross-
sectional data.
3. Third Essay Objectives
Because the use of cross-sectional data ignores the possibility of heterogeneity of
the parameters in the stochastic production function approach, the technical writer inefficiency
estimates in the second essay may be biased. Hence, in this essay the objective is to
obtain more powerful estimates by:
1. simultaneous estimation of the parameters of the stochastic production function and
technical writer inefficiency assuming district-specific effects vary over time, and
2. testing the hypothesis that district-specific effects are stochastic.
It is likely that there are biases in the measure of technical writer efficiency when
parameter heterogeneities among cross-sectional units are ignored. Biases also may arise
if the intercepts and slopes vary through time, even though for a given time period they
are identical for all individuals. Hence, the assumption of time-varying technical writer
efficiency and the use of panel data offers more consistent estimates of technical writer
efficiency. Whether the district-specific characteristics affecting inefficiency are
stochastic or nonstochastic is tested using a generalized likelihood test. The uniqueness
of this essay is that the parameters of the production function as well as inefficiency
effects model are estimated simultaneously.
6
4. Appendix B Objectives
The objective of this extension of the second essay is to model a school district as
a multi-output rather than a single output producing unit, in order to
1. measure technical writer efficiency explicitly conditioned on the socioeconomic and
environmental factors that are beyond the control of the school districts,
2. to disaggregate the scale, congestion, and technological innovation components in
total factor productivity.
Most of the studies that estimate technical writer efficiency in public education using a
parametric or nonparametric approach Ml to consider the fixed and variable nature
socioeconomic and environmental variables appropriately. This results in biased
estimates of technical writer efficiency. The parametric approach models these variables as
inputs in the production function, while the nonparametric two-stage DEA approach
treats than as factors influencing the efficiency index in the second-stage regression
analysis. This essay models socioeconomic and environmental factors nonparametrically
using a modified form of DEA (subvector efficiency) model where efficiency of the
school districts is defined over a subvector of endogenous variables. Further, instead of a
conventional use of an input based measure of technical writer efficiency, an output based
measure is used.
7
CHAPTER2
ECONOMIES OF SCALE IN PUBLIC EDUCATION:
AN ECONOMETRIC ANALYSIS
1. Introduction
The consolidation, merger, and elimination of small rural school districts has been
a controversial issue in the United States for many years. For most of this 20th century,
the elimination of small rural school districts mainly was done to rationalize increased
state control (Wiles, 1994). The pro-consolidation literature has proclaimed the virtues of
consolidation in bringing about more effective schools by: increasing the tax base; quality
of professional personnel; expanding the breadth of educational program, special
services, and transportation services; and reducing overall educational costs per student
(Butler and Monk, 1985; Omstein, 1989).
A strong and persuasive counterattack against consolidation started in the mid-
1970s. It began with Summers and Wolfe (1975), who found higher achievement with
smaller class size, and was followed by Sher (1977), who found that the efficiency and
size of rural schools in Nebraska were unrelated to student performance. A
comprehensive review by Monk (1990) suggests that any gain achieved by consolidation
may be quickly exhausted when size interacts with socioeconomic variables. His view is
that there is no evidence that increased size has resulted in a more comprehensive
curriculum or any financial gain from economies of scale.
Most of the pro-consolidation literature rationalizes consolidation of school
districts on the basis of reducing costs due to economies of scale arising out of larger
8 districts. In contrast, Lewis and Chakraborty (1996) have shown that in Utah, economies
of scale exist at the school level rather than at the district level, and to capture economies
of scale, average school size must be increased. The study estimated cost and
expenditure functions using fixed and random effect models within the framework of
classical linear regression assumptions. While strong evidence of economies of scale is
found at the school level, the robustness of those results were not verified under
conditions of heteroscedasticity and autocorrelation, the two major problems of time
series data. This study offers a robust estimate of the parameters of both cost and
expenditure functions correcting for all probable sources of heteroscedasticity and
autocorrelation using fixed and random effect models.
This essay is organized as follows. The next section provides the background for
the study. Next, a formal model is developed in order to estimate the education cost
function directly and indirectly. Then, the data set is described and the statistical results
are presented and discussed. Summary and concluding remarks are reported in the final
section.
2. Background
The term economies of scale is defined as the reduction of the long-run average
cost as the rate of output increases when the “state of the art” is given. Economies and
diseconomies of scale are referred to as scale effects. Starting with the earliest work of
Hirsch (1959), a substantial research has been done on this topic to address two basic
questions in the production of education: (i) whether economies of scale exist in the
production of education, and (ii) if so, do they exist at school or district level or both?
9 A scientific analysis of these issues is important given the general belief that
consolidation of school districts always leads to lower cost per student. Most of the
research is consistent regarding the existence of economies of scale in the production of
education.1 However, there are conflicting empirical results about the second issue. For
example, Riew (1966) found economies of scale exist for Wisconsin high schools up to a
level of 1,675 students, and his 1986 study found larger economies for secondary schools
compared to elementary schools. Cohn’s (1968) study of high schools in Iowa also found
economies of scale up to 1,500 students. Using the data from public schools in New
York state, Butler and Monk (1985) found that school districts with 2,500 or fewer
students show greater economies of scale than did the larger school districts. Kumar’s
study (1983), of Canadian schools, shows the possibility of capturing economies of scale
by school consolidation. The study by Bee and Dolton (1985), using data from English
schools, suggests a negative relationship between school size and average cost. The issue
of economies associated with school district size is not clear. (See Callan and Santerre,
1990; Tholkes and Sederberg, 1991.)
Monk (1990) argues that consolidation has been pushed so far that any further
attempt to reduce the number of school districts may be counterproductive, and that there
will be only limited gains in productivity. However, he advocates capturing the benefits
of consolidation without eliminating school districts by developing regional service units,
and sharing facilities and administrative services, so that all community social services be
shared by neighboring districts. For example, Minnesota encourages this by providing up
to 75% of the cost of shared secondary-school facilities and programs, and Iowa provides
10 between 5 to 50% additional funding to local school districts that share course offerings,
teachers, administrators, and school buildings. (See Ornstein, 1992.)
Finally, most of the earlier research on economies of scale in education
production were done using cross-sectional data. In contrast, this study uses panel data to
answer the two primary questions on economies of scale.
3. Developing the Cost Function
The model described below was developed by Downes and Pogue (1994) in
estimating cost functions using data from Arizona’s elementary and secondary education
systems. The authors started with the standard cost function (i.e., the dual of neoclassical
production function). The conventional specification is the log-linear relation between
the total cost as the dependent variable and output, input prices, measures of attributes of
the school district, and a stochastic disturbance term. A problem arises when public
sector output cannot be measured satisfactorily. A method suggested by Chambers
(1978), Bradbury et al. (1984), Ladd and Yinger (1989), and Ratcliffe, Riddle, and
Yinger (1990) is that equilibrium output depends on some exogenous variables, including
the median income of families. The expenditure function is derived by substituting these
variables for output in the cost function. The basic underlying assumption in this case is
that there is no lag in the adjustment process between a community’s actual and preferred
output; otherwise the expenditure function will suffer from specification error.
Consistent estimates of the parameters in both cost and expenditure functions are
dependent on the correct measure of output and correct specification of the expenditure
function.
11 In this study, cost function parameters are estimated both directly and indirectly
(by using expenditure function) and compared. It is found that economies of scale exist
both at the school and district level although the effects are stronger at the school level.
The comparison is important because the two methods require different information and
impose different degrees of structure. Direct estimation of the cost function requires
measurement of public sector outputs, while identification of these parameters from an
estimated expenditure function does not. However, it does require specific assumptions
about how a community’s spending is determined.
Each school district2 can be thought of as producing a vector of outputs, Q, using
a vector of inputs, X. Hence, the underlying cost function in this production relationship
for community j at time t is:
C(.)= M inpX = c(Q,p,S), (2.1) X
where C is total cost, p is a vector of input prices, and S is a vector of variables that
measure those attributes of the school district that influence cost. Given data on outputs
(0 , input prices, and school district characteristics, the cost function can be estimated
directly and will show the cost of producing each output vector dependent on input prices
and the community specific characteristics.
While the output of an educational production function reasonably may be
measured by standardized test score, a satisfactory measure of output for most public
sector production units are not available. Estimating the cost function from a reduced
form expenditure function generally offers a solution to that problem.
12 Assume that each school district’s output vector, Q, depends on a set of demand
variables, D, as well as on each cost factor (i.e., variables p and S ) and expressed as:
Q = f(D,p,S) (2.2)
Expenditure functions for the school district are obtained by substituting (2.2) in (2.1), as:
C=d[f(D,p,S),p,S] =e(D,p,S) (2.3)
The estimation of this expenditure function does not require data on outputs.
However, the parameter estimates of each cost factor (i.e., each element of p and S) in
Equation (2.3) underestimate their true effect on cost because these parameters reflect
effects on cost as well as demand. As has shown by Downes and Pogue (1994) and
Gertler and Waldman (1992), cost function parameters may be identified from the
reduced form expenditure function by restricting the demand variables (D) that influence
school district demand but not cost.
If we assume that the production function is homothetic and cost per unit of
output aggregate q(Q) is constant, then the cost function may be written as the product of
that output aggregate and the per unit cost of output g(p, S) and written as:
C = c(Q,p,S) = q(Q) * giPyS), (2.4)
This specific form leads to a log-linear specification of the cost function g(.) that shows
the proportion by which the cost of a given combination of output changes when p and S
change. If we assume the underlying production function is of the Cobb-Douglas type
with constant returns to scale, then the cost function is linear in output In this case,
average cost, average variable cost, and the marginal cost function are all the same. (See
Varian, 1992.)
13 The specific log-linear form of Equation (2.4), in terms of cost per student served
for school district i at time r, may be written as :
InC = cto + Saw hiQ: + hip* + S « 3* hiS* +e (2.5) j k k
where Q} are elements of aggregate output, a0 is the intercept term and denotes the
locality-specific effect capturing the influence of omitted variables, and fis white noise.
The omitted variables may include the socioeconomic conditions of the local population
that affect cost through their effect on the productivity of the inputs. For example, higher
teacher productivity is observed in districts where most students come from a high
socioeconomic background. Conversely, the cost of providing education is higher in
districts with a high proportion of students needing special attention. If the outputs, Qjf
can be measured, Equation (2.5) can be estimated directly. As indicated above, if outputs
cannot be measured directly, the cost function parameters can be estimated indirectly
from an expenditure function.
Following Downes and Pogue, the expenditure function is estimated using median
voter theory, under which each community provides the median preferred output (i.e.,
that output preferred by the median income voter). This theory is widely used for the
estimation of individual demand or public expenditure functions from cross-sectional
studies (Barr and Davis, 1966). Bergstrom and Goodman (1973) give this theory specific
empirical content by showing that, subject to certain strong assumptions, majority rule
implies that one can treat an observation of expenditure levels in a given jurisdiction as a
point on the “demand curve” of a median income citizen of that community.
14 If it is assumed that T is the median voter’s share of taxes and / is the median
voter’s income, then the equilibrium output, Q, will depend on the median voter’s
income, his share of the per-unit cost of output aggregate (T*g(p, S)), and other
characteristics of the median voter, X.
Assuming the median voter’s demand function for output j is log-linear, it may be
expressed as: InQ=fojX + 02, In/ + 03y- ln[r*g(/>,S)] + v
or InQ = 0Xj X + 02, In/ + 03,- In T + 03,g(p,5)] + v (2.6)
where v is a zero mean error which has an identical, independent normal distribution.
Substituting (2.6) into (2.5) gives the community expenditure function:
lnC = a*+0*ln/ + y/ln7’+ S (l + Y*)a2ifcln/,ifc + Z (l + Y*)<*3ifc lnS*. +p k k
V-T)
where 0* = Z<xiy02y> Y* = S a iyp3y > a * = a0 + A ^ Z a^ i,). j J J
The error term, p, represents the unobserved random components in the demand
and cost functions. It also accounts for any sort of lagged adjustments between desired
and actual expenditure for a preferred output of the district that varies across location.
The location-specific effect (a^) plays two roles. In the cost function (2.5), it controls for
the effect of output (Q) and district-specific effects (S) that are excluded and time
invariant, anHt in expenditure function (2.7), it does the same for the demand variables
(£>) and the district-specific effects (5). However, the magnitude of its effect on cost and
expenditure functions differs.
15 The demand for educational output is directly affected by the median voter’s share
of the tax. Hence, the coefficients of T in equation (2.7) may be used to indirectly
identify the parameters of the cost function (a^ and a3k) in Equation (2.5). But we cannot
measure the direct influence of output on cost (i.e., parameter a t in Equation [2.5]) from
Equation (2.7). Estimates of parameters, y*, P\ a21c, and a3k can be identified from
reduced form estimates of Equation (2.7).
4. The Data Set
This study uses input and output data from all of the 40 school districts in Utah.
However, because of the log-linear specification of our model, the number of school
districts (i.e., observations) were restricted only to those districts that utilize positive and
non-zero inputs. This has led to the total number of observations used in this study to 33
school districts.
The cost of providing current educational services (C) is measured by operating
expenditure per student in the school district. Since there is no uniform measure of
educational output (Q), most studies of educational production relationships measure
output by standardized achievement test scores, although some have used other
qualitative measures such as student attitude, school attendance rates, and college
continuation or drop-out rates. However, because test scores were not available for all
years under study, this analysis uses the proportion of students graduating in each school
district as a measure of educational output (Cohn, 1968). An average 20-year-salary for a
teacher having a B.A. degree is used as the input price (p) in the cost and expenditure
functions. Due to nonavailability of the median voter’s share of taxes, only the median
16 voter income, approximated by per capita income in the community, is used as a demand
variable (D) in the expenditure function. Variables accounting for district and the
student-specific characteristics (S) include total students in the district (district size) and
average students per school (school size). Unlike our earlier study, we restricted the
choice of independent variables to: the proportion of students graduating, per capita
income, teacher’s salary, school size, and district size in both cost and expenditure
functions, because these are considered as the most important variables in estimating
economies of scale.
The explanatory variables in the cost function include the proportion of students
graduating (a measure of output, 0 , teacher’s salary (a measure of input prices, p), and
district and school sizes as measures of community or district-specific characteristics (S).
The expenditure function includes all explanatory variables listed above except the
measure of output, and it also includes per capita income as a measure of demand (D).
The coefficients on total students in the school district and average students per school
indicate the nature of scale economies. A negative sign (positive) indicates economies
(diseconomies) of scale.
The data set was developed from various publications of the Utah State Office of
Education (1983a/b/c, 1988a/b/c, and 1993a/b/c) and the Utah Education Association
(1983a/b, 1988a/b, and 1993a/b). Table 2-1 shows that as the size increases, average
spending per student decreases and vice versa. Generally, cost per student is about 40 %
lower in larger and growing districts than the smaller ones. An initial explanation is that
average cost will be lowest in a large, growing district due to the operation of economies
of scale; of course, the net effect of each component (i.e., size and growth) is not
17 indicated by these means nor are the effects of quality accounted for. It is hypothesized
that there are economies of scale in both school and district size and, therefore, the
coefficients on district size and school size should be negative. Finally, it is hypothesized
that the demand measures should have a positive effect on expenditures.
5. Parameter Estimation and Analysis of Results
Examination of the raw data suggests that the variance may be different for the 40
time series; this is due to districtwise variation in the scale of all variables in the model.
Hence, we would expect groupwise heteroscedasticity. We do not expect timewise
autocorrelation. This is because each school district is observed at five-year intervals,
and, thus, any disturbance that occurred in one year should not be serially correlated with
the observations in the fifth or tenth year. However, we may expect cross-sectional
correlation of the disturbances across districts. This is likely because the macroeconomic
factors affect these districts in varying degrees.
The estimation of the parameters follows two distinct stages. In the first stage, we
test the hypothesis about the basic assumption concerning the behavior of the stochastic
disturbance term (e). That is, we test for timewise heteroscedasticity, timewise
autocorrelation, groupwise heteroscedasticity, and cross-sectional autocorrelation.
Second we correct for these violations of the classical OLS model, reestimate the model,
and apply the Hausman test to determine which of the models, fixed or random effects,
would be appropriate. Three procedures are used in the first stage: (i) pooled regression,
(ii) timewise heteroscedastic regression, and (iii) groupwise heteroscedastic and cross-
sectionally autocorrelated regression.
18
5.1 Procedure 1
Initially we use White’s test for heteroscedasticity and the Durbin Watson test for
autocorrelation separately for the cost and expenditure functions on the pooled regression.
While we do not find any evidence of autocorrelation in either function, the expenditure
function proved to be heteroscedastic. The results from the OLS estimation of the cost
and expenditure functions corrected for the heteroscedastic covariance matrix are reported
in Table 2-II. All coefficients for both functions have the expected signs and, except for
district size, all are significantly different from zero at the 5% or lower probability level.
5.2 Procedure 2
Treating all cross-sectional units in each year under study as a group, we tested
the cost and expenditure functions for groupwise heteroscedasticity. Using the Lagrange
multiplier, Wald, and likelihood ratio tests, we fail to reject the null hypothesis of
homoscedasticity in case of cost function while the expenditure function is found to be
heteroscedastic. The LM test for autocorrelation suggests that there is no yearwise
autocorrelation in either function. Groupwise least-square estimates of the parameters
(i.e., 40 cross-sectional units observed in years 1982, 1987, and 1992) are reported in
Table 2-III. All coefficients in both functions have the expected signs. The coefficients
on teacher’s salary for 1987 and 1992 and school size for each year are significantly
different from zero at or below the 5% level for both functions. However, the coefficients
on per-capita income in the expenditure function and proportion of students graduating in
the cost function are insignificant. The feasible generalized least square (FGLS) results
for yearwise homoscedastic and nonautocorrelated cost functions and groupwise
19 heteroscedastic corrected and nonautocorrelated expenditure functions are reported in
Table 2-IV. As expected, after correcting for yearwise hetreroscedasticity the magnitude
of the coefficients in the expenditure function (Table 2-IV) are almost the same as the
coefficients in Table 2-EL
5.3 Procedure 3
Treating n cross-sectional units (/ = 1, 2, 3…40 school districts) observed at each
time period (f = 1,2,3) as n groups, cost and expenditure functions are tested for
groupwise heteroscedasticity and cross-sectional correlation while assuming they are
uncorrelated across time. Using OLS parameter estimates, we obtain the estimates of a,2
for each school district. These estimates suggest that the variances do not differ
substantially across school districts for the cost function while they do for the expenditure
function. We use LM, Wald, and LR tests for heteroscedasticity. While the LM statistics
are based on pooled regression, the Wald statistics for the common estimate of o2 used
the total sum of squared GLS residuals, and the LR statistics are based on the FGLS
estimates. For the cost function, we fail to reject the null hypothesis of homoscedasticity,
and the expenditure function is found to be heteroscedastic by the Wald and LR tests and
homoscedastic by the LM test. We reestimate the expenditure function allowing for
groupwise heteroscedasticity as shown in the FGLS results reported in Table 2-V.
Column 2 of that table reports groupwise heteroscedastic regression estimates with
nonautocorrelated disturbance.
Allowing correlation of the disturbances across the school districts, we extend the
9 model for cross-sectional correlation which is is[e,sy ] = otyI but continue to assume
20 that observations are uncorrelated across time. To test the hypothesis that the off
diagonal elements of £ are zero (i.e., there is no correlation across units), we used the LM
test, which has a x.2 distribution with n(n-l)/2 degrees of freedom. For the cost function,
we found that there is no such correlation across school districts, but we fail to reject the
null hypothesis of nonautocorrelation for the expenditure function. Hence, allowing
cross-sectional correlation across units, we used different AR(1) processes for each
school district and reestimated the expenditure function. From the initial least square
estimates, the district-specific autocorrelations and variances of uit and sjt based on the
transformed data are estimated. The FGLS parameter estimates based on the transformed
data are reported in column 3 of Table 2-V. Finally, the estimates are recomputed using
the full model with cross-sectional covariances giving the results shown in column 4 of
Table 2-V.
The results suggest that the direct estimation of the cost function generates mostly
insignificant coefficients on the output measure except in Table 2-IV, but they have the
right sign (i.e., negative). In the expenditure function, the coefficient on the income
generally is significant and positive, except in Table 2-V, where it is negative and
insignificant. This suggests that a one dollar increase in the median voter’s income
would increase expenditures by more than a dollar.
The positive and highly significant coefficients on teacher salary imply that
additional spending on resources will be productive. The negative and highly significant
coefficients on the variable “average student per school” in all the tables suggest strong
economies of scale at the school level. This suggests that the per-student cost decreases
as enrollment increases. In fact, when school size is included along with the district size,
21 the coefficient on the latter becomes insignificant. In order to test the combined effect of
these two explanatory variables which might have resulted in the reduction of the level of
significance of district size, we used the decomposition test suggested by Kmenta (1986).
The joint effect of these two variables is separated by decomposing the residual sum of
squares (SSR). While individual contribution of school size appeared stronger than
district size, their joint contribution to SSR is found to be weak.
In the second stage of our estimation process, the parameters of the cost and
expenditure functions are estimated using the fixed-effect and random-effect models (also
known as covariance and error component models, respectively). In the covariance
model, the specific characteristic of a cross-sectional unit is a parameter (i.e., a separate
intercept term for its own); for the error component model, the specific characteristic of a
cross-sectional unit is a normally distributed random variable (Kmenta, 1986).
The results are reported in Table 2-VI. The fixed and random-effect estimates
differ mainly in the case of input price (i.e., teacher’s salary), where the coefficients differ
by more than one standard deviation. The estimated effects of other variables are similar
in fixed-and random-effect specifications. The large values of the Hausman test statistics
for random-versus-fixed-effect models also suggest the use of a fixed-effect model. For
both cost and expenditure functions we cannot to reject the null hypothesis that the
district-specific effects are fixed against the alternative hypothesis of random effect based
on the F-test. The sign of the coefficients in both cost and expenditure functions for fixed-
effect models are as expected and are highly significant These results confirm our earlier
findings that the average school size is more important than district size in expanding
economies of scale.
22
6. Summary and Conclusion
The results of this analysis confirm our earlier findings that show economies of
scale exists at the individual school level. Hence, consolidation of schools could reduce
costs per student substantially. The most important determinants of the cost of public
education are average school size, teacher salary, and per capita income of the district
population. As the coefficient of district size is not significantly different from zero at the
5% or below level, we conclude that economies of scale are not important at the district
level. Thus, consolidation of school districts in an attempt to reduce per-unit cost may not
be successful unless average school size can be increased at the same time. As there
often is political resistance to district consolidation, if average school size is the driving
force behind cost reduction, it may be possible to consolidate schools within a district and
capture most of the scale economies.
Table 2-1. Average operating expenditure per student and size and growth of Utah school
districts: 1982-83, 1987-88, and 1992-93. 1982-83 1987-88 1992-93 Average Average Average operating operating operating % District size (students) Dist. expenditure ($) Dist. expenditure ($) Dist. expenditure ($) change 1982-93 <1000 8 3,170 7 3,389 6 5,639 43.8 1,001-2,000 7 2,129 7 2,743 7 3,814 44.2 2,001-5,000 9 2,164 8 2,741 9 3,890 44.3 5,001-10,000 7 1,991 6 2,476 6 3,394 41.3 10,001-25,000 5 1,999 8 2,409 6 3,281 39.1 >25,000 4 1,858 4 2,191 6 3,129 40.6 Median 2,060 2,536 3,594 District growth rate <0% 6 2,909 6 3,624 6 5,198 44.0 0-10% 8 2,485 8 3,104 8 4,277 41.9 10-20% 12 2,094 12 2,467 12 3,474 39.7 >20% 14 2,044 14 2,505 14 3,377 9.53
Source: Utah State Office of Education (1983a/b/c, 1988a/b/c, 1993a/b/c).
Table 2-EL Pooled regression estimates with 120 observations.
24
Variables Cost function
Expenditure function Constant -15.321 -12.575 (-12.594)* (-8.141)* Ln(% of students graduating) -0.095 (-1.933)* Ln(per capita income) 0.196 (3.091)* Ln(average 20-yr-teacher salary) 1.941 1.578 (20.215)* (10.433)* Ln(students in district) -0.027 -0.022 (-1.735) (-1.814) Ln(students in school) -0.248 -0.260 (-7.139)* (-7.303)* Variance (e‘e) 1.8084 1.6693 R-squared 0.8452 0.8571
t-statistics are shown in parentheses. * Indicates that the estimated coefficient is significant at or below the 5% level.
25
Table 2-III. Groupwise least square estimates Cost function Expenditure function Variables 1982 1987 1992 1982 1987 1992 Constant -3.384 (-0.622) -8.499 (-1.698) -4.286 (-0.873) -0.280 (-0.005) -6.893 (-1.363) -5272 (-1.054) Ln(% of students graduating) -0.051 (-0.459) -0.101 (-1591) -0.183 (-1.469) Ln(per capita income) 0237 (2.676)* 0.103 (1.035) 0.036 (0.353) Ln(average 20-yr- teacher salary) 0.473 (1-112) 1.415 (3.612)* 1.114 (3.000)* 0.581 (1.491) 1.197 (2.964)* 1.131 (2.923)* Ln(students in district) -0.006 (-0.226) -0.018 (-0.675) -0.017 (-0.653) 0.005 (0.181) -0.029 (-1.122) -0.016 (-0.593) Ln(students in school) -0216 (-4.388)* -0.266 (4.663)* -0268 (4.836)* -0.325 (5.391)* -0233 (4.442)* -0.253 (4.394)* Variance (e‘e) R-Squared 0.5100 0.7163 0.5274 0.7579 0.4505 0.7900 0.4259 0.7631 0.5488 0.7481 0.4765 0.7778
t-statisties are shown in parentheses. ♦-indicates that the estimated coefficient is significant at or below the 5% level. Values of s‘e in parenthesis are based on the pooled slope estimator.
26
**<!-****-nextpage-****->**
Table 2-IV. FGLS estimates of groupwise homoscedastic cost function and heteroscedastic expenditure function with nonautocorrelated disturbance
Variables Cost function Expenditure function Constant -15.321 -12.537 (-12.865)* (-9.540)* Ln(% of students graduating) -0.095 (-1.974) Ln(per capita income) 0.200 (3.858)* Ln(average 20-yr-teacher salary) 1.941 1.572 (20.650)* (12.726)* Ln(students in district) -0.027 -0.022 (-1.771) (-1.473) Ln(students in school) -0.248 -0.262 (-7.292)* (-8.090)* Log of likelihood function 81.32 86.28
t-statistics are shown in parentheses. * Indicates that the estimated coefficient is significant at or below the 5% level.
27
Table 2-V. Groupwise heteroscedastic FGLS estimates of the expenditure function parameters
Variables
Groupwise het. & corr. With nonautocorrelated disturbances
Groupwise het. & group specific autocorrelation
Groupwise het.& corr. with group specific autocorrelation Constant -17.8020 -17.9200 -18.540 (-30.022)* (-39.926)* (-61.230)* Ln(per capita income) -0.0041 -0.0050 -0.0068 (-0.835) (-0.832) (-2.347)* Ln(average 20-yr-teacher 2.1064 2.1310 2.1852 salary) (44.212)* (56.073)* (85.977)* Ln(students in school district) -0.0325 -0.0263 -0.0266 (-4.058)* (-2.911)* (-3.732)* Ln(students in school) -0.2066 -0.2457 -0.2561 (-10.051)* (-10.463)* (-12.776)*
t-statistics are shown in parentheses. * Indicates that the estimated coefficient is significant at or below the 5% level.
28
Table 2-VI. Estimates of the cost and expenditure function parameters dependent variable: Ln(operating expenditure per student) Cost function Expenditure function Variables Fixed Random Fixed Random Constant -14.2250 (-10.782)* -12.3200 (-8.868)* Ln(% of students graduating) -0.1312 (-2.484)* -0.0955 (-2.107)* NA NA Ln(per capita income) NA NA 0.0986 (1.669) 0.1819 (3.634)* Ln(20-yr-teacher salary) 1.1301 (5.223)* 1.8537 (17.954)* 1.1799 (5.374)* 1.5668 (12.486)* Ln(students in district) -0.0178 (-0.999) -0.0317 (-2.148)* -0.0307 (-1-791) -0.0276 (-1.882)* Lnfstudents in school) -0.2455 (-6.686)* -0.2371 (-7.427)* -0.2265 (-6.303)* -0.2495 (-7.926)* R-squared 0.8452 0.8439 0.9239 F-statistic 157.03 21.47 Hausman test statistic 16.84 9.51
t-statisties are shown in parentheses. * Indicates that the estimated coefficient is significant at or below the 5% level. The Hausman statistic is based on fixed vs random effect results from the corresponding models.
29
Notes: 1 For a comprehensive literature review, please see Lewis and Chakraborty, 1996. 2 The term school district and community are used interchangeably here.
30
CHAPTER 3 MEASURING technical writer EFFICIENCY IN PUBLIC EDUCATION: A STOCHASTIC PRODUCTION FUNCTION APPROACH
1. Introduction
The pioneering work by Farrell in 1957 provided the definition and conceptual
framework for both technical writer and allocative efficiency. While technical writer efficiency refers
to failure to operate on the production frontier, allocative efficiency generally refers to the
failure to meet the marginal conditions for profit maximization. Since that time,
considerable effort has been directed at refining the measurement of technical writer efficiency.
The literature is broadly divided into deterministic and stochastic frontier methodologies.1
The deterministic nonparametric approach that developed out of mathematical
programming is commonly known as data envelopment analysis (DEA), and the
parametric approach that estimates technical writer efficiency within a stochastic production,
cost, or profit function model is called the stochastic frontier method. Both approaches
have advantages and disadvantages, which are discussed in Forsund, Lovell, and Schmidt
(1980).
DEA has been used extensively in measuring efficiency in the public sector (e.g.,
education) where market prices for output are not available. For example, Levin (1974),
Bessent and Bessent (1980), Bessent et al. (1982), and Fare et al. (1989b) used this
method to estimate efficiency in public education. The stochastic frontier methodology
was used by Barrow (1991) to estimate a stochastic cost frontier using data from schools
in England. Wyckoff and Lavinge (1991) and Cooper and Cohn (1997) estimate
technical writer efficiency using school district data from New York and South Carolina,
31 respectively. Grosskopf et al. (1991) use the parametric approach to estimate allocative
and technical writer efficiency in Texas school districts. The recent literature has seen a
convergence of the two approaches and their complementarity is being recognized.2
Here, the technical writer efficiency estimates are made for each school district using the
stochastic frontier method. The analysis uses data from the 40 school districts in Utah for
the academic year 1992-93. Using standardized test scores of the 11th grader as school
output, two classes of inputs are included. The first, considered to be subject to control
by school administrators, includes the student-teacher ratio, percentage of teachers having
an advanced degree, and percentage of teachers with more than IS years of experience.
The second class includes such uncontrollable factors as the socioeconomic status of the
students (SES), education level of the local population, and net assessed real property
value per student. The objective of the study is to measure technical writer efficiency at the
individual school district level, using both the DEA and the stochastic frontier estimation
methods, and to identify the sources of inefficiency.
This essay is organized as follows. First, the relevant literature is reviewed, and
this is followed by a definition of educational production function. Next, the stochastic
specifications of technical writer inefficiency are discussed. This is followed by a review of the
stochastic estimation methods outlined by Jondrow et al. (1982). Finally the data set is
discussed and the empirical results presented.
2. Background For a given technology and set of input prices, the production frontier defines the
maximum output forthcoming from a given combination of inputs. Similarly, the cost
32 frontier defines the minimum level of cost for providing a specified rate of output given
input prices, and the profit frontier defines the maximum profit attainable given input and
output prices. Inefficiency is measured by the extent that a firm lies below its production
and profit frontiers and above its cost frontier. Koopmans (1951) defined a technical writerly
efficient producer as one who cannot increase the production of any one output rate
without decreasing another or without increasing some input. Debreu (1951) and Farrell
(1957) offered a measure of technical writer efficiency as one minus the maximum
equiproportionate reduction in all inputs that still allows continuous production of a given
output rate (Lovell, 1993).
The earliest study that measured technical writer inefficiency in education production
was Levin (1974, 1976). He used the Aigner and Chu (1968) parametric nonstochastic
linear programming model to estimate the coefficients of the production frontier and
found that parameter estimation by ordinary least squares (OLS) does not provide correct
estimates of the relationship between inputs and output for technical writerly efficient schools; it
only determines an average relationship. Klitgaard and Hall (1975) used OLS techniques
to conclude that schools with smaller classes and better paid and more experienced
teachers produce higher achievement scores. Their study also estimates an average
relationship, rather than an individual school specific relationship between inputs and
output
The concept of a stochastic production frontier was developed and extended by
Aigner, Lovell, and Schmidt (1977), Meeusen and van den Broeck (1977), Battese and
Corra (1977), Battese and Coelli (1988), Lee and Tyler (1978), Pitt and Lee (1981),
Jondrow et al. (1982), Kalirajan and Flinn (1983), Bagi and Huang (1983), Schmidt and
33 Sickles (1984), and Waldman (1984). The basic idea behind the stochastic frontier model
as stated by Forsund, Lovell, and Schmidt in 1980 is that the error term is composed of
two parts: (i) a systematic component that captures the effect of measurement error, other
statistical noise, the random shocks, and (ii) a one-sided component that captures the
effects of inefficiency.
Frontier production models have been analyzed either in the framework of the
production function or by using duality in the form of a cost minimizing or profit
maximizing framework. Barrow’s study (1991) of schools in England tested various
forms of the cost frontier and found the level of efficiency to be sensitive to the method
of estimation. In their study of technical writer inefficiency in elementary schools in New York,
Wyckoff and Lavinge (1991) estimated the production function directly and found that
the index of technical writer inefficiency depends on the definition of educational output. For
example, if output is measured by the level of cognitive skill of students rather than their
college entrance test score (i.e., the ACT or SAT or any other type of composite test score
based on reading, writing and mathematics skills), the index of technical writer inefficiency
based on each output measure will be different. Grosskopf et al. (1991) used a stochastic
frontier and distance function to measure technical writer and allocative efficiency in Texas
school districts and concluded that they were technical writerly efficient but allocatively
inefficient.
3. Defining the Educational Production Function
In the production of education, school districts use various school and non-school
inputs to produce multiple outputs, generally measured by achievement test scores. As
34 the purpose of education is to develop the student’s basic cognitive skills, these skills
often are measured by scores on reading, writing, and mathematics tests. However, there
are references in the literature where output is measured either by the number of students
graduating per year, student success in gaining admission into the higher education,
and/or future earning potential. In most of these studies, the measure of output is limited
by the availability of data. School inputs that are associated with students’ achievement
scores typically include the student-teacher ratio, teacher’s educational qualification and
teaching experience, and various instructional and non-instructional expenditures per
student Non-school inputs include the socioeconomic status (SES) of the students and
other environmental factors that influence student productivity. While family income,
number of parents in the home, parental education, and ethnic background measure the
SES of the students, geographic location (i.e., rural/urban) and net assessed value per
student capture the environmental factors.
School inputs that basically are associated with instructional and non-instructional
activities are under the control of the school management. Most studies in educational
production find an insignificant relationship between most of the school inputs and
outputs. For example, see Walberg and Fowler (1987), Hanushek (1971), Defier and
Rudnicki (1993) and Cooper and Cohn (1997). Studies by Hanushek (1986) and Fare,
Grosskpof, and Weber. (1989a) find a significant influence of SES and environmental
factors on student’s achievement scores.
A school district is technical writerly efficient if it is observed to produce the maximum
level of output from a given bundle of resources used or, conversely, uses minimum
resources to produce a given level of output. Here, the single output of our educational
35 production function is measured by the average test score of I lth-grade students on the
standardized battery test.
4. Stochastic Specification of technical writer Efficiency
In the stochastic frontier model, a nonnegative error term representing technical writer
inefficiency is added to the classical linear model. The general formulation of the model
where y; is output and the x ’s are inputs. It is postulated that e, = vf — ut where v,
independent. The error term (ej is the difference between the standard white noise
disturbance (v*) and the one sided component (u*). The term v, allows for randomness
across firms and captures the effect of measurement error, other statistical noise, and
random shocks outside the firm’s control. The one-sided component «, captures the effect
of inefficiency (Forsund, Lovell, and Scmidt, 1980). Most of the earlier stochastic
production frontier studies could only calculate mean technical writer inefficiency of firms
because they could not decompose the residual for individual observations into the two
components. Jondrow et al. (1982) solved the problem by defining the functional form of
the distribution of the one-sided inefficiency component and derived the conditional
distribution of [u, | vr«J for two popular distribution cases (i.e., the half normal and
exponential) to estimate firm-specific technical writer inefficiency.
For this study, let the production function for the z’th school district be written as:
is: y, = Vi + + P3*z3+………………..+e.- (3.1)
~N(0,o^) and uf — |iV(0,o^)|, i. e., «, >0 , and the u, and v, are assumed to be
/«1
(3.2)
36 where y is output, and Xj are exogenous inputs. A is the efficiency parameter and v is the
stochastic disturbance term. The production function in (3.2) is related to the stochastic
frontier model by Aigner, Lovell, and Schmidt (1977), who specify A as:
where Oq is a parameter common to all districts and u is the degree of technical writer
inefficiency that varies across school districts. Units for which u=0 are most efficient. A
district is said to be technical writerly inefficient if output is less than the maximum possible
rate defined by the frontier. The term v is the usual two-sided error term that represents
shifts in the frontier due to favorable and unfavorable external factors and measurement
error.
After including the component of inefficiency (i.e., e “), the actual production
function is written as:
If there is no inefficiency and potential output is denoted by Y, then the production
function is written as:
A = aQe “ u > 0
(3.3)
(3.4)
Hence, the appropriate measure of technical writer efficiency is:
actual output >7 _ j=i potential output
= e (3.5)
37 Potential output is the maximum possible when u = 0 in equation (3.3). A technical writerly
efficient school district produces output (i.e., standardized test scores) that is on the
However, because of differences in managerial efficiency, actual performance deviates
from the frontier.
Since, u > 0, 0 < e~“ < 1, and e* are measures of technical writer efficiency, the mean
technical writer efficiency is E(e»). Thus, technical writer inefficiency is measured by l-e“ where e’“
and 0, and technical writer inefficiency is bounded between 0 and 1.
5. Method of Estimation This study uses the method of estimation suggested by Jondrow et al. (1982) to
estimate technical writer inefficiency in each school district. Here, the technical writer inefficiency
error term u is alternatively assumed to be of the half-normal and exponential type.
Results based on these both distributional assumptions are reported.
To estimate Equation (3.3) by the maximum likelihood method, we need to know
the probability density function (pdf) of e{, which is composed of if, and vf. The pdf,
mean, and variance of «, and v, are written as:
stochastic production frontier, which is subject to random fluctuations captured by v.
is technical writer efficiency bounded by 0 and I. That is, technical writer efficiency lies between 1
(3.6)
E(v£) = 0 and Var(v) = ct2v
pdf of iff —f(Uj) = = — \Z.n<Ju
, jL e if, > 0. (3.7)
38 The half normal distribution (3.7) has the following mean and variance (Maddala, 1977;
Aigner, Lovell, and Schmidt, 1977):
£(“«) = ~i==Gu 30(1 Var(u) = a£ . V 7l *
Assuming k, follows a one-parameter exponential distribution with pdf written as:
—U
pdf of Uf =f(u) = /(«) = — . (3.8)
To formulate the log-likelihood function, the density function of the composite
residual (y-u) (i.e., the joint density function of f[y, u)) is formulated and then transformed
into a joint density of e and u by integrating u from 0 to oc. Following Maddala (1977),
the pdf of e , which is a composite of v and u, is written as:
/( E) = — r [l-F (f)]e ** , (3.9) OV 7t °
where a2 =o^ + a^ , X=aJ<jw, and F(^) are the cumulative density fanction^cdf) of
the normal variable evaluated at ^ .
If we have a sample of n observations, we can form the relevant log-likelihood
function as:
IniO-lp,….ps,X,o!)=Bln^ + nln-!- + < 310) Vti cr f t «• J 2a w
eX x* It should be noted that the meaning of 1-FQ, where F is evaluated at — = ——-, is the a a*
a2e a^a2 probability that a N(p.a.2) variable with and cl = is positive.
39 Differentiating Equation (3.10) with respect to the unknown parameters and setting the
partial derivatives equal to zero, we solve for an estimate of P„.., (35, X and a2.
Individual specific estimates of inefficiency measured by Jondrow et al. (1982),
using the conditional mean of u given s, involve the following steps. First find the joint
density of u and v, which may be written as:
/(«,v) = f(u)f(v) = — exn —~ w 2 7iauCTv |_ 2a~ 2c^
, u> 0. (3-11)
For detail on the derivation, see Appendix A. Then, using the relationship of e = v-u,
the joint density of u and e is
/ ( a ‘6) — _ i (“2+e2+2ke)
(3.12)
Therefore, following Jondrow et al. (1982), the conditional density of u given e, for half
normal distribution is:
/( « .e) ____ /(e) l-FS/27«7CCT* -exp
2o2 I o2
u> 0 (3.13)
and the mean of this conditional density function is written as:
E[u\zA — ———— (3.14) L *’ *J ^ 1-F(p./CT.)
where/.) and F(.) represent the standard normal density and cumulative distribution
functions, respectively. In the exponential case the technical writer inefficiency component of the error term, u,
follows a one-parameter distribution. The density function of u is given in (3.8).
40
e <jv CTV Gu Let A = — + — . Then, the mean of the conditional distribution of u given e is » ~u AA) £(u/e) = tr3 -A (3-15)
6. The Data Set
Again, data for the 40 school districts in Utah were collected. The output (y) is
measured by the llth-grade battery test score, a composite of reading, writing, and
mathematics skills. The school inputs used in this study are: the student-teacher ratio
(x,), percentage of teachers with an advanced degree (xj, and percentage of teachers with
over 15 years of experience (x3). Non-school inputs consist of percentage of students
who qualify for Aid to Families with Dependent Children (AFDC) subsidized lunch (x4),
percentage of district population having completed high school (xs), and net assessed
value per student (x*). While x, is a proxy for the level of instructional input, Xj and x3
measure quality of teaching inputs and x4, xs, and x6 measure the SES and the
environmental factors. In the single equation model, the first three inputs (x„ Xj, x3) are
subject to control by management, whereas x4 through x6 are beyond such control. The
summary statistics for both inputs and output are reported in Table 3-1.
Following Schmidt and Lovell (1979) and Battese and Coelli (1988), a Cobb-
Douglas functional form of the production function in log linear form is postulated:
lny, = a 0 +Pj lnxt +P2 + 03 + p4 ^*4 + Ps lnx5 + v-n (3.16)
where yx is the educational output (i.e., average test score), the x, are the inputs described
above, vf ~~N(0,o^) and u{ ~ |W(0,o^)| . The condition that m, > 0 allows production to
occur below the stochastic production frontier.
41 The following relationships between output and each explanatory variables are
hypothesized:
Variable Coefficient Hypothesized Sign Student-teacher ratio P, <0 Percentage of teachers with advanced degree P2 >0 Percentage of teachers with over 15 years of experience P3 >0 Percentage of students receiving subsidized lunch P< <0 Percentage of population with high school education Ps >0 Net assessed value per student P* >0
7. Empirical Results
Maximum-likelihood estimates3 of the parameters based on half normal and
exponential distributions of u are reported in Table 3-H. Except for the net assessed value
per student, all the coefficients have the correct sign but only the coefficient on
percentage of population with high school education is significant at the 0.05 or lower
level. The possible reason for a negative sign on net assessed value per student input is
the presence of strong multicollinearity with other socioeconomic inputs. The highly
significant coefficient on the education level of the district population implies a 1%
change in population with high school diploma is associated with a 0.96% to 0.91%
change in test score. This indicates the importance of the environment for learning
provided in the home. The negative sign on the student-teacher ratio is as expected and
confirms the conventional wisdom that smaller classes are more conducive to better
learning. Positive coefficients on teaching experience and teacher’s educational
qualification imply positive contributions of these inputs toward students’ learning
42 process. Finally, the welfare variable has the expected negative sign but the coefficient is
not statistically significant.
These results are consistent with those obtained by Walberg and Fowler (1987)
and Cooper and Cohn (1997), who found a positive relationship between quality of
instructional staff and a weak and negative relationship of student-teacher ratio (i.e., the
class size) with achievement test score. The coefficient on variable X, which is equal to
indicates the presence of inefficiency in the production process. A highly
significant coefficient on X, implying higher values of ctu than crv, suggest a high degree
of inefficiency. The insignificant coefficient on X in Table 3-II implies that on the
average, school districts in Utah are utilizing their resources efficiently.
The technical writer efficiency (e’“) estimates based on half normal and exponential
distributions of the one-sided component of the disturbance are compared and contrasted
in Table 3-III. While there are differences in the measures of technical writer efficiency
between the half normal and exponential distributions, the rankings are very similar. The
correlation coefficient for the two rankings is 0.964. The mean efficiency is 0.8607 for
the half normal estimates and 0.8852 for the exponential function. The size of the district
(i.e., number of students) also is shown in column 3 of Table 3-III. There is no obvious
relationship between size and efficiency discernible from these data.
In case of half normal distribution, the most and the least efficient school districts
are Grand and San Juan whose technical writer efficiency scores are 0.9698 and 0.5252,
respectively. While analyzing the sources of inefficiency in case of the latter, it is
revealed that a high student-teacher ratio is mainly responsible for the poor performance,
which amounts to managerial inefficiency. Depending on the measure used, 18 to 24 of
43 the school districts have efficiency measures of 0.90 or more, which probably should be
construed as being good performance given the nature of the production system and the
constraints within which resource allocation decisions can be made, especially with
regard to personnel, many of whom have rather strong employment security.
44
8. Summary and Conclusion
This study measures technical writer efficiency in each of the 40 school districts in Utah
using stochastic estimation methods. Substantial variation of technical writer efficiency among
school districts is observed, though it is invariant as to the distributional assumption of
the one-sided component of the error term s. The results of this study suggest that most
of the school districts in Utah are technical writerly efficient with mean efficiency scores
86.07% and 88.52% for the half normal and exponential distributions, respectively. The
empirical results also indicate that the single most important factor explaining student
performance is the level of parental education. There does not appear to be systematic
similarity among the groups of most efficient and least efficient school districts. In terms
of size, ten districts at each end of the efficiency scale include both large and small
districts and they are geographically dispersed. There also is no apparent correlation
between efficiency and the local economic base. Both groups include districts located in
areas where agriculture, mineral extraction, or tourism is the predominant economic
activity.
While the thrust of the essay is primarily methodological, the results do have
policy implications. For example, districts with high SES students might improve
efficiency by better management of controllable inputs (i.e., teaching and other staff,
student work load, etc.) and/or adoption of programs that link part of teacher
compensation to student performance. Districts with a large number of low SES students
face a more difficult challenge as they deal with students who have less intellectual
support at home. In such districts, efficiency might be enhanced by some resource
45 allocation: (i) to pre-kindergarten programs to better prepare young children for entering
school, (ii) adult education, and/or (iii) greater teacher-parent interaction designed to
encourage parental support of the student’s educational activity.
46
Table 3-L Summary statistics for Utah school districts, 1992-93 (obs=40).
Variable Mean
Standard deviation Minimum Maximum Average 11* grade test score 52.100 7.523 30.000 68.000 Student-teacher ratio 20.165 3.285 10.590 30.030 Percentage of teachers with advanced degree 26.043 10.019 2.780 43.590 Percentage of teachers with over 15 years experience 17.361 4.016 5.880 25.81 Percentage of population with high school diploma 82.818 6.137 59.700 91.600 Percentage of students receiving subsidized lunch 25.650 10.618 5.000 51.000 Net assessed value per student 191,290 162,970 56,700 702,800
47
Table 3-El. Stochastic frontier parameter estimates dependent variable: Ln(test score). Variable MLE (half normal) MLE (exponential)
Constant 0.877 0.766 (0.402) (0.463) Ln(Student-teacher ratio) -0.289 -0.196 (-1.546) (-1.380) Ln(percentage of teachers with 0.024 0.057 advanced degree) (0.382) (1.195) Ln(percent of teachers with 0.032 -0.016 experience over 15 years) (0.234) (-0.206) Ln(percentage of students -0.039 -0.041 receiving subsidized lunch) (-0.430) (-0.739) Ln(percentage of population with 0.959* 0.909* high school diploma) (2.215) (3.029) Ln(net assessed value per student) -0.015 -0.011 (-0.329) (-0.308) A. 12.705 (0.468) 0° 8.832* (3.247) Log of the likelihood function 32.969 32.622
t-statistics are in parentheses. * Indicates coefficients are significant at 5% or lower probability. a„<b(z/ay) „ , ©-0 is defined as : z + ———:—— where z = e—9cr. <D(z/c7V)
48 Table 3-HI technical writer efficiency measure using half normal and exponential distributions.
District District Size
HalfNormal Efficiency Rank
Exponential Efficiency Rank 1 Alpine 40,322 0.9489 8 0.9644 6 2 Beaver 1,396 0.8521 25 0.8880 27 3 Box Elder 11,190 0.9033 9 0.9357 15 4 Cache 12,593 0.9224 19 0.9478 20 5 Carbon 5,150 0.8194 21 0.8528 17 6 Daggett 191 0.9419 14 0.9523 10 7 Davis 57,116 0.9148 20 0.9444 23 8 Duchesne 4,411 0.7958 23 0.8263 25 9 Emery 3,400 0.8569 28 0.8915 28 10 Garfield 1,097 0.7628 35 0.7885 34 11 Grand* 1,576 0.9698 1 0.9745 1 12 Granite 79,575 0.8756 18 0.9108 21 13 Iron 5,475 0.9416 13 0.9587 13 14 Jordan 68,843 0.8806 16 0.9175 16 15 Juab 1,644 0.7540 33 0.7870 33 16 Kane 1,415 0.9547 17 0.9667 9 17 Millard 3,861 0.7606 36 0.7902 36 18 Morgan 1,889 0.8774 24 0.9156 26 19 Nebo 17,161 0.8491 26 0.8877 29 20 No.Sanpete 2,352 0.6603 40 0.6821 40 21 No.Summit 944 0.7590 37 0.7882 37 22 Park City 2,540 0.9576 5 0.9677 8 23 Piute 385 0.9713 4 0.9745 3 24 Rich 549 0.8763 29 0.9128 24 25 San Juan* 3,400 0.5252 39 0.5288 39 26 Sevier 4,859 0.8289 31 0.8633 32 27 So.Sanpete 2,899 0.9058 22 0.9378 18 28 So.Summit 1,106 0.7151 38 0.7379 38 29 Tintic 241 0.8060 32 0.8232 30 30 Tooele 7,355 0.8668 11 0.9016 14 31 Uintah 6,795 0.9009 7 0.9327 4 32 Wasatch 3,137 0.9485 3 0.9638 2 33 Washington 14,596 0.9545 2 0.9669 7 34 Wayne 580 0.9248 30 0.9520 22 35 Weber 26,832 0.7711 27 0.8011 31 36 Salt Lake 25,538 0.9139 12 0.9378 12 37 Ogden 12,589 0.7560 34 0.7804 35 38 Provo 13,565 0.9824 6 0.9820 5 39 Logan 5,894 0.9489 10 0.9628 11 40 Murray 6,799 0.8737 15 0.9092 19 Mean 11,531 0.8607 0.8852 ^Indicates the most and the least efficient school districts.
permission of the copyright owner. Further reproduction prohibited without permission.
49
Notes: 1 See Ali and Byerlee (1991) for a detailed discussion on the methods for analyzing technical writer efficiency. 2 The Journal ofE conometrics (1990) devoted an entire supplemental issue to parametric and nonparametric approaches to frontier analysis. 3 Parameter estimates of the stochastic frontier production function and the technical writer efficiency estimates are made using LIMDEP.
50
CHAPTER 4 technical writer INEFFICIENCY EFFECTS IN STOCHASTIC FRONTIER PRODUCTION FUNCTION: AN APPLICATION TO PUBLIC EDUCATION
1. Introduction
Between 1960-90, public school enrollment in the United States grew only 19%
but real spending in public education increased over 200%. Despite the rising real
expenditures per student, scores in the Scholastic Achievement Test (SAT) have declined
from 980 to 900 between 1960-90. At the state level,
on average, Utah’s public school students scored at the 53rd percentile, meaning that they performed 3 percentile points above the national average. However, because of the relatively low number of Utah students who are below the poverty level, Utah students would be expected to score at the 57th percentile (given the usual effects of poverty on national averages). It should be noted, that even ranking above average in the U.S. would not be as great a distinction as it might seem, given that American eighth graders only scored 17th in science and 28th in math in a recent comparison among 41 industrialized nations. (Utah Schools: A Consumer’s Guide, 1997, p. 3.)
Hanushek’s (1986) review of literature on educational production functions found
no evidence of any significant relationship between traditional school inputs and
achievement scores. Educational researchers in education, economics, and sociology are
concerned with the negative impact of this trend on the economic competitiveness of the
country. Many have unequivocally concluded that the public education system in most
parts of this country has become inefficient (see Deller and Rudnicki, 1993; Dertouzos,
Lester, and Solon, 1989). Unfortunately, policies aimed at achieving higher expenditure
per student, smaller class size, more educated teachers, and/or higher teacher salaries as
51 measures to enhance student achievement scores often are based on incomplete
understanding of the educational production process. If current resources are not being
utilized efficiently, additional resource commitments may not generate better student
achievement but simply may be wasted through greater inefficiency.
A production frontier provides the basis for defining efficient performance. A
school district is technical writerly efficient if it is observed to produce the maximum level of
output for a given bundle of resources. The degree of departure from the frontier is the
measure of inefficiency. Use of conventional stochastic frontier methodology as
developed by Aigner, Lovell, and Schmidt (1977) involves two random components, one
associated with the technical writer inefficiency and the other being the traditional random error.
If the level of inefficiency is maintained by the firm (i.e., the school district) over time,
then it is possible that the choice of inputs will be affected by the inefficiency component.
For example, if the high student-teacher ratio is found to be a source of inefficiency over
the years, the school district may hire more teachers, which will decrease the ratio and
increase efficiency. Hence, the effects of technical writer inefficiency on the productivity of
inputs will be different. For example, a school district may acquire more knowledge,
information, and experience about the productivity of one input than another. As a result,
input productivity as well as the marginal rate of technical writer substitution will change,
which will cause a shift in the production frontier. This phenomenon is called technical writer
inefficiency effects in the stochastic frontier production function.
There has been limited use of the stochastic frontier methodology for analyzing
technical writer efficiency in public education. Modeling technical writer inefficiency of a school
district as a function of observed factors, these studies ignored the simultaneity of the
52 unobserved factors influencing inefficiency (see Cooper and Cohn, 1997; Deller and
Rudnicki, 1993). Further, these studies measure technical writer inefficiency using cross-
sectional data which suffer from omitted variable bias because inputs not included in the
model are forced to held constant over the cross-sectional units. Use of panel data offers
a solution to that problem where the intercept terms capture the unobserved heterogeneity
across repeated observations for each units over time. The objective of this study is: (i)
empirically apply the model outlined by Battese and Coelli (1995) for estimating the
parameters of the production function and the inefficiency effects model simultaneously,
and (ii) to measure time varying technical writer inefficiency of each school district in Utah
using panel data.
The organization of this essay is as follows: the next section reviews the literature
and theoretical problems in the field; the model is developed in the next section; then the
data set is described; the statistical results are discussed in the next section; and summary
remarks are made in the last section.
2. Literature Review and Problems
There have been a number of articles that extend the basic stochastic frontier
production model developed by Aigner, Lovell, and Schmidt (1977) and Meeusen and
van den Broeck (1977). An extensive review of such studies has been provided by
Forsund, Lovell, and Schmidt (1980), Bauer (1990), and Battese and Coelli, (1992).
Most of the earlier applications of stochastic frontier production functions estimated time-
invariant technical writer inefficiency using cross-section data. A recent development is the use
of stochastic frontier production functions for estimating time-varying technical writer
53 inefficiency using panel data (see Kumbhakar, 1990; Cornwell, Schmidt, and Sickles,
1990; Battese and Coelli, 1992).
There have been attempts in the past to explain technical writer inefficiency effects by
some observed factors. For example, in agricultural production a farmer’s educational
level may contribute to inefficiency as noted by Pitt and Lee (1981), Kalirajan (1981),
Kalirajan and Flinn (1983), and Kalirajan and Shand (1986), all of which followed a two-
stage procedure. First, technical writer inefficiency is estimated from the stochastic frontier
production function. In the second stage, these inefficiencies are regressed on a set of
explanatory variables to identify the differences in technical writer efficiency among firms.
Problems associated with two-stage procedure can arise because: (i) technical writer
inefficiency may be correlated with the inputs (see Kumbhakar, 1987); and (ii) use of
OLS in the second-stage conflicts with the assumption that the technical writer inefficiencies are
independent Hence, the technical writer inefficiency estimates using a two-stage procedure
may be inconsistent
A single-stage procedure incorporating technical writer inefficiency effects has been
proposed by Kumbhakar, Ghosh, and McGuckin (1991), Reifschneider and Stevenson
(1991), Huang and Liu (1994), and Battese and Coelli (1995). The basic idea behind
incorporating technical writer inefficiency effects into the frontier production function is that the
firm’s characteristics or non-physical inputs, such as experience, knowledge, and
information, influence its ability to use the available production technology. Kumbhakar,
Ghosh, and McGuckin (1991) assume that technical writer inefficiency effects are nonnegative
truncations of a normal distribution, the mean of which is a linear function of exogenous
factors with unknown coefficients and an unknown variance. In their study of estimating
54 farm-level technical writer and allocative efficiency of U.S. dairy fanners, they found technical writer
inefficiency effects are significantly related to the level of education of the farmers and
the size of the farm.
Reifschneider and Stevenson (1991) modeled technical writer inefficiency effects as a
function of relevant explanatory variables and a nonnegative random variable, which is
assumed to have a half-normal, exponential, or gamma distribution. Using data on
electricity generation in the U.S. for three different periods, they found that inclusion of
an inefficiency function does affect the estimates of the frontier function parameters.
Huang and Liu (1994) modeled technical writer inefficiency effects as a linear function
of firm characteristics and a normally distributed error term with mode zero, truncated
from above, that depends on firm characteristics. Combining a stochastic frontier
regression and a truncated regression to estimate production frontier assumes that the
explanatory variables in the inefficiency model are a function of firm characteristics as
well as input variables. This interaction between firm’s characteristics and input usage
results in a nonneutral shift of the average production function with respect to efficiency.
Using cross-section data from Taiwan’s electronic industry, they found evidence of non
neutral effects of the age of the firms, the export ratio, and expenditures on research and
development, on the frontier production function and productive efficiency.
Further development of the model incorporating technical writer inefficiency effects was
proposed by Battese and Coelli (1995), who extended Huang and Liu’s (1994) model and
applied it to panel data. Modeling negative technical writer inefficiency effects as a function of
firm specific variables and time, inefficiency effects are assumed to be independently
distributed as truncation of a normal distribution with constant variance, whose mean is a
55 linear function of firm specific variables. Empirical application of their model for rice
producing firms in an Indian village indicates that the model for technical writer inefficiency
effects is a significant component in the stochastic frontier production function.
This study uses the technical writer inefficiency effects model proposed by Battese and
Coelli (1995) and applies it to the public education system in Utah to estimate technical writer
change and time varying technical writer inefficiency. The panel nature of the data will allow
measurement of technical writer inefficiency in each school districts for each year under study,
and to investigate the causes of technical writer change, if any.
3. technical writer Inefficiency Effects Model
Consider the stochastic frontier production function as:
Y,-X„ p + r .- u , (4.1)
where Yit denote the output of zth the firm (i = 1, 2, …N) at /th observation (t = 1, 2, …T)
in log form; X* is a (lx&) vector of inputs of production in log form; p is a (&xl) vector of
unknown parameters to be estimated; Vft is a random error term identically and
independently distributed N(0, a2,), and independent of the U*; Uh > 0, is associated with
technical writer inefficiency of production which is independently distributed as a normal
distribution truncated at zero with Uit ~ N (Zit8, or2); Zh is a (lxm) vector of firm specific
characteristics associated with technical writer inefficiency effects over time; 8 is a(mxl) vector
of unknown coefficients.
Following Battese and Coelli (1995), the technical writer inefficiency effects Uit in (4.1)
could be specified as having two components: a deterministic component, which may be
56 explained by a vector of observable qualitative factors (Zit), and a random component,
which may be written as:
Uu = Zith + Wil (4.2)
where, random variable Wjt is defined as truncation of a distribution N(0, cr2), such that
the point of truncation is -Zh8, i.e., Wit > — Zjt8. This assumption is consistent with Uit ~
N(Zh8, a2).
The parameters of the stochastic fiontier production fimction and the model for
technical writer inefficiency effects are simultaneously estimated using the maximum likelihood
method. Again, following Battese and Coelli (1993), the log of the likelihood function is
in terms of the variance parameters, crj =<y^+a2 and y =o2 /c^, and the parameter
estimates are expressed by taking partial derivatives of the likelihood function with
respect to p, 8, oj, andy. technical writer efficiency of production for the rth firm at rth
observation is defined as:
TEU = exp(-Uu ) = exp(-Za8 — W„ ) (4.3)
Predictions of technical writer efficiencies are based on the conditional expectation, as
formulated by Battese and Coelli (1993).
4. The Data Set This study uses input and output data from all of the 40 school districts in Utah.
These data were collected from various reports and bulletins prepared by the Utah State
Office of Education (1993-95), Utah Education Association (1993-95), and the Utah
Foundation (1993-95). The single output of our educational production function is
measured by the basic battery test score at the 11th grade, which is a composite of
57 reading, writing of language/English, and mathematics skills. Inputs used in the
stochastic frontier production function include: (i) percentage of teachers with MA and/or
Ph.D degree (Degree), (ii) percentage of teachers with over 15 years of experience
(Expr), (iii) net assessed value per student (Value), and (iv) year of observation (Year).
The first two input variables account for the quality of teaching input, which is expected
to be positively related with the measure of output. Intuitively, the input variable, net
assessed value per student, is expected to have a positive impact on educational outcomes
because school districts endowed with greater real property wealth per student will be
able to spend more dollars per student for any given tax rate. However, because of
conflicting evidence in the literature about the impact of higher expenditure on student
achievement scores, the coefficient on net assessed value per student may turn out to be
negative. District specific characteristics that are expected to influence technical writer efficiency
and that are used as explanatory variables in the technical writer inefficiency effects model
include: (i) percentage of students receiving subsidized lunch (Lunch), (ii) dropout rate
(Drate), and (iii) the year of observation (Year). The first variable reflects the
socioeconomic status of the student population, and the second represents the
environment or the neighborhood within which the school district operates. Generally,
dropouts are expected to be high if the economic/employment opportunities for unskilled
and uneducated workers are high in the locality of the school district (see Bickel and
Papagiannis, 1988). The general hypothesis is that the higher the percentage of student
population receiving subsidized lunch, the higher will be the level of inefficiency, and the
higher the dropout rates, the lower will be the inefficiency of the school district.
58 A log-linear specification of the stochastic frontier model, Cobb-Douglas type,
may be written as follows:
LnY„ = P0 + P, Degree* + p 1Expru + $%Valueu + p4Yeara + Vit — Uit (4.4)
Uit =50 +8lLurtchit +82Drateu +82Yearit +fYit (4.5)
5. Analysis of Results
The stochastic frontier function incorporates the inefficiency effects, which are
linearly related to the percentage of students receiving a subsidized lunch, the dropout
rate, and the year of observation by adding the intercept term into it (see Battese and
Coelli 1995). hi addition to the intercept parameters po and 80, identification of the
technical writer change and time-varying behavior of the inefficiency effects are possible due to
the specific distributional assumptions on the inefficiency effects in this model.
technical writer change and time-varying technical writer inefficiency effects are incorporated into the
inefficiency frontier model through Equations (4.4) and (4.5). The variable Year in the
stochastic frontier production function (4.4) accounts for Hicksian neutral technological
change, and it accounts for change in inefficiency effects over time in model (4.5).
Parameters of this model are estimated by the maximum likelihood technique
using the computer program FRONTIER Version 4.1, developed by Coelli (1994). Signs
of the estimated parameters of the stochastic frontier function (Table 4-1) are as expected
except for the coefficient on the variable “net assessed value per student,” which is
negative. As we discussed in the previous section, there is no definite relationship
between expenditure per student and student achievement score. Several articles on
educational production have shown that expenditure per student is either uncorrelated
59 (e.g., Hanushek, 1986) or negatively related to student achievement scores (e.g., Deller
andRudnicki, 1993).
The positive coefficient on teacher’s experience confirms our hypothesis that
more experienced teachers will increase achievement scores. The coefficient on teacher’s
qualification is positive, as hypothesized, but it is not statistically significant. Therefore,
a weak confirmation of the hypothesis that qualification leads to higher test scores is
provided. A negative but insignificant coefficient on Year indicates that the test scores
decreased by a small but insignificant rate over the three-year period.
In the inefficiency model, the coefficients on the Lunch and Drate have the
expected sign; however, the latter is not significant. The positive coefficient on Lunch
implies that a school district’s inefficiency increases with an increase in student
population with low socioeconomic status. The negative coefficient on Drate suggests
that inefficiency decreases with an increase in the dropout rate. Intuitively, this is correct
because school districts with low achievement scores often are those with a large number
of potential dropouts. Since these potential dropouts are less concerned about their class
grades and future return on educational investment, the larger their number the greater is
the probability that the district’s achievement score will be low. However, once they
drop out, the average achievement score for the rest of the students in the district goes up,
and efficiency increases (or the district becomes less inefficient). The estimate for the
variance parameter, y, generally is close to one, which indicates that the inefficiency
effects are likely to be highly significant in the analysis of public education.
Generalized likelihood ratio tests for the hypothesis that technical writer inefficiency
effects are absent or that they have simpler distribution are presented in Table 4-II. The
60 first null hypothesis that inefficiency effects are absent from the model is rejected. The
second null hypothesis, that inefficiency effects are nonstochastic (i.e., y = 0), also is
rejected. The underlying idea in the second null hypothesis is that the variance of the
inefficiency effect is zero. This reduces the model to a mean response function in which,
the variables Lunch and Drate are included in the production function. If the parameter y
is zero, then 8„ and 5, must be zero since the production function involves an intercept
parameter and year of observation (see Battese and Coelli, 1993). The third null
hypothesis, that inefficiency effects are not a linear function of the percentage of students
receiving subsidized lunch, dropout rates, and year, also is rejected. This means that
although the individual effects of two out of three explanatory variables in the
inefficiency effects model are insignificant, their joint effect is statistically significant
The fourth null hypothesis that inefficiency effects do not have an intercept parameter
also is rejected. Thus, it is evident that the inefficiency effects in the stochastic frontier
are stochastic and related to the school district characteristics and year of observation.
Estimates of technical writer efficiencies for individual school districts are reported in
Table 4-EQ. It is seen that there is wide range of variability in these estimates across
school districts. However, variability over time is less evident The most efficient school
district for 1993 is Grand, and for 1994 and 1995 the Logan District is the most efficient
The least efficient school district for all years under study is San Juan. Though it is
apparent from Table 4-DD[ that there is a general trend of declining technical writer efficiency
over time for most of these districts, the efficiency level did not change for the Davis,
North Sanpete, and Washington districts. Mean technical writer efficiency decreased from
0.8909 in 1993 to 0.8773 in 1995.
61
6. Summary and Conclusion
The results obtained from this study are consistent with our earlier study on
estimating time-invariant technical writer efficiency using stochastic frontier production
approach and its extension dynamic productivity analysis using data envelopment
analysis for Utah school districts (see Chakraborty, Biswas, and Lewis, 1996;
Chakraborty and Mohapatra, 1997). While our study on time-invariant technical writer
efficiency found Grand and San Jaun as the most and the least efficient school districts,
respectively, the dynamic productivity analysis found declining technical writer efficiency for
all Utah school districts.
The major advantage of this study is that the model specification allows for both
estimation of time-varying technical writer efficiency and technical writer change, assuming
inefficiency effects are stochastic, and whose distribution is known. The results of the
present study indicate that, for most of the school districts, technical writer change has made no
positive contribution towards productivity increase.
Table 4-1. Parameter estimates of the inefficiency stochastic frontier production functions: Dependent variable Ln(test score). Variable Parameter Coefficients
Constant P* 4.112* (22.40) Ln(Percentage of teachers with Advanced degree) P, 0.019 (0.79) Ln(Percentage of teachers with Experience over 15 years) Pz 0.068** (1.87) Ln(Net assessed value per student) Pz -0.059* (-2.52) Year of observation P< -0.001 (-0.06) Constant 50 -0.537** (-174) Percentage of students receiving Subsidized lunch 5, 0.018* (2-67) Dropout rates 5z -0.002 (-0.10) Year of observation §3 0.018 (0.33) o.2 0.030* (2-61) Y 0.849* (8.09) LLF 78.369
t — Statistics are in parentheses. * — indicates coefficients are significant at 5% or lower level. ** — indicates coefficients are significant at 10% or lower level. LLF — Log of the Likelihood Function.
63
Table 4-II. Hypothesis tests for parameters of the inefficiency stochastic frontier production function. Null hypothesis Logflikelihood) Test statistics % 20.9J Decision 1. H0: y = 50 = 8, = 52 = fh =0 57.435 41.868 11.07 Reject 2. Hq: y = 50 = 5, = 8j =0 73.977 8.782 7.82 Reject 3. Ho: 8, = = 83 =0 64.314 28.109 7.82 Reject 4. Ho: 80 = 0 74.701 7.334 3.84 Reject
LR test statistics A. = -2[Log{Likelihoog) Ho — Log(Likelihood) Ha ] ~ xL/
64
Table 4-HL Yearwise technical writer efficiency estimates. School districts 1993 1994 1995
1. Alpine 0.963275 0.949526 0.967654 2. Beaver 0.947460 0.908374 0.847348 3. Box Elder 0.962572 0.930342 0.967726 4. Cache 0.944603 0.943225 0.926853 5. Carbon 0.833215 0.793882 0.768005 6. Davis 0.956880 0.958199 0.958048 7. Duchesne 0.860487 0.853762 0.824659 8. Emery 0.915031 0.853128 0.834710 9. Garfield 0.786048 0.927124 0.914745 10. Grand 0.970039 0.911379 0.915509 11. Granite 0.936548 0.919018 0.944876 12. Iron 0.944045 0.933077 0.923110 13.Jordan 0.953004 0.956706 0.966002 14. Juab 0.757004 0.925480 0.797190 IS. Kane 0.937671 0.900059 0.793932 16. Millard 0.881113 0.944630 0.937753 17. Nebo 0.901650 0.903431 0.943360 18. NoSanpete 0.720355 0.728107 0.724588 19. Rich 0.890439 0.923678 0.774957 20. San Juan 0.543418 0.564500 0.590375 21. Sevier 0.807414 0.753820 0.819246 22. So.Sanpete 0.864966 0.805750 0.736181 23. So.Suxnmit 0.873591 0.858399 0.968051 24. Tooele 0.924268 0.916967 0.853752 25. Uintah 0.898303 0.857653 0.863408 26. Wasatch 0.963912 0.953180 0.914828 27. Washington 0.942704 0.941877 0.946404 28. Weber 0.907021 0.901402 0.923081 29. Salt Lake 0.948799 0.957355 0.962486 30. Ogden 0.774776 0.736928 0.795269 31. Provo 0.966507 0.944961 0.940977 32. Logan 0.968830 0.977524 0.977196 33. Murray 0.955784 0.943737 0.931441 Mean 0.890962 0.887187 0.877386
65
CHAPTERS SUMMARY AND CONCLUSIONS
The results of this analysis provide several important insights into educational
efficiency and the basis for correctional policy action. This study finds strong evidence
of technical writer efficiency for most Utah school districts with mean efficiency scores of 87,
90, and 88%, respectively, as outlined in the second, third, and fourth essay. Even with
the stricter nonradial measures, mean efficiency levels for Utah schools remain at the
90% level in the third essay. Thus, the analysis supports the conclusion that these school
districts are technical writerly efficient and that additional productivity can be secured only
through technical writer innovations (perhaps involving increases in overall expenditures).
The results also provide strong evidence of large socioeconomic and
environmental influences on technical writer efficiency. The measure of scale inefficiency
identifies those school districts that are producing an inefficiently small output in a phase
of increasing returns to scale. Scale inefficiencies are observed for a fairly large number
of schools, suggesting policy action with regard to input and output uses for those
schools. In the third essay, Jordan, Juab, Sevier, Weber, and Murray school districts are
found to be overutilizing their highly qualified teacher input, leading to a congestion of
student learning. The dynamic analysis results in the third essay support the earlier
conclusions. School districts that have had net increases in productivity over the period
1993-95 have done so primarily as a result of increases in efficiency. Surprisingly,
technical writer progress in Utah school districts has been slow, with many units experiencing
inward shifts in their technical writer frontiers over the time period. Both the static estimates
66 from the second essay and the dynamic estimates from Appendix B, and the third essay
strongly suggest policies geared toward educational research rather than extension.
Educational research involves innovative methods of teaching while extensions involve
introducing early childhood and adult education programs. In order to secure increases in
educational productivity in technical writerly efficient Utah school districts, policies should
focus on research expenditures for the introduction of technological innovations.
While the thrust of this dissertation is primarily methodological, the results do
have policy implications. For example, districts with high socioeconomic student status
(SES) could improve efficiency by better management of controllable inputs (i.e., better
teaching personnel, higher student work load) and/or adoption of programs that link part
of teacher compensation to student performance. Another policy measure may be
promoting students at the 3rd, 6th, 8th, and 9th grade based on academic capability rather
than social promotion based on attainment of a certain age. Making homework
mandatory at all grades and at all levels of school education is a positive step toward
improving the educational standard. Districts with a large number of students from low
socioeconomic backgrounds face a more difficult challenge as they deal with students
who have less intellectual support at home. In such districts, efficiency might be
enhanced by some resource allocation to: (i) pre-kindergarten programs to better prepare
children for entering school and (ii) adult education, and/or (iii) to foster greater teacher-
parent interaction designed to encourage parental support of the student’s educational
activity.
67
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75
APPENDICES
76
Appendix A Derivation of the Joint Density of u and e
Let 8j = + Vj where v ~ N (0,crv2), u ~ |N(0,au2)|, ux < 0 where, v, is the white
noise, two-sided error component; u{ is the technical writer inefficiency, one sided error term.
Write the joint density of v and u, transform this to a joint density in e and u and integrate
u from — oc to zero.
The joint density of u and v is the product of the individual densities. Since they
are independent,
/(«.v)= /( u)/(v)=
rU2l£)’
(A.1)
Let us substitute v = e — u, (A. 1) can be written as
To derive the f (s), u must be integrated out of f («,e),
/(e)= f f{u£)du —oc
du
(A.2)
Let of = of of of +at
substituting in (A.2) we get: ■Jlntnauav tAexp I 2 at u2- 2suat e2at
2 P 1 1
/
eof ’ s!a:f
-J2navau J rz— “* V27t 2of
u — \ of J ‘ of I
2 P 1 [ ‘ I eof >2 e2of r y[2navau J /—— exp “* v 27t 2of j uof > + of 1
eofl I of
2 4 e a.
du
du
1
o2o2 1 w»’/« 1 a2 +02 a2 V U V ■
du
2 p 1 1 f eof 2 e2of ‘of + of-ofV rz— J rz—GXP V27t avau \2% 2of r J ‘ of L of +af ) du 2 P 1 _ J 1J 1 ( \2 e2 ■V2^.a. ‘ £ , & eXFi 2 k 1″ ^ + <^ + iu, where ]iu = e~j or.
V2^< 7CCTvCTh
+ k du, where k —
at +of
-J2navau
r \2 of >
‘du • exp(——)
V2^< 2 , ks P 1 -e x p (—K • J~ exp<j 7rova„ ; -Jlna.
f V “~Pu I of </u
2 J e2 1 auav p 1 ^ ————exPI — 7 7 3 — 7: f * i , , • L l ^ e x P l ‘7r<TvCTH 2(of+of)j Jot +of «~V2ito.
/A-—-expf-^-jl- f . *■■ — exp ■yfeHa \ 2c?) «*72710.
z’ V “~P„ of J
du
rt — “-P, . of
17
du (A-3)
78
U — |A 1 Let s = ds = —du, du = dsa, substituting in (A.3) we get:
V2tcct
2 v s . exp( — ^ ) ‘t v i b : exp(jIK<fa
2 V2tk I ( 82 ) LI ==- • exp ——j -<|>(—— , because u is truncated Tier v 2cr >/ a. at zero.
eo? — u t1. ^ 1Note ]iu = -3 — , hence, — = -3——-= , r r — T“ &’ <*. <K <*. a t^ + a t <*» <* sc a <t 1 K V U _ _ 1 _ — l — E • • — E • A. • CT
Jlna exf( — i <b(-—), where X = — . cr cr„
79
Appendix B Dynamic Productivity, Efficiency, and technical writer Innovation in Education: A Mathematical Programming Approach Using Data Envelopment Analysis’
I. Introduction
The efficacy of public education in the United States has been a source of
considerable concern and debate over several decades. The accountability of public
education is especially important now in a milieu of spiraling costs, increasing
population, and the apparent inability of administrators to improve educational outcomes
by augmenting expenditures. In the wake of widespread disenchantment regarding public
education, and public services in general, policy makers have vigorously renewed their
commitment towards the performance assessment of public school units. Historically, the
Coleman report (Coleman et al., 1966) advanced equalization policy reforms to smooth
heterogeneity in expenditure budgets and endorsed equal educational opportunities across
school districts. The ensuing reforms in property tax revenues, which largely forge
public school expenditures, secured homogenous funding and expenditure structures.
However, with increasing evidence that expenditures were not the driving variable behind
educational performance (e.g., Hanushek, 1986; Riew, 1986; Walberg and Fowler, 1987),
it was not entirely clear that adjustments in expenditures necessarily map into
performance increases. Consequently, the homogeneous structure only shifted the burden
of adjustment from expenditure alignments to the assessment of relative performance
80 among the homogenous school units. The publication of A Nation at Risk in 1983 by the
National Commission on Excellence in Education heralded these concerns and reignited
the debate surrounding public education in the United States. The renewed dialogues
emphasized the importance of productivity and efficiency as key factors in the public
education reform movement.
Thus, the primary question facing managers and policy makers in education is,
whether the U.S. public education system efficiently develops students’ cognitive skills
as measured by standardized test scores. In response to this question, there has spawned
a fairly large amount of literature on the estimation of relative performance across school
districts in different states (e.g., Levin, 1974; Bessent and Bessent, 1980; Barrow, 1991).
However, although the efficiency and productivity of public education raises important
concerns and needs for policy action, there have been no comprehensive empirical studies
that effectively model these diverse concerns.
In the light of large increases in incomes and employment and under pressure in
recent years from a growing population, the state of Utah faces a significant increase in
the demand for a better and larger public education system. Given that property taxes
remain the dominant source of funding for public education in Utah, policy makers must
carefully consider the efficiency of existing schools to investigate possible increases in
performance before sanctioning additional funding. Only one study (Chakraborty,
Biswas, and Lewis, 1996), assessing performances across Utah school districts, could be
identified. However, this study has not adequately modeled the diverse concerns related
to the public education system in Utah, and many important issues remain unexplored.
First, the production function for education may be misspecified in this conventional
81 single-output model as the realization of skills from education is not singular. Second,
the existing efficiency model is overly restrictive in its parametrization of the production
technology and in its assumptions concerning returns to scale. Third, the existing study
fails to model fixed and weakly disposable inputs correctly, which yields biased
efficiency estimates. Such considerations are especially important in the context of
education as efficiencies of schools are highly conditioned on the socioeconomic
environment of students and the qualifications of teachers (Walberg and Fowler, 1987;
Ray, 1991, McCarty and Yaiswargn, 1993). Finally, there exists clear theoretical
evidence in a broader development literature that efficiency issues are inherently dynamic
and that dynamic structure offers important policy insights for extension or research
activities. Despite this, no studies, in the general educational efficiency literature could
be identified that use a dynamic treatment of educational efficiency issues. This study is
a first attempt at assessing the performance of Utah school districts using a
comprehensive approach that addresses the concerns listed above.
II. Key Issues School units convert various instructional and noninstructional inputs into
multiple learning outputs as measured by student achievement test scores (e.g.,
standardized math, verbal, or science test scores). Inputs, such as the availability of
teachers per student, the proportion of teachers with an advanced degree, and
expenditures per student affect output directly and often are endogenous to the school
unit These inputs are called discretionary or controllable inputs. Nondiscretionary
inputs, which include various socioeconomic or environmental factors such as family
82 income and the assessed property value per student, affect learning indirectly and usually
are beyond the school unit’s control. The multiple-learning outputs produced by schools
have been observed, both analytically and empirically, to be increasing functions of both
the instructional and noninstructional inputs. The key issues surrounding efficiency or
relative performance assessment of school units are primarily driven by stylized facts of
the educational production function. These considerations, which motivate the empirical
methodology adopted in this paper, are discussed below.
PRODUCTION FRONTIER
First, theoretical production functions are extremal and define the maximum
output possible from given inputs or, alternately, the minimum amount of inputs
necessary to produce a given level of outputs. The concept of maximality or minimality
inherent in these functions sets a limit to the range of feasible observations. However,
conventional ordinary least squares (OLS) type conditional expectation estimators
approximate the average production correspondence instead of estimating the production
frontier surface. In order to estimate the extremal production function we use the
mathematical programming approach based on the activity analysis model of production.
This nonparametric approach is due to the pioneering works of Farrell (1957) and
Koopmans (1951) and was formalized as the DEA approach by works of Chames,
Cooper, and Rhodes (1978) and Fare, Grosskopf, and Lovell (1985).
SOCIOECONOMIC FACTORS (FIXED INPUTS)
A second consideration in modeling efficiency in school districts is the
appropriate treatment of socioeconomic factors, which have important effects on
83 productivity. While socioeconomic and environmental variables are exogenous to the
school unit, excluding them from the model leads to specification errors and a biased
technical writer efficiency component in the productivity measure. However, these variables,
when included, need to be treated differently from endogenous variables in the system.
Ideally, efficient school units should be able to bring forth high educational outcomes for
students even in relatively low socioeconomic conditions and poor environments. These
concerns are similar to those faced when modeling fixed factors. Fixed factors, when
treated as variables, generate upward biases in technical writer efficiency estimates and need to
be modeled carefully. We model socioeconomic factors nonparametrically by using a
modified form of the DEA. model where efficiency of schools is defined over a subvector
of the (endogenous) variables for given levels of the vector of socioeconomic and
environmental variables. Additionally, parametric estimates, using a second-stage
regression of the DEA efficiency scores on the socioeconomic and environmental
variables, are derived to provide insights into the nature of socioeconomic and
environmental influences.
RETURNS TO SCALE Priors on the scales of production are often arbitrary and impose a large amount of
structure, hence, leading to biased efficiency measures. This destroys one of the basic
virtues of the nonparametric approach and leads us back to the parametric problem of
“incredible” restrictions. Moreover, given the fact that conditional efficiency estimates
identify technical writer efficiency and socioeconomic components separately, a further
confounding component in the technical writer efficiency measure could be scale inefficiencies.
In order to identify scale inefficiencies, we relax the constant retums-to-scale assumption
84 on the frontier and allow for a highly flexible technology that accommodates variable,
decreasing, and nonincreasing returns to scale
ECONOMICS OF OVEREDUCATION: CONGESTING INPUTS
A third key issue in modeling educational frontiers relates to the typical
restrictions imposed on production technologies. Standard assumptions specify inputs as
freely disposable, implying that increases in any of the inputs do not lead to decreases in
output. Thus, input-based efficiency targets are always approached by input reduction.
However, in the context of this study, we suspect that overutilization of teaching
personnel with advanced degrees may hamper student learning. These inputs then have
to be modeled and evaluated in their contribution towards efficiency under weak
disposability assumptions, such as, increases in this input, holding all other input usage
constant, may generate decreases in output.
TOTAL FACTOR PRODUCTIVITY: DYNAMIC PERFORMANCE EVALUATION
Moreover, consider the technical writerly efficient expansion path of a school unit as a
series of movements from within an output set which finally push it on to the frontier
where it is considered efficient. In a multiperiod time horizon, these snapshot
descriptions may be combined to redefine the efficient school as one that undergoes the
continuous motion of moving from its initial location within the output set onto a frontier
which itself shifts over time due to the salutary effects of technical writer innovations. In a
dynamic world with changing technical writer and economic environments, efficient school
units constantly need to adjust for new equilibria, and high payoffs exist for managerial
85
effectiveness and increases in information. Thus, efficiency measures should provide
policy information on both the statics and dynamics of school performance.
III. Methods2
THE BASIC DEA MODEL
The input-based frontier estimators in DEA construct a nonparametric, piecewise
linear surface by enveloping the sample data with a convex hull consisting of a series of
linear segments. The constructed reference surface provides an upper bound for technical writer
efficiency as school units hitting this bound would be fully efficient. Although, the
constructed technology is well behaved and satisfies general axioms of production theory
(Fare, Grosskopf, and Lovell, 1994), it is not differentiable everywhere due to the
presence of linear segments that connect the best practice units. However, asymptotic
smoothness is achieved as the number of activities increase and the piecewise
representation converges to the smooth neoclassical function.
Consider the activities of / school units, each employing N inputs to produce M
outputs. Let N denote the (I*N) matrix of N inputs used by I school units, with typical
element denoting the nth input utilized by the ith school unit. Let M represent the
(/*Af) matrix of M outputs of / different school units, where the typical element
denotes the mth output of the ith school unit. Outputs and inputs are hypothesized to
obey the usual nonnegativity restrictions. The piecewise linear input set, constructed
under standard assumptions of constant returns to scale and free disposability of inputs,
denotes all input vectors capable of producing at least output vector y:
L(y |C,F) = {x:y < zM, zN £ x, z 6 <R$.},y e 9?$* (B.l)
86 where z denotes an (1XI) intensity vector that forms convex combinations (with variable
returns) of observed input and output vectors. The technical writer efficiency of an observed
school unit is measured by a Shephard distance function (19S3; 1970) measured from the
candidate input vector towards the constructed piecewise linear, convex isoquant The
distance function measure seeks out a parameter of technical writer efficiency 4 such that, when
multiplied to an inefficient school’s input bundle, renders that school efficient. The
input-based distance function measure, bounded by 0 and 1, can be written as:
Di(yi,xi|C,F) = Minfe4xie Uyi|C,F)}. (B.2)
This input-oriented measure considers the equiproportionate shrinkage in inputs required
to project a school back onto the frontier, while still maintaining the production of its
given level of outputs. Solution of the following linear programming model consolidates
Equations (B.l) and (B.2) and obtains school-specific technical writer efficiency measures
relative to the bounding technology,
Bi(yi,xi|C,S) = M in4 0-3) S.t y*<zM zN < 4x* z e » i
Figure B-I below, describes the input requirement set for a sample of 6 (/ = 6) schools
situated at points A, B, C, D, E, and F.
Application of the DEA approach finds schools A, B, C. and D to be efficient
These efficient schools are used to construct an envelopment surface over the sample
schools in a manner that ensures that all schools are either on or below the envelopment
87 frontier. A close observation of Figure B-L, immediately identifies schools E and F to be
inefficient.
In order to have individual efficient target locations for each unit, DEA constructs
i virtual “super*’ schools which are efficient As an example, we illustrate the
construction of a super school F* for inefficient unit F. Illustrating for the first input,
efficiency in input use of the super school is guaranteed by the inequality in (B.l)
ZAXA1+ZB*B1+ + zjxji< xji. (B-4)
The left-hand side of the previous inequality, F*, is a composite input bundle formed as a
linear combination of input bundles of all other relevant school units. F* represents the
efficient reference point for F and serves as a direct efficiency comparator. In (4.4), F*
is constructed as a linear combination of other relevant schools with weights (zj assigned
on the basis of this relevancy. Thus, F* serves as an efficiency target location onto which
school unit F can be projected by an equiproportionate shrinkage in its input vector.
This is given by (B.3) where q, evaluated at the optimum, is a number less than unity by
which inputs of F can be multiplied while still producing at least as much of the given
level of outputs. £,, thus provides a relative measure of technical writer efficiency as the
maximum feasible shrinkage of the input vector possible while maintaining production at
given output level y.
Although the model discussed above saved as the basis of this efficiency
analysis, in order to derive robust efficiency estimates, the basic DEA model is modified
to take into account the idiosyncrasies of educational inputs. Towards this end, we
construct several submodels to address die key issues (Section 2) surrounding the
educational production function.
88
CONDITIONAL EFFICIENCY ESTIMATES: THE SUB VECTOR EFFICIENCY MODEL WITH FIXED INPUTS
Given the nature of exogenous socioeconomic inputs, we use estimation methods
which allow for unbiased efficiency estimates while considering the impact of
socioeconomic factors that are fixed and beyond the school units control. Additionally,
the more conventional two-step method of regressing the efficiency scores from the basic
DEA model on the exogenous socioeconomic inputs is used to give an account of the
exogenous influences. The subvector DEA efficiency measure allows a subvector of
(endogenous) inputs to be scaled back onto the frontier while recognizing the fixed nature
of the socioeconomic variables.
Letting jcv represent variable inputs under the school’s control and xK represent the
fixed socioeconomic inputs, we partition the input matrix N into variable and fixed input
vectors (TV = A/v, A* ). The subvector technology under variable returns and free input
disposability may be formed as:
U y |V,F) = {xv,xK:y ^ zM, zNK < x, zNv< x, Ezi= 1, z 6 51^.}, y e
(B-5)
The radial subvector technical writer efficiency measure is given by:
SiCyUkxtlV,F) = Ming : (5x{,,xy e L^IV.F)}. (B.6)
The parametric approach towards accounting for socioeconomic and
environmental influences utilized a censored regression (as the input based efficiency
scores were bounded between 0 and 1) of DEA efficiency scores on the socioeconomic
factors for consistent and unbiased estimation. The residuals obtained from this
regression are the “pure” technical writer efficiency scores after eliminating the effects of the
89 uncontrollable inputs. While, the zero values for the residuals imply that a school
district’s performance is as good as the average school district with the same
socioeconomic variables, the nonzero values signify differences in the performance from
the average school district with the same set of uncontrollable factors. Higher values of
the censored residuals signal higher technical writer efficiencies given the socioeconomic and
environmental factors.
SCALE ASSUMPTIONS: DEPARTURES FROM CONSTANT RETURNS
Restrictive retums-to-scale assumptions prevent the separate identification of
scale and technical writer inefficiencies. Given the importance of uncovering different types
and sources of inefficiency for policy making, efficiency estimates need to be
decomposed into scale, input congestion, and pure technical writer inefficiency components. For
example, constraining the sum of the intensity vector, z, to be equal to unity allows for
variable returns to scale. The earlier nonnegativity constraint on z in (B.3) allowed all
observations to be scaled up or down, effectively imposing constant returns to scale.
INPUT OVERUTILIZATION: THE CONGESTION MODEL
The standard frontier technology is constructed as the constant returns to scale,
convex, free disposable hull of observed input and output vectors. Free disposability
implies that increases in any of the inputs do not lead to a decrease in output. In terms of
(B.l) this can be written as:
x > x’ e L(y|C,F) => x e L(yJC,F). (B.7)
90 However, given the earlier hypothesis about the overutilization of qualified teachers, we
need to define the technology under the weaker restriction of weak disposability, i.e., if
increases in inputs are not proportional then output may decrease:
x e L(y|C,W) => ax e L(y|C,F), a > 1. (B.8)
The failure to correctly identify the nature of disposability of an input would incorrectly
attribute the school’s deviation from the frontier, caused by overutilization of highly
qualified teachers, to technical writer inefficiency. This is demonstrated in Figure B-I, where
the efficiency of school W is evaluated relative to the lower bound ABCD of the input set
constructed under free disposability assumptions. Now suppose X2 represents the
teachers’ degree input, then the constructed technology implies that increases in teachers
with advanced degrees, holding other inputs constant, allows the school to remain on
isoquant ABCD producing the same level of output. Thus, school W, which lies outside
the free disposable isoquant, is found to be technical writerly inefficient as measured by a ray
from the origin. However, this is incorrect if teachers with advanced degrees cause
intellectual overcrowding and would lead to identification of congestion as technical writer
inefficiency.
We use an input-based technical writer efficiency measure relative to a technology that
exhibits variable returns to scale and two types of input disposability behavior. The
technical writer efficiency relative to this technology is defined over the subvector of inputs
under the schools control as before. Let xy represent the freely disposable variable inputs
under the school’s control, x*. represent the free disposable socioeconomic variables, and
xa represent the weakly disposable input vector (teacher’s degree). Thus, the input matrix
N is partitioned into 3 subvectors {N — N’, N*, N°). Using the variable returns to scale,
91 piecewise linear input set for the endogenous inputs xv and exogenous inputs xx, and
with free disposability of all input vectors excepting for input subvector x„, the school-
specific radial subvector technical writer efficiency estimates can be derived by solving the
following linear program / times:
Sfri(yi,x*,,4,xLir’JrK’V’fra)= Afinz (B.9) 5* S.t yl < zM zNk < xKl ZNV < ^Xy 2N«=§xi z e 91+ Zzj =1
92
INPUT SLACKS: THE NONRADIAL CONDITIONAL EFFICIENCY MODEL
In the models presented above, efficiency is measured radially. Technology is
modeled with the input correspondence y —> L(y). Given that inputs x are feasible, i.e., x
e L(y), input-based technical writer efficiency is measured by determining the location of x in
the input requirement set, L(y). Radial efficiency measures seek out the maximum
feasible shrinkage necessary to project an observed input vector onto the isoquant
However, this equiproportionate shrinkage along the radial path from feasible x* back
towards the origin does not take into account input slacks and is unable to reflect optimal
input usage. For example, in Figure 4-1, consider the input bundle S (i.e., school S). A
radial shrinkage measure projects S onto the efficiency target, A, on the isoquant
However, the true efficiency target for S is B as it exists on the same isoquant but uses AB
amount (slack) less of input XL To be able to project back input vectors onto such
efficient subsets of the isoquant one needs disproportionate reductions in inputs. This is
achieved by the nonradial Russell efficiency measure, which assigns efficiency labels
only to vectors which belong to the efficient subset. The non-radial Russell measure for
constant returns to scale and free disposability assumption is . . . N p NRi(y1,xt,x^|C,K) = M in (B-10) n=lN S.t. y1 < zM zNk < x*. zNv£§ct zefti,
where % is a (N X 1) vector.
93 In Figure B-I, radial and nonradial efficiency measures for school F coincide as
there is no slack in the efficiency target point F*. For school S, with slack in input XI,
the nonradial measure gets past the slack by comparing jc* to (£,’ x*,, E, 2* xu>… 4 N* x^)
PRODUCTIVITY CHANGE, EFFICIENCY CHANGE, AND TECHNOLOGICAL INNOVATIONS
The above measures give only a static description of efficiency. Dynamic
productivity and efficiency estimates can be derived using multiperiod analysis.
However, standard productivity indexes, such as the Fisher and Tomqvist indexes,
require information on prices and impose profit maximization behavioral assumptions,
which are incredible for the educational firm. We use Malmquist productivity indexes,
which use only information on the input and output quantities, to estimate productivity
and productivity growth over time. The Malmquist productivity index, proposed by
Caves, Christensen, and Diewert (1982), measures productivity differences across time
periods and, in its modified version (Fare, et al. 1989b) describes sources of dynamic
productivity changes. These indexes are based on distance functions, which rely on the
primal description of technology. At each time period, t = 1,2 … T, technology is
modeled as:
The Malmquist input-based productivity index measures changes in performance during
two time periods and can be expressed as:
Lt(yt|C,F) = {xt:yt < zMt,zNt < xl,z e ttih y 1 e m? (B.ll)
SkyVlQF) „ s r I(yt,xt|C,F) 1/2
(y ,x ,y ,x |C,F) = _t+l „t „t
(B.12)
94 The Malmquist index is computed by running each of the four linear programs /
times. The same period efficiency measures, e.g., numerator of the first expression and
the denominator of the second expression, are the same as (1) with observations dated in
order to specify year-specific input correspondences. Using this school-specific
productivity index, it is possible to attribute the productivity growth to either changes in
efficiency across time or shifts in the frontier due to technical writer innovations. Further
estimations of productivity under various returns to scale and disposability assumptions
help find the sources and components of dynamic productivity changes, i.e., pure
technical writer efficiency changes, scale efficiency changes, congestion changes, and technical writer
innovations.
IV. The Data
This study uses input and output data for secondary school in the 36 Utah school
districts for the period 1993-95. The variables used as school inputs in our study include:
(1) student teacher ratio based on average daily membership, (2) percentage of teachers
with an MA or Ph.D. degree, (3) expenditure other than staff salary per “average daily
membership” (ADM); (4) net assessed value per ADM, and (5) percentage of student
population buying their own lunch. While variables (1) through (3) measure instructional
inputs, variable (4) is a proxy for environmental input (by measuring the economic
condition of the neighborhood), and variable (5) is a proxy for family income of the
students. All are aggregated over the school districts.
Our output measure is based on the standardized test administered by the state at
the 11th grade in each of the school districts. This test consists of two parts, the basic
95 battery test (a composite score of mathematics, language/English, and reading), and
subject tests. Hence, the outputs are defined as the average scores obtained by each
school district in (1) the basic battery test, (2) mathematics, (3) reading, (4)
language/English, (5) science, and (6) social science (Utah Foundation Research Report
1995).
V. Analysis of Results3
Although the primary empirical approach utilized in this study is nonparametric,
we utilize parametric estimates of the educational production function as a starting point
for the DEA specification. Towards this end, we estimated a translog production function
relating educational output (battery test score as a proxy for multiple-leaming outputs)
and the multiple inputs. Due to severe multicollinearity, we reestimated the model using
a stepwise regression technique.4 The parametric estimates supported the model
specification chosen for the DEA analysis. We specify school units as firms producing
the five learning outputs (as measured by test scores of reading, language/English,
science, mathematics, and social science) from five inputs (student teacher ratio,
percentage of teachers with an M.A. or Ph.D. degree, expenditure per student other than
teacher’s salary, net assessed value per student, and percentage of the student population
buying their own lunch).
96
CONDITIONAL AND UNCONDITIONAL technical writer EFFICIENCY ESTIMATES: IDENTIFYING THE IMPACT OF SOCIOECONOMIC AND ENVIRONMENTAL FACTORS
We initially estimated the basic DEA model using the five learning outputs and
only the endogenous inputs in the system. The technical writer efficiency scores from this basic
model, assuming variable returns to scale, are reported in column 3 of Table B-I (these
are the unconditional technical writer efficiency or T.E scores). Following these unconditional
T.E. scores, we find 14 districts (Alpine through Granite) to be fully efficient, while
school districts such as Kane and Tintic appear on the other end of the spectrum with
efficiency levels below the mean of0.937.
However, these efficiency scores from the controllable input DEA model do not
exactly reflect inefficient management and resource wastage scenarios that would initiate
correctional policy action. These scores provide little insight into the optimization
behavior of school performances, which, in the real world, are highly conditioned on
socioeconomic and environmental factors. Unconditional technical writer efficiency estimates
would identify schools struggling with students from low socioeconomic backgrounds as
being technical writerly inefficient. That would lead to biased policy implications, moreso as
these variables are exogenous to the school. In order to derive a robust ordering of
relative conditional technical writer efficiencies, we estimate a DEA model incorporating both
controllable and uncontrollable inputs. This model computes efficiency over the
controllable factors only, but this efficiency is explicitly conditioned on school-specific
socioeconomic constraints. The ordering and results from this subvector efficiency
model assuming variable returns to scale is given in columns 5 and 6 of Table B-I (i.e.,
97 the conditional T.E. scores). In comparing these final conditional technical writer efficiency
scores with the unconditional scores, several important patterns emerge. The conditional
ordering of schools in column 5 reveals that school districts Alpine through Iron are fully
efficient while school districts Weber through Kane are technical writerly inefficient. Average
efficiencies over all the school districts increase by 0.036, compared to the conditional
measure.
A classic example of the importance of socioeconomic factors in efficiency
measures is the Tintic District, which ranks lowest in the unconditional measure but is
among the most efficient in the conditional measure. Schools marked with an asterisk
represent “inefficient” schools, which are rendered efficient once one incorporates the
richness of socioeconomic and environmental factors.
In order to get additional insights on the effects of socioeconomic factors, we
analysed the residuals from a censored regression of the DEA efficiency scores (column
4) on socioeconomic and environmental factors. The censored residuals depict how a
school’s efficiency measure compares with an average district facing the same
socioeconomic factors. The unconditional orderings are shown in column 2. School
districts with larger censored residuals signal higher technical writer efficiencies given
socioeconomic and environmental variables. School districts, such as Carbon, Duchesne,
Provo, Salt Lake, South Sanpete, and San Juan, which were found to be inefficient under
the unconditional measure and efficient under the conditional measure, have relatively
larger residuals associated with them, signaling that socioeconomic factors play a major
role in their deviation from the frontier.
SCALE ERRORS: THE SCALE COMPONENT IN CONDITIONAL
98
technical writer EFFICIENCY ESTIMATES
We estimated the conditional DEA model under varying retums-to-scale
structures. Table B-II (columns 2 through 7) reports the conditional efficiency scores and
rankings of school districts as observed under constant, nonincreasing, variable returns to
scale (CS, NS, and VS, respectively), and free disposability (of inputs) assumptions. A
brief description of these results follow.
A comparison of efficiency scores between CS and NS reveals no significant
changes except for Daggett school district, which is one of the most efficient district
under NS. As expected, almost all of the school districts improved their efficiency scores
under the VS assumption. However, substantial improvements are evident only in cases
of Piute and Sevier, whose efficiency scores increased by 35% and 31%, respectively.
Under the VS assumption, the least efficient (Kane) school district is about 24%
inefficient This implies that a student would have a 24% higher chance of making better
achievement scores if the school district would have used resources more efficiently
given the socioeconomic status of students and the environmental factors. Overall,
efficiency rankings change as the retums-to-scale assumptions are varied.
In order to identify scale components from the conditional technical writer efficiency
measure, we construct a scale efficiency index, p, which is the ratio of the values of the
objective functions evaluated at the optimum, from the CS and VS efficiency measures.
The results are reported in column 8 of Table B-II. For example, districts such as Alpine
and Logan, with a value of p = 1, are scale efficient as they are equally efficient
regardless of the scale assumptions. School districts with a value of p < 1 (e.g., Box
Elder and Daggett) are not scale efficient. Once the scale-inefficient units have been
99 identified, in order to enforce any kind of corrective actions, we need to know if the input
scale-inefficiency is due to production of an inefficiently small output in the realm of
increasing returns to scale or due to the production of an inefficiently large output in the
realm of a decreasing returns to scale. For this purpose, we construct an intermediate
index 5, the ratio of the efficiency scores from optimally evaluated objective functions
under the constant and nonincreasing retums-to-scale assumptions. Thus, for any district
for which the relationship (p < 1 and 5=1) holds, input scale inefficiency is due to
increasing returns to scale. Such districts are marked with * in Table B-II. Thus, the Box
Elder school district is scale inefficient due to increasing returns to scale. The policy
implications would be to restrict output increasing input changes in that school district If
for any district p < 1 and 8 < 1, then the input scale inefficiency is due to decreasing
returns to scale. For example, the Provo, Carbon, Daggett, and Emery districts are
operating where decreasing returns to scale is in effect Those school districts that are
scale inefficient are marked with ** in Table B-II.
CONGESTING INPUTS: DOES TEACHER OVEREDUCATION REALLY DECREASE STUDENT EDUCATION?
There is ample evidence in the literature (Tsang and Levin 1985) that shows that
overutilization of teachers with advanced degrees might actually reduce student
achievement scores. According to these theoretical tunes, we test the hypotheses that
increases in the percentage of teachers with a Ph D or M A might lead to input congestion
instf»aH of a monotonic increase in output for given scales. To this end, we specify the
input) teachers with advanced degrees, as weakly disposable. The conditional technical writer
100 efficiency scores obtained under the assumptions of weak disposability of the teaching
input, free disposability of all other inputs, and variable returns to scale are reported in
columns 10 and 11 of Table B-II. However, it is evident from column 10 that there have
been marginal improvements in the efficiency scores of a few school districts (when
compared with the free disposable VRS measure in column 6). However, substantial
improvements are noticeable in the cases of Jordan and Weber, whose efficiency
increased from 0.940 and 0.968 to 0.972 and 1, respectively. This suggests that the
presence of input congestion, which is responsible for lower efficiency scores. On the
other hand, poor efficiency scores in the Kane district, which do not change under weak
disposability assumptions, cannot be blamed on overqualified teachers.
In order to assess school-specific congestion we construct an input congestion
index p, which yields a comparison of feasible input shrinkage under weak and freely
disposable inputs. It is formed from the optimally evaluated VF and VW conditional
measures of technical writer efficiency. Thus, if the congestion index, is p = 1, then the input
subvector (i.e., teachers with advanced degrees) does not congest the output vector of
student’s achievement scores in that school. But if p < 1, then the input subvector
congests output. In our study, the Jordan, Juab, Sevier, Weber, and Murray school
districts are overcrowded with M.A and Ph.Ds which congest student’s productivity.
Schools for which teachers’ overeducation decreases students learning are marked with
the © symbol in Table 4-EL
101
INPUT SLACKS: NONRADIAL CONDITIONAL technical writer EFFICIENCY ESTIMATES
In order to eliminate technical writer efficiency estimate biases due to the presence of
input slacks, we estimate a nonradial conditional technical writer efficiency measure. The
efficiency scores obtained from the conditional radial and nonradial measures (CF) are
reported in Table B-DI. Most school districts become less efficient under the nonradial
measure, and the mean efficiency goes down by 0.038. This is not surprising since
nonradial measures are constructed under a more restrictive definition of technical writer
efficiency. Schools that were not radially projected back onto the efficient subset of the
isoquant due to the presence of input slacks are marked with an asterisk in Table B-III,
and their true conditional technical writer efficiencies after eliminating input slacks are reported
in column (S).
TOTAL FACTOR PRODUCTIVITY
Using multiperiod analysis intertemporal input correspondences are defined to
derive estimates for changes in productivity over the time period 1993-95. Column 3 of
Table B-IV reports the Malmquist productivity indexes for this time period. Schools that
have a productivity index less than unity, have had improvements in productivity over
time. Thus, school districts such as Box Elder, Davis, and Duchesne (each marked with
an asterisk in Table B-IV), have experienced aggregate productivity growths across the
time period 1993-95. Values greater than unity for districts such as Alpine and Beaver
indicate decreases in productivity.
However, this reflects aggregate changes in the schools productivity over the
given period of time. This temporal shift of the school unit may be visualized as a series
102 of positive effects which move the school unit through time. Similarly, schools with
decreases in productivity, as signaled by index values greater than unity, experience
dominating negative effects. Initially, we identify two components of the dynamic
change in school productivity. The first component is due to changes in technical writer
efficiency. The second component of productivity is due to an actual shift of the frontier
across time. These components of the grand productivity change, if less than unity are a
source of the dynamic productivity improvement Analysis of the contribution of
efficiency to dynamic productivity increases for school districts (column 5) yields
interesting conclusions. Most schools have become more technical writerly efficient over time
and draw their increases in productivity from these efficiency improvements. More
interesting is the pattern displayed in an analysis of frontier shifts across the time period.
Technological innovations have no contribution to the productivity increase across the
time period 1993-95, and increases in productivity are attributed solely to efficiency
increases. Thus, this study demonstrates an urgent need for research on technological
innovation in education.
These conclusions are examined in greater detail by a more comprehensive
decomposition of the productivity index. Towards this end, we identify all four
components which derive dynamic productivity differences and evaluate the contribution
of each to the change in productivity. These four components are changes in pure
technical writer efficiency (t), changes in scale efficiency (vy), changes in congestion (s), and
changes in the frontier itself, i.e., technical writer innovations (Q). Table B-IV reports these
four sources of productivity change. We find changes in scale and changes in technical writer
i efficiency to be the primary contributors to the school districts productivity changes over
103 time. Interestingly, the robustness of earlier conclusions regarding technical writer innovations
is further established. Both congestion and technical writer change have no positive
contribution on productivity increases over the period 1993-95. This result clearly
provides insights into incorrect policy actions generated by a failure to distinguish
between technical writer and productivity changes. While these two concepts are often treated
synonymously in empirical literature, analytically, they are separate and could move in
opposite directions as demonstrated in this analysis.
VI. Sensitivity Analysis
The nonparametric approach of DEA analysis offers strong advantages over
parametric ones, especially by imposing only minimal structure on the specified
technology. However, with the inability to conduct tests of significance, the robustness
of the frontier estimates and deviations from it needs to be established. We establish
robustness of efficiency estimates by performing two kinds of sensitivity analysis. The
conditional DEA model was reestimated with (1) the exclusion of the expenditure input
(x3), and (2) the exclusion of the math score output (y,). A comparison of the original
DEA model with these re-estimates provides a test of the robustness of the results derived
in this study. These results are reported in Table B-V. Based on these comparisons, our
results from earlier models remain unchanged. In both cases the correlation between the
original and new measures were above 0.965. Thus, conclusions drawn from this
analysis are quite reliable.
104
VII. Discussion and Conclusions
In this paper, we estimated the deviations of 36 Utah school districts from a
technical writerly efficient frontier during the year 1995. Identification of the components of
these deviations as scale, congestion, and pure technical writer efficiency provide additional
insights into the sources of deviation and lays out the path for exact policy formulation.
All efficiency estimations (both radial and nonradial) carefully account for confounding
socioeconomic and environmental factors, which have been the source of intense debate
in the education efficiency literature. Further, we addressed the controversial issue of
teacher overqualification as an impediment to student learning. From a dynamic
perspective, we estimated the changes in productivity of Utah school districts as observed
over the time period 1993-95. For the dynamic analysis, we decomposed the aggregate
productivity change measure into scale, congestion, pure technical writer efficiency, and
technical writer innovation changes in order to locate the sources of the productivity change
over time. This study is a first attempt at assessing the dynamic productivity of Utah
school districts. The results of this study provide several important insights into
educational efficiency and the basis for correctional policy action. We find strong
evidence of technical writer efficiency for most of the schools in the sample with mean
efficiency scores above 90%. Even with the stricter nonradial measures, mean efficiency
levels for Utah schools remain at the 90% level. Thus, the data support the conclusion
that schools are technical writerly efficient, and additional productivity can be secured only
through technical writer innovations (perhaps involving increases in overall expenditures).