i
AN ANALYSIS OF THE NO CHILD LEFT BEHIND ACT USING GRADUAL
SWITCHING REGRESSIONS
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
___________________________
Martin Dunbar Smith
Certificate of Approval:
_________________________ ________________________
Jose R. Llanes Henry W. Kinnucan, Chair
Professor Professor
Educational Foundations, Agricultural Economics
Leadership, and Technology and Rural Sociology
_________________________ ________________________
James L. Novak George T. Flowers
Extension Specialist Professor Dean
Agricultural Economics Graduate School
and Rural Sociology
ii
AN ANALYSIS OF THE NO CHILD LEFT BEHIND ACT USING GRADUAL
SWITCHING REGRESSIONS
Martin Dunbar Smith
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Masters of Science
Auburn, Alabama
August 10, 2009
iii
AN ANALYSIS OF THE NO CHILD LEFT BEHIND ACT USING GRADUAL
SWITCHING REGRESSIONS
Martin Dunbar Smith
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon request of individuals or institutions and at their expense. The author reserves all
publication rights.
________________________
Signature of Author
________________________
Date of Graduation
iv
VITA
Martin Dunbar Smith, son of Wilburn A. Smith Jr. and Ellyn Smith of
Montgomery, AL, was born on February 13, 1985. He graduated from The Montgomery
Academy in May 2003. He attended Auburn University for his undergraduate career, and
graduated with a degree in Agricultural Economics and Business in May 2008. He
continued on at Auburn University for a Master of Science in Agricultural Economics in
the Fall of 2008. He married Rachel Danielle Kichler, daughter of Leonard and Susan
Kichler, on May 31, 2008.
v
THESIS ABSTRACT
AN ANALYSIS OF THE NO CHILD LEFT BEHIND ACT USING GRADUAL
SWITCHING REGRESSIONS
Martin Smith
Master of Science, August 10, 2009
(B.S., Auburn University, 2008)
77 Typed Pages
Directed by Henry W. Kinnucan
Using a rich panel data set, this paper conducts unique analysis on the structural
and overall effects of the No Child Left Behind Act on the sixty-seven county school
systems in Alabama. A system of equations is specified to test the effects of NCLB on
education production, quality, and cost. Gradual change to the new policy regime is
modeled using both linear and non-linear specifications. Wald tests firmly reject the null
hypothesis that NCLB had no effect on the rural education market. However, the effects
appear to be confined to structural change, as the intercept shifters across the equations
were jointly zero. A key and robust finding is that the county school system in Alabama
exhibits constant returns to scale.
vi
ACKNOWLEDGEMENTS
The author would like to thank Dr. Henry W. Kinnucan for his time and
generosity with the thesis analysis and as his academic adviser. The author would also
like to thank Dr. James L. Novak and Dr. Jose R. Llanes for their work as committee
members. The author would also like to thank his wonderful wife for the countless hours
spent listening to him talking about each new result produced by his research.
vii
Style manual or journal used American Journal of Agricultural Economics
Computer software used Microsoft Office, Microsoft Excel, and Eviews Standard 6.0
viii
TABLE OF CONTENTS
LIST OF TABLES ....................................................................................................... ix
I. INTRODUCTION .............................................................................................1
II. REVIEW OF LITERATURE ........................................................................... 4
III. DATA AND MODELS ...................................................................................12
Data Summary .................................................................................................12
Production Function .........................................................................................14
Cost Function ...................................................................................................17
Test Score Equilibrium ....................................................................................20
Gradual Switching Regressions .......................................................................23
IV. RESULTS ........................................................................................................28
V. CONCLUSIONS..............................................................................................43
BIBLIOGRAPHY ........................................................................................................46
APPENDIX A ..............................................................................................................49
APPENDIX B ..............................................................................................................57
APPENDIX C ..............................................................................................................61
APPENDIX D ..............................................................................................................65
ix
LIST OF TABLES
Table 1.0 Summary of Estimated Production Function Effects ..........................................6
Table 2.0 Variable Definitions and Descriptive Statistics .................................................13
Table 3.0 Restriction Tests ................................................................................................25
Table 4.0Wald Tests ..........................................................................................................26
Table 5.0 Logged Production Function Estimates .............................................................30
Table 6.0 Logged Cost Function Estimates .......................................................................33
Table 7.0 Logged Test Score Equilibrium Estimates ........................................................35
Table 8.0 Differenced Production Function Estimates ......................................................37
Table 9.0 Differenced Cost Function Estimates ................................................................39
Table 10.0 Differenced Test Score Equilibrium Estimates ...............................................41
Table 11.0 Result Restriction Tests ...................................................................................42
Table 12.0Time Path Vector Values ..................................................................................62
Table 13.0Time Path Vector Rankings ..............................................................................66
1
I. INTRODUCTION
Often hailed as one of the most significant pieces of legislation introduced by
President George W. Bush, the No Child Left Behind Act of 2001 (NCLB) was a nation-
wide attempt to combat the problem of substandard primary and secondary school test
scores across the United States of America. The act contained a variety of economic
?carrots? designed to provide positive incentives for schools to improve their testing
proficiency rates, as well as a number of economic ?sticks? to address the issue
underachieving schools. These incentives included the promise of better facilities and
environments to schools achieving the NCLB?s objectives, but they also threatened
underachieving schools with the loss of jobs and governmental intervention (Executive
Summary). The NCLB act provides a unique opportunity for social research. This paper
will test the effectiveness of NCLB in achieving improved test scores for Alabama school
systems over a period of 9 years using a system of equations modified by the gradual
switching regressions technique.
In addition to examining the effect of the No Child Left Behind Act on school
systems in Alabama, this paper will also attempt to answer the question of the rate of
returns to scale within the state. A search of this topic did not find the question addressed
in the Alabama educational economic literature. The rate of returns to scale of Alabama
school systems, but also the manner in which the NCLB may have affected the returns to
scale will be explored in this study. While this normally would be a difficult procedure,
2
the method in which it was analyzed in this paper will present an analysis unique to the
field of educational economics.
While many papers have attempted to discern the overall impact of the No Child
Left Behind Act on educational system components, such as test score quality and cost,
very few have examined the educational system with the intent of looking for structural
effects as well. While it is easy to add a dummy variable representing a policy to an
equation, such a technique only evaluates whether or not the level of the entire equation
changed. This paper implements a gradual switching regression technique utilized by
Bacon and Watts (1971), Tsurumi (1983), Ohtani and Katayama (1986), and Moschini
and Meilke (1989) to analyze possible intra equation effects of the NCLB act. A thorough
search of the educational economic literature revealed that this technique has never been
used to evaluate educational economic policy, indicating that many of the structural
effects of the NCLB act may have yet to be evaluated. The most recent work by Moschini
and Meilke titled ?Modeling the Pattern of Structural Change in U.S. Meat Demand?
used a gradual switching regression framework the identify specific changes that
occurred in U.S. meat demand so that the industry might tailor marketing and production
to the new climate. First using likelihood ratio tests to make global statements regarding
the structural inconsistencies in U.S. meat demand, they moved on to identify specific
structural shifts in the market (Moschini and Meilke).The question of structural changes
occurring due to the NCLB act will be addressed in the same manner.
Specifically, this paper will examine the effect (if any) of the No Child Left
Behind Act on the 67 county school systems of the state of Alabama. While the intent of
this paper is to focus primarily on the effect of the NCLB on rural school systems, it
3
should be noted that some counties in Alabama have separate schools systems for the
large cities within the county. For example, in Lee County, a relatively small county in
East Alabama containing 32 schools, there are separate school systems for the city of
Auburn, the city of Opelika, and the county itself. However in Montgomery County, one
of the state?s larger counties, there is only one school system encompassing both the rural
and urban schools (ALSDE).
The data used for this analysis was time-series and cross-sectional, covering the
67 county systems using annual data for the period 1999 to 2007. Data were converted to
natural logarithms (excluding dummy variables) in order to produce elasticities from the
coefficient estimates. For the estimates, a system of equations was used, including a
production function based on Robert Solow?s 1956 growth model (Solow), a cost
function derived from the Solow model, and a reduced form quality equilibrium test score
derived from Kinnucan, Zheng, and Brehmer (2004 work entitled ?State Aid and Student
Performance, a Supply-Demand Analysis?).
For this analysis, this paper will consist of 5 sections. The first section is an
introduction to NCLB and a general overview of the problem. The second section is a
review of current and pertinent literature. Third is a section detailing the data and model
specification used. Section four presents the regression results, and finally the fifth
presents conclusions and recommendations.
4
II. REVIEW OF LITERATURE
A vast literature regarding educational determinants, effective resource allocation,
and the No Child Left Behind Act can be found with a simple search on the internet, so it
is important to highlight some of the more significant pieces as they relate to the analysis
presented in this paper. Beginning with works relevant to the formation of the system of
equations, this section will then proceed to review a selection of articles crucial to the
understanding of the current educational economics literature.
As was mentioned in the introduction, this paper will attempt to estimate a system
of equations in order to analyze the effect of the No Child Left Behind Act on various
aspects of the educational system. The first equation in the system is a production
function modeled after the one Solow proposed in his 1956 article ?A Contribution to the
Theory of Economic Growth?. Solow proposed a model of long run growth, appearing in
its initial form as equation (1).
(1) Y = F(K,L)
This simple production function models output (Y) as a function of capital (K) and labor
(L). In the extensions section of the same article, Solow modifies the equation to reflect
neutral technological change, indicated by equation (2)(Solow).
(2) Y=A(t)F(K,L)
With the addition of the technological change parameter A(t), the production function?s
isoquants are allowed to shift in and out without changing the structure of the
5
function itself. This ?increasing scale factor? (Solow) will be represented in this paper by
the various county attributes which affect the given system?s Average Daily Admissions
per school
The next work with significance to both this paper and the field in general is
Kinnucan, Zheng, and Brehmer?s article ?State Aid and Student Performance: A Supply-
Demand Analysis.? In this piece, the authors use a supply-demand framework to specify
a six-equation model in order to examine the relationship between governmental aid and
pupils? academic performance (Kinnucan et al). Using econometric techniques, the
authors analyzed the effect of government aid on student performance and they also
examined alternative determinants of student performance using variables such as
income, poverty, property value, and parental education levels. The authors? work
concluded with several key findings, including statistical evidence from their data which
indicated that increases in state aid led to a reduction in local funding (Kinnucan et al).
Additionally, Kinnucan, Zheng, and Brehmer posited in their concluding comments that
the same results of increasing state aid might also be achieved by the expansion of
programs aimed at reducing county poverty or increasing average family income. The
parsimonious demand and supply equations for educational quality estimated by
Kinnucan, Zheng, and Brehmer serve as the basis for the equilibrium reduced form
equation for test score quality used in this paper.
Another author whose work vastly contributes to an understanding of the state of
the field of educational economics is Eric A. Hanushek. Specifically, Hanushek has
produced several key works identifying problems in the area of academic achievement
production functions. In his 2004 article ?What if There Are No ?Best Practices???,
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Hanushek indicates that there may be several obstacles preventing robust factor estimates
from being obtained. One possibility he alludes to is the notion that ?the achievement
process is a complicated interactive one such that simple linear additive formulations
break down? (Hanushek 2004). He illustrates this same point in his 1986 article ?The
Economics of Schooling: Production and Efficiency in Public Schools? where he creates
tables indicating the wide ranging results for some of the more commonly appearing
variables in the literature (Hanushek 1986). One such table is Table 1.0, included below.
Table 1. Summary of estimated effects from 147 education production function
studies
Significant
School Input
Number
of studies Positive Negative Insignificant
Teacher-pupil ratio 112
9
14
89
Teacher education 106
6
5
95
Teacher experience 109
33
7
69
Teacher Salary 60
9
1
50
Expenditures per pupil 65 13 3 49
Source: Hanushek
(1986)
One can quickly see the wide ranging sets of results for some of the more common
variables. Not only were many of the variables found to be statistically insignificant well
over 50% of the time, but some of the estimates that actually were statistically significant
turned out to have inconsistent signs. In a 2005 article by Rivkin, Hanushek, and Kain,
the authors indicate that some of the blame for inconsistent and conflicting research
regarding academic achievement may lie with incomplete or improperly measured data
sets (Rivkin et al). Finally, in a 2003 article, ?The Failure of Input-Based Schooling
Policies?, Hanushek notes the massive importance that governments across the globe
7
have placed on pouring resources into schools while achieving little success from these
policies (Hanushek 2003). As one will discover upon reading in the model specification
and results sections, this paper attempts to circumvent some of the problems noted by
Hanushek.
One of the key techniques making this paper unique in the field of educational
economics is the utilization of the gradual switching regressions technique as
implemented by Bacon and Watts (1971), Tsurumi (1983), Ohtani and Katayama (1986),
and Moschini and Meilke (1989), in analyzing the No Child Left Behind Act. Gradual
switching regressions is a revolutionary technique affecting the manner in which one
analyzes the effect of a change in policy or regime. After determining a basic structural
specification, the following substitution is then made:
(3) ?k = ?k + ?k??t
?k represents all coefficients in a given function with K number of variables. ?k?
represents the variable shift coefficient while ?t serves as time path vector reflecting the
following values:
(4.1) ?t = 0 | t=0 ? t 1
(4.2) ?t = (t-t1)/(t2-t1) | t=(t1+1) ?(t 2-1)
(4.3) ?t = 1 | t= t2 ? T
The variable t represents the current time while t1 represents the starting point of a
policies? implementation, and t2 represents the point in time that the policy has reached
one hundred percent implementation. As is readily apparent and was mentioned by
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Moschini and Meilke, if t2= t1 + 1, then the policy implementation is abrupt and the time
path vector assumes the appearance of a standard dummy variable attached to each shift
coefficient. Whereas a standard dummy variable appearing a single instance in a function
only indicates whether or not there has been an overall change in the rate of production
given a set level of inputs, gradual switching regressions allows the structure of the
equation in question to change despite the output remaining the same. Thus, if a given
policy affected the ratio of inputs, but not the level of output in a production function, a
standard dummy variable would appear insignificant while the gradual switching
regressions? shift coefficients would illuminate the effect. Using this technique, this paper
is equipped to discover any structural changes occurring in the system of equations as a
result of the No Child Left Behind Act.
In 1990, Ohtani, Kakimoto, and Abe used the aforementioned transition path but
allowed it to shift as a polynomial of time in their article ?A Gradual Switching
Regression Model with a Flexible Transition Path.? In this paper, the author tested
various polynomials and selected the optimal one comparing Akaike?s information
criterion and Schwarz?s criterion values. This paper will follow the technique as it was
used by Konno and Fukushige (2002) in that it will test convex and concave time path
vectors. However, this paper will not test for the optimal non-linear function. Rather, the
aim of this paper will be to find the optimal t1 and t2 values in conjunction with the
optimal functional shift form, be that a step up (abrupt), linear, concave, or convex
function. The convex and concave vectors are formed by manipulating equation 4.2 into
equation (5):
(5) ?t = (t-t1)/(t2-t1)Z | t=(t1+1) ?(t 2-1)
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In equation (5), the Z variable assumes the value of 0.5 to test a concave time path, 1 to
test a linear time path, and 2 to test a convex time path.
Literature should also be examined regarding the No Child Left Behind Act itself
in order to give the reader a sense of the policy actually being tested. According to the
Department of Education Executive Summary (2002), the No Child Left Behind Act?s
(NCLB) overarching aim is to improve standardized test scores in the United States for
primary school students and to specifically close the education gap between certain
minority groups (Kim and Sunderman). The NLCB sets the commendable goal of 100%
of students in grades three through eight reaching ?proficient? levels of academic
achievement in standardized testing within 12 years of the bill being signed into law
(2014, as the 2001 act was actually signed into law in 2002). Part of the ?teeth? of NCLB
is the section regarding adequate yearly progress (AYP). According to the 2002 article by
Linn, Baker, and Betebenner titled ?Accountability Systems: Implications of
Requirements of the No Child Left Behind Act of 2001,? the definition of AYP was
initially set by the House and Senate to mean an increase in percentage proficiency of at
least one point per year. While this was later changed to allow states to set their own
AYP rates it brings a particularly problem to light; states have different definitions of
what it means to be?proficient?. As mentioned by Linn et al, Louisiana, Mississippi, and
Texas reported proficiency rates on the Grade 8 mathematics assessment as 7%, 38%,
and 92% respectively in 2001. Given the initial required AYP rate of 1% a year, a state
could meet AYP each year, yet not meet the overall goal of 100% unless its proficiency
rate was already at 88% or higher (Linn et al). Therefore, not only is it a problem that
10
states are allowed to possess different definitions of proficient, but they are allowed to
define different AYP rates, making it virtually impossible to meet the 2014 goal of 100%
proficiency (barring a change of standards).
However, despite the ambiguity regarding the AYP rates and proficiency levels, it
certainly behooves a state and school district to meet AYP. The second section of the
NCLB Executive Summary (2002) states:
School districts and schools that fail to make adequate yearly
progress (AYP) toward statewide proficiency goals will, over
time, be subject to improvement, corrective action, and
restructuring measures aimed at getting them back on course to
meet State standards. Schools that meet or exceed AYP objectives
or close achievement gaps will be eligible for State Academic
Achievement Awards.
Thus, the NCLB has significant funding and job implications at the municipal and county
levels. The ?restructuring? of a school due to failure to meet AYP could mean the loss of
jobs. Because of the significant positive incentives to meet or exceed AYP and the
negative incentives discouraging failure, this paper has the potential to illuminate
important shifts in various aspects of the educational system. In addition, this paper will
be able to conjecture as to the effectiveness of the NCLB in meeting its stated goal of
100% proficiency in 2014.
Finally, a 2001 article, ?Response to Skrla et al. The illusion of educational equity
in Texas: a commentary on ?accountability for equity??, Walt Haney examines possible
explanations of the ?Texas Miracle? (Haney). The ?Texas Miracle? was a phenomenon
11
that occurred in the late 1990?s regarding a near miraculous jump in the level of
standardized achievement test proficiency within that state. Haney explained that such a
jump could have occurred due to several reasons, but specifically focused on the manner
in which the percentage of correct questions required to achieve proficiency was changed
(Haney). He also points at that the change in percentage was made without sound
academic explanation, indicating that it was indeed an unfounded manipulation of the
system. While this did not affect the actual numerical results of the score, it did affect the
levels of proficiency as reported in compliance with the NCLB act.
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III. DATA AND THE MODELS
As was explained in the introduction, the purpose of this paper is to ascertain
some of the effects of the No Child Left Behind Act (NCLB) using a system of equations
and the gradual switching regressions technique. This paper is not an attempt to explain
every single aspect of the educational system, nor is it an attempt to verify previous
analysis of the NCLB. Rather, it is an exploratory effort to analyze the effects of the
NCLB on some of the more critical components of the educational system, while at the
same time obtaining robust coefficient estimates. To that end, the system of equations
was estimated in the most parsimonious manner possible, leaving out some of the more
popular variables in education literature.
Data Summary
Prior to an examination of the actual system of equations, it is useful to view a
table of summary statistics in order to understand trends in the data that may be appear in
estimation. As the NCLB act was first implemented in 2002 (Executive Summary), the
data is split in two halves; 1999 ? 2002 and 2003 ? 2007. The results are displayed below
in table 2.0. The first variable, Q, exhibits a very interesting trend over the split. While
the standard deviations of the data remain relatively unchanged, the test score quality
experiences a 3 point decrease on average from the first period to the second.
Additionally, the per school expenditure experiences a significant increase, though it is
largely explained by the near doubling of the maximum per school expenditure. Though
13
Table 2.0
Variable definitions and descriptive statistics (Upper number is for 1999-2002
(nobs = 268); lower number is for 2003-2007 (nobs = 335). Dollar amounts
are expressed in constant 1982-84 dollars.)
Variable Definition Mean SD Min Max
Q Eighth-grade SAT test
score, national percentilea
48 7.9 24 67
45 8.3 24 67
PG Total education
expenditures, dollars per
pupil
3,801 426 3,053 5,891
4,318 845 3,185 10,222
TSR Student-teacher ratio 15.6 0.78 13.4 18.1
14.3 1.53 9.5 17.6
TPAY Teacher salary, dollars
per year
22,463 818 20,262 26,090
19,899 2,479 13,199 25,781
ENROL Average daily attendance
in county system, per
school
454 127 209 770
443 126 225 824
INC Median family income,
dollars
17,556 3,726 10,258 34,524
17,179 3,682 9,895 33,095
RACE Nonwhite student
enrollment, percent
38.4 31 0.1 100
39.1 30.9 0.3 100
POV Poverty rate, percent 22.6 5.9 7.3 41.8
24.8 7.2 7.4 43.3
UNEMP Unemployment rate,
percent
5.8 2 1.5 13
5 1.7 2.3 11.3
RURAL Rural county, dummy
variable
0.74 0.44 0 1
0.72 0.45 0 1
many of the other variable remained relatively unchanged, the other interesting finding
yielded by the summary statistics is the drop in deflated TPAY over the two periods. All
of these signs indicate a decrease in test score quality coupled with an increase in cost as
14
time progresses over the 9 year set. Specific definitions of data manipulation and sources
can be found in appendix A.
Production Function
The first equation estimated in the system is a slightly modified version of Robert
Solow?s 1956 growth model. This equation was previously mentioned in the literature
review as equation (1). Much of the previous educational economics literature regarding
test scores has viewed academic achievement as an output of a production function. Table
1.0 from Hanushek?s 1986 article was a study of 147 production functions where some
variation of academic achievement served as the output variable. With one goal being an
elucidation of the effects of NCLB on the educational system, it is felt that a more
accurate use of the production function was one that examined the actual unit of school
output, namely enrollment (Enrol). Thus, the ENROL variable as used as the dependent
variable in our production function is defined to be average daily admissions per school
within the county level school system. Average daily admission is the term used to label
the 30 day, K-12 attendance average which is used on a system by system basis to
quantify yearly enrollment (ALSDE). All appropriate variables have been divided by the
number of schools contained within the county system in order to obtain a per school
average for each variable. This is the best method for a variety of reasons. First, it allows
for the system of equations to make statements regarding the school as a firm, which is
the proper implementation of Solow?s 1956 growth model. Second, it should be noted
that test scores as reported in the Alabama State Department of Education Annual
Reports (ALSDE) are averages for the county and city systems. Finally, the NCLB?s
rewards and corrective actions are based on a per school basis (Executive Summary), not
15
on a per pupil basis. Thus, a larger unit of analysis would not capture as much detail
while a smaller unit of analysis would misconstrue reaction to the NCLB as being on a
per pupil basis rather than per firm basis.
Next, variable selection was requisite for the two components of a production
function, namely capital and labor. The capital variable is a particularly difficult one to
define given the aggregated nature in which school expenditures are reported. Thus, this
paper chose to define capital as being non-instructionally related expenditure, to clearly
differentiate it from the educational labor variable. Solow, in his 1956 piece, does not
refer to capital using the current financial definition of the word capital, but instead noted
that ?Output is produced with the help of two factors of production, capital and labor?
(Solow). The sum of all non-instructional expenditures was gathered for each county
system and divided by the number of schools within that system to generate a non-
instructional expenditure variable (NIE). The NIE variable was then deflated using
consumer price indexes (CPI) from the Bureau of Labor and Statistics (BLS CPI), using
1982 ? 1984 as a base value of 100. All financial calculations from this point on will be
deflated in the same manner.
The second important factor of production utilized in Solow?s growth model is
labor, a much easier variable to identify and implement. While it can be argued that all
laborers at the school level can have an effect on the number of students produced by a
school, this paper believes that the obvious choice for this variable is number of full time
equivalent teachers. This statistic was gathered on a county system basis from the
National Center for Education Statistics (NCES) Common Core of Data (CCD), and was
then divided by the number of schools within the system to produce an average number
16
of full time equivalent teachers per school (NCES). This variable (TEACH) was then
utilized as the ?L? in Solow?s growth model.
The other important aspect of Solow?s growth model is found in the previously
mentioned extension, which is the addition of a technological shift function A(t). This
paper felt that the most crucial variables with which to analyze base levels of technology
were average standardized test scores (TS) and poverty level within the county (POV).
For this paper, eighth grade standardized test scores were selected as the measure of TS,
taking the average of the county systems? scores for reading, mathematics, and language
garnered from the ALSDE annual reports. The POV variable was gathered using national
census data and the government census website?s SAIPE function, generating values for
the percentage of children aged 5-17 per county in families in poverty (Census Bureau).
A lagged dependent variable was also added to the equation in order to test it as a
form of partial adjustment model in which the system attempts to adjust each year toward
an optimal equilibrium level (King and Thomas). Furthermore, the use of a lagged
dependent variable allows this paper to examine the amount of memory within the system
regarding enrollment per school. Thus, the final growth model with new variables
replacing Solow?s original ones results in the specification of equation (6.1)
(6.1) ENROL = F(NIE,TEACH)A(TS,POV,ENROL-1)
Prior to estimation, the natural logarithm was taken of all data excepting dummy
variables in order to obtain elasticities with the final coefficient estimates. In order to
examine some of the cross-sectional and time-series effects of the panel data set, a time
trend variable was added to all three equations as was a set of district dummy variables.
According to the ALSDE website, the state of Alabama is divided into seven different
17
school districts containing all 67 counties (ALSDE). District 5 was withheld as a control
variable. Thus a final double log model with a time trend and district dummy variables is
represented in equation (6.2):
(6.2) ln(Enrol)= ?0 + ?1ln(NIE) + ?2ln(Teach) + ?3ln(TS) + ?4ln(POV) +
?5ln(Enrol-1) + ?6-11(D1-D7) + ?12(t) + ?
Cost Function
The second equation in the system of equations is the cost function. Using
standard economic theory, the cost function can be derived from Solow?s production
function to yield equation (7). The mathematical procedure behind this derivation can be
found in appendix B of this paper.
(7.1) C = (r,w,Q,t)
This function contains the same output label as the dependent variable of equation (6.2),
namely Enrol, as well as the same vector of variables ?t? located in the technological shift
function. However, the equation differs exogenously with the appearance of the variables
r and w, which represent the input factor prices of NIE and TEACH respectively. For the
sake of this paper, the price of NIE will be represented by the percentage of non-
instructional expenditure (PNIE) per school relative to its total expenditure. This is an
accurate representation of the price of NIE because a higher value of PNIE indicates that
a larger portion of the given system?s budget is being consumed by NIE, versus a
different system in which NIE is the same, but PNIE is lower due to a larger budget.
PNIE was gathered from the ALSDE annual report by dividing the non-instructional
system expenditures by the total expenditures for that system (ALSDE).
18
The input factor price used in the cost function relative to labor is average
instructional expenditure per teacher (TPAY). This variable was generated by taking the
sum of each county system?s yearly instructional and instructional support expenditures
(found in the ALSDE annual reports) and dividing them by the total number of full time
equivalent teachers within that county (ALSDE). Those familiar with Alabama?s
educational system might object to this characterization on the grounds that teacher?s
salary In Alabama has been determined by a pre-set pay matrix since the late 1990?s
(ALSDE). However, the identification of the cost of labor as this paper?s definition of
TPAY is superior for three reasons. First, while Alabama teacher pay is set by a pay
matrix based on degree achieved and number of years in service, the pay matrix does not
account for temporary incentives used to hire teachers to different systems. Second, while
this paper used the number of full time equivalent teachers as a definition for labor,
instructional and instructional support expenditures also encompass the number of
temporary and substitute teachers hired to bolster a given systems instructional labor
pool. These may be viewed as a subsidy, in that the full time equivalent laborers have
their labor load reduced while not having their pay cut. Therefore, the expenditures on
additional resources and temporary laborers should be added to the full time equivalent
teachers? average pay in order to reflect the additional benefit. Finally, using this
definition gives the variable the necessary variation required to generate testable
hypotheses, which otherwise would have been ignored by a procedure considering
teacher pay to be strictly defined by the pay matrix. The numbers generated by this
variable were also deflated using the same method as the financial variables in the
production function.
19
The endogenous variable representing cost in this function is average per school
expenditure per county system (PG). This variable was acquired by taking the sum of the
State Funding, Federal Funding, Local Funding, and Other Funding values in the ALSDE
annual reports and dividing the sum by the number of schools within that system
(ALSDE). As with other financial variables, the cost variable was deflated using BLS
CPI?s (BLS CPI), resulting in the final computed variable PG. In the same manner as the
production function, the cost function followed the partial adjustment model framework
by added a lagged dependent variable, PG-1. After computing the natural logarithm of all
non-dummy variables as well as inserting a time trend and a set of district dummy
variables, the final cost function equation can be written as equation (7.2).
(7.2) ln(PG)= ?0 + ? 1ln(PNIE) + ? 2ln(Tpay) + ? 3ln(TS) + ? 4ln(POV) +
?5ln(Enrol) + ? 6ln(PG-1) + ?7-12(D1-D7) + ?13(t) + ?
This specification yields several interesting results and testable hypotheses which will be
explored in the results section of this paper. It should be noted that the coefficient of the
production function endogenous variable found in the cost function will denote the short
run returns to scale in elasticity form. A cursory exploration of the literature indicates that
this statistic has never been discovered in reference to the Alabama county school
systems. Additionally, upon obtaining the short run returns to scale, the following
restriction may be tested to examine long run returns to scale:
(8) 1.0 = ? 5/(1- ? 6)
In theory, the long run returns to scale of the firm should result in a one to one ratio of
production increases to cost increases. An additional testable hypothesis per the
mathematical theory found in appendix 1 would be the notion that the coefficient of
20
output in the cost function is equal and opposite in sign to the sum of the coefficients of
the technological shift parameters carried from the production function. Equation (9), if
true, would indicate that county school systems are operating as cost minimizing firms.
(9) ? 5 = ? 3 + ? 4 + ?7-12 + ?13
These theories will be examined using Wald coefficient tests upon the estimation of the
final system of equations.
Test Score Equilibrium
The test score equilibrium is largely based on the theory and technique of
Kinnucan, Zheng, and Brehmer?s 2002 article ?State Aid and Student Performance: A
Supply-Demand Analysis?. Initially, the test score equation was specified as the demand
function equation (10), reflecting the counties? demand for a certain test score quality,
TS, as indicated by a vector of demand variables X and a price for that quality of test
score, P.
(10) TSd = d(P,X)
Using the supply equation for test scores, equation (11), and a definitional equation for
price, equation (12), the equilibrium price was then computed by setting the supply and
demand equations? values for TS equal to each other. Substituting the equilibrium price
back into either test score equation then yielded the equilibrium test score, equation (13),
which is a reduced form function of the exogenous variables found in equations (10) and
(11).
(11) TSs = s(P,Z)
(12) P = C/Q
(13) TS = f(X,Z)
21
By using a reduced form equation to indicate the test score resulting from equilibrium,
this paper captures both the supply of test score quality by the system as well as the
demand for test score quality by the citizens of the respective counties.
For the purpose of this paper, the endogenous test score variable (TS) is defined
as the average of eighth grade students? scores in reading, language, and mathematics by
county system. The grades used to measure school system AYP are 3 through 8
(Executive Summary 2002), and I selected the last grade used in the AYP evaluation was
selected as the level at which to examine the test score equilibrium.
The variable vectors X and Z were established using the same theory and
reasoning as Kinnucan et al, with the supply variables being teacher student ratio (TSR),
average teacher pay (TPAY), a rural county dummy (Rural), and the same poverty
variable used in the production and cost functions (POV). The demand variables used
were average county income (INC), county unemployment levels (UNEMP), and a racial
demographic variable (RACE).
The TSR variable is a statistic obtained from the NCES CCD, reported
specifically as the number of pupils per teacher within the specific county system
(NCES). The rural county dummy variable is a standard dummy variable denoting the
different overall classifications of the counties as being ?rural? or ?urban? counties as
defined by the University of Alabama?s Center for Business and Economic Research
(University of Alabama). The TPAY and POV variables appearing in the supply vector
maintain the same definitions as used in the cost and production functions.
Again, using the same selections as Kinnucan et al, the first demand vector
variable is income. The INC variable was generated using the United States? government
22
census website?s SAIPE function to estimate median family income per county per year
(Census Bureau). This variable was then deflated in the same manner as the previously
mentioned financial variables. The second demand vector variable used is
unemployment. This variable was obtained via the BLS website (BLS LAES), recording
values for county level, annual unemployment. The final demand vector variable used by
Kinnucan et al was a percentage non-white racial demographic variable. Until 2007, these
statistics were reported by the ALSDE annual reports for each county and city school
system (ALSDE). Due to the small level of variance over the previous 8 years, the final
year was estimated using a weighted average. However, one must remember that this
variable continues to be very important, especially with regards to the NCLB act. Not
only must each overall system make appropriate AYP gains towards 100% proficiency,
but the specific demographic groups identified by the NCLB legislation must make
appropriate AYP as well (Kim and Sunderman).
Thus, with the demand and supply variable vectors identified, the final linear
equation will be estimated as equation (14) following the addition of cross sectional
dummies and a trend variable. As with the production function and cost functions, a
lagged dependent variable has been added to maintain the partial adjustment model
framework.
(14) ln(TS)= ?0 + ? 1ln(Race) + ? 2ln(Inc) + ? 3ln(Unemp) + ? 4ln(TSR) +
?5ln(Tpay) + ?6(Rural) + ? 7ln(Pov) + ? 8ln(TS-1) + ?9-14(D1-D7) + ?15(t)
+ ?
23
Gradual Switching Regressions
As was previously mentioned in the introduction and literature review, the heart
of this paper?s analytical power regarding the No Child Left Behind Act is the use of
linear and non linear gradual switching regressions. However, one must first find the
optimal time path vector to use for each equation prior to estimating a final set of results.
The equations (6.2), (7.2), and (14) that were previously specified in this section were
then subjected to a substitution using equation (3) that was explained in the literature
review section. This gives each constant and variable its own corresponding shift
coefficient, though all shift terms in each equation are multiplied by the same time path
vector. Using these new equations, the system was then tested for the optimal time path
vector for each equation.
To find the optimal time path vector, all possible combinations of t1 and t2 were
tested beginning in 2001, when the bill was first passed (Executive Summary). Though it
could be argued that schools could begin altering their educational systems in anticipation
of the passage of the bill, it was deemed unlikely due to the tendency of the data to prefer
the latter time points. Each possible t2 value was tested in conjunction with each possible
t1 value until the year 2008. It should be noted that the year 2008 is beyond the scope of
the available data set, which allows for the data to indicate that the full effect of the
NCLB has not yet been achieved. The different possible vectors of ?t, as defined by
equations (4) and (5), are listed in appendix C. The different combinations resulted in 84
testable time path vectors. The explanatory power of the different time paths was then
compared by ranking the resulting R2 values for each regression. The rankings produced
from these tests are available in appendix D. The rankings resulted in optimal t1 and t2
24
values of 5 and 7 respectively for the production function, using the non-linear convex
time path. These values indicated that the NCLB began to have an effect on the
production function in 2003, reached 25% of that effect by 2004, and had taken its full
effect by the end of 2005. However, the rankings resulted in optimal t1 and t2 values of 6
and 7 for both the cost function and the test score quality equilibrium. These values
indicate that the full system adjustment caused by the NCLB occurred from 2004 to
2005. These values correspond with the 2002 article by Linn, Baker, and Betebenner
which noted that the NCLB act would have achieved full legislative implementation by
the end of the following year (Linn et al). It can then be observed that while the effects of
the NCLB on the production function were felt at an accelerating rate over the course of 2
years beginning in 2003, the effects on the cost function and test score equilibrium took
the form of a standard dummy variable between the years 2004 and 2005. By testing the
multitude of different combinations of time path vectors, this paper allowed the data set
to indicate the proper time at which the impact of the NCLB act was felt, rather than
assigning such a time capriciously.
Using the newly found optimal time path vectors for the respective equations, the
constant shift and slope shift coefficients were subjected to a series of Wald tests, the
results of which are found in table 3.0. The execution of these Wald coefficient tests
allowed several very important statements to be made. The first test, restricting the all
constant and slope shift coefficients to zero, was rejected at the 1% level. This rejection
allows the blanket statement to be made that the NCLB act definitely created statistically
significant changes in the structure of the education system. The next two tests
25
specifically tested all of the slope shift coefficients and all of the constant shift
coefficients in two separate groups.
Table 3.0
Restriction =
Zero
Test
Stastistic Value DoF Probability
All Shift
Coefficients Chi-square 180.473 22 0
All Slope
Coefficients Chi-square 88.13429 19 0
All Constant
Terms Chi-square 7.148532 3 0.0673
Production
Slopes Chi-square 40.91271 5 0
Cost Slopes Chi-square 29.94037 6 0
TS Slopes Chi-square 17.28121 8 0.0273
Production
Constant Chi-square 0.710104 1 0.3994
Cost Constant Chi-square 5.941915 1 0.0148
TS Constant Chi-square 0.496513 1 0.481
While the restriction of the slope shift coefficients to zero was rejected at the 1% level,
the constant shift coefficients failed to reject at the 5% level, indicating that some of the
equations might not have experienced an overall shift, but only a structural shift. The
slope shift coefficients were then jointly tested for each individual equation, and all
rejected the restriction to 0 at the 5% level or better, indicating definite evidence of
statistically significant structural change. However, of the constant term coefficients, only
the cost function rejected the restriction at the 5% level. This indicates that while there
was an overall shift in the behavior of the cost function due to reaction to the NCLB act,
there was only structural change within the production and test score functions (which
presumably would not have been recognized with the use of a standard dummy variable).
With the knowledge that the constant shift coefficients for the production function and
26
test score equilibrium function failed to reject the restriction to zero, they were then
dropped from the final system of equations.
Next tested was the system of equations for insignificant and jointly insignificant
slope coefficients. One of the difficulties of using the gradual switching regressions
technique is that its initial implementation effectively doubles the number of variables in
each equation. Thus, a high degree of multicollinearity and insignificant t values begin to
occur. Testing the insignificant slope shift coefficients for joint significance yielded
interesting results, located in table 4.0. While only one slope shift coefficient was
insignificant in the production function, there were several insignificant slope shift
coefficients in the cost function and test score equilibrium function. The joint Wald
coefficient tests of these sets of slope shift coefficients resulted in a failure to reject either
restriction at the 5% level. Thus, with these coefficients resulting in insignificant values
on both the individual and the joint levels, they were dropped from the final equation.
Table 4.0
Restriction to zero
Test
Stastistic Value DoF Probability
Insignificant Production
Slope Shifters Chi-square 4.827597 4 0.3054
Insignificant Test Score
Slope Shifters Chi-square 4.324057 6 0.6329
Utilizing the Wald test results to created the most parsimonious system of
equations possible, the final system of equations was specified as equations (15), (16),
and (17).
27
(15) ln(Enrol)= ?0 + ?1ln(NIE) + ?1? ?nlv15 ln(NIE) + ?2ln(Teach) + ?2? ?nlv15
ln(Teach) + ?3ln(TS) + ?4ln(POV) + ?4? ?nlv15 ln(POV) + ?5ln(Enrol-1) +
?6-11(D1-D7) + ?12(t) + ?12? ?nlv15 (t) + ?
(16) ln(PG)= ?0 + ?0? ?nlv19 + ? 1ln(PNIE) + ? 2ln(Tpay) + ?2? ?nlv19ln(Tpay) + ?
3ln(TS) + ? 4ln(POV) + ? 5ln(Enrol) + ? 6ln(PG-1) + ?7-12(D1-D7) + ?13(t) +
?13?? ?nlv19ln(t) + ?
(17) ln(TS)= ?0 + ? 1ln(Race) + ? 2ln(Inc) + ? 3ln(Unemp) + ?3??nlv19ln(Unemp)
+ ? 4ln(TSR) + ? 5ln(Tpay) + ?6(Rural) + ? 7ln(Pov) + ? 8ln(TS-1) + ?9-
14(D1-D7) + ?15(t) + ?15? ?nlv19ln(t) + ?
A quick point should be mentioned regarding these final three specifications. The
differing values of ?t (?nlv15 and ?nlv19) indicate the differing optimal time paths for the
production function versus the cost function and test score equilibrium.
28
IV. RESULTS
Equations (15), (16), and (17) that were specified in the data and models section
were then estimated using Eviews standard version 6.0 in a variety of different
specifications. The system of equations was estimated using ordinary least squares,
seemingly unrelated regressions, and three stages least squares to test for the systems
sensitivity to different methods of estimation. As mentioned in the data and models
section, the natural logarithm of the data was taken for the first set of estimations. The
system was estimated an additional three times using the same respective estimation
techniques, but after having taking the first difference of the logged data. Though this
sacrificed an additional year of the data set, the degrees of freedom remained adequate
while allowing for an examination of the sensitivity of the data to manipulation. The
results have been organized by data type and equation, yielding the following 6 tables;
two for each equation using the two different data manipulations. One must also keep in
mind the fact that these coefficient estimates are elasticities. The elasticities indicate the
manner in which the dependent variable will respond to a percentage change in the
exogenous variables.
For those unfamiliar with the gradual switching regression technique, it is
important to take note of the proper manner in which to interpret the variable ?Shifter?
results. The shift coefficient only affects the initial coefficient in the years following t1,
which the reader will remember as the starting point of a policy?s effect used in
29
constructing the time path vector ?t. The manner of the shift coefficient?s effect is
evaluated by adding the shift coefficient (multiplied by the value of ?t) to the initial
coefficient. When the policy has taken its full effect, that is when ?t = 1 or t2 = t, the full
value of the shift coefficient may be added to the initial coefficient to demonstrate the
total effect. However, when t2 ? t1 > 1, one must remember that the policy (and thus the
shift coefficient) only takes a partial effect in the years between t1 and t2, as mitigated by
the factional value of ?t.
The first estimation of the system of equations used simple logged data to
generate results in elasticity form. The results found on the next page in table 4.0 show
the parameter estimates for the production function using the three different estimation
techniques. The t-statistic for each respective estimate is shown immediately below in
parentheses. The number of asterisks denotes the level of significance, with three
indicating significance at the 1% level, two indicating the 5% level, and one indicating
the 10% level.
While there were several points of consistency between the three estimation
methods, the three stage least squares estimation technique created several results that
were inconsistent with the other two methods of estimation. Though this could be a
testament to the fact that coefficient estimates were not as robust as expected, it could
also be attributed to model misspecification error due to the instruments select for the
3SLS procedure. Therefore, the primary focus of this result analysis will be with the OLS
and SUR results in mind. The first result of note in table 5.0 is the coefficient attached to
the expenditure variable (NIE), which was used to denote capital in Solow?s 1956 growth
30
model (Solow). It is positive, as expected, and significant at the 5% level in both the SUR
and OLS results, though it appears to be negative in the 3SLS results.
Table 5.0 - Parameter estimates of the Production Function using logged data
Variable/ statistic OLS SUR 3SLS
Constant 0.956565*** 0.820972*** 1.450054**
(-6.206649) (5.439402) (2.183284)
NIE 0.024194** 0.044066*** -0.250229***
(2.555719) (4.771872) (-3.99992)
NIE Shifter 0.045953*** 0.044086*** 0.668971***
(5.113418) (5.113912) (5.333546)
TEACH 0.520986*** 0.534549*** 1.41098***
(18.17275) (19.40147) (6.837442)
TEACH Shifter -0.120459*** -0.114261*** -1.625993***
(-5.025552) (-4.974109) (-5.21512)
TS 0.021692 0.025066 0.043661
(1.269695) (1.49114) (0.574705)
POV 0.016748 0.019026 0.547698***
(1.14392) (1.331484) (3.847472)
POV Shifter -0.067615*** -0.065321*** -1.203331***
(-4.091614) (-4.133209) (-4.444053)
Lagged ENROL 0.477895*** 0.446067*** 0.229763**
(19.25274) (18.66777) (2.28813)
D1 -0.009884 -0.010392 -0.029632
(-0.907145) (-0.969346) (-0.779014)
D2 0.013951** 0.015271** 0.02862
(2.236375) (2.48801) (1.313637)
D3 0.023531*** 0.024814*** 0.058649**
(3.085967) (3.308449) (2.143443)
D4 0.008547 0.010828 -0.0097
(0.781904) (1.006861) (-0.252605)
D6 0.002726 0.004605 0.0506
(0.265886) (0.456724) (1.365251)
D7 -0.018942*** -0.018437*** -0.012228
(-2.75959) (-2.729913) (-0.509923)
Trend 0.007023*** 0.006749*** -0.003799
(3.176632) (3.105076) (-0.429527)
Trend Shifter -0.014185*** -0.015346*** 0.038035
(-3.248285) (-3.575671) (1.445726)
R Squared 0.968478 0.968115 0.611748
Adj. R Squared 0.967506 0.967132 0.599779
The SUR result indicates that a 10% increase in the level of capital would create a 0.44%
increase in the number of students per school within that county system, which would
suggest that capital levels are fairly inelastic relative to enrollment. Furthermore, the NIE
shifter is positive and significant at the 1% level using all three estimating techniques,
31
which indicated that the NCLB act increased the importance of NIE as it affected
ENROL.
The other primary factor used in the production function, labor or the number of
teachers per school, produced a coefficient estimate greater than .5 and significant at the
1% level using all three methods of estimation. Compared to facilities and non-
instructional expenditure, it would appear that the instructional labor is vastly more
important. However, the universal result following this estimate is that the NCLB act
mitigated the impact of TEACH as it affects the number of students per school. Thus, the
tradeoff between the production factors would appear to be an increased importance on
capital resulting from the NCLB with a decreased importance on the number of teachers
per school.
As for the technology function affecting the level of ENROL, there was mixed
significance among the district dummies, but consistent positive results for the lagged
dependent variable, indicating a moderate degree of memory within the system. In
addition, the trend variable alluded to a positive trend prior to the impact of the NCLB
act, but with a change toward a negative trend utilizing the shift variable. These results
for the changes to the trend variable were significant at the 1% level in both the SUR and
OLS methods of estimation. In addition, it was surprising to see that test score quality
had no effect on the number of students per school. This would seem to say that a
school?s enrollment is not affected by the standardized achievement test results of the
students.
Table 6.0 examines the coefficient estimates as provided by the 3 methods of
estimation on the double log model with respect to the cost function. Most interesting in
32
this set of results are the first two results, indicating a positive constant and constant shift
term at the 5% level using both OLS and SUR. As was indicated by the Wald tests
conducted in the data and models sections, the cost function was the only equation to
exhibit an overall change due to a shift in the constant term. However, the shift was not
an intuitive one, nor one that the U.S. government would appreciate. The constant shift
coefficient of the cost function was 3.62, indicating that base costs increased as a result of
the NCLB legislation. Coupled with the result of an insignificant effect on the test score
quality equilibrium, this would point to a series of unexpected results coming from the
NCLB.
However, as expected, the input factor prices PNIE and TPAY both exhibited
positive signs with economic and statistical significance using all three methods of
estimation. While the level of the factors did not have equal importance relative to the
production function, the level of the prices of the factors have almost equal elasticities.
Furthermore, the OLS and SUR results indicate a negative shift in the elasticity of TPAY,
significant at the 1% level. This can be interpreted as teacher?s wages becoming less
contributory to the level of per school expenditure following the passage of the NCLB,
which correlates with the drop in importance exhibited by the production function
coefficient results for TEACH.
Again, one must be cognizant of the fact that the coefficients estimated by these
regressions are elasticities. While the blanket statement may be made that the average
teacher salary decreased in importance relative to the per school expenditure, the actual
cause of this effect could have been brought about in many ways. It could be the case that
33
PG experienced and increase while TPAY remained the same, which would cause the
absolute value of the coefficient of TPAY to decrease. Conversely, average teacher
Table 6.0 - Parameter estimates of the Cost Function using logged data
Variable/ statistic OLS SUR 3SLS
Constant 3.336715** 3.251405** 3.303595
(2.448731) (2.491728) (1.146231)
Constant Shifter 3.624889** 2.80551** 2.98544
(2.456522) (1.98617) (0.787628)
PNIE 0.500932*** 0.493962*** 0.521096***
(14.84039) (15.01227) (14.56061)
TPAY 0.601847*** 0.688032*** 0.662643**
(4.658093) (5.567387) (2.380237)
TPAY Shifter -0.380992*** -0.301908** -0.369103
(-2.588709) (-2.142354) (-1.000537)
ENROL 0.937191*** 1.039623*** 0.99328***
(24.9582) (28.81375) (23.57448)
TS -0.05292 -0.084735** -0.066623
(-1.464571) (-2.385232) (-1.364486)
POV -0.0237 -0.027327 -0.022947
(-0.911799) (-1.068965) (-0.762301)
Lagged PG 0.002239 -0.086897** -0.054147
(0.06276) (-2.54769) (-1.352329)
D1 0.054284** 0.05597** 0.057188**
(2.367689) (2.481338) (2.364506)
D2 -0.046565*** -0.049524*** -0.048173***
(-3.526639) (-3.81446) (-3.453693)
D3 -0.031441** -0.034183** -0.031729*
(-1.967977) (-2.175404) (-1.879332)
D4 -0.027072 -0.0378* -0.031555
(-1.178422) (-1.672812) (-1.291277)
D6 -0.059296*** -0.05761*** -0.058852***
(-2.775262) (-2.740606) (-2.626849)
D7 -0.012571 -0.008368 -0.010412
(-0.869772) (-0.588423) (-0.684445)
Trend 0.019785*** 0.02147*** 0.007575
(4.810428) (5.309977) (1.256227)
Trend Shifter 0.039525*** 0.047904*** 0.123275***
(3.892197) (4.812781) (4.524076)
R Squared 0.876775 0.873379 0.860313
Adj. R Squared 0.872976 0.869476 0.856006
34
salaries could have experienced a real decrease over the period while PG remained
constant, which would also diminish the value of the elasticity. It could have also been a
combination of the two factors. The important fact to remember is that this elasticity
represents the contributory power of a change in TPAY relative to the overall cost per
school, PG.
One of the additional objects of this paper was to examine the rate of returns to
scale in Alabama county school systems. The coefficient of ENROL, which represents
the production function output as it appears in the cost function, is not only significant at
the 1% level using all three methods of estimation, but it is very close to one. This
indicates that, in the short run, a 1% increase in enrollment per school creates almost a
perfect 1% increase in the expenditure per school, or constant returns to scale.
This result alludes to the notion that county school systems in Alabama are very close to,
if not already at, the point where the marginal cost of adding an additional student is
equal to the average cost of all students. This level is the point at which a cost minimizing
firm would optimally operate.
One of the more surprising results, again, was the lack of impact by the TS
variable on cost. Though the NCLB act in theory would force TS to have an effect on
cost, the data did not indicate such a reaction. However, the data did indicate a
statistically significant increasing cost trend at the 1% level with increase in the rate made
by the Trend Shifter variable after the impact of the NCLB act began to be felt. The
shifter was significant at the 1% level using all three techniques while the trend was
significant in only two.
35
Some of the more disappointing results were those coefficient estimates yielded
by the test score quality equilibrium equation in Table 7.0, though the testable ones were
Table 7.0 - Parameter estimates of the Test Score Equilibrium using logged data
Variable/ statistic OLS SUR 3SLS
Constant 0.224959 0.33324 0.389978
(0.256138) (0.386253) (0.417537)
RACE -0.024875*** -0.024914*** -0.024949***
(-5.126557) (-5.229089) (-5.223791)
INC 0.106527* 0.104445* 0.103285*
(1.922974) (1.920052) (1.874572)
UNEMP -0.003118 -0.004406 -0.006523
(-0.13734) (-0.197658) (-0.240902)
UNEMP Shifter -0.082255*** -0.082008*** -0.080098*
(-4.049024) (-4.109011) (-1.748937)
Lagged TS 0.674526*** 0.673741*** 0.672546***
(22.2129) (22.58616) (22.46186)
TSR -0.02558 -0.010205 0.001567
(-0.357657) (-0.145291) (0.021694)
TPAY 0.023691 0.011521 0.004683
(0.338705) (0.167649) (0.063752)
RURAL -0.004648 -0.00412 -0.003928
(-0.524294) (-0.473262) (-0.449358)
POV -0.022042 -0.023048 -0.023452
(-0.623785) (-0.664065) (-0.6534)
D1 -0.018507 -0.018328 -0.018255
(-1.059712) (-1.06759) (-1.062154)
D2 -0.019968* -0.020486** -0.020964**
(-1.927857) (-2.012276) (-2.043679)
D3 -0.019801 -0.020734 -0.021514
(-1.49291) (-1.590489) (-1.634407)
D4 -0.039862** -0.039978** -0.040246**
(-2.291951) (-2.338352) (-2.352571)
D6 -0.050851*** -0.051445*** -0.051971***
(-2.77489) (-2.85654) (-2.883987)
D7 -0.030859** -0.030679** -0.030439**
(-2.393758) (-2.421584) (-2.400721)
Trend -0.002761 -0.002778 -0.00265
(-0.867416) (-0.887753) (-0.686881)
Trend Shifter 0.013848*** 0.013754*** 0.013257
(2.585229) (2.613307) (1.128655)
R Squared 0.824931 0.824914 0.824877
Adj. R Squared 0.819186 0.819168 0.819129
36
generally consistent with the results found by Kinnucan et al in 2004. As was discovered
in ?State Aid and Student Performance, a Supply-Demand Analysis?, the Race variable
yielded a slight negative coefficient using all three methods of estimation, significant at
the 1% level. Additionally consistent was the estimation of a positive coefficient attached
to the income variable, significant at the 10% level using all three estimation methods.
This would confirm the recommendation made by the authors that one of the most
effective manners in which to boost test scores would be to stimulate county economies,
rather than to pour money directly into schools (Kinnucan et al). However, this could be
interpreted as a genetic effect; scilicet that the progeny of high income earners are likely
to succeed in the same manner as their parents.
With the exception of some varying statistically significant cross sectional results
yielded by the dummy variables, the final notable result of the test score quality
equilibrium equation was the elasticity of 0.67 attached to the lagged dependent variable,
which was found to be significant at the 1% level using all three methods of estimation.
This, again, indicates a high degree of memory in the system. While it may not be a result
that pleases policy makers, it would seem to indicate that test scores for different districts
have remaining relatively consistent over the 9 year period.
Table 8.0 marks the first of the three sets of tables in which the first difference of
the logged data was taken prior to estimation. The results in these three table serve as
confirmations of the robustness of some of the variables, while it casts doubt on others.
The first equation estimated was the production function.
37
This set of estimations yielded some consistent results as well as some
inconsistent results. The first results were the confirmation of the positive sign and 5% or
better levels of significance of the two primary production factors, NIE and TEACH.
Table 8.0 - Parameter estimates of the Production Function using differenced data
Variable/ statistic OLS SUR 3SLS
Constant -0.007582** -0.007462** -0.017591**
(-2.018254) (-2.019227) (-2.267886) NIE 0.025113** 0.032289*** -0.194723**
(2.051473) (2.688294) (-2.346521) NIE Shifter 0.01892 0.020543 0.510446***
(1.059963) (1.174948) (3.035636) TEACH 0.749805*** 0.745858*** 1.520644***
(16.94831) (17.17216) (12.09249)
TEACH Shifter -0.564828*** -0.561919*** -1.572283***
(-10.81307) (-10.95996) (-9.809108) TS -0.018171 -0.007596 -0.404835***
(-0.66676) (-0.283401) (-2.948979) POV 0.024034 0.022687 1.063175***
(0.517148) (0.497546) (5.246066) POV Shifter -0.141355** -0.143538*** -1.729573***
(-2.520932) (-2.612534) (-5.746083) Lagged ENROL -0.098256*** -0.082853*** -0.008176
(-3.098695) (-2.665935) (-0.130155) D1 -0.000595 -0.001011 -0.011982
(-0.054338) (-0.093909) (-0.574737) D2 0.007052 0.006812 0.001244
(1.145974) (1.125144) (0.105335) D3 0.007371 0.006973 0.007936
(1.053444) (1.012911) (0.598943) D4 -0.003772 -0.004021 -0.010817
(-0.343356) (-0.372084) (-0.498815) D6 -0.00748 -0.00694 -0.001132
(-0.775438) (-0.731253) (-0.062531) D7 -0.005402 -0.005269 0.005146
(-0.804274) (-0.797267) (0.406177)
R Squared 0.502207 0.500869 -1.492833
Adj. R Squared 0.486856 0.485477 -1.569705
Furthermore, all three methods of estimation confirmed the negative shift in the
TEACH variable following the passage and effect of the NCLB.
38
Perhaps one of the more puzzling results presented in Table 8.0 is the appearance
of a negatively signed coefficient for the lagged dependent variable at the 1% level of
significance in both the OLS and SUR estimations. Given the theory of a partial
adjustment model, the fact that the lagged dependent variable has a magnitude less than
one indicates that the system is gradually converging to an optimal level of enrollment
per school. However, the negative sign of the lagged dependent variable alludes to the
idea that the path of adjustment is an oscillatory one. So again, it indicates that the system
has a degree of memory (albeit a small one), but that the equation reacts in the opposite
fashion each following year. It should be noted that this change only occurred after the
first differencing of the data, which is an issue to be examined in further research.
One will most likely note the disappearance of the time trend variable in the
production function in the new data form. This is due to the fact that such a trend is
eliminated by taking the logged first difference, making the addition of such a variable
unnecessary.
Table 9.0 contains coefficient estimates generated by the estimation of the cost
function using data in which the first differences of the logged data were computed.
Though the sign of both the constant and the constant shift term remained the same (as
well as significant in all three methods of estimation), the magnitudes decreased in a
proportional manner. In all three instances, the NCLB effect appears to have been a
statistically significant increase in the level of funding per school.
The data also confirmed both the sign and apparent magnitude of the input factor
prices, with PNIE and TPAY containing positive signs and 1% level significance using
OLS, SUR, and 3SLS. Furthermore, the TPAY shifter coefficient again appears to be of a
39
large magnitude and significant at the 5% level or better. This confirms the finding in
table 5.0 that the NCLB act decreased the effect of price on per school spending.
Table 9.0 - Parameter estimates of the Cost Function using differenced data
Variable/ statistic OLS SUR 3SLS
Constant 0.020205* 0.023724** 0.026949**
(1.88483) (2.251476) (2.470371)
Constant Shifter 0.055099*** 0.053259*** 0.051258***
(4.516332) (4.456308) (4.170431)
PNIE 0.47438*** 0.459216*** 0.471803***
(11.94053) (11.77597) (12.01995)
TPAY 0.526221*** 0.621314*** 0.787063***
(3.145321) (3.789576) (3.875056)
TPAY Shifter -0.45642** -0.51865*** -0.78518***
(-2.504711) (-2.903745) (-3.459496)
ENROL 0.966104*** 1.092654*** 1.099593***
(10.97202) (12.65365) (12.13626)
TS 0.04744 0.097064 0.319545*
(0.692028) (1.440201) (1.882478)
POV 0.095899 0.113047* 0.11541
(1.373589) (1.646005) (1.601695)
Lagged PG -0.406113*** -0.403902*** -0.400615***
(-11.26669) (-11.43612) (-11.28775)
D1 0.009789 0.008912 0.006732
(0.347735) (0.32175) (0.237926)
D2 -0.005379 -0.007299 -0.008814
(-0.340341) (-0.46945) (-0.55384)
D3 0.00508 0.003166 0.001
(0.282585) (0.179009) (0.055201)
D4 0.031434 0.032563 0.032501
(1.115204) (1.174216) (1.149455)
D6 -0.020368 -0.019028 -0.019609
(-0.819558) (-0.778171) (-0.786498)
D7 -0.012251 -0.010687 -0.012169
(-0.709317) (-0.628913) (-0.700926)
R Squared 0.502207 0.502177 0.482914
Adj. R Squared 0.486856 0.486826 0.466969
40
Very reassuringly, the sign and magnitude of the production function output
variable ENROL retained both its magnitude of near 1.0 and 1% level of significance.
This further confirms the conclusion that Alabama county school systems are operating at
levels of constant returns to scale, which can have interesting policy implications for the
state. It is interesting to note that the magnitude of the returns to scale coefficient appears
to have increased in all three estimation techniques.
As with the production function, the lagged dependent variable yielded a
negatively signed coefficient of economic and statistical significance. This again points to
an oscillating convergence path per the partial adjustment model framework, which is not
an intuitive result.
The final results table is table 10.0, which provides the coefficient estimates for
the test score quality equilibrium using logged, first-differenced data with the three
aforementioned regression techniques. These results tended to be the most inconsistent
of the three equations, with the RACE and INC variables no longer resulting in
significant coefficient estimates. Though the idea of TPAY having a positive effect on
test score quality is a positive assumption to make (Hanushek 1986), the fact that it was
not significant in the previous set of regressions using the simply logged data makes it a
questionable statistic to give credence to. While TPAY might have an intuitive
coefficient result, the estimation of the teacher student ratio (TSR) coefficient defies
logic. The notion that a smaller teacher student ratio negatively affects test scores is a
very difficult one to logically accept. Therefore, unlike the results in table 6.0 which
seemed to confirm several of the findings from Kinnucan et al?s work in 2002, this table
seems to be victim to an unknown statistical error.
41
Table 10.0 - Parameter estimates of the Test Score Equilibrium using differenced data
Variable/
statistic OLS SUR 3SLS
Constant -0.000413 -0.000235 0.006572
(-0.045186) (-0.026189) (0.736212) RACE -0.084676 -0.081587 -0.063179
(-1.365674) (-1.340353) (-1.082016) INC 0.146085 0.158639 0.117401
(1.290101) (1.427038) (1.073149) UNEMP -0.013997 -0.012814 -0.077302*
(-0.359768) (-0.335423) (-1.663357) UNEMP Shifter 0.110289** 0.110017** 0.228199***
(2.051874) (2.084599) (3.173266) Lagged TS -0.323116*** -0.324274*** -0.301179***
(-7.754792) (-7.926923) (-7.418173) TSR -0.188885* -0.179938* -0.352032***
(-1.867357) (-1.81203) (-3.683482) TPAY 0.250693*** 0.249044*** 0.370516***
(2.579999) (2.610454) (4.034914) RURAL -0.013581* -0.013638* -0.017031**
(-1.676008) (-1.71447) (-2.236762) POV -0.029309 -0.02757 -0.012221
(-0.639902) (-0.612545) (-0.268038) D1 0.010609 0.010612 0.012699
(0.595058) (0.605655) (0.722936) D2 0.004007 0.003968 0.003138
(0.398413) (0.401417) (0.316976) D3 0.014052 0.014 0.014243
(1.237076) (1.254044) (1.274292) D4 -0.000253 -0.000393 -0.000357
(-0.014211) (-0.022453) (-0.020319) D6 0.00666 0.006556 0.005372
(0.413484) (0.414163) (0.339592) D7 0.006108 0.006072 0.005392
(0.556872) (0.563244) (0.499579)
R Squared 0.1793 0.1792 0.165892
Adj. R Squared 0.152124 0.152021 0.138273
The final table in the results section, table 11.0, presents the results of the two
tests mentioned in the data and the models sections regarding long run returns to scale
and the relationship of the coefficient values in the cost function. The theory behind these
tests was also addressed in the derivations presented in appendix 1. The first restriction
tested was the one regarding the long run returns to scale being equivalent to 1. This
restriction was rejected at the 1% level. The second restriction tested was the hypothesis
formulated in appendix 1 which stated that the coefficient of the production output
42
variable as it appears in the cost function will be equal and opposite in sign to the sum of
the coefficients of the technology shift parameters in the cost function.
Table 11.0 - Restriction testing
Restriction Test Stastistic Value DoF Probability
Long Run
RTS Chi-square 9.322459 1 0.0023
Coef Q = - ?
Coef A(t) Chi-square 483.415 1 0
This restriction was also rejected at the 1% level, yielding inconclusive results.
43
V. CONCLUSION
This paper began with three primary objectives that were set forth in the
introduction section. The first objective was to develop a system of equations examining
important components of the Alabama county school system, and to produce robust
coefficient estimates for several of these determinants. The second objective was to
determine whether the No Child Left Behind Act of 2001 resulted in a statistically and
economically significant impact on these coefficient estimates over the nine year period
using the gradual switching regressions technique. The final objective was to analyze the
rate of returns to scale of Alabama schools, and to attempt to determine the rate at which
the systems were currently operating. The objectives have all been partially or wholly
achieved, utilizing a technique heretofore unused by the educational economics
community.
For the first objective, the system of equations was specified using rigorous
economic theory, and produced a set of mixed results regarding the significance of the
estimated determinants. While the primary factors and prices of the production and cost
equations were of the proper sign, magnitude, and significance that intuition would
suggest, the equations seemed to suffer slightly from specification errors relating to the
use of the three stages least squares technique. However, the production factors and
prices did prove to be fairly robust irregardless of the data manipulation conducted in
44
taking the first difference of the set. The test score quality equilibrium proved to have less
explanatory power than the other two equations in the system, yet the significant
coefficient estimates produced matched the results obtained by Kinnucan et al in 2004.
This would seem to indicate, as Kinnucan et al noted, that one manner in which to
approach the issue of test score quality might be to stimulate income growth and poverty
reduction within the counties, letting the indirect effects filter into the schools. However,
it was disappointing to note that the TS variable seemed to have little if any effect on
enrollment per school and expenditure per school, indicating that schools may not be as
reactive toward test score quality as is commonly assumed.
The second objective was met extremely well, with the No Child Left Behind Act
creating structural impact in all three equations and an overall impact in the cost
equation. The shift of the constant term in the cost function illuminated by the gradual
switching regressions seems to indicate an increased level of expenditure per school due
to the impact of the No Child Left behind Act. This result, though counterintuitive, would
point to inadvertent effects of the NCLB act, providing more evidence for Hanushek?s
2003 piece ?The Failure of Input-Based School Policies.? However, the Wald tests noted
in the data and models section unequivocally indicated the NCLB act had statistically
significant effects within the educational system. It merely appears that the effects were
absorbed primary through structural shifts rather than overall shifts. The analysis of the
No Child Left Behind Act yields some interesting policy implications. As previously
mentioned, it highlights the inefficient manner in which the resources of this act were
absorbed by the system without producing desired effects. However, this also could be a
45
result created by manipulation of this testing system and its results as noted in the
literature review section (Haney).
The final objective of examining the level of constant returns to scale of the
Alabama county school system yielded the finding that the systems are presently
operating at constant returns to scale. The coefficient of production function output as it
appeared in the cost function maintained a value close to 1 and significant at the 5% level
or greater in all six regressions. This statistic has apparently not been evaluated in
educational economic literature regarding Alabama schools systems, and provides an
interesting basis for policy making regarding the creation of new facilities.
Opportunities for further study regarding this topic abound, including the
investigation as to the cause of the consistently significant and negative coefficient of the
lagged dependent variables using logged first differenced data. The application of this
same system to another state would be an excellent test for the robustness of the
coefficient estimates, as would a comparison of another state?s results contrasted with
Alabama?s. Finally, it would be beneficial to test not only whether the optimal functional
form had been selected for the time path vectors, but the optimal rate as well.
46
BIBLIOGRAPHY
Alabama State Department of Education. http://www.alsde.edu/html/home.asp. 2009.
Accessed on July 5th, 2009.
Bacon, D.W. and D. G. Watts. ?Estimating the Transition Between Two Intersecting
Straight Lines.? Biometrika. 58 (1971) 525 ? 534.
Haney, W. ?Response to Skrla et al. The Illusion of Educational Equity in Texas: a
Commentary on ?Accountability for Equity??. International Journal of Leadership
in Education. 4, 3 (2001): 267-275.
Hanushek, E. A. ?The Economics of Schooling, Production, and Efficiency in Public
Schools.? The Journal of Economic Literature. 24, 3 (1986): 1141-1177.
Hanushek, E. A. ?The Failure of Input-Based Schooling Policies.? The Economic
Journal. 113, 485 (2003): F64 ? F98.
Hanushek, E. A. ?What If There Are No ?Best Practices??? Scottish Journal of Political
Economy, 51 (2004) 156-172.
Kim, J. S. and G. L. Sunderman. ?Measuring Academic Proficiency Under the No Child
Left Behind Act: Implications for Educational Equity.? Educational Researcher.
34,8 (2005): 3-13.
King, R. G. and J. K. Thomas. ?Partial Adjustment Without Apology?. International
Economic Review. 47, 3 (2006): 779 - 809
Kinnucan, H. W, Y. Zheng, and G. Bremer.?State Aid and Student Performance: A
Supply-Demand Analysis. Education Economics. Vol 14 No. 4 (2006) 487-509.
Linn, R. L., E. L. Baker, and D. W. Betebenner. ?Accountability Systems: Implications
of Requirements of the No Child Left Behind Act of 2001.? Educational
Research. 31, 6 (2002): 3-16.
Moschini, G. and K. D. Meilke. ?Modeling the Pattern of Structural Change in U.S. Meat
Demand.? American Journal of Agricultural Economics. 71, 2 (1989): 253 ? 261.
47
National Center for Education Statistics. ?Common Core of Data?.
http://nces.ed.gov/ccd/bat/. 2009. Accessed July 5th, 2009.
Ohtani, K., S. Kakimoto and K. Abe. ?A Gradual Switching Model with a Flexible
Transition Path.? Economic Letters, 32 (1990): 43 - 48
Ohtani, K., and S. Katayama. ?A Gradual Switching Regression Model with
Autocorrelated Errors.? Econonmic Letters. 21 (1986): 169-72
Rivkin, S. G., E. A. Hanushek, and J. F. Kain. ?Teachers, Schools, and Academic
Achievement.? Econometrica. 73, 2 (2005): 417-458.
Solow, R. M. ?A Contribution to the Theory of Economic Growth.? The Quarterly
Journal of Economics. 70, 1. (1956): 65-94.
Tsurumi, H. ?A Bayesian Estimation of Structural Shift by Gradual Switching
Regressions with an Application to the US Gasoline Market?. A. Zellner (ed.).
Bayesian Analysis in Econometrics and Statistics: In Honor of Harold Jeffreys.
North-Holland, Amsterdam. 1983.
United States Census Bureau. ?Small Area Income and Poverty Estimates.?
http://www.census.gov//did/www/saipe/. 2009. Accessed July 5th, 2009.
United States Department of Education. ?Executive Summary: No Child Left Behind
Act? http://www.ed.gov/nclb/overview/intro/execsumm.html. 2002. Accessed on
July 5, 2009.
United States Department of Labor: Bureau of Labor Statistics. ?Consumer Price Index?.
http://www.bls.gov/cpi/. 2009. Accessed July 5th, 2009.
Untied States Department of Labor: Bureau of Labor Statistics. ?Local Area
Unemployment Statistics?. http://www.bls.gov/lau/. 2009. Accessed July 5th,
2009.
University of Alabama Center for Business and Economic Research. ?Alabama Maps?.
http://cber.cba.ua.edu/edata/maps/AlabamaMaps1.html Accessed July 5th, 2009.
48
APPENDICES
49
APPENDIX A
Data Definitions and Manipulations
50
Before starting, a few words to the wise should be mentioned about using the
National Center for Education Statistics (NCES) Common Core of Data (CCD) Table
Builder function. 1) One must remember that the definition of ?district? as referred to by
the NCES is equivalent to this paper?s use of the word ?system?. This paper?s use of the
word ?district? refers to the ALSDE designation of 7 districts grouping counties within
the State of Alabama. Thus, whenever selecting the ?Row? designator using Table
Builder, always select ?district?, NOT ?county?. Selecting ?county? will provide
statistics relevant to the entire county, which lumps together county systems as well as
city systems within the county as a whole. This will provide erroneous results, when
compared to the ALSDE annual reports? references to county systems. 2) A small, but
important point is the fact that this paper lists ?Saint Clair? county alphabetically before
?Shelby? county. However, many other sources, such as the Department of Examiners of
Public Accounts and the NCES list ?Shelby? county alphabetically before ?St. Clair?
county. This should always be checked before adding values to the data set, or the two
values will often be juxtaposed.
Q8R, Q8M, Q8L
These variables are Stanford Achievement Test Scores by Alabama County for
the 8th grade. Each subject is annotated by R for reading, M for Math, and L for language.
It should be noted that test scores prior to 2003 were in the Stanford 9 format as opposed
to the Stanford 10 currently used format. They can be converted using the ?Percentile
Rank Conversion Tables? provided by Harcourt Assessment. The data were obtained via
the ALSDE accountability reporting system, at
http://www.alsde.edu/Accountability/preAccountability.asp. The file containing the
51
scores is the ?YEAR Stanford 10 test results complete (zip file)?. The scores used were
the ones for ?all students? and ?entire system?. In addition, at the time of gathering, the
2007 zip file was not available, so it should be noted that these scores are also available
through the ?YEAR Chief State School Officer?s Report For Stanford 10? where it may
be copied value by value.
Q8 or TS
These variables are the simple average of each grades score in the three selected
subjects.
Q8-1 or TS-1
These variables are the once lagged values of Q3, Q4, and Q8.
SF, PN, FF, OF
These variables were gathered from the ALSDE Annual Reports, on the page
titled ?Local Education Agencies (LEA)?. The annual reports are found at
http://www.alsde.edu/html/annual_reports.asp?menu=none&footer=general . The top of
each column is titled ?State Revenue?, ?Local Revenue?, ?Federal Revenue?, and ?Other
Revenue? respectively. The data were then converted to a per school value by dividing
each figure by its county?s number of schools, which is found in the NCES CCD. The
numbers are then deflated, using 1982 ? 1984 as a base value of 100, and then the
respective year?s CPI, found at ftp://ftp.bls.gov/pub/special.requests/cpi/cpiai.txt . An
example of the location of the values would be page 38 of the 2004 annual report for
Revenues.
52
PG
This variable is simply the sum of the per school, deflated values of SF, PN, FF,
and OF.
PG-1
This variable is the once lagged value of the PG variable.
NIE
This variable was constructed by dividing the sum of the ?Instructional Support?
and ?Instructional Services? Expenditure figures by the ?Total Expenditure? figure per
county ?LEA?. The three figures needed to compute this percentage are found in each
year?s annual report, located at
http://www.alsde.edu/html/annual_reports.asp?menu=none&footer=general . The
Instructional Support and Instructional Services figures in the 2004 report can be found
on page 44, while Total Expenditures can be found on page 45. The pages are titled
?System Expenditures by Function FY 2004?. It was then converted to non instructional
expenditures by subtracting the aforementioned percentage from one. Finally, this
percentage was multiplied by total expenditures per school to create a value for dollars of
non instructional expenditure dollars per school.
PNIE
The percentage of non instructional expenditures by a system, created by dividing
the sum of all expenditures excluding instructional support and instructional expenditure
by total expenditure. These values are found in the ALSDE annual reports at
http://www.alsde.edu/html/annual_reports.asp?menu=none&footer=general
TSR
53
The number of pupils per teacher (Teacher-Student Ratio) per county can be
found at the National Center for Education Statistics, using the Common Core of Data
Facility. This webpage is located at ?http://nces.ed.gov/ccd/?. At this webpage, one can
use the ?build a table? function listed under ?CCD Data Tools? to build a table listing the
?Pupil/Teacher Ratio (School)? per district per year per state. The number reported by the
table is the number of pupils per teacher per system. It is very important to note that
when using the build a table function, the first column of the table must be selected as
?district? and NOT ?county?. Using county will account for the TSR in the county, and
NOT the TSR in the county system. To get the TSR in the county system, all districts
must be pulled up.
ENROL
These variables denote Average Daily Membership (ADM) and the square root of
the ADM respectively. These numbers are listed in the ALSDE Annual Reports, found at
http://www.alsde.edu/html/annual_reports.asp?menu=none&footer=general , and
recorded as totals per county. The statistics for the 2004 ADM?s could be found on page
21 of the 2004 Annual Report, under the column ?TOTAL? on the page titled ?Average
Daily Membership (ADM)?. This was then converted to average ADM per school by
dividing enrollment by the number of schools in the county system.
ENROL-1
This variable is the once lagged value of Enrol.
POV and INC
Found using census data?s SAIPE function at :
http://www.census.gov/hhes/www/saipe/tables.html . One can use the SAIPE table
54
creation function to create an excel table by county by year. POV was defined as the
percentage of children ages 5-17 per county in families in poverty relative to the counties
children of the same age. INC was the SAIPE?s estimate for Median Family Income per
county per year. The INC numbers were deflated in the same manner as the funding
variables PG, SF, PN, FF, and OF , using the CPI base of ?82 ? ?84 as 100 and then
corresponding BLS CPI figures for each year located on the BLS website.
UNEMP
The yearly, county level, annual averages for unemployment were obtained at the
BLS website: http://www.bls.gov/lau/#tables . From this website, one can scroll down to
the county level data section and open text tables for each year?s unemployment statistics.
RACE
The RACE variable is a percentage defining the percentage of non-white students
per county system per year. These variables are found in each year?s respective ALSDE
Annual Report, found at
http://www.alsde.edu/html/annual_reports.asp?menu=none&footer=general. However,
demographic information ceased to be reported as of the 2006 annual report. Thus, the
observations for 2006 and 2007 were generated using a diminishing weighted average.
The immediate prior year carries a weight of .5, the second previous year?s weight is .33,
and the third previous year?s weight is .17. It can be computed also as (3*Rt-1 + 2*Rt-2 +
1*Rt-3)/6 = Rt .
TPAY
The TPAY variable is the deflated average salary per full time equivalent teacher
in each system per year. The variable is first created using the NCES?s Common Core of
55
Data located at http://nces.ed.gov/ccd/ . The two statistics needed to create the undeflated
TPAY numbers are ?FTE Teachers (District)?, located under the ?Teacher/ Staff
Information? section as well as ?Salary-Instruction Expenditures (District ? Fin)? located
under the ?Current Expenditure Details? section. Remember, these numbers must be
computed per District (under the first Rows selection choice) and NOT per county. The
numbers were deflated in the same manner, using the same factors as the funding
numbers. Finally, data was not available at the time for the 2007 ?Salary-Instruction
Expenditures?, so observations were generated using a weighted average of the previous
three years in the same manner as the missing yearly RACE variables.
Rural
This is a dummy variable used to denote rural counties in the state of Alabama,
found at http://cber.cba.ua.edu/edata/maps/AlabamaMaps1.html , the University of
Alabama?s Center for Business and Economic Research.
D1 ? D7
This collection of 6 dummy variables is used to denote which ALSDE school
district each system belongs to, using District 5 as a default base district. The map
denoting school districts can be found on the ALSDE website, at
http://www.alsde.edu/html/school_info.asp?menu=school_info&footer=general&sort=co
unty.
TEACH
This value represents the number of teachers per school within a given county
system, as provided by the NCES?s Common Core of Data located at
http://nces.ed.gov/ccd/ . However, the CCD only provided the number of FTE teachers
56
per system, so in order to create an average number of teachers per school, the number of
teachers was divided by the number of schools in the system.
l1 ? l28
These dummy variables were created to test different rates of linear change using
the gradual switching regressions technique. The GSR shift variables takes one of three
sets of values depending on the current time period relative to pre set values of t1 and t2.
If current t is less than or equal to t1, the shift variable is 0. If the current t is greater than
or equal to t2, the shift variable is 1. If the current t is between t1 and t2, then the shift
variable is equal to ?(current t ? t1)/(t2-t1)?. T1 represents when a policy first began
taking effect and t2 represents when it finishes taking effect. The versions of the shift
variable tested represent every possible combination of t1 and t2 beginning with policy
implementation in 2001 and every value up to it taking full effect by 2008.
nlv1 ? nlv28
These dummy variables are non linear convex gradual switching regression sets,
created by taking the values of l1-l28 and squaring them. This creating dummies that
increased at an increasing rate.
nlc1 ? nlc28
These dummy variables are non linear concave gradual switching regression sets,
created by taking the values of l1-l28 and taking the square root of them. This created
dummies that increased at a decreasing rate.
t
This variable is simply a time trend variable starting at 1 and increasing by 1 each
year.
57
APPENDIX B
Cost Function Derivation
58
The purpose of appendix B is to describe the derivation of Robert Solow?s 1956
growth model into a cost function using standard economic theory. A Cobb-Douglas
version of Solow?s growth model (including technology shift parameter) results in
equation (1). This equation is examined in conjunction with a standard cost function (cost
being equal to sum of the input factors multiplied by their respective prices) in equation
(2), and the restriction found in equation (3).
(1) Y = K?L?A(t)
(2) C=rK + wL
(3) minimize: rK + wL, subject to Y - K?L?A(t)=0
Y represents output, K represents a capital factor input, L represents a labor factor input.
C is the total cost, with r representing the factor price of K and w the factor price of L
(rental rate of capital and wages, respectively, for the purpose of this appendix). A(t) is an
unknown function of technology shifting the production isoquant.
Using the dual nature of the production function, the cost function (2) can be
examined subject to the restriction (3). This yields the Lagrangian in equation (4).
(4) L (K,L,?) = rK + wL + ?(Y - K?L?A(t))
Taking the derivative of this Lagrangian with respect to the choice variables yields
equation (5), (6), and (7).
(5) ?L /?K = r - ??AK?-1L? = 0
(6) ?L /?L = w - ??AK?L?-1 = 0
(7) ?L /?? = Y - K?L?A(t) = 0
Dividing equation (5) by equation (6) yields ratio of factor prices, (8).
(8) r/w = ?L/?K
59
Solving (8) for L and substituting into the constraint (7) yields (9), allowing K to be
solved for in terms of parameters and output. The symmetric nature of the function
allows for the same to be done for L, yielding equation (10). The substitution of both (9)
and (10) into (2) yields (11), or the minimized cost using optimal demand for L and K.
(9) K = (Y/A)1/(?+?)*(?w/?r)?/(?+?)
(10) L = (Y/A)1/(?+?)*( ?r/?w) ? /(?+?)
(11) C = r[(Y/A)1/(?+?)*(?w/?r)?/(?+?)] + w[(Y/A)1/(?+?)*( ?r/?w) ? /(?+?)]
With a bit of factoring, (11) can be reduced to (12), with z defined by (13)
(12) C=z (Y/A)1/(?+?)[r(w/r) ?/(?+?) + w(r/w) ? /(?+?)]
(13) z = (?/?) ?/(?+?) + (?/?) ? /(?+?))
Further factoring and algebraic manipulation of equation (12) yields equation (14)
(14) C = z * A-1/(?+?) * Y1/(?+?) * w ?/(?+?) * r ? /(?+?)
Taking the natural logarithm of equation (14) yields equation (15), which is the linear
cost equation used in this paper. It can be further simplified into the more aesthetically
appealing equation (16).
(15) ln(C) = ln(z) ? (1/(?+?))ln(A) + (1/(?+?))ln(Y) + (?/(?+?))ln(w) + (?
/(?+?))ln(r)
(16) ln(C) = ?0 - ?1ln(A) + ?1ln(Y) + ?2ln(w) + ?3ln(r)
Substituting the generic variables from Solow?s growth function with variables used in
this paper and adding the lagged dependent variable to maintain the partial adjustment
model framework, would result in equation (20):
(20) ln(PG) = ?0 - ?1ln(POV, TS) + ?1ln(Enrol) + ?2ln(TPAY) + ?3ln(PNIE) +
?4ln(PG-1)
60
Equation (20) yields some interesting and testable results in addition to being the cost
function used in this paper. The first testable result is that the coefficient of ln(POV, TS)
should be equal and opposite in sign to the coefficient of ln(Enrol), if the firm in question
is indeed a cost minimizing firm. The second interesting result is that of the coefficient of
ln(Enrol), which in this case represents the short run returns to scale of the respective
Alabama county school system using logged data. In theory, the long run returns to scale
should be constant. The test of this theory would be represented by equation (21)
(21) 1.0 = ?1/(1- ?4)
These tests are evaluated using Wald coefficient tests in the results section of this paper
61
APPENDIX C
Time Path Vector Values
62
This appendix lists the 84 different time path vector values of ?t, as defined by
equation (4) and (5) in the literature review and used to test for the optimal fit. The
starting point of the effect is indicated by the value of t1 and the point at which the full
effect has taken place is indicated by t2. All possible linear, concave, and convex time
paths were tested beginning with t1 = 3, indicating that the No Child Left Behind Act first
began affecting the educational system in school year 2001, which was the year of the
bill?s passage. The current period is indicated by t, which begins at 1 with the year 1999.
Though the ?t values have been reduced to three decimal places for the sake of fitting into
this page, the actual values used were carried to six decimal places. In addition, the labels
for ?t, lnumber, nlvnumber, and nlcnumber, represent linear time path, non-linear convex time
path, and non-linear concave time path respectively.
Table
12.0
?t t1 t2 99 00 01 02 03 04 05 06 07
l1 3 4 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 1.000
l2 3 5 0.000 0.000 0.000 0.500 1.000 1.000 1.000 1.000 1.000
l3 3 6 0.000 0.000 0.000 0.333 0.667 1.000 1.000 1.000 1.000
l4 3 7 0.000 0.000 0.000 0.250 0.500 0.750 1.000 1.000 1.000
l5 3 8 0.000 0.000 0.000 0.200 0.400 0.600 0.800 1.000 1.000
l6 3 9 0.000 0.000 0.000 0.167 0.333 0.500 0.667 0.833 1.000
l7 3 10 0.000 0.000 0.000 0.143 0.286 0.429 0.571 0.714 0.857
l8 4 5 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000
l9 4 6 0.000 0.000 0.000 0.000 0.500 1.000 1.000 1.000 1.000
l10 4 7 0.000 0.000 0.000 0.000 0.333 0.667 1.000 1.000 1.000
l11 4 8 0.000 0.000 0.000 0.000 0.250 0.500 0.750 1.000 1.000
l12 4 9 0.000 0.000 0.000 0.000 0.200 0.400 0.600 0.800 1.000
l13 4 10 0.000 0.000 0.000 0.000 0.167 0.333 0.500 0.667 0.833
l14 5 6 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000
l15 5 7 0.000 0.000 0.000 0.000 0.000 0.500 1.000 1.000 1.000
l16 5 8 0.000 0.000 0.000 0.000 0.000 0.333 0.667 1.000 1.000
l17 5 9 0.000 0.000 0.000 0.000 0.000 0.250 0.500 0.750 1.000
l18 5 10 0.000 0.000 0.000 0.000 0.000 0.200 0.400 0.600 0.800
l19 6 7 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000
63
l20 6 8 0.000 0.000 0.000 0.000 0.000 0.000 0.500 1.000 1.000
l21 6 9 0.000 0.000 0.000 0.000 0.000 0.000 0.333 0.667 1.000
l22 6 10 0.000 0.000 0.000 0.000 0.000 0.000 0.250 0.500 0.750
l23 7 8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000
l24 7 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.500 1.000
l25 7 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.333 0.667
l26 8 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000
l27 8 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.500
l28 9 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
nlv1 3 4 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 1.000
nlv2 3 5 0.000 0.000 0.000 0.250 1.000 1.000 1.000 1.000 1.000
nlv3 3 6 0.000 0.000 0.000 0.111 0.444 1.000 1.000 1.000 1.000
nlv4 3 7 0.000 0.000 0.000 0.063 0.250 0.563 1.000 1.000 1.000
nlv5 3 8 0.000 0.000 0.000 0.040 0.160 0.360 0.640 1.000 1.000
nlv6 3 9 0.000 0.000 0.000 0.028 0.111 0.250 0.444 0.694 1.000
nlv7 3 10 0.000 0.000 0.000 0.020 0.082 0.184 0.327 0.510 0.735
nlv8 4 5 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000
nlv9 4 6 0.000 0.000 0.000 0.000 0.250 1.000 1.000 1.000 1.000
nlv10 4 7 0.000 0.000 0.000 0.000 0.111 0.444 1.000 1.000 1.000
nlv11 4 8 0.000 0.000 0.000 0.000 0.063 0.250 0.563 1.000 1.000
nlv12 4 9 0.000 0.000 0.000 0.000 0.040 0.160 0.360 0.640 1.000
nlv13 4 10 0.000 0.000 0.000 0.000 0.028 0.111 0.250 0.444 0.694
nlv14 5 6 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000
nlv15 5 7 0.000 0.000 0.000 0.000 0.000 0.250 1.000 1.000 1.000
nlv16 5 8 0.000 0.000 0.000 0.000 0.000 0.111 0.444 1.000 1.000
nlv17 5 9 0.000 0.000 0.000 0.000 0.000 0.063 0.250 0.563 1.000
nlv18 5 10 0.000 0.000 0.000 0.000 0.000 0.040 0.160 0.360 0.640
nlv19 6 7 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000
nlv20 6 8 0.000 0.000 0.000 0.000 0.000 0.000 0.250 1.000 1.000
nlv21 6 9 0.000 0.000 0.000 0.000 0.000 0.000 0.111 0.444 1.000
nlv22 6 10 0.000 0.000 0.000 0.000 0.000 0.000 0.063 0.250 0.563
nlv23 7 8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000
nlv24 7 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.250 1.000
nlv25 7 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.111 0.444
nlv26 8 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000
nlv27 8 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.250
nlv28 9 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
nlc1 3 4 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 1.000
nlc2 3 5 0.000 0.000 0.000 0.707 1.000 1.000 1.000 1.000 1.000
nlc3 3 6 0.000 0.000 0.000 0.577 0.816 1.000 1.000 1.000 1.000
nlc4 3 7 0.000 0.000 0.000 0.500 0.707 0.866 1.000 1.000 1.000
nlc5 3 8 0.000 0.000 0.000 0.447 0.632 0.775 0.894 1.000 1.000
64
nlc6 3 9 0.000 0.000 0.000 0.408 0.577 0.707 0.816 0.913 1.000
nlc7 3 10 0.000 0.000 0.000 0.378 0.535 0.655 0.756 0.845 0.926
nlc8 4 5 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000
nlc9 4 6 0.000 0.000 0.000 0.000 0.707 1.000 1.000 1.000 1.000
nlc10 4 7 0.000 0.000 0.000 0.000 0.577 0.816 1.000 1.000 1.000
nlc11 4 8 0.000 0.000 0.000 0.000 0.500 0.707 0.866 1.000 1.000
nlc12 4 9 0.000 0.000 0.000 0.000 0.447 0.632 0.775 0.894 1.000
nlc13 4 10 0.000 0.000 0.000 0.000 0.408 0.577 0.707 0.816 0.913
nlc14 5 6 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000
nlc15 5 7 0.000 0.000 0.000 0.000 0.000 0.707 1.000 1.000 1.000
nlc16 5 8 0.000 0.000 0.000 0.000 0.000 0.577 0.816 1.000 1.000
nlc17 5 9 0.000 0.000 0.000 0.000 0.000 0.500 0.707 0.866 1.000
nlc18 5 10 0.000 0.000 0.000 0.000 0.000 0.447 0.632 0.775 0.894
nlc19 6 7 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000
nlc20 6 8 0.000 0.000 0.000 0.000 0.000 0.000 0.707 1.000 1.000
nlc21 6 9 0.000 0.000 0.000 0.000 0.000 0.000 0.577 0.816 1.000
nlc22 6 10 0.000 0.000 0.000 0.000 0.000 0.000 0.500 0.707 0.866
nlc23 7 8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 1.000
nlc24 7 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.707 1.000
nlc25 7 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.577 0.816
nlc26 8 9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000
nlc27 8 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.707
nlc28 9 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
65
APPENDIX D
Time Path Vector Rankings
66
The following are the results produced by testing the 84 possible linear, convex,
and concave time path vector values of ?t. The first column represents the time path
vector tested (the value of which can be found in Table 10 of appendix 2), then the
corresponding R2 value for each of the three functions followed by its overall ranking.
The value of ?NSM? for the R2 represents a near-singular matrix error in Eviews 6.0
which prevented the particular regression from being estimated.
Table 13.0
?t Production R2 Rank Cost R2 Rank TS R2 Rank
l1 0.965603 60 0.872865 73 0.824684 31
l2 0.966232 47 0.874822 65 0.825574 14
l3 0.966607 35 0.876366 24 0.823752 52
l4 0.967596 18 0.87646 23 0.824193 46
l5 0.967035 26 0.876172 33 0.824283 40
l6 0.965051 65 0.876358 25 0.823643 55
l7 0.965051 66 0.876358 26 0.823643 56
l8 0.965877 55 0.874994 59 0.825571 15
l9 0.966445 41 0.875754 46 0.82289 67
l10 0.967917 11 0.876594 22 0.823101 66
l11 0.967392 21 0.875988 37 0.824404 38
l12 0.965552 63 0.876119 34 0.824628 35
l13 0.965552 64 0.876119 35 0.824628 36
l14 0.966587 36 0.877856 4 0.822239 73
l15 0.968328 8 0.877764 9 0.825191 25
l16 0.967873 13 0.87608 36 0.8238 51
l17 0.966676 31 0.875832 40 0.823644 53
l18 0.966676 32 0.875832 41 0.823644 54
l19 0.968485 3 0.87792 1 0.82641 1
l20 0.967906 12 0.87564 49 0.82541 22
l21 0.967684 14 0.877252 13 0.824252 41
l22 0.967684 15 0.877252 14 0.824252 42
l23 0.96383 67 0.874424 69 0.825798 5
l24 0.963749 74 0.874977 62 0.825403 23
l25 0.963749 75 0.874977 63 0.825403 24
l26 NSM
NSM
NSM
l27 NSM
NSM
NSM
l28 NSM
NSM
NSM
nlv1 0.965603 61 0.872865 74 0.824684 32
67
nlv2 0.966083 49 0.875115 56 0.825797 8
nlv3 0.96661 33 0.876797 19 0.822824 70
nlv4 0.968349 7 0.877162 15 0.824242 43
nlv5 0.967517 19 0.875813 45 0.824148 47
nlv6 0.965968 53 0.875827 42 0.823615 60
nlv7 0.965968 54 0.875827 43 0.823615 61
nlv8 0.965877 56 0.874994 60 0.825571 16
nlv9 0.966579 39 0.877036 16 0.822292 72
nlv10 0.968576 2 0.877479 12 0.824722 28
nlv11 0.967655 16 0.875532 52 0.824145 48
nlv12 0.96686 27 0.875644 47 0.823327 64
nlv13 0.96686 28 0.875644 48 0.823327 65
nlv14 0.966587 37 0.877856 5 0.822239 74
nlv15 0.968715 1 0.877828 7 0.826325 4
nlv16 0.967413 20 0.875096 57 0.824931 27
nlv17 0.96737 22 0.875896 38 0.823459 62
nlv18 0.96737 23 0.875896 39 0.823459 63
nlv19 0.968485 4 0.87792 2 0.82641 2
nlv20 0.966028 50 0.874534 68 0.825427 20
nlv21 0.966255 42 0.876271 29 0.823636 57
nlv22 0.966255 43 0.876271 30 0.823636 58
nlv23 0.96383 68 0.874424 70 0.825798 6
nlv24 0.963774 70 0.875297 53 0.824717 29
nlv25 0.963774 71 0.875297 54 0.824717 30
nlv26 NSM
NSM
NSM
nlv27 NSM
NSM
NSM
nlv28 NSM
NSM
NSM
nlc1 0.965603 62 0.872865 75 0.824684 33
nlc2 0.96616 48 0.874108 72 0.825066 26
nlc3 0.966461 40 0.875056 58 0.824402 39
nlc4 0.966843 29 0.875217 55 0.824675 34
nlc5 0.966608 34 0.875557 51 0.824513 37
nlc6 0.966009 51 0.876224 31 0.824226 44
nlc7 0.966009 52 0.876224 32 0.824226 45
nlc8 0.965877 57 0.874994 61 0.825571 17
nlc9 0.966236 46 0.874851 64 0.823952 49
nlc10 0.967077 25 0.875623 50 0.823628 59
nlc11 0.96675 30 0.875817 44 0.825424 21
nlc12 0.96562 58 0.876285 27 0.825579 12
nlc13 0.96562 59 0.876285 28 0.825579 13
nlc14 0.966587 38 0.877856 6 0.822239 75
nlc15 0.967607 17 0.877776 8 0.823841 50
68
nlc16 0.967303 24 0.876957 17 0.82277 71
nlc17 0.96625 44 0.876759 20 0.822864 68
nlc18 0.96625 45 0.876759 21 0.822864 69
nlc19 0.968485 5 0.87792 3 0.82641 3
nlc20 0.968482 6 0.876869 18 0.825765 9
nlc21 0.968075 9 0.877741 10 0.825431 18
nlc22 0.968075 10 0.877741 11 0.825431 19
nlc23 0.96383 69 0.874424 71 0.825798 7
nlc24 0.963774 72 0.874728 66 0.825708 10
nlc25 0.963774 73 0.874728 67 0.825708 11
nlc26 NSM
NSM
NSM
nlc27 NSM
NSM
NSM
nlc28 NSM
NSM
NSM