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???, 6 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 8 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) 9 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. 12 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