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11100 cICIRCID‘OmmS-p.“

THREE ESSAYS IN APPLIED ECONOMICS

By

Te-Fen Lo

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Economics

2000

ABSTRACT

THREE ESSAYS IN APPLIED ECONOMICS

By

Te-F en L0

This dissertation consists of three essays. Chapter 1 is “Charter School Location”,
Chapter 2 is “Are Urban Districts Inefﬁcient?” and Chapter 3 is “Migration and

Economic Growth”.

Chapter 1: Charter schools represent one part of the larger movement toward
parental choice in the US. school system. The intention of these programs is to use
market mechanisms to improve school efﬁciency, innovation, and program variety. This
study provides evidence on these issues using data on Michigan and California. We
regress the number of charter schools in each school district on the characteristics of
students and public schools in those same districts. The results indicate that more charter
schools locate where populations are diverse in terms of race, income, and adult
education levels. We interpret this as demand for horizontal differentiation, reﬂecting
both preferences for homogeneous student populations and preferences for speciﬁc
education programs. In both states, charter school location is negatively related with
public school test scores. This implies that parents pay attention to test scores (vertical
differentiation), even though these scores may be imperfect signals for school

performance. Most differences in results across the two states can be attributed to

differences in state policies regarding the granting of charters and the funding of public

schools.

Chapter 2: We use the stochastic production frontier model to estimate efﬁciency
for public school districts in Michigan. Three different speciﬁcations for the efﬁciency
term including the half-normal, truncated normal and exponential are used. We ﬁnd that
urban school districts are signiﬁcantly less efﬁcient than the non-urban school districts.
Besides, the school districts with higher percentage of children in poverty, higher
percentage of children who are black, larger enrollment (large districts), and larger total
expenditure are less efﬁcient. Moreover, the school districts with less schooling of the
adult population are also less efﬁcient. The by far biggest urban school district, Detroit,
is relatively inefﬁcient in the production of schooling in our model, especially for science
test, however, it is not the worst one. The average measure of efﬁciency of Detroit is

roughly twice as large as the least efﬁcient one.

Chapter 3: We use the dynamic panel data model to explore the relationship
between the migration rate and the economic growth. We ﬁnd that high level income
instead of high growth rate of income attracts people move in, and the inﬂow of migrants
also contributes to the economic growth. In addition, the level of per capita income have
signiﬁcant dynamic properties, that is, its current level is signiﬁcantly related with its
previous value. However, we do not ﬁnd signiﬁcant dynamic property for the migration
rate. We also ﬁnd the urban population and the crime rate have signiﬁcantly negative

impact on the migration rate.

C0pyright by
Te-Fen Lo
2000

Dedicated to My Dearest Parents, Chien-Nan Lo and Ju-Hua Tsai,

for their love, encouragement, and all kinds of support.

ACKNOWLEDGMENTS

I would like to begin by thanking my adviser, Professor Gerhard Glomm for the
completion of this dissertation. I always think my self as very fortunate to have been
under his instruction. His guidance and encouragement have helped me through his
program. Moreover, he has not only been my advisor in economics, but also been an
advisor in my life. He has given me the warmest concern and advice I have ever had
during my study in the United States. I also thank Professor Steven Matusz, who has
given me constructive guidance during in study of economics, and helped improve the
quality of my dissertation. I also appreciate Professor John Giles’s insightful comments.
In addition, my thanks go to my co-author, Douglas Harris, for his invaluable comments

and warm ﬁ'iendship.

I want to give my special thanks to my parents, Chien-Nan Lo and Ju-Hua Tsai.
Without their love, encouragement, and ﬁnancial support, I would never have arrived at

this point.

Many of my friends and fellow graduate colleagues have also contributed in
different ways. I thank Jenhei Chen, Yi-Yi Chen, Bih-Shiow Chen, Chiung-Ying Cheng,
Hung-Jen Wang, and Chung-Jung Lee for sharing the happiness and life/academic
experiences with me. There are still a lot of people who have helped me, but not

mentioned in the list. I appreciated what they have done for me with deep gratitude.

vi

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. ix

LIST OF FIGURES ........................................................................................................... xi
CHAPTER 1

CHARTER SCHOOL LOCATION ................................................................................... 1

I. Introduction ............................................................................................................. 1

II. An Informal Theory ............................................................................................... 6

III. Data and Methodology ....................................................................................... 10

IV. Results ................................................................................................................ 18

V. Conclusion ........................................................................................................... 36

Appendix A: Summary Statistics (Michigan) ........................................................... 38

Appendix B: Summary Statistics (California) .......................................................... 39

Appendix C: The Educational Production Frontier Function (Michigan) ................ 40

References ................................................................................................................. 42
CHAPTER 2

ARE URBAN DISTRICTS INEFFICIENT? ................................................................... 46

I. Introduction ............................................................................................................ 46

II. Methodology ........................................................................................................ 50

111. Result .................................................................................................................. 56

IV. Conclusion .......................................................................................................... 78

Appendix A: Public School Districts with Missing Data ......................................... 80

Appendix B: MLE Estimates for Different Models for Math Test ........................... 81

Appendix C: 20 Most Efﬁcient Public School Districts ........................................... 82

Appendix D: Counts for Appearing in the 20 Least Public School Districts ........... 85

References ................................................................................................................. 89
CHAPTER 3

MIGRATION AND ECONOMIC GROWTH ................................................................. 92

I. Introduction ........................................................................................................... 92

II. Literature Review ................................................................................................ 94

III. Methodology of the Estimation ........................................................................... 99

3.1 Dynamic Panel Data Model ............................................................................ 99

vii

3.2 Endogeneity Problem .................................................................................... 101

3.3 Choices of Instrumental Variables ................................................................ 103

3.4 Statistical Tests ............................................................................................. 105
IV. Speciﬁcation of the Model ............................................................................... 106
V. Regression Results ............................................................................................ 111
VI. Conclusion ....................................................................................................... 126
Appendix: Data Deﬁnition and Source ................................................................... 128
References ............................................................................................................... 13 1

viii

LIST OF TABLES

CHAPTER 1: CHARTER SCHOOL LOCATION

Table 1: Charter School Growth in Michigan ..................................................................... 3
Table 2: Mean of Independent Variables Categorized by Number of Charter Schools in
School Districts (Michigan) .................................................................................. 12
Table 2 (cont’d) ................................................................................................................. 13
Table 3: Mean of Independent Variables Categorized by Number of Charter Schools in
School Districts (California) ................................................................................. 14
Table 3 (cont’d) ................................................................................................................. 15
Table 4-1: Regression Results for Michigan .................................................................... 21
Table 4-1 (cont’d) ............................................................................................................. 22
Table 4-2: Regression Results for California .................................................................... 23
Table 4-2 (cont’d) ...... ' ....................................................................................................... 24
Table 5: Regression Results Omitting Potential Outliers for Michigan ........................... 33
Table 6: Regression Result with Border Districts for Michigan ....................................... 35
Table A: Summary Statistics (Michigan) ......................................................................... 38
Table B: Summary Statistics (California) ......................................................................... 39
Table C: The Educational Production Frontier Function (Michigan) ............................... 40

CHAPTER 2: ARE URBAN DISTRICTS INEFFICIENT?

Table 1: Average Test Score along Different Dimensions ............................................... 48
Table 2: Data Statistics Summary ..................................................................................... 58
Table 3: OLS Estimates for Different Models .................................................................. 61
Table 4: Summary of Five Types of Regressions for Model 1, 2, and 3 .......................... 62
Table 5: Estimated Efﬁciency Distribution for Different Models .................................... 65
Table 6A: 20 Least Efficient Public School Districts for Math ........................................ 67
Table 6B: 20 Least Efﬁcient Public School Districts for Science .................................... 68
Table 6C: 20 Least Efﬁcient Public School Districts for Graduation Rate ...................... 69
Table 7: Relative Measure of Efﬁciency for Detroit ........................................................ 71

Table 8: Correlation Coefﬁcients between Educational Achievement and Efﬁciency 73

Table 9: Comparison of Estimated Efﬁciency (HST Math) ............................................. 75
TablelO: MLE Estimates for Technical Efficiency Models for Math Test ...................... 77
Table A: Public School Districts with Missing Data ........................................................ 80
Table B: MLE Estimates for Different Models for Math Test ......................................... 81
Table Cl: 20 Most Efﬁcient Public School Districts for Math ....................................... 82
Table C2: 20 Most Efﬁcient Public School Districts for Science ................................... 83
Table C3: 20 Most Efﬁcient Public School Districts for Graduation Rate ...................... 84
Table D: Counts for Appearing in the 20 Least Public School Districts .......................... 85
Table D (cont’d) ................................................................................................................ 86
Table D (cont’d) ................................................................................................................ 87

CHAPTER 3: MIGRATION AND ECONOMIC GROWTH

Table 1: Data Statistics Summary by Decades ............................................................... 108
Table 2A: Migration Equation in Different Estimation Methods .................................. 113
Table 2B: Growth Equation in Different Estimation Methods ....................................... 114
Table 3A: GMM Estimations with Different IVs for Migration Equation .................... 118
Table 3B: GMM Estimations with Different IVs for Growth Equation ........................ 119
Table 4A: Alternative Speciﬁcations for Migration Equation ....................................... 121
Table 48: Alternative Speciﬁcations for Growth Equation ............................................ 122
Table 5: “Quasi-Simultaneous” Migration and Growth Equation .................................. 125

LIST OF FIGURES

Figure 1: Horizontal and Vertical Differentiation of Schools ............................................ 8

xi

CHAPTER 1

CHARTER SCHOOL LOCATIONl

I. Introduction

Before 1990, choice in the American education system was limited to assigned
public schools and private schools. Since then, there has been a signiﬁcant restructuring.
Many states have allowed public school choice, allowing students to attend the public
schools outside the districts in which they reside. Voucher plans in Milwaukee,
Cleveland, San Antonio, and Florida have further expanded these choices to include
private schools. These small-scale programs appear to be precursors to widespread

choice programs, such as those proposed in Michigan, California, and other states.

A third instrument of providing greater school choice is the charter school system.
Charter schools are publicly ﬁnanced and often subject to less regulation than traditional
public schools. Some oversight is usually administered by third parties, including
universities or state government agencies, rather than local school boards. Charters

receive a ﬁxed amount per student enrolled. However, in contrast to public schools, they

 

' We thank participants in association meetings of the ABA, Econometric Society, and Public
Choice, as well as seminars at the Federal Reserve Bank of Chicago, Michigan State University,
the University of Michigan, and the University of Kentucky. We especially thank Bih-Shiow
Chen, Julie Cullen, Tom Downes, David Figlio, Larry Kenny, Bob Rasche, Peter Schmidt, and
John Strauss for useful comments. Financial support for this project from the Business College at
MSU is gratefully acknowledged.

do not receive a separate allotment for capital expenditures.2 In addition, operating
revenues in charter schools are sometimes as low as 50 percent of neighborhood public
schools with an average of about also 80 percent (Finn, Manno, and Vanourek, 2000).
Therefore, total spending per student is usually much lower in charter schools compared

with nearby public schools.

The charter school movement is the most rapidly developing part of the US.
school choice movement (Paris, 1998). By the end of 1994, eleven states had adopted
charter programs. By 1999, thirty-three states had charter policies, yielding over 1,700
charter schools and 350,000 enrolled students (Finn et al, 2000). In Michigan, the charter

school growth was similarly rapid, as indicated in Table 1.

 

2 Most public school districts can raise capital funding through local property tax levies. Arizona
is a slight exception to the rule for charter policies, allowing charter start-up grants up to
$100,000. However, this is quite small compared with total required capital costs for most
schools.

Table 1: Charter School Growth in Michigan

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number of Charter YEAR
30110015 in Dim“ 94-95 95-96 96-97 97-98 98-99
0 545 527 508 495 487
1 10 21 35 43 47
2 o 3 6 9 12
3 0 2 3 4
4 O 1 1 1 2
5 0 0 1 2 2
7 0 1 0 0 0
1 3 O O 1 0 0
21 0 0 0 1 0
36 0 O O 0 1
Total Chalrthelrl Schools 10 44 78 108 137
Total ScholsilI District in 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

 

 

 

 

 

 

 

 

Perhaps the most compelling argument for school choice in general, and charter
schools in particular, is that it will improve schooling by increased reliance on market
mechanisms. One argument here is that students will “vote with their feet” for a better
alternative school if charter school productivity is not sufficiently high. The market thus
imposes discipline on the charter schools. Only good schools will survive. Peterson
(1999) ﬁnds some support for this view in his review of research on recent, small-scale
voucher programs. However, Bettinger (2000) ﬁnds that charter schools are not any

more productive than public schools.

A second argument is that charter schools may improve productivity in the
traditional public schools since they stand to lose enrollment, and hence state ftmding, to
the charter schools. If this efﬁciency argument is valid, we expect to ﬁnd a negative
relationship between the number of charter schools and public school performance as
measured by productive efﬁciency within each school district. Hoxby (1994, 1997) ﬁnds
some evidence for this hypothesis in her studies of competition among private schools

and public school districts.

A third aspect of efﬁciency relates to the Tiebout hypothesis, which roughly states
that pe0p1e will receive their optimal bundles of government-provided goods if there are
many options to choose from and free mobility across jurisdictions. These bundles
include the level of local taxation, private school tuition, school performance (vertical
differentiation), the type of education being provided (horizontal differentiation), and
other amenities. In the case of education, the less perfect is the sorting, the more likely it

is that parents will seek other bundles when given alternatives.

There is anecdotal evidence that school choice produces schools with specialized
curricula and homogeneous student bodies within schools. For instance, the El-Hajj
Malik EL-Shabazz Academy in Lansing, Michigan describes its mission as to "serve
students using a holistic, Afrocentric curriculum." More generally, 70 percent of all
charter school students in Michigan are minority compared with 22 percent of public
school enrollment.3 In California, Grutzik, et al (1995) show that “the communities

surrounding charter schools are primarily white and have income levels at or above the

city and county averages.” Wells et. al. (1996) comes to a similar conclusion, as do other
survey-based studies. These outcomes are apparently similar in other countries that have

implemented expansive school choice programs.4

In this paper, we address the question whether charter schools do indeed locate in
those district where the lack of Tiebout sorting has provided for inefﬁcient outcomes.
What are the criteria which charter schools use to make their location decisions? Why
might the demand for these schools be greater in some districts than others? For a given
demand, what factors inﬂuence the supply of charter schools? Do charter schools indeed

enter in districts where there are few existing alternatives and imperfect Tiebout sorting?

To help answer these questions, we assemble data for all charter schools in
Michigan and California, and match each charter school with the public school districts in
which they locate. We then regress the number of charter schools in a district on the
student and public school characteristics of those same districts. The econometric
methodology we use is similar to the one used by Downes and Greenstein (1996), using a
statistical model of count data advanced by Hausman, Hall and Griliches (1984) and

Cameron and Trivedi (1986).

 

3 These results come from a study commissioned by the State of Michigan: Public Sector
Consultants and MAXIMUS (1999). The results are similar in Arizona, where the numbers are 6
percent and 4.3 percent respectively (Gifford, Ogle and Solmon, 1998).

‘ England, Chile, China, Sweden, New Zealand, South Africa, and the Czech Republic are a few
examples. See Harris, Oliver, and Plank (2000) for discussion of these countries. For a
description of differences across US. states, see Wohlstetter et. al (1995), Nathan (1996), and
Mintrom (1998).

Downes and Greenstein study entry (location) by privately ﬁmded schools in
California. Bettinger (2000) and Filer and Munich (2000) estimate regressions similar in
form to those estimated here. The Bettinger paper also relates to charter schools in
Michigan, but his main purpose is to obtain instruments for other regressions that study
the impact of charter schools on public school performance. Most of the variables we
ﬁnd to be important are excluded in Bettinger’s regressions. Filer and Munich’s
speciﬁcation is more similar to those used here, but he studies new choice-based schools
in the Czech Republic. These authors also exclude a large number of variables that we
find to be important, but they do ﬁnd that charter school location is negatively related

with a measure of public school efﬁciency.

In Section II, we provide an informal theory of charter school location based on
product differentiation. We describe our data in Section III. Empirical tests of our

hypotheses are reported and discussed in Section IV.

II. An Informal Theory

The evidence above suggests that charter school entry is closely related to product
differentiation, including both horizontal (h) and vertical (v) dimensions. Common
examples of vertical differentiation (quality) in education include graduation rates, value-

added, and the proportion of kids going on to college.

In most markets, inputs would not be considered an aspect of quality because
more productive inputs are reﬂected in better outputs and higher prices. However, there

are many reasons to believe that the price-quality relationship is rather weak in education.

First, dependence on the tax system means that non-consumers are paying most of the
cost. Second, the movement toward state level funding may be weakening the
connection between local taxation and local school funding. Third, information problems
may prevent parents from being able to observe actual school quality. These three facts
imply that it may be reasonable to interpret input levels as signals of public school quality.
We consider student-teacher ratios, expenditure per student, teacher salaries, and special

education expenditures.

There are also various dimensions of horizontal product differentiation. These are
dimensions along which preference heterogeneity generates disagreement among
consumers over what is best. Some consumers may prefer an academic curriculum in
high school, while others favor more vocational training. Other consumers may favor
schools with racial and ethnic diversity, while others may favor homogeneity. Some
favor authoritarian schooling by Catholic nuns, while others favor schooling in which

children are free to determine their own rules of conduct.

Ideally, the location of schools ought to be considered as the outcome of a
location game such as in Hotelling (1929) or Prescott and Visscher (1977). In these
papers, location is an outcome of a game played by proﬁt-maximizing ﬁrms.
Unfortunately, the objective functions for public and private schools are controversial.5
Instead of fully specifying such a location game and characterizing the equilibrium

location, we only illustrate potential outcomes of such games.

 

5 See, for example, Manski (1992) and Nechyba (1996).

In Figure 1, product differentiation is shown in h-v product space. This space is
n-dimensional, where n is the number of product differentiation dimensions. Household
preferences are distributed over the h-v space. The small open circle represents the
preferred bundle of an individual person, assuming that person has to pay for the full cost
of education. Small closed circles represent schools of which there are three types:

public schools (1,, private schools P,, and charter schools C ,.

Figure 1: Horizontal and Vertical Differentiation of Schools

 

 

 

O O
c.
O 0
P2
0 C
U’ U’ 713 \ /
O
‘\ 0 C3

 

The parent illustrated in Figure 1 can choose any school in the space, but C ,, P2,
and U 3 most closely match the parent’s preferences. If all schools did charge full tuition,
then we would expect the person to locate at the nearest school in the h-v space.
However, the actual funding system includes tuition only at private schools. Therefore, if
P, charged high tuition, this option would be eliminated from consideration, leaving C ,

and U 3, which would provide similar satisfaction at a much lower price.

While the above discussion is framed in the language of horizontal product
differentiation, we will carry out our econometric work using heterogeneity of tastes.
Actual measures of product differentiation, such as the amount of time spent teaching
foreign languages, are not available. Therefore, we assume that variation in tastes will

give rise to variation in education programs.

Public school education programs are chosen by school boards and
superintendents for entire districts, therefore, we expect relatively minor variations in
product differentiation across public schools within a district. Other models in which
public and private schools co-exist, such as the one studied by Epple and Romano (1996),
predict that private school quality is higher than public school quality. In addition,
horizontal differentiation among the private schools is much greater than among the
public schools, providing for the possibility of elite schools with high high-tuition and

religious schools with low tuition.

As in Prescott and Visscher (1977), we assume that a charter school needs to
attract a sufﬁcient number of customers in order to cover ﬁxed costs. It appears in Figure
1 that this is best accomplished by locating away from public and private schools in the
h-v product space. However, if only private schools charge tuition, then charter schools
may seek to provide education similar to private schools, but free of charge to consumers.
Therefore, charter schools may instead try to locate very close to private schools in the
product space. In any case, the location of charter schools in the h-v space is dependent

upon the characteristics of both public and private schools in the district.

III. Data and Methodology

In this section we describe our data and the hypothesized relationships between
the independent variables and charter school location. All of our data describing public
school districts comes from the Departments of Education in Michigan and California.
The data on charter schools in Michigan come from the Michigan Association of Public
School Academies (MAPSA). The data on California charter schools comes from the
Califomia Department of Education. The data on demographic variables comes from the
School District Data Book (SDDB), which includes US. Census Bureau data organized

by school district.

The Michigan sample includes more than 500 observations in each regression
speciﬁcation, out of a total of 555 districts in the state. None of the missing district
observations contains a charter school. However, our samples in California are in the
range of 288-350 district observations, which is substantially lower than the total number
of districts.‘5 Approximately 60 of the 142 districts that contain charter schools are among
the list of missing observations, though none of the missing observations contain more
than one charter school. All missing observations for both states are due to missing data

in the original sources listed above.

The unit of observation is the school district. In addition to simplifying the
analysis, this approach is appropriate because school policies are set at the school district

level. In addition, it is difﬁcult to obtain data at the school level. We use the following

 

‘5 The actual total number of districts in California varies from year to year, but is approximately
1,000.

10

variables: 1) wealth/income as measured by median household income and the poverty
rate; 2) ethnic composition of the population; 3) adult (parent) educational attainment; 4)
geographic characteristics, such as area and the enrollment density of the district; 5)
performance of the public schools measured by outputs, such as test scores, graduation
rates, and productive efﬁciency; 6) public school inputs, such as student-teacher ratios; 7)
charter school revenue (state grants) and costs (teacher salaries); 8) the degree of
competition from private schools.7 Summary statistics are provided in appendices A and
B. In Table 2 and Table 3 below, we exhibit public school characteristics by the number

of charter schools per district.

 

7 At least three of the variables discussed above are measured with error: teacher salaries,
graduation rates, and the number of private schools. We tried various symmetric truncations, but
found that the results were unaffected.

ll

Table 2: Mean of Independent Variables Categorized by Number of Charter Schools in
School Districts (Michigan)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number of Charter Schools 0 1 2 3
Number of Observations 487 47 12 4
Med. House Value 54.4400 62.6596 53.1362 59.4318
Per Capita Income 12.4307 13.6903 12.83 83 15.7198
Med. Household Income 29.2252 31.6506 27.7818 33.3838
Percentage of Children in Poverty 14.8747 13.1036 21.0569 16.3991
Percentage of Children Black 2.6253 4.9382 16.5913 34.9468
Percentage of Children Hispanic 2.2202 2.8294 4.9671 1.9740
Herﬁndahl Index for Race 88.5989 85.6163 74.4901 56.2649
Percentage of Adult with 12 Grade 23.6570 20.9840 23.1694 19.3347
Percentage of Adult with High School 38.1048 35.7185 30.1879 26.2820
Percentage of Adult with Some College 26.2057 28.5187 30.5313 31.5413
Eigﬁ‘r‘age 0f Ad‘m With Bacmlo’ Degree °’ 12.0325 14.7739 16.1115 22.8420
Average Years of Schooling 13.0095 13.2473 13.3456 13.8080
Herﬁndahl Index for Education 30.5036 29.2565 28.2192 27.2992
MEAP Math Score 4th Grade 43.2593 44.5915 38.8917 34.0250
Graduation Rate 90.2330 80.8489 83.6250 65.4500
Total Expenditure Per Student 4.3025 4.5489 4.9903 6.3618
Sgregangxeiesnpecial Educational Expenditure 0.3 1 62 0.3 57 6 0. 5 5.7 6 0.4449
Pupil-Teacher Ratio 20.1948 19.5957 20.8333 17.7500
Average Teacher Salary 30.5072 33.1776 35.1267 39.1403
K-12 Enrollment Density 51.6359 45.5082 122.7717 189.7679
Number of Private Schools 1.4262 1.9574 5.5000 10.5000
Total K-12 Enrollment 2.2858 3.7169 7.5739 10.1568
Gr. Rate of K-12 Enrollment -0.1121 0.5445 -0.9812 0.4128
Percentage of Public School located in City 3.3223 10.5910 36.4355 25.0000

 

 

Table 2 (cont’d)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number of Charter Schools 4 5 36
Number of Observations 2 2 1
Med. House Value 77.1520 52.9955 25.2940
Per Capita Income 14.0100 12.2060 9.4430
Med. Household Income 26.7145 26.7460 18.7420
Percentage of Children in Poverty 24.3284 25.2006 44.2895
Percentage of Children Black 30.4542 25.9369 82.0884
Percentage of Children Hispanic 8.3750 9.6727 3.2848
Herﬁndahl Index for Race 47.8377 44.9284 69.2901
Percentage of Adult with 12 Grade 17.8227 21.7313 37.1489
Percentage of Adult with High School 23.0214 26.9645 28.3550
Percentage of Adult with Some College 27.9106 32.7880 25.5750
:eirglzgrage of Adult with Bachelor Degree or 312453 18.51 62 8.9211
Average Years of Schooling 14.2547 13.5494 12.6753
Herﬁndahl Index for Education 34.2620 26.2948 29.1771
MEAP Math Score 4th Grade 42.5500 30.3000 27.3000
Graduation Rate 69.6500 34.6000 71.6000
Total Expenditure Per Student 5.9510 5.0685 5.2930
Sgregzﬁceiesnpecial Educational Expenditure 0. 5234 0.71 68 0.61 09
Pupil-Teacher Ratio 18.5000 21.5000 22.0000
Average Teacher Salary 34.0570 31.7525 36.2440
K-12 Enrollment Density 114.6518 227.7712 510.1699
Number of Private Schools 14.0000 23.0000 98.0000
Total K-12 Enrollment 14.6515 26.4940 183.1510
Gr. Rate of K-12 Enrollment -0.6721 -2.1684 -0.5748
Percentage of Public School located in City 98.3333 69.4444 94.5946

 

 

 

 

13

 

Table 3: Mean of Independent Variables Categorized by Number of Charter Schools in

School Districts (California)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number of Charter Schools 0 l 2 3 4 5
Number of Observations 683 73 1 1 7 2 2
Med. House Value 161.0317 155.5765 193.4620 166.0547 194.9715 143.8660
PerCapitaIncome 15.9204 15.1404 16.0471 14.9779 16.7700 14.0115
Med. Householdlncome 35.8001 33.1040 36.3837 34.8543 35.6245 35.3025
Pe’cemag" °fCh"d'e"‘" 16.5882 15.6422 12.5887 17.3796 13.0240 10.5404
Poverty

Percentage of Children Black 2.8713 4.9998 3.4797 6.7075 1.8736 2.3503
Pirceniageomh‘m'e“ 28.9470 24.3305 19.1421 26.8825 31.8585 34.0138
Hispamc

Herﬁndahl Index forRace 60.0797 56.1949 56.4036 59.4534 58.5670 51.6981
gi'ﬁ‘mge OfAduhw'ﬂ‘” 27.0420 23.9646 17.4619 24.5834 21.0595 23.3758
Pircemage “Ad“"w‘th 24.9958 25.5062 24.9778 25.4451 21.6060 28.4146
Hrgh School

PercentagedAduhw‘m 29.9629 33.0880 36.8122 32.7108 34.1032 34.1096
Some College

Percentage °fAd“"""."h 17.9994 17.4412 20.7481 17.2607 23.2313 14.1000
Bachelor Degree or H1 gher

Average Years of Schooling 13.4088 13.4686 13.8065 13.4440 13.8654 13.2944
He'ﬁmIah'me" f“ 31.5873 29.4507 28.2123 28.3918 27.2748 27.3166
Educat1on

MEAP Math Score 4th Grade 618.8822 615.2763 611.5000 614.1600 618.6500 616.9000
Graduation Rate 1.4072 2.3918 1.6143 3.4500 3.6000 1.9333
TotalExpenditurePerStudent 18.0109 42.8215 37.5880 33.2919 147.0195 37.8225
Average Special Educational

Expenditure PerSmdem 0.0989 0.1128 0.0993 0.0950 0.1080 0.1164
Pupil-Teacher Ratio 22.9183 23.4600 24.8571 25.4125 23.3500 23.7000
Average Teacher Salary 25.1110 24.8379 24.5185 26.1930 23.9448 25.0770
K-12 EnrollmentDensity 81.5364 87.7917 40.9671 67.3475 146.1783 57.4615
NumberofPrivate Schools 2.2568 5.5631 4.6667 4.7500 26.0000 45.0000
Total K-12 Enrollment 3.6456 8.9111 7.0964 6.9376 21.1140 27.6347
Gr. Rate ofK-12 Enrollment 1.9272 2.0919 3.2202 6.0345 1.7747 3.3940
Percen‘agCOfpub'” SChO‘” 13.2095 20.2143 27.8592 18.5183 0.5814 6.2500

located in City

 

 

 

 

 

 

 

14

 

Table 3 (cont’d)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Number of Charter Schools 6 1 1 12 14 34
Number of Observations 2 1 l 1 1
Med. House Value 81.4525 87.0370 172.1070 183.2150 226.4900
Per Capita Income 11.5705 10.2930 14.6750 16.1970 15.3170
Med. Household Income 26.4850 22.7000 27.0900 3 1.9910 30.7950
Percentage of Children in Poverty 29.9362 32.0099 29.7194 20.9674 27.3885
Percentage of Children Black 4.9249 2.7295 51.2762 14.0586 13.6627
Percentage of Children Hispanic 39.1633 10.6700 17.7937 28.4801 59.1756
Herﬁndahl Index for Race 38.8973 76.2095 34.1082 29.1420 41.0275
Percentage of Adult with 12 Grade 31.3894 18.9516 25.4587 17.0489 35.6124
gzgﬂmge 0f Ad“ With High 24.8657 20.3629 20.9801 21.4979 19.6746
2:15:23“ °f Ad“" With some 29.0823 39.7177 27.9328 35.0387 25.0559
5881;621:1323?“ With BaChe'm 14.6626 20.9677 25.6284 26.4145 19.6571
Average Years of Schooling 13.1475 13.8629 13.8418 14.1152 13.3244
Herﬁndahl Index for Education 26.8309 27.9096 25.2537 26.7826 26.6953
MEAP Math Score 4th Grade 614.1000 629.9000 594.0000 616.1000 601.3000
Graduation Rate 5.8500 0.0000 8.8000 4.4000 13.2000
Total Expenditure Per Student 158.5740 2.5060 231.5000 571.2540 3756.8060
nggﬁeitﬁfeeg‘ 3:118:20“! 0.1160 0.1968 0.1056 0.1153 0.1081
Pupil-Teacher Ratio 24.7500 16.2000 22.6000 23.2000 24.5000
Average Teacher Salary 25.4180 23.8540 27.4040 24.8810 28.7040
K-12 Enrollment Density 203.8404 0.6787 350.9178 243.4163 430.8289
Number of Private Schools 16.5000 1.0000 54.0000 94.0000 593.0000
Total K-12 Enrollment 38.8115 0.3760 51.2340 125.1160 639.7810
Gr. Rate of K-12 Enrollment 1.4416 8.0251 0.4031 1.6966 1.2283
Percentage 0f ”"1“ SChO‘” '“ated 36.4706 0.0000 94.4444 100.0000 79.6825

in City

 

 

 

 

 

 

15

 

The empirical analysis involves regressing the number of charter schools in each
district on the characteristics of the schools and students in those same districts.
Therefore, the dependent variable occurs in non-negative integer amounts and OLS is
inconsistent. One of the methods created to deal with such issues is Poisson regression,
developed by Hausman, Hall and Girliches (1984) and Cameron and Trivedi (1986).
Poisson regressions have been used also by Papke (1991) in the context of manufacturing
ﬁrm start-up and by Downes and Greenstein (1996) in the context of private school start-
ups in California. Two of the potential weaknesses of this approach are: 1) violation of
the independence assumption; and 2) "overdispersion" in the data. We use the Huber-

White robust standard errors to correct for dispersion. [See Huber (1967), White (1982).]

There are two potential simultaneity problems in this framework. Charter schools,
private schools, and public schools of choice are substitutes for each other. Entry of
charter schools in a district is determined simultaneously with entry of private schools
and/or public schools of choice within the same district. Moreover, public schools may
respond to charter school entry by changing their behavior. We deal with both these
problems by regressing the number of charter schools in 1998 on school district

characteristics in 1992, the year before charter school policies began.8

The second simultaneity issue stems from the fact that students can cross district
boundaries to attend charter schools.9 This means that the number of charter schools in

district 1' , C ,, depends not only on the characteristics of that district, X, , but also on the

 

8 See footnote 6 for information about speciﬁc years.

16

characteristics of neighboring districts, X], including the number of charter schools, Q.

This yields

CI=f(Xl’Xj’Cj) (1)

District j may be a composite of information for many districts because there may
be multiple districts nearby. This deﬁnition is important because it deﬁnes market size,
or the geographic area over which it is possible to attract students. We start by excluding
neighboring district data altogether, focusing only on home district characteristics. Next,
we substitute in for Q in (1) to obtain a reduced form that is a function only of X, and X].
In later parts of the paper, we expand the market deﬁnition to include nearby districts that

do not border the home district.

In most cases where nearby districts are included, composite variables are created
that account both for the number of students and physical distance of the border districts
relative to the home district. For variables that are hypothesized to be positively
(negatively) related to the number of charter schools, increasing the distance and
decreasing the proportion of students in the border districts is expected to decrease

(increase) the composite variable.

A common econometric issue is identifying the simultaneous equations of supply
and demand. The usual simple model is not appropriate in this context because the price

is exogenously ﬁxed by the government, rather than being endogenously determined by

 

9 Kelejian and Prucha (1998) state that “cross-sectional spatial models frequently contain a spatial
lag of the dependent variable as a regressor or a disturbance term that is spatially autocorrelated.
The ﬁrst of these topics is discussed here. We assume there is no spatial autocorrelation in our
model.

markets. This yields two equations, but only one endogenous variable: the number of
charter schools in a district. Therefore, we combine them into a single equation, which

can be estimated consistently without additional changes to the estimation procedure. ‘0

IV. Results

In this section, speciﬁc variables are introduced that relate to various aspects of
horizontal and vertical product differentiation. We hypothesize speciﬁc relationships and
interpret these results for both Michigan and California, which are presented below in

Tables 4 through 7.

There are two reasons to expect that the results will be different across the two
states. Both relate to state education policy. First, California education spending on
traditional public schools is signiﬁcantly constrained in wealthier school district to be
below actual desired levels (Fernandez and Rogerson, 1999). Michigan also limits
spending in high income districts; however, the limits appear to allow wealthier districts
to come closer to their desired spending levels. In addition, Michigan has recently
redistributed substantial funding to low income districts, resulting in a substantially

higher average spending level (Papke, 2000).

 

'0 Consider the following structural equations: and C, = 7,h + 72v + 6;, and C, = ,Blp + 8,. The
demand for charter schools Cd is a ﬁinction of the variation in preferences for various horizontal
characteristics h, the strength of preferences for quality v, and a disturbance 8d. The supply of
charter schools C, is a function the price of inputs and output, and a disturbance 8,. Again, prices
do not show up on the demand side as they usually would because price is ﬁxed at zero. Aﬂer
imposing the equilibrium condition C = C, = C,, it now appears that we have two separate
equations trying to explain the same phenomenon. Therefore, we instead estimate C = 7r,h + 7r,v
+ mp + a.

A second key policy difference is that California’s charter school policy limits
chartering authority to local school districts, implying that charters cannot start without
some support from the public schools. In Michigan, school districts may serve as
chartering authorities, but this permission is also extended to universities and other

11

organizations outside the district in which the charter schools reside. The possible

implications of these policy differences for our results are discussed below.

The demand for charter schooling is related to the size of the market. We
measure the size of the market by the number of children enrolled in public schools or by
the number of school-age children in the school district (ages 5-17). We would expect
the number of charter schools to be positively related with this measure of "market-size,"
other things being equal. If entry decisions by charter schools are forward looking, trends
in market size might matter as well. We therefore include the growth rate of enrollment
as an independent variable. Many districts have been growing at a rapid rate, especially
in California. In addition, the fact that only school districts can authorize charters in
California implies that they are most likely to occur in growing districts that are adding
schools. Building new charter schools, instead of traditional schools, allows the district

to expand while decreasing regulatory burdens.

All of these hypotheses about market size are supported by the results in Table 4-
1 (Michigan) and 4-2 (California). The coefﬁcient on the number of students is
consistently positive and signiﬁcant in both states. Enrollment growth is positive and
signiﬁcant in California, but insigniﬁcant in Michigan. This provides support for the

impact of differences in state charter policy.

 

" The vast majority of schools are authorized by universities.
I9

We use various measures of income.12 Assuming education is a normal good, we
might expect both median family income and median house value to be positively related
with charter school entry.13 This is especially true in California where state equalization
policy has constrained public school spending in high-income districts. On the other
hand, low income households may have fewer opportunities to move their residences,
implying a negative relationship between income and charter entry. In other words, low-
income households may demand less of a normal good, but they may also be further

away from their most preferred bundle.

 

'2 Chambers (1999) calculates school district-level cost indexes using a hedonic wage model.
Many variables in our regressions are denominated in dollars, however, deﬂating them has very
little impact on the results. All reported results are not deﬂated.

'3 The assumption that education is a normal good relates only to the vertical dimension of
education characteristics. We have no theory about the relationship between income and any
horizontal characteristic.

20

Table 4-1: Regression Results for Michigan

Dependent Variable: Number of Charter Schools for Each School District in Michigan, 1998-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1999
(Robust standard errors in parentheses)
(1) (2) (3) (4) (5)
Med. Household Income -0.0329 -0.0182 -0.0248 -0.0413 -0.0276
(0.0123) ” (0.0182) (0.0182) (0.0199) " (0.0212)
Percentage of Children in 0.0059 -0.0004 -0.0067 -0.0028
Poverty " (0.0253) (0.024 1L (0.0246) (00247)
Average Yrs. of Schooling 1.1902 1.0257 0.9872 0.9954 0.9729
(0.1918 " (0.2398) ** (0.2455) "”" (0.2438) "”" (0.2594) **
Herf. Index for Ad. -0. 1248 -0.1233 -0.0960 -0.1084
Education -- (0.0487) ** (0.0472) " (0.0505) * (0.0503) **
Herf. Index for Race -0.0326 -0.0312 -0.0265 -0.0249
-- (0.0108) *“‘ (0.0106) " (0.0111) ” (0.0112) "
MEAP Math Score for 4th -0.0431 -0.0190 -0.0149 -0.0176 -0.0161
Grade (0.0123 ” (00103)" (0.0120) @0128L (0.0125)
Productive Efficiency -3.9482 -3.8725 -3.9039
-- -_ (4.5992) (5.0275) (4.8826)
Graduation Rate -0.0065 -0.0026 -0.0024 -0.0024 -0.0021
(0.0044L 40.0032) (0.0030) (0.0029) (0.0029)
Tot. Exp. Per Student 0.1997 0.0326 0.0886 -0.0745 0.0485
(0.0948) " (0.0899) (0.0931) (0.1295) (0.1160)
Avg. Special Educational 1.5656 1.5624
Expenditure Per Student " " " (0.8082) * (0.8056) *
Pupil-Teacher Ratio -0.0151
“ " " " @0363)
Avg. Teacher Salary __ -- __ 0.0469 __
(0.0260) "'
Expenditure minus
Foundation Grant -- -- -- -- ..
Number of Priv. Schools __ __ -- -- --
Total Enrollment 0.0267 0.0263 0.0274 0.0254 0.0264
(0.0010) " (0.0029) " (0.0030) "”" (0.0033) " (0.0032) **
Gr. Rate of Enrollment -0.0379 -0.0196 -0.0201 0.0202 -0.0003
(0.0608) (0.0622) (0.0632) (0.0700) (0.0750)
City __ __ __ __ __
Constant -15.1547 -7.7136 -3.8374 -5.8819 -4.6103
(1.9957) ** (3.6071) *"‘ (5.4191) (5.6816) (5.7761)

 

 

 

 

 

 

Note: One asterisk (") indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (") indicates signiﬁcance at the 95 percent conﬁdence level.

21

 

Table 4-1 (cont’d)

Dependent Variable: Number of Charter Schools for Each School District in Michigan, 1998-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1999
(Robust standard errors in parentheses)
(6) (7) (3) (9) (10)
Med. Household Income -0.0303 -0.0374 -0.0208 -0.023 1 -0.0134
(0.0214) (0.0213) * (0.0188) 40.0179) (0.0175)
Percentage of Children in —0.0042 -0.0050 -0.0001 -0.0151 -0.0064
Poverty (0.0243) (0.0242) (0.0242) (0.0260) (0.0245)
Average Yrs. of Schooling 0.9721 0.9543 1.0215 0.9105 0.7771
(0.2538 ** (0.2624) " (0.2465) ** @2430) ** (0.2552) *"
Herf. Index for Ad. -0.1126 -0. 1025 -0.1183 -0.1188 -0.1150
Education (0.0509) ** (0.0519) ** (0.0471) "”" (0.0352) " (0.0340) "”"
Herf. Index for Race -0.0340 -0.0244 -0.0319 -0.0265 -0.0232
(0.0108) ** (0.0102) " (0.0099) " (0.0099) " @0094 **
MEAP Math Score for 4th -0.0173 -0.0174 -0.0145 —0.0148 -0.0115
Grade (0.0122) (0.0127L (0.0120) (0.0121) (0.0121)
Productive Efficiency 3.3967 -3.7854 -3.9480 -2.4928 -1.6921
(4.8807) (4.8789) (4.6431) (5.3160) (5.4030)
Graduation Rate -0.0027 -0.0022 -0.0029 -0.0038 -0.0012
(0.0030) (0.002 8) (0.003 1) (0.0037) 40.0022)
Tot. Exp. Per Student -0. 1275 __ -- -- --
(0.1596)
Avg. Special Educational 1.5336
Expenditure Per Student -- (0.8081) * -- -- --
Pupil-Teacher Ratio -0.03 72 -0.0175
(0.0418) (0.0310) " “ "
Avg. Teacher Salary 0.0526 0.0371
(0.0268) H (0.0193) * " " "
Expenditure minus -- __ 0.1853 0.2440 0.3354
Foundation Grant (0.2596) (0.2502) (0.2573)
Number of Priv. Schools 0.0461
"' " " “ (0.0054) H
Total Enrollment 0.0273 0.0255 0.0276 0.0253
(0.0031) H (0.0033) H (0.0029) H (0.0030) H "
Gr. Rate of Enrollment -0.0436 -0.0020 -0.0141 -0.0096 ~0.0157
(0.0768) (0.0772) (0.0632) (0.0662) (0.068%
City __ __ __ 0.0093 0.0091
(0.0031) " (0.0033) “
Constant -3.8675 -5.2281 -3.9424 -3.9508 -4.0476
(5.4394) (5.7439) (5.3989) (5.6821) (5.7591)

 

 

 

 

 

 

Note: One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

22

 

Table 4-2: Regression Results for California

Dependent Variable: Number of Charter Schools for Each School District in California, 1998-99

(Robust standard errors in parentheses)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1) (2) (3) (4) (5)
Med. Household Income -0.0861 -0.0739 -0.0731 -0.0695 -0.0702
(0.0300) "“" (0.0212) " (0.0232) " (0.0240) " (00245)"
Percentage of Children in 0.0127 0.0274 0.0329 0.0259
Poverty (0.0234) (0.0200) (0.0210) (0.0214)
Average Yrs. of Schooling 0.8558 0.7799 0.9871 0.9842 0.9486
(0.2737)" (0.3307) ** (0.3707) " (0.4839) "”" (0.4117) "
Herf. Index for Ad. -0.2144 -0.2137 -0.2031 -0.2159
Education (0.0698) ** (0.0646) " (0.0784) ” (0.0728) "
Alternative Herf. Index for 0.0131 0.0193 0.0146 0.0196
Race (0.0082) (0.0111) * (0.0112) (0.0113) "'
STAR Math Score for 4th -0.0052 -0.0083 -0.0078 -0.0071 -0.0087
Grade (0.0158) (0.0177) (0.0185) (0.0184) (0.0181)
Productive Efficiency -3.3665 -2.9579 -3.1755
(1.1353) ** (3.7429) (3.4613)
Dropout Rate 0.0176 -0.0026 -0.0042 0.0038 -0.0039
(0.0328) (0.0379) (0.0370) (0.0350) (0.0358)
Tot. Exp. Per Student -0.0170 -0.0163 -0.0150 -0.0138 -0.0153
(0.0014) " (0.0022) ** (@025) " (0.0031) ** (0.0032) "
Avg. Special Educational -1 .0158 -0.6271
Expenditure Per Student (8.3636) (7.1063)
Pupil-Teacher Ratio -0.0262
(0.0671)
Min. Teacher Salary -0. 1276
(0.0969)
Expenditure minus
Foundation Grant
Number of Priv. Schools
Total Enrollment 0.1076 0.1028 0.0960 0.0894 0.0976
(0.0089) " (0.0131) ** (0.0151) " (0.0182) “ (0.0190) "
Gr. Rate of Enrollment 0.1276 0.1444 0.1325 0.1363 0.1297
(0.0557)” (0.0504) *"‘ (0.0486) ** (0.0473) ** (0.0482) **
City
Constant -7.4269 0.4003 -0.6781 1.6724 0.8964
(7.9253) (9.4383) (8.9617) (9.7679) (9.9190)

 

Note: One asterisk (*) indicates significance at the 90 percent conﬁdence level.

Two asterisks ("‘*) indicates signiﬁcance at the 95 percent conﬁdence level.

23

 

Table 4-2 (cont’d)

Dependent Variable: Number of Charter Schools for Each School District in California, 1998-99
(Robust standard errors in parentheses)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(6) (7) (8) (9) (10)
Med. Household Income -0.0648 -0.0903 -0.0995 -0.0997 -0.0864
(0.0260) ” (0.0301) ** (0.0292) " (0.0278) ‘* (0.0301) *"
Percentage of Children in 0.0277 0.0549 0.0191 0.0191 0.0330
Poverty (0.0194) (0.0176) ** (0.0195) (0.0194) (0.0191) "'
Average Yrs. of Schooling 0.8936 1.4456 0.9530 0.9575 0.8956
(0.4198) "”" (0.4045) ** (0.3109) " (02931)" (0.3394) *"
Herf. Index for Ad. -0. 1994 -0.2582 -0.1864 -0.1865 -0.2127
Education (0.0656) *" (0.0697) "“" (0.0565) *"' (0.0568) ** (0.0572) "
Alternative Herf. Index for 0.0149 -0.0045 0.0018 0.0017 -0.0005
Race (0.0112) (0.0138) (0.0151) (0.0143) (0.0158)
STAR Math Score for 4th -0.0095 0.0036 0.0100 0.0100 0.0091
Grade (0.0206) (0.0184) (0.0175) (0.0178) (0.0179)
Productive Efﬁciency -1.7487 -6.8301 -3.6933 -3.6840 -3.3845
(1.7257) (1.8641)* (1.1975) ** (1.2445) *"‘ (1.2546) *"‘
Dropout Rate 0.0059 0.0271 -0.0703 -0.0704 -0.0105
(0.0376) (0.0446) (0.0607) (0.0602) (0.0498)
Tot. Exp. Per Student -0.0146
(0.0028) **
Avg. Special Educational -8.4139
Expenditure Per Student (6.9365)
Pupil-Teacher Ratio -0.0452 0.0619
(0.0671 ) (0.0644)
Min. Teacher Salary -0.1329 -0. 1528
(0.0982) (0.0800) *
Expenditure minus -0. 1037 -0.1044 -0.0574
Foundation Grant (0.0204) " (0.0205) ** (0.0243) **
Number of Priv. Schools 0.0237
(0.0071) *"'
Total Enrollment 0.0939 0.0080 0.0357 0.0359
(0.0167) ** (0.0012) ** (0.0062) ** (0.0060) **
Gr. Rate of Enrollment 0.1316 0.1408 0.1228 0.1228 0.1278
(0.0459) *" (0.0477) ** (0.0434) ** (0.0434) *" (0.0443) **
City -0.0003 0.0032
(0.0042) (0.0049)
Constant 4.33 52 -5 .0846 -9.6069 -9.6076 -8.3214
(10.7343) (10.4561) (9.1573) (9.1597) (9.3238)

 

Note: One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

24

 

We use both median household income and median property values, but the
results are unaffected by this choice. We use household income and ﬁnd that the
coefficient is negative for both states. It is occasionally signiﬁcant for Michigan, and
uniformly signiﬁcant for California. The negative coefﬁcients contradict the evidence for
California from Grutzik et a1 (1995) who ﬁnds a positive relationship between income
and charter location. However, Grutzik excludes adult education levels, which is highly
correlated with income. The magnitude of the effect for California (Table 4-2, column 7)
suggests that a one standard deviation increase in median income ($7,781) decreases the
expected number of charter schools by 65 percent. In Michigan, the effect magnitude is

smaller.

We include education of the adult population as an indicator of family preferences.
Parents with higher education levels might receive greater utility from the education
quality of their children, making them more eager to choose a charter school if traditional
public schools perform below par. The measures we use are fractions of the population
with: (1) high school degrees only, (ii) some college, and (iii) college degree. From these,
we calculate a measure of average parent education level. The coefﬁcient on this
variable is consistently positive and signiﬁcant in both states, which is consistent with the
above hypothesis and the results of Downes and Greestein (1996). The magnitude for
Michigan (Table 4-1, column 7) suggests that a one standard deviation increase in
average years of schooling (0.6 years) raises the expected number of charter schools by

61 percent.

There are at least three ways in which population diversity may be associated with

parents’ demand for charter schools. First, diversity may imply greater dispersion of

25

preferences and student needs. Second, parents may desire schools whose students have
characteristics similar to those of their own children. Third, the median voter model
suggests that if any group has greater than 50 percent of the population, then this group
may be able to control schools through voting. Populations with less than 50 percent of
the population may, therefore, seek to open schools that more closely match their

preferences.

We include three measures of population diversity, based on race, income, and
education of the adult population. Herﬁndahl indices are used both for race and
education. A higher (lower) Herﬁndahl index implies a more (less) homogeneous
population. Therefore, if diversity leads to more charter schools, then the coefﬁcients on
these variables should be negative. This hypothesis is generally supported by Tables 4-1
and 4-2, except that racial mix does not appear to be a factor in California. One reason
for this may be the more restrictive charter-granting policies in California. If minority
groups cannot affect school district polices, then these groups can seek charters on their
own in Michigan. In California, they must work through the same school districts that
have apparently already failed them. In Michigan, the magnitude of the effect suggests
that a one standard deviation increase in either Herﬁndahl index yields a 29 percent

decrease in the number of charter schools.

Available data does not allow for easy calculations of a Herﬁndahl index for
income at the school district level. As an alternative, we include a measure of lower tail
of the income distribution, in addition to median household income. Controlling for

median income, a larger portion of the population in poverty implies a less equal income

26

distribution. The coefﬁcient on poverty in Tables 4-1 and 4-2 are generally positive, as

expected, but they are usually insigniﬁcant.

The geographic size of the district may be important due to transportation costs.
The number of students per square mile, controlling for the number of schools, indicates
the average distance to school, which we would expect to be negatively related to charter
school entry. Similarly, the number of public schools is expected to be negatively related
with charter school location, since a large number of public schools would be associated
with lower transportation costs and greater horizontal differentiation, controlling for the
number of students.” It turns out that these variables are consistently insigniﬁcant,
therefore, we omit them from the results reported here.” However, whether the schools
are in cities, versus suburbs and rural areas, is important. The coefﬁcient on the
proportion of schools located in a central city is insigniﬁcant for California, but positive

and signiﬁcant in Michigan, regardless of whether density is included.

The intent of creating charter schools was to increase choice and competition in
schooling. We expect charter schools to enter more frequently where school choice is
limited. A larger number of private schools implies a larger number of substitutes to
public schooling and, therefore, a higher degree of competition. We might expect then
that the number of private schools (and public schools) will be negatively related with
charter school entry."5 However, charter schools could also locate near private schools

and provide a similar type of education without charging tuition. In this case, charter

 

'4 It may also be the case that people prefer schools with few students.
‘5 They were also highly correlated, therefore, we did not include both in any given speciﬁcation.

‘6 Cross-district schools-of-choice programs, which were described earlier, also measure choice,
however, these do not have a signiﬁcant impact.

27

school location might be an increasing function of the size of the private sector.'7 Our
estimation results provide a test of which effect is dominant. The results in Table 4-1 and
Table 4-2 indicate that more private schools are associated with more charter schools,

providing support for the second hypothesis.

The quality of public schools might be an important determinant of charter school
entry. If public school quality is low, dissatisﬁed parents might be more inclined to
choose the charter school alternative. Two of our measures of public school quality are
measures of output: graduation rates and student test scores.18 Survey research suggests
that when test scores are low, parents might consider alternatives to public education and
we therefore expect these scores to be negatively related with charter school entry (F inn,
2000). The negative point estimates here support this hypothesis, but they are generally

not signiﬁcant.

Student and family characteristics vary widely across districts. Previous research
indicates that these differences have an important impact on education outcomes.l9
Therefore, test scores and drop out rates alone will not accurately indicate the
contribution to education made by the schools themselves. The ideal measure of school
quality (vertical differentiation) is value-added. We calculate a measure of public school

efﬁciency using frontier regressions of educational production functions in which the

 

'7 Unfortunately, we do not have data on private school tuition and other private school
characteristics.

'8 For Michigan, the student test is the 4th grade Michigan Educational Assessment Program
(MEAP). For California, the student test is the 4th grade Standardized Testing and Reporting
(STAR). We use the test scores for math since there is some evidence that math scores are more
sensitive to school quality, whereas reading scores are more dependent on interaction with parents
in the home. The results are unaffected by the choice of reading and math tests.

'9 See, for example, Coleman (1966) and Harris (2000a).

28

dependent variable is 7th grade math scores.20 The estimated functions are shown in
appendix C. The best any district can do is produce on the production frontier. The
further the actual test score is from the production frontier, the more inefﬁcient is the
district. If markets are relatively efﬁcient, then we would expect a negative relationship
between this measure of efﬁciency and the number of charter schools. However, we ﬁnd
just the opposite. As with the coefﬁcients on test scores, the efﬁciency coefﬁcients are
consistently negative and insigniﬁcant. However, the low signiﬁcance levels are at least
partially caused by the high correlation between test scores and this measure of

efﬁciency.”

Our data set includes many variables that measure the inputs of public school
districts, including expenditure levels, student-teacher ratios, teacher salaries, and special
education spending. To the extent that parents residing in a district do not pay the costs

of education, we expect that fewer inputs will induce more charter schools to enter.

The interpretation of two input coefﬁcients requires additional discussion. First,
controlling for class size, teacher salaries, and special education spending, higher
expenditures per pupil implies more spending on other inputs, such as after-school
programs and administrators.22 Second, variation in teacher salaries may reﬂect
differences in compensating differentials across districts (e. g. crime) that are not

accounted for by other included variables.

 

2° For Michigan, these scores are from 1993. For California, the scores are from 1998, which is
the oldest available.

2' The correlation range is 07-09, depending on the frontier speciﬁcation.

22 We use two measures of teacher salaries, namely average teacher salary and the contractual
starting salary for a teacher with a bachelor’s degree and zero teaching experience.

29

The results for school inputs are different across the two states. The positive
relationship between charters and special education spending in Michigan is especially
interesting, given the survey evidence that charter schools attract a disproportionate

number of students with special needs compared with public schools.23

The special
education programs at traditional public schools often involve labeling kids and placing
them in “pull-out” programs that separate kids from the mainstream. Also, Cullen (1999)
ﬁnds that ﬁscal incentives lead traditional public schools to include more students in
special programs. Charter schools, in contrast, have far lower funding levels and rarely

offer pull-out programs. Therefore, one interpretation is that many parents prefer to keep

their kids in mainstream classrooms and programs.

In Michigan, the coefﬁcient magnitude for special education expenditures
suggests that a one standard deviation increase in this category ($156 per pupil) increases
the expected number of charter schools by 24 percent. For teacher salaries, a one
standard deviation increase ($6,079) raises the number of charter schools by 23 percent.

A similar increase in class size decreases the number by 6 percent.

The discussion of coefﬁcient magnitudes above focuses on Michigan. In
California, the effect magnitudes are larger for the demographic variables. For example,
a one standard deviation increase in median household income is associated with an 88
percent drop in the number of charter schools. This may be due to differences in state

equalization policy, as discussed earlier. The one demographic variable that has a

 

One possible exception is teacher salaries, as indicated earlier.

30

smaller impact in California is the Herfmdahl index for race, which here implies only a
12 percent decrease in the number of charters, compared with 30 percent in Michigan.

The effects of most other variables are quite similar across the two states.

On the supply side, revenue will be a key factor regardless of a school’s objective
function. In both states, schools receive the per pupil foundation grant for the district in
which the school is located. This grant also indicates the districts' minimum total
spending for public schools, as guaranteed by the state government. Without going into
detail about how this is calculated, the important characteristic of the funding system is
that public school districts with a low foundation grant also tend to have total spending
that is exactly equal or slightly above the foundation grant. This means that charter
schools will have an easier time competing with public schools in low-spending
districts.24 We incorporate the revenue effect by including a variable that measures
expenditures minus foundation grant, which should reveal a negative relationship with
the number of charters. For California, the coefﬁcient is negative and signiﬁcant, as

expected. The same coefficients for Michigan are positive and insigniﬁcant.

In many ways, Detroit is an unusual school district in Michigan. It is by far the
largest district in Michigan with an enrollment of about 180,000, which exceeds mean

enrollment by a factor of 60! The ethnic composition of Detroit is much different from

 

23 Firm (2000, p.79) states that “many charter schools attract youngsters with more problems and
deﬁcits than the conventional schools to which they are compared.” Also, 20 percent of their
survey respondents indicated that they chose a charter school because “my child’s special needs
[were] not met at [the] previous school.”

2" The overall advantage/disadvantage is somewhat difﬁcult to establish for two other reasons: 1)
charter schools do not receive capital funds from the government; 2) the foundation grant does
not account for grade level, allowing charter schools to focus on "cheaper" student populations.
However, these differences should affect all districts in relatively equal ways.

31

the rest of Michigan, the poverty rate is much higher and test scores are much lower.
Perhaps most importantly, a very large proportion of all charter schools is located in
Detroit. In order to check to what extent our results are driven by this single observation,
we re-run the regression from Table 4-1 column 7, omitting Detroit. We also run
regressions omitting some other potential outliers, namely the very small school districts. .

The results are shown in Table 5 and are fairly robust to these changes.

32

Table 5: Regression Results Omitting Potential Outliers for Michigan

Dependent Variable: Number of Charter Schools for Each School District in Michigan, 1998-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1999
With Detroit Delete Detroit Delete 5% Delete 10%
smallest Districts smallest Districts
(with Detroit) (with Detroit)
Number of Observations 517 516 492 466
Med. Household Income -0.03 74 -0.0447 -0.0485 —0.0493
(0.0213) "' (0.0216) " (0.0220) ** (0.0219) "
Poverty Rate -0.0050 -0.0074 -0.0121 -0.0161
(0.0242) (0.0244) (0.0262) (0.0262)
Average Yrs. of Schooling 0.9543 0.8418 0.9494 0.9164
(0.2624) ** (0.2511) ** (0.2642) " (0.2630) "
Herf. Index for Ad. -0.1025 -0.0826 -0.1140 -0.1095
Education (0.0519) ** (0.0451) * (0.0523) " (0.0510) **
Herf. Index for Race -0.0244 -0.0186 -0.0264 -0.0280
(0.0102) “' (0.0096) * (0.0107) ** (0.0109) **
MEAP Math Score 4th -0.0174 -0.0169 -0.0111 -0.0101
Grade (0.0127) (0.0130) (0.0118) (0.0123)
Production Efﬁciency -3.7854 -1.2361 -5.8932 -6.3623
(4.8789) (5.5450) (4.8209) (4.9279)
Graduation Rate -0.0022 -0.0004 -0.0020 -0.0019
(0.0028) (0.0010) (0.0029) (0.0027)
Avg. Special Educational 1.5336 1.2851 1.5302 1.4613
Expenditure Per Student (0.8081) * (0.9139) (0.8325) * (0.8321) *
Pupil-Teacher Ratio -0.0175 -0.0457 -0.0237 -0.0258
(0.0310) (0.0321) (0.0340) (0.0338)
Average Teacher Salary 0.0371 0.0399 0.0396 0.0350
(0.0193) * (0.0198) ** (0.0200) ** (0.0202) "'
Total Enrollment 0.0255 0.0704 0.0267 0.0272
(0.0033) ** (0.0207) ** (0.0035) *"' (0.0035) **
Gr. Rate of Enrollment -0.0020 0.0108 0.0138 0.0083
(0.0772) (0.0802) (0.0826) , (0.0841)
Constant -5.2281 -6.7197 -2.5142 -1.3709
(5.7439) (5.9850) (5.6140) (5.6048)
Note: One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (") indicates signiﬁcance at the 95 percent conﬁdence level.

33

Many charter schools attract students not only from the district in which they are
located but also from neighboring districts. In order to account for this, we regress the
number of charter schools not only on characteristics of the district in which the charter
school is located, but also on characteristics of neighboring (contiguous) districts. There
are two possible effects of neighboring districts: First, the parents may send their children
to charter schools in other districts. If this were the only effect, then we would expect the
neighboring coefﬁcients to be of the same sign as the home district coefﬁcients.
Furthermore, if the charter schools expected most of their students to come from the

home district, then the border-district coefﬁcients should have lower magnitudes.

The home district coefﬁcients in Table 6 reveal results quite similar to those in
Table 4-1. Also, unlike Downes and Greenstein (1996), we ﬁnd that some characteristics

of neighboring districts have a signiﬁcant impact on the number of charter schools.

34

Table 6: Regression Result with Border Districts for Michigan

Dependent Variable: Number of Charter Schools for Each School District in Michigan, 1998-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1999
(With Contiguous Districts and with Detroit)
Home ] Contiguous
Number of Observations 516
Med. Household Income -0.03 76 0.0158
(0.0279) (0.0639)
Poverty Rate 0.0110 -0.0833
(0.0247) (0.0535)
Average Yrs. of Schooling 1.0497 0.2305
(0.2887) ** (0.413 I)
Herf. Index for Ad. Education -0.1 187 -0.1243
(004382“ (0.1224L
Herf. Index for Race -0.0238 0.0409
(0.0110 ** (0.0253)
MEAP Math Score 4th Grade -0.0076 -0.0914
(0.0135) (0.0375) **
Production Efﬁciency -1.2660 -9.3094
(5.7479) (4.8134L“
Graduation Rate -0.0032 -0.0043
(0.0034) (0.0066)
Avg. Special Educational 0.3621 5.7005
Expenditure Per Student (0.8689) (1.6936) **
Pupil-Teacher Ratio -0.1105 0.4073
(0.0381) ** (0.0956) **
Average Teacher Salary 0.0157 -0.0783
(0.0217) (0.0526)
Total Enrollment 0.0194 0.0088
(0.0036L" (0.0071)
Gr. Rate of Enrollment -0.0985 0.3104
(0.0878) (0.1903) *
Constant -3.9033
(4.9852) "
Note: One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

35

 

V. Conclusion

This paper provides evidence about the equity and efﬁciency effects of charter
schools by studying their location patterns. The results reinforce the conclusions of
existing research that ﬁnds a signiﬁcant impact of racial diversity. However, we clarify
this result by including variables that measure other forms of diversity, including income
and adult education levels. The effect of population diversity on the demand for charter
schools may reﬂect differences in preferences for education programs, not just

preferences for homogeneous student populations.

We ﬁnd mixed support for the effect that charter schools have on school
efﬁciency. Charter schools do appear to locate in more school districts with less efﬁcient
public schools. However, they also locate in districts that should already be competitive
by way of private school entry. Therefore, instead of providing competition, charters

may simply shift resources to students who previously went to private schools.

The results here also suggest that state policies toward charter and public schools
do impact the location of charter schools. This means that state education policy at least
partially determines the effects that charter schools have on equity and efﬁciency.
Several other issues are left for future research. One is the impact that charter schools
have on public school performance. Also, Table 6 here indicates that the characteristics
of border districts seem to impact charter location. While this speciﬁcation does not
seem to alter the general role of home district characteristics, the general issue of

geography and market size warrants further attention.

36

APPENDICES

37

Appendix A: Summary Statistics (Michigan)

Table A: Summary Statistics (Michigan)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Variable Mean Std. Dev. Min Max
Med. House Value 55.1641 23.5722 19.0000 230.3750
Per Capita Income 12.5686 3.9566 6.2200 48.3400
Med. Household Income 29.3916 9.3980 9.8050 88.9180
Percentage of Children in Poverty 14.9930 9.6126 0.0000 51.7051
Percentage of Children in Black 3.6778 11.6971 0.0000 97.0390
Percentage of Children in Hispanic 2.3 795 3.2701 0.0000 35.2941
Herﬁndahl Index for Race 87.4753 12.4457 37.7571 100.0000
Percentage of Adult with 12 Grade 23.3867 7.5265 2.5726 51.0638
Percentage of Adult with High School 37.5373 7.1426 6.6098 60.1852
Percentage of Adult with Some College 26.5604 5.901 1 3.1915 50.7564
Percentage of Adult with Bachelor 12.5156 8.4886 0.0000 60.1718
Degree or Higher

Average Years of Schooling 13.0483 0.6371 11.9362 16.1 119
Herﬁndahl Index for Education 30.3224 2.7340 25.3326 44.43 70
MEAP Math Score 4th Grade 43.1317 13.5106 4.3000 100.0000
Graduation Rate 88.7272 115.0292 0.0000 2570.4000
Total Expenditure Per Student 4.3633 1.1428 0.0000 14.4420
Average Special Educational 0.3286 0.1559 0.0000 0.8590
EQenditure Per Student

Pupil-Teacher Ratio 20.1391 3.6000 8.0000 41.0000
Average Teacher Salary 30.9208 6.0789 0.0000 51.4430
K-12 Enrollment Density 55.3187 124.0824 0.0194 1068.5000
Number of Private Schools 1.9209 5.0483 0.0000 98.0000
Total K-12 Enrollment 3.0317 8.3631 0.0040 183.1510
Gr. Rate of K-12 Enrollment -0.0817 2.4367 -11.9489 15.0722
Percentage of Public School located in 5.5431 21.7519 0.0000 100.0000

City

 

 

 

 

 

38

 

Appendix B: Summary Statistics (California)

Table B: Summary Statistics (California)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Variable Mean Std. Dev. Min Max
Med. House Value 160.8949 99.4099 0.0000 490.9770
Per Capita Income 15.8181 7.7811 4.0510 64.9930
Med. Household Income 35.4842 13.5953 11.4060 122.9080
Percentage of Children in Poverty 16.5164 12.6396 0.0000 76.1134
Percentage of Children in Black 3.2033 6.3613 0.0000 71.5090
Percentage of Children in Hispanic 28.4079 23.7525 0.0000 100.0000
Herﬁndahl Index for Race 59.5052 18.2437 25.9773 100.0000
Percentage of Adult with 12 Grade 26.5714 16.1165 2.1212 90.1883
Percentage of Adult with High School 25.0245 7.2968 0.0000 87.5000
Percentage of Adult with Some College 30.4035 7.8658 0.0000 47.8919
Percentage of Adult with Bachelor 18.0006 13.0581 0.0000 70.5273
Degree or Higher

Average Years of Schooling 13.4224 0.9858 11.1992 16.6736
Herﬁndahl Index for Education 31.2533 6.7032 25.2537 81.8317
STAR Math Score 618.2699 19.6486 555.2000 681.9000
Drop-Out Rate 1.7616 4.2021 0.0000 68.5000
Total Expenditure Per Student 27.1975 139.5741 0.0000 3756.8060
Average Special Educational 0.2453 0.8829 0.0021 16.3793
Expenditure Per Student

Pupil-Teacher Ratio 22.1008 5.3378 0.0000 60.6000
Entry Teacher Salary 25.0853 2.4229 15.9120 44.3570
K-12 Enrollment Density 81.4625 149.2684 0.0000 1450.7500
Number of Private Schools 3.4679 19.2437 0.0000 593.0000
Total K-12 Enrollment 4.8513 21.1820 0.0000 639.7810
Gr. Rate of K-12 Enrollment 2.3278 4.9000 -14.4514 67.1028
Percentage of Public School located in 14.4063 33.1666 0.0000 100.0000

City

 

 

 

 

 

39

 

Appendix C: The Educational Production Frontier Function (Michigan)

Table C: The Educational Production Frontier Function (Michigan)

Dependent Variable: MEAP Math Test Score in 7th, 1993

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Variable MLE Estimates
Per Capita Income 0.1762
(Deﬂated by Teacher Cost Index) (0.1126)
Percentage of Children in Poverty 03925
(0.1509) **
. . 0.0219
Percentage of Children 1n Black (0.0442)
. . . . 0.0015
Percentage of Chrldren 1n Hispanic (0.0040)
-0.4558
Herﬁndahl Index for Race (0.1338) H
. -0.2908
Average Years of Schooling (0.0379) 4*
. . 0.2453
Puprl-Teacher Ratio (0.0384) H
Total Expenditure Per Student 2.0855
(Deﬂated by Teacher Cost Index) (3.1521)
0.2079
Total K-12 Enrollment (00357) H
. -2.1753
K-12 Enrollment Densrty (27757)
Average Teacher Salary 0.0051
(Deﬂated by Teacher Cost Index) (0.0032) *
Average Special Educational Expenditure Per 0.2974
Student (Deﬂated by Teacher Cost Index) (0.2445)
60.1321
Consul” (9.6782) H

 

 

Note: One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

40

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Wohlstetter, P., Wenning, R., Briggs, KL. (1995). “Charter Schools in the United States:
The Question of Autonomy.” Educational Policy, Vol. 9, December, 331-358.

Wooldridge, J.M. (1992). "Quasi Likelihood Methods for Count Data." Journal of
Economics, pp. 352-406.

45

CHAPTERZ

ARE URBAN DISTRICTS INEFFICIENT?

I. Introduction

“In a move virtually unprecedented in American education, the state of
Missouri is stripping away the accreditation of all [Kansas City]
schools because of poor academic performance?”

“The children failed the test. There are arguments about the way in
which standards are imposed, but I suppose those kinds of arguments
could go on forever about any test. The fact is, it's a statewide test.”

Many people believe American schools are failing. In the case of urban districts,
the public perception is one of complete disaster. Chicago, Detroit, Cleveland, Baltimore,
Kansas City, and Los Angeles school districts have all seen serious attempts by city or
state governments to impose special controls, reforms, or outright takeover. For instance,
the state of Ohio has implemented an audit program that applies only to urban districts.
In Michigan, Wisconsin, Florida, and Ohio, recent proposals have sought to fund school

vouchers targeted almost exclusively to school children in large urban districts like

Cleveland, Milwaukee or Detroit.

Such proposals are certainly not new. Local governments have often been subject
to special provisions and controls, usually due to extreme ﬁnancial crisis or

mismanagement. What is new is that the controls are now being justiﬁed based on

 

25 Christian Science Monitor.

46

general performance measures — i.e. student test scores. Table 1 below provides a

summary of average district test scores for Michigan along several dimensions.

47

Table 1: Average Test Score along Different Dimensions

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Difference Mean
Urban“ Non-Urban
Math Grade 11 High School Test, 1999 -1 1.3457 54.0688 65.4144
(2.6300) **
Math Grade 11 High School Test, 1998 -l4.0979 49.0750 63.1729
(2.8332) "
Science Grade 11 High School Test, 1999 -12.5547 41.3844 53.9390
(2.6348) **
Science Grade 11 High School Test, 1998 -11.6276 42.7281 54.3557
(2.7693) **
Rich27 Poor
Math Grade 11 High School Test, 1999 10.1151 69.7714 59.6523
(1.2176) "
Math Grade 11 High School Test, 1998 8.7049 66.6510 57.9461
(1.3519) "
Science Grade 11 High School Test, 1999 9.3811 57.8526 48.4715
(1.2369) *"'
Science Grade 11 High School Test, 1998 8.5954 57.9337 49.3383
(1.3107) **
Nonblack28 Black
Math Grade 11 High School Test, 1999 3.1358 66.2749 63.1391
(1.2900) “”"
Math Grade 11 High School Test, 1998 4.2681 64.4282 60.1602
(1.3931) **
Science Grade 11 High School Test, 1999 2.9967 54.6541 51.6574
(1.2981) "
Science Grade 11 High School Test, 1998 4.2988 55.7812 51.4824
(1.3516) "

 

 

Note: Number in parenthesis presents standard error.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

 

2" Public school districts are deﬁned as urban districts in Table 1 when half of the public schools
in this district locate in large and mid-size central cities.

27 Public school districts are deﬁned as rich districts in Table 1 when the percentage of children in
poverty in a speciﬁc school district is less than 12.88 percent. We also deﬁne rich districts when
the per capita income (deﬂated by the teacher cost index) in a speciﬁc school district is greater
than 12406 dollars. The result is similar as the ﬁrst deﬁnition of rich districts.

2" Public school districts are deﬁned as black districts in Table 1 when the percentage of children
in a speciﬁc school district is greater than 0.348 percent.

48

In Table 1, the difference between rich and poor districts is about 8~10 percentage
points”, the difference between non-black and black districts is about 3~4 percentage
points, and ﬁnally the difference between urban districts and non-urban districts is about
11~14 percentage points. Those differences are statistically signiﬁcant. The distribution
of scores in Table 1 comes as little surprise. The relatively low level of scores in urban
districts is well known. What appears to be unknown, however, is the degree to which
this is caused by low school inputs (e.g. large classes), student characteristics, or low
efﬁciency of teachers and/or administrators. Existing research provides little, if any,
evidence, even though the distinction is fundamental to evaluating and reforming schools.
If the distribution of student ability or background is not equal across schools, as appears
to be the case, then test scores should vary even if school perfonnance/efﬁciency is
identical in every school. If different schooling outcomes are caused by differential
school inputs, changing the amount of school inputs might be the desired policy. In

neither case is institutional reform such as a state take over called for.

The above discussion implies a need for some type of “value-added” measure of
school performance that indicates the contribution of the school to each child’s learning.
Ideally, this would involve students taking comparable tests on the ﬁrst and last days of
class. However, few districts actually collect such data. Another option is using student
demographic data to form a proxy for student ability, and using funding levels to control
for the other resources that schools have to work with. The objective is to determine how

close districts are to being efﬁcient — i.e. how close they are to the production frontier.

 

29 The MEAP and HSP results are reported in percentage of meeting some requirements. See
Section III. Data and Model for detailed description.

49

There are at least three ways to implement the above approach. The ﬁrst is to
regress test scores on student characteristics and funding levels, using the error terms as
measures of productive efﬁciency. However, this approach is inconsistent with economic
theory because this allows schools to be above the production frontier. Two general
approaches to ﬁxing this problem are: data envelopment analysis and frontier regression.
The former uses mathematical optimization to ﬁnd the frontier. Frontier regression, as

the name suggests, relies on statistical estimation of the same function.

This paper uses frontier regression to estimate the productive efﬁciency of
schools using data on Michigan school districts. The methodology is described in
Section II. Estimation results are presented in Section III, yielding a measure of
efﬁciency for each district. Besides, Section III also includes comparisons of these
efﬁciency measures across district types. While comparisons are possible on various
dimensions, the primary issue is whether large urban districts, such as Detroit, are indeed
less efﬁcient than other school districts and, therefore, whether governments are justiﬁed
in targeting those districts with special controls/institutional reform to improve school

performance.

11. Methodology

The production frontier methodology has been developed fairly recently. Its
application is increasingly widespread in many different ﬁelds, such as air force
maintenance units [Charnes, Clark Cooper, and Golany (1985), Bowlin (1987)],
education [McCarty and Yaisawamg (1990, 1992), Ray (1991), Wyckoff and Lavigne

(1991)], national parks [Rhodes (1986)], employment ofﬁces [Cavin and Stafford (1985)],

50

and health clinics [Huang and McLaughlin (1989), Johnson and Lahiri (1992)]. The ﬁrst
empirical estimation of production functions begins with the papers of Cobb and Douglas
(1928). However, it was largely used to study the functional distribution of income
between capital and labor at the macroeconomic level. The production frontier model in

disaggregated level hasn’t drawn much empirical attention until recent decades.

The analysis of productive efﬁciency is a branch of the production frontier model
because deviations from a production frontier have a natural interpretation as a measure
of the efﬁciency with which economic units pursue their technical or behavioral
objectives. Greene (1993) has a short deﬁnition for “productive efﬁcient” that producers
will be characterized as efﬁcient if they have produced as much as possible with the
inputs they have actually employed or they have produced that output at minimum inputs.
Lovell (1993) deﬁned the efﬁciency of a production unit as a comparison between

observed and optimal values of its output and input.

Koopmans (1951) provides a formal deﬁnition of technical efﬁciency. A
producer is technically efﬁcient if an increase in any output requires reduction in at least
one other output or an increase in at least one input, and if a reduction in any input
requires an increase in at least one other input or a reduction in at least one output. Thus
a technically inefﬁcient producer could produce the same outputs with less of at least one

input, or could use the same inputs to produce more of at least one output.

Debreu (1951) and Farrell (1957) introduced a measure of technical efﬁciency.
Their measure is deﬁned as one minus the maximum equiproportionate reduction in all

inputs that still allows continued production of given outputs. A score of unity indicates

51

technical efﬁciency because no equiproportionate input reduction is feasible, and a score

less than unity indicates the severity of technical inefﬁciency.

There are two competing paradigms on how to construct the frontier efﬁciency
model. One uses mathematical programming techniques, the other employs econometric
techniques. The mathematical programming techniques is also called “Data
Envelopment Analysis” (DEA), which is a body of techniques for analyzing production,
cost, revenue, and proﬁt data without parameterizing the technology. By wrapping a hull
around the observed data, we can calculate the distance of each observed producer to that
frontier. Presumably, the larger is the sample, the more precisely this information will be
revealed. The major advantage of “Data Envelopment Analysis” approach is that no
explicit functional form need be imposed on the data. However, since it lumps noise and
inefﬁciency together, the calculated frontier may be warped if the data are contaminated
by statistical noise. The econometric approach can handle statistical noise, but it imposes
an explicit, and possibly overly restrictive, functional form for technology. This kind of
parametric approach is likely to confound the effects of misspeciﬁcation of functional
form (of both technology and inefﬁciency) with inefﬁciency estimation. However, the
more structure we impose on a model the better our estimates — provided the structure we

impose is correct. This is a trade-off between structure and ﬂexibility.

For the econometric approach, let us construct a single production equation cross-
sectional model to illustrate the basic concepts. Suppose producers use inputs x e R f to

produce scalar output y e R, with technology

y, =f(x.;ﬂ)€Xp{V. +u.}- (2-1)

52

The econometric version of the Debreu-Farrell output-oriented technical

efﬁciency (TE) measure is written

TE, 2 y.
[f(xr ;ﬂ)eXP{V. }l

 

= exp {u,} , (2.2)

where [3 is a vector of technology parameters to be estimated and i =1, ..., I
indexes producers. The random disturbance term v, is intended to capture the effects of

statistical noise and is assumed to be independently and identically distributed as

N (0,0',2 ) . The disturbance term u, is assumed to be distributed independently of v, and

to satisfy u, S 0.

Early studies adopted the deterministic frontier model, that is, assume v, = 0.
Then the frontier becomes f (x,; )6). The residual is also adjusted to make sure that

0 < TE, 31. Winsten (1957) proposed corrected ordinary least squares (COLS), which

corrects the downward bias in the estimated OLS intercept by shifting it up until all
corrected residuals are nonpositive and at least one is zero. Richmond (1974) introduced

modiﬁed ordinary least squares (MOLS), which makes an assumption about the
functional form of the nonpositive efﬁciency component u,. The most popular
assumption is half normal and exponential. Two-parameter distributions, including the
truncated normal and the gamma distribution, are also proposed by Stevenson (1980) and
Greene (1980), respectively. Finally, MLE is an implemented method that
simultaneously estimates all the technology parameters and the parameters of the

distribution of u,.

53

Recent econometric approaches usually select the stochastic frontier model
because it allows for statistical noise resulting from events outside the ﬁrrn’s control,
such as luck and weather. Employing a stochastic frontier can also be seen as allowing
for some types of speciﬁcation error and for omitted variables uncorrelated with the
included regressors. In equation (2.1), the stochastic production frontier is
f (x, ; ,6) exp{v,} and the nonpositive error component u, represents technical inefﬁciency.
The degree of technical efﬁciency of a producer is given by the ratio of observed to

maximum feasible output, where maximum feasible output is given by the stochastic

production frontier.

To calculate TE in equation (2.2), we need to estimate equation (2.1). Moreover,
we need to decompose the residuals into separate estimates of noise and technical
inefﬁciency, which is a difﬁcult task. COLS is no longer a feasible technique. MOLS
and MLE proceed roughly as in the deterministic frontier strategy, however, the resulting

residuals contain both noise and inefﬁciency, and they adjusts the OLS intercept by
minus the mean of u, , which is extracted from the moments of the OLS residuals. The
distribution of the compound error term a, = v, + u, has been derived by Weinstein (1964)

and is discussed in Aigner, Lovell, and Schmidt (1977). Jondrow, Lovell, Materov, and

Schmidt (1982) suggested a technique to decompose the noise term and inefﬁciency term.

For half-normally distributed inefﬁciency term, the log-likelihood function for the

model is

 

0' 20'

Log(a,,6,0', )1) = -—Nan' - constant + Z[ln (D[— 8’1 J — 1(5) J, (2.3)

54

and the explicit form of the estimated inefﬁciency component u, is

 

 

Elu. 14.1 =[ “12] z.+ ¢—’—(z ) (2.4)
1+ ,1 <I>(z )
2 2 2 '— 8,1 . .
where /1 = a", /0', , a = a", + 0"“ , z, = , ¢(-) rs the densrty of the standard normal
0'

distribution, and <D(-) is the cumulative density function of the standard normal

distribution.

Meeusen and van den Broeck (1977) and Aigner, et. A1. (1977) presented the log-

likelihood function for an exponentially distributed disturbance,

Log(,6, 0",,6)= Nl:ln6+% (60' )2]+ﬁ[ln¢[——60, ]+05,], (2.5)

1:161!

and the expression of the estimated inefﬁciency term u ,. is

¢<z./o.)
9(4/0.) ’

EIut I 8!] : Zr + 0v (2'6)

where z, = 8, 490,2.

Stevenson (1980) argued that the zero mean assumed in the Aigner, Lovell, and
Schmidt (1977) model was an unnecessary restriction. Therefore, he proposed a

truncated normal distribution. The log-likelihood function is

55

(=1

Log(a,,B,0',/l,,u°) = —N[ln0' + %1n27r + ln<D(,u°):l - i[%(ﬂ) — 1n<l>(,u0 -f-'-/I-)] 9
0'
(2.7)

and the explicit form of the inefﬁcient term u, is similar with the equation (2.4), however,

replace 2,. with z, - ,u".

The above three approaches are employed in our model to estimate efﬁciency for

public school production in Michigan.

III. Result

Our analysis is mainly based on the Michigan public school students’ academic
achievement on mathematics and science, which is measured by MEAP (Michigan
Educational Assessment Program) satisfaction percentage. A cross-sectional data set of
511 public school districts in Michigan is used in our estimation”. Selected data statistics
are summarized in Table 2. To investigate the relationship between the student’s
performance, the school inputs and the characteristic of the speciﬁc public school district,
we employ the following linear stochastic frontier model to estimate the efﬁciency level

for individual public school district.

MEAPHSTScore=a+ﬂX,+yZ,+(v,+u,). (3.1)

 

3° There are 555 public school districts in Michigan, however, 44 public school districts are
removed from our dataset because of missing data. These 44 public school districts were listed in
Appendix A.

56

The educational achievement is measured by the MEAP (Michigan Educational
Assessment Program) and HST31 (High School Test) results on Mathematics and Science
for the public school students in Michigan. MEAP tested students of 4th and 7th grade in
Mathematics, and the result is reported in percentage of students achieving a satisfactory
performance. MEAP also tested students of 5th and 8th grade in Science, and the result

is reported in percentage of satisfaction.

 

3' The result of HST for 11th grade in both Mathematics and Science is reported in percentage of
students achieving level 1 or 2, which was adopted begins from 1998 academic year, replacing
the original MEAP test for 10th graders. Level 1 implies that students exceed Michigan standard,
and level 2 implies that students meet Michigan standard. We sum up the percentage of level 1
and 2 to construct a comparable measure to the original MEAP result for 10th grade.

57

Table 2: Data Statistics Summary

Observation: 511 public school districts in Michigan

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Variable Mean Std. Dev. Min Max
Math Grade 11 High School Test, 199932 64.7039 14.6509 0.0000 100.1000
Math Grade 11 High School Test, 199810 62.2900 15.8744 0.0000 93.8000
Science Grade 11 High School Test, 1999'0 53.1528 14.7343 2.9000 96.9000
Science Grade 11 High School Test, 1998'° 53.6276 15.4124 0.0000 88.7000
Math Grade 4 MEAP result, 199210 43.0955 12.8369 4.3000 89.0000
Science Grade 5 MEAP result, 1993'° 75.3025 11.0131 25.6000 100.0000
Per Capita Income33 12.9272 3.2200 6.3981 41.1796
Percentage of Children in Poverty 14.7225 9.2616 0.8441 51.7051
Percentage of Children in Black 3.9806 12.1739 0.0000 97.0390
Percentage of Children in Hispanic 2.4143 2.9180 0.0000 26.5808
Racial Herf. Index 87.0414 12.4616 37.7571 100.0000
Average Schooling Years 13.0717 0.6385 12.0772 16.1119
Educational Herf. Index 30.1435 2.4663 25 .3326 44.4370
Enrollment Density, 1998" 53.9974 101.9657 0.1466 758.5000
Average Enrollment Density, 1990-199812 55.6475 110.7128 0.1405 868.1875
K-12 Enrollment, 199835 3.2188 8.3143 0.0930 174.7300
Average K-12 Enrollment, 1990-199813 3.2061 8.4432 0.0878 177.9384
Total Expenditure, 199810 6.3248 1.0456 4.4017 17.0296
Average Total Expenditure, 1990-1998'° 5.3944 0.8739 3.2534 11.2415
Average Teacher Salary, 199810 46.1317 4.8170 32.6905 62.0232
Average Teacher Salary, 1990-1998'o 38.9330 3.6639 28.3104 50.6924
Specilgl Educational Expenditure per student, 0.4193 0.1534 0.0000 1.0793
Aiii'age Special Educational Expenditure per 0.3859 0.1391 0.0000 0.8993
student, 1990-1998'°

Pupil-Teacher Ratio, 1998:"5 21.1898 2.4926 10.0000 29.0000
Average Pupil-Teacher Ratio, 1990-1998“ 21.3116 2.4068 9.7500 28.3750

 

 

3’ 1n percentage points.

33 In thousand dollars, and deﬂated by Teacher Cost Index.

3‘ In persons per square kilometers.
35 In thousand persons.
3" In number of students per teacher.

58

 

Here, X, is a vector of characteristics of public school districts, including per
capita income, percentage of children in poverty, percentage of children in Black,
percentage of children in Hispanic, educational Herﬁndahl index”, average schooling
years, K-12 enrollment, and student density,...etc38. In equation (3.1), Z, is a vector of
school input variables, includes pupil-teacher ratio, total expenditure, average teacher
salary, and special educational expenditure per pupil..etc. The random disturbance term

is v, , which captures the effects of statistical noise, is assumed to be independently and

identically distributed as N (0, of). The other disturbance term u, is assumed to be

distributed independently of v, and to satisfy u, _<_ 0.

Three different functional speciﬁcations are used. The output variable is MEAP
satisfaction percentage for all models, however, we change the characteristic variables
and school input variables for different models. In Model 1, the characteristics variables
includes per capita income, percentage of children in poverty, percentage of children
which are Black, percentage of children which are Hispanic, educational Herﬁndahl
index, average schooling years, K-12 enrollment, and student density. The school input
variables includes pupil-teacher ratio, total expenditure, average teacher salary, and
special educational expenditure per pupil. In Model 2, we use racial Herﬁndahl index

instead of percentage of children in Black and percentage of children in Hispanic in the

 

37 The educational Herﬁndahl index is constituted by summing up the square term of the
percentage of adult educational attainment including 12 grade, high school, some college, and
bachelor or higher. When the adult educational attainment is homogenous, say all are with high
school education, the value of index is at its maximum, which equals 1. The smaller the value of
educational Herﬁndahl index, the more heterogenous the adult educational attainment is in that
school district.

33 Per capita income, average teacher salary, total expenditure, and special educational
expenditure per pupil are deﬂated by teaching cost index (TCI).

59

characteristic variables. In Model 3, we add square term of school input variables based

on Model 2’s speciﬁcation.

Ordinary least squares estimates of the education production of Model 1, 2, and 3
appear in Table 3. Greene (1993) suggested that although the OLS coefﬁcients
themselves are of somewhat limited usefulness in the frontier model, it is worthwhile to
note how different assumptions and estimators sometimes only produce minor variation
from the OLS estimates. We will also show MLE estimates for different speciﬁcation for
the efﬁciency term later for comparison. The consistency of the OLS estimates does not
require the normal distribution of the compound error term, instead, the only requirement
is the zero mean of the error term. Therefore, we can always get the consistent OLS
estimates by subtracting the mean of compound error term from both of the constant term
and the compound error term. Since the OLS estimates are consistent here, exploring the
impacts of each input variable on the educational performance for the public school
districts in Table 3 is a good starting point. The MLE results reported in Appendix B are

remarkably similar to the OLS results for all those speciﬁcations.

Table 3 shows that the lower the socioeconomic status (higher percentage of
children in poverty, higher percentage of children who are minority, and fewer years of
adult’s schooling) and the poorer the characteristics of school districts (higher enrollment

density), the lower the HST test result is.

60

Table 3: OLS Estimates for Different Models

Observation: 511 public school districts in Michigan
Dependent Variable: Math Grade 11 High School Test, 1999

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model 1 Model 2 Model 3
Per Capita Income, 1989 -0. 1659 -0.2769 -0.2790
(0.3003) (0.3067) (0.3017)
Percentage of Children in Poverty, 1989 -0.1889" -0.3397** -0.2483"‘*
(0.0923) (0.0876) (0.0919)
Percentage of Children Black, 1989 -0.3395** -- --
(0.0600) -- --
Percentage of Children Hispanic, 1989 -0.3649** -- --
(0.1724) -- --
Racial Herf. Index, 1989 -- 0.2021" 0.2006"
-- (0.0526) (0.0520)
Educational Herf. Index, 1989 -0.2552 -0.1994 -0.2220
(0.2191) (0.2236) (0.2204)
Avg. Schooling Years, 1989 8.5937" 8.7564" 9.3384”
(1.3783) (1.4343) (1.4253)
K-12 Enrollment, 1998 0.0581 -0.0224 0.0150
(0.0645) (0.0641) (0.0636)
Enrollment Density, 1998 -0.0114*" -0.0155** -0.0176**
(0.0055) (0.0055) (0.0055)
Pupil-Teacher Ratio, 1998 4.0099" 4.3681" -9.7294**
(0.3246) (0.3198) (2.2359)
Pupil-Teacher ratio (Square), 1998 -- -- 0.1908"
-- -- (0.0516)
Total Expenditure, 1998 -1.0312 -1.5417* -3.7346
(0.8337) (0.841(3) (2.4016)
Total Expenditure (Square), 1998 -- -- 0.0225
-- -- (0.1271)
Average Teacher Salary, 1998 0.6634” 0.7850" 5.4527"
(0.1337) (0.1330) (1.4488)
Average Teacher Salary (Square), 1998 -- -- -0.0484"
-- -- (0.0153)
Special Educational Expenditure, 1998 -0.8303 -0.2903 5.1694
(3.5165) (3.5926) (12.3680)
Special Educational Expenditure -- -- -4.9503
(Square), 1998 -- -- (12.5879)
Constant -34.6820* 49.2028" -66.6397
(20.2290) (22.5234) (43.0470)

 

Note: Number in parenthesis presents standard error.

 

One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level. Two asterisks (**)
indicates signiﬁcance at the 95 percent conﬁdence level.

61

In addition to regressing Math High School Test in 1999 on the characteristic and
school input variables in 1998 (Type 1 showed in Table 3), there are four more
alternative speciﬁcations for each model. We regress Math High School Test in 1999 on
the average characteristic and school input variables from 1990 to 1998. We also regress
Math High School Test in 1998 on the characteristic and school input variables in 1998.
Moreover, we regress Math High School Test in 1998 on the average characteristic and
school input variables from 1990 to 1998. Finally, we regress Math High School Test in
1999 on the Math MEAP in which grade results in 1992, and average characteristic and
school input variables from 1992 to 1998”. Table 4 summarizes all of ﬁve types of

regressions as follows:

Table 4: Summary of Five Types of Regressions for Model 1, 2, and 3

 

-329..- _
‘ 1 2.-. 3 -,4 5
, ,. . -- . __ .‘ _ ,_ PsesndsatVariablc-_. _
Math/ Science Grade 11 High School Test, 1999 x x x
Math / Science Grade 11 High School Test, 1998 x x
I M F ' Explanatory Variables
Socioeconomic Status of the Students, 1989 x x x x x
Characteristics of Public School Districts, 1998 f x x x
Characteristics of Public School Districts, Average x x
1991-98 '
Schools’ Inputs, 1998 E x x x
Schools’ Inputs, Average 1991-98 1‘ x x
MEAP Math 4th Grade, 1992 / Science 5th Grade, x
1993 %

 

 

’9 This is analogous to the value-added approach in Fox and Taylor (1991) and Grosskopf, Hayes,
and Weber (1991).

62

We found the regression results are robust across all these different speciﬁcations

within the same model.

Table 5 presents method of moments estimators for the parameters of our
stochastic frontier models. The basis for the calculations appears in equation (2.3) — (2.7).
All the analysis in Table 5 is carried out using the LIMDEP (Greene, 1991) computer
program. However, it did not automatically calculate the efﬁciency measure deﬁned in
equation (2.2), which limits the value of efﬁciency between 0 and 1. The estimated

outcomes by LIMDEP includes observed Y, predicted Y, residual (efﬁciency term u,),

xﬂﬂ , and y — xué . In order to calculate the measure of efﬁciency deﬁned in equation

(2.2), we adopt Coelli’s (1996) deﬁnition as follows:
EFF, = E(y,.' Iu,,x,)/E(y: lu, = 0,x,) = (x,,3+t?,)/x,,é. (3.2)
Comparing equation (3.2) to equation (2.2)

TE z y, z f(x';ﬂ)exP{v’ +u') =exp{u} (2 2)
' [I’m/”6mm [f(x,;,6)eXP{V.}] ' ’ -

 

 

we obtain TE, = y, = f(x,;,6)exp(u,) =exp{ﬁ,}, (2.2’)

f (mm f(x.;[3)

 

 

where the numerator in equation (3.2) presents the sum of the predicted HST result and
the efﬁciency residual term (negative in our model), which is similar to the numerator in
equation (22’), except that equation (2.2’) adopts a Cobb-Douglas production form.

Remember y, = f (x,; ,B)exp{v, +u,), and predicted value of y, becomes

63

j», = f (x, ; ﬁg) exp{zi,). The denominator presents the predicted test score excluding the

efﬁciency residual term, which is the optimal test score when no inefﬁciency occurred,
and which is similar to the denominator in equation (2.2’). Table 5 shows that the
estimated average efﬁciency are similar across different speciﬁcation of the functional

form for efﬁciency term (half-normal, truncated-normal, and exponential).

64

Table 5: Estimated Efﬁciency Distribution for Different Models

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model 1 1 Model 2 1 Model 3
Half-Normal
x 1.0140 1.0345 1.0154
0 12.9528 13.2754 12.9215
E(u,) 7.3173 7.5740 7.3107
E(TE,.) 0.8953 0.8925 0.8956
Truncated-Normal40
u/ 0;, 0.0218 0.0170 0.009]
X 1.1161 1.1366 1.0869
0 13.5568 13.7876 13.3317
E(u,.) 7.9558 8.1465 7.7472
E(TE,) 0.8873 0.8852 0.8901
Exponential
0 0.1833 0.1757 0.1804
0', 8.9961 9.1237 8.9572
E(u,) 5.4544 5.6928 5.5424
E(TE,) 0.9196 0.9166 0.9185

 

 

 

 

 

 

‘0 Greene (1993) claimed that although truncated-nonnal beneﬁts by relaxation of a possibly
erroneous restriction (,u =0), the cost appears to be that the log-likelihood is relatively ﬂat in the
dimension of p.. Therefore, estimation of a nonzero ,1: often inﬂates the standard errors of the
other parameters considerably, and quite frequently impedes or prevents convergence. We also
encounter the same problem when estimating the individual value of efficiency, which is assumed
as truncated-normally distributed. The results produced by LIMDEP for truncated normal model
routinely denied convergence and, as often as not, produced nonsense estimates according to the
warning message showed by program. Therefore, we also run the same estimation by using the
FRONTIER 4.1 (Coelli, 1995) computer software. It did not produce any error message for the
truncated normal model. Besides, we found that the values of the individual efﬁciencies are the
same in the result carried out by LIMDEP and FRONT 4.1 (accurate to 4 decimal points). Hence
we still report the truncated normal result for our model.

65

There are totally 45 regressions for Mathematics and Science, respectively“.
However, only 18 regression results for Science are reported since OLS residuals do not
pass the skewness check. This implies the model is probably not well speciﬁed or the

data are inconsistent with the model.

Average efﬁciency is calculated by summing up all the efﬁciency calculated from
each of the regressions, then divided by total number of regressions estimated. We rank
all the public school districts from least efﬁcient to most efficient according to the value
of average efﬁciency, and report the 20 least efﬁcient public school districts in Table 6.
Table 6A is for Mathematics, and Table 6B is for Science. Besides, Hanushek (1986)
discussed the drawbacks of test scores as a single measure of educational output and
claimed that the perceived importance of schooling is to affect the ability of students to
perform in and cope with society after they leave school. Therefore, the percentage of
continuation of schooling (graduation rate) would be a better measure for school
performance in some extent. We also use the graduation rate as the dependent variable in
our production model for comparison, and the estimated result for efﬁciency measure is
reported in Table 6C. Finally, the 20 most efﬁcient school districts are also reported in

Appendix C.

 

“ There are 5 types of regressions for each of three models, moreover, each of them has three
alternative functional form of efﬁciency term

66

Table 6A: 20 Least Efﬁcient Public School Districts for Math

 

Avg. Relative District Percent of Schl. Relative Math

SChOOI District Name Eﬁ‘iciency"2 Size“3 In City Test Score“

 

Engadine Consolidated Schs 0.4150 0.1053 0.00 0.0000
Burr Oak Comm School Dist 0.5192 0.1084 0.00 0.2922
Waldron Area Schools 0.5501 0.1345 0.00 0.7192
City Of Muskegon Heights Sd 0.5657 0.7695 0.00 0.1445
Highland Park City Schools 0.6039 1.1274 0.00 0.2312
Benton Harbor Area Schools 0.6149 1.8050 100.00 0.1589
Fulton Schools 0.6217 0.3007 0.00 0.3211
Buena Vista School District 0.6273 0.5120 100.00 0.1445
River Rouge City Schools 0.6935 0.7910 0.00 0.2504
Deerﬁeld Public Schools 0.6951 0.1277 0.00 0.3211
Arenac Eastern School Dist 0.6997 0.1488 0.00 0.5940
Dearbom Hgts Sch Dist No. 7 0.7065 0.7528 0.00 0.2569
Baldwin Community Schools 0.7092 0.2461 0.00 0.3773
Ecorse Public School Dist 0.7128 0.3902 0.00 0.4495
Britton Macon Area Sch Dist 0.7371 0.1516 0.00 0.5153
Oak Park City School Dist 0.7446 1.1231 0.00 0.4013
Pontiac City School District 0.7569 3.9676 93.55 0.3757
Memphis Community Schools 0.7633 0.3116 0.00 1.0708
Springport Public Schools 0.7656 0.3389 0.00 0.4913
Westwood Community Schools 0.7659 0.6822 0.00 0.4880

 

 

‘2 The mean of average efficiency for math test is 0.9001, and with 10% percentile of 0.8099,
25% percentile of 0.8627.

‘3 Numbers are reported as K-12 enrollment in 1998 of each district divided by average K-12
enrollment in 1998.

‘4 Numbers are reported as HST Math test score in 1998 of each district divided by average HST
Math test score in 1998.

67

Table 6B: 20 Least Efﬁcient Public School Districts for Science

 

 

School District Name Eﬂ'léi‘gcy” RelatgiggsMCt PercitrllthfySchl. Relagsiyg 3:11:3C6
Highland Park City Schools 0.0000 1.1274 0.00 0.0000
Burr Oak Comm School Dist 0.3099 0.1084 0.00 0.0000
Engadine Consolidated Schs 0.3100 0.1053 0.00 0.0000
City Of Muskegon Heights Sd 0.4163 0.7695 0.00 0.0839
Inkster City School District 0.4343 0.5589 0.00 0.1175
Godfrey Lee Public Sch Dist 0.4498 0.3871 0.00 0.1548
Coloma Community Schools 0.4928 0.7298 0.00 0.2667
Marenisco School District 0.4937 0.0367 0.00 0.2667
Willow Run Community Schools 0.5347 1.0361 0.00 0.2387
Webberville Community Schs 0.5448 0.2426 0.00 0.3897
Fulton Schools 0.5471 0.3007 0.00 0.3729
Buena Vista School District 0.5542 0.5120 100.00 0.1883
Akron Fairgrove Schools 0.5775 0.1553 0.00 0.4084
Benton Harbor Area Schools 0.5902 1.8050 100.00 0.2200
Mason Cons School District 0.5928 0.4837 0.00 0.3767
Detroit City School District 0.6160 54.2843 94.59 0.2461
Kingsley Area School 0.6230 0.3974 0.00 0.4662
Oak Park City School Dist 0.6254 1.1231 0.00 0.2816
T ahquamenon Area Schools 0.6429 0.3821 0.00 0.4736
Dearbom City School Dist 0.6475 4.9329 96.30 0.5128

 

 

4’ The mean of average efﬁciency for science test is 0.8771, and with 10% percentile of 0.7462,
25% percentile of 0.8292.

4° Numbers are reported as K-12 enrollment in 1998 of each district divided by average K-l2
enrollment in 1998.

‘7 Numbers are reported as HST Science test score in 1998 of each district divided by average
HST Science test score in 1998.

68

Table 6C: 20 Least Efﬁcient Public School Districts for Graduation Rate

 

 

 

School District Name Avg. Relative District Percent of Schl. Relative
Efﬁcrency"8 Size49 In City Graduation Rate50

Lansing Public School Dist ' 0.4569 5.9181 166.067 ‘ " ”W6 136?”
Inkster City School District 0.5022 0.5589 0.00 0.4590
Galien Township School Dist 0.5930 0.1417 0.00 0.6428
Colon Community School Dist 0.6030 0.2986 0.00 0.6475
Eau Claire Public Schools 0.6348 0.2774 0.00 0.6760
Ecorse Public School Dist 0.6357 0.3902 0.00 0.6025
Mancelona Public Schools 0.6360 0.3408 0.00 0.6819
Jackson Public Schools 0.6600 2.3788 100.00 0.6736
River Rouge City Schools 0.6615 0.7910 0.00 0.6392
Fennville Public Schools 0.6784 0.5008 0.00 0.7009
River Valley School District 0.6805 0.4110 0.00 0.7507
Lakeville Comm School Dist 0.6871 0.7105 0.00 0.7353
Hale Area Schools 0.6971 0.2554 0.00 0.7566
Grand Haven City School Dist 0.7046 1.8836 0.00 0.7673
Alcona Community Schools 0.7060 0.3287 0.00 0.7732
Orlaway Area Comm School 0.7081 0.2973 0.00 0.7626
giaslievue Comm Sch Dist 0.7121 0.3203 0.00 0.7768
Huron School District 0.7181 0.6036 0.00 0.7697
Saranac Community Schools 0.7187 0.3821 0.00 0.7780
South Haven Public Schools 0.7217 0.8727 0.00 0.7448

 

 

‘8 The mean of average efﬁciency for graduation rate is 0.9135, and with 10% percentile of
0.7965, 25% percentile of 0.8555.

‘9 Numbers are reported as K-12 enrollment in 1998 of each district divided by average K-12
enrollment in 1998.

5° Numbers are reported as graduation rate in 1998 of each district divided by average graduation
rate in 1998.

69

Detroit is the largest urban school district in Michigan and has been the focus of
much policy debate, hence it may worth our additional attention. Departing from the
common perception that Detroit is in a complete disaster for education, Detroit does not
show up in the list of 20 least efﬁcient public school for HST math result. The average
efficiency measure of Detroit for Mathematics is 0.7898, and with the rank of 33. For
science test score, Detroit performs much worse with an average efﬁciency measure of
0.6160, and a rank of 16. For both the HST math and science results, Detroit’s measure
of efﬁciency is roughly twice as large as the least efﬁcient one. Besides, when we measure
district performance by using graduation rate, Detroit even performs very well, the average

efﬁciency measure for graduation rate is 0.9450, and with a rank of 357. Besides, from
Appendix D we see that Detroit enter the list of 20 least efﬁcient school districts for HST
math result only once in total 45 regressions. However, it shows up most of time in the
list of 20 least efﬁcient school districts for HST science result (17 out of 18 regressions).
Finally, Detroit show good performance in graduation rate again. It did not enter the list
of 20 least efﬁcient school districts for graduation rate, and with an average ranking of

343.74. We report the relative measure of efﬁciency for Detroit in Table 7.

70

Table 7: Relative Measure of Efﬁciency for Detroit

 

 

 

 

 

 

Regression Half Normal Truncated- Exponential
Normal
HST Mathematics, 1998
Model 1
Type 3 0.8399 0.8385 0.8747
Type 4 0.8396 0.8349 0.8694
Model 2
Type 3 0.8222 0.8208 0.8690
Type 4 0.8243 0.8234 0.8656
Model 3
Type 3 0.8300 0.8299 0.8705
Type 4 0.8288 0.8162 0.8679
HST Science, 1998
Model 1
Type 3 0.6903 0.6994 0.7405
Type 4 0.6756 -4.8621 0.7441
Model 2
Type 3 0.7110 0.7100 0.7735
Type 4 0.7030 0.6992 0.7680
Model 3
Type 3 0.7049 0.7019 0.7660
Type 4 0.7080 0.7045 0.7706
Graduation Rate, 1998
Model 1
Type 3 1.0665 1.0668 1.0336
Type 4 1.0682 1.0688 1.0361
Model 2
Type 3 1.0714 1.0721 1.0310
Type 4 1.0720 1.0710 1.0345
Model 3
Type 3 1.0666 1.0710 1.0296
Type 4 1.0728 1.0786 1.0342

 

 

Note: Number reported as measure of efﬁciency divided by average measure of

efﬁciency in each regression.

71

Roughly speaking, Detroit is relatively inefﬁcient in the production function of
schooling in our model, however, it did not perform as bad as people think. The
difference between the regression result and the common knowledge maybe comes from
the different measure of academic performance. Usually, people use the test score to
justify whether a speciﬁc school district performs well or bad. However, as we
mentioned in section I, if the characteristics of students are different, then the
performance of test score may not only result from the school inputs. We examine the
correlation coefﬁcient between the test result and the efﬁciency measure, as well as the
correlation coefﬁcient between the graduation rate and the efﬁciency measure. Table 8
shows the correlation coefﬁcient between the HST mathematics result in 1998 and the
estimated efﬁciency is around 90 percent when we use half-normal or truncated normal
model. However, the correlation coefﬁcient decreases when an exponential model is
used (up to 12 percentage points). This is also the case for HST mathematics result in
1998, and for most of the HST science result. On the other hand, we found the
correlation coefﬁcient is roughly 95 percent between the graduation rate and efﬁciency
measure, and this highly correlated property may suggest that graduation rate is a better

index for district’s academic performance then test score.

72

Table 8: Correlation Coefﬁcients between Educational Achievement and Efficiency

 

 

 

 

 

 

 

Efﬁciency Half Normal Truncated-Normal Exponential
HST Mathematics, 1998
Model 1
Type 3 0.9120 * 0.9100 * 0.6908 *
Type 4 0.9136 * 0.9125 * 0.8428 *
Model 2
Type 3 0.9087 "‘ 0.9061 * 0.8454 *
Type 4 0.9108 * 0.9075 * 0.8526 *
Model 3
Type 3 0.9061 "‘ 0.9034 * 0.8438 *
Type 4 0.9084 * 0.9085 * 0.8517 *
HST Science, 1998
Model 1
Type 3 0.4939 * 0.5483 * 0.7398 *
Type 4 0.0370 0.1352 * 0.6995 *
Model 2
Type 3 0.9106 * 0.9094 * 0.8394 *
Type 4 0.9195 * 0.9149 * 0.8427 *
Model 3
Type 3 0.9104 * 0.9105 * 0.8366 *
Type 4 0.9105 * 0.9109 * 0.8412 *
Graduation Rate, 1998
Model 1
Type 3 0.9436 * 0.9425 "‘ 0.9099 *
Type 4 0.9441 0.9423 * 0.9125 *
Model 2
Type 3 0.9554 * 0.9541 * 0.9460 *
Type 4 0.9560 * 0.9543 * 0.9207 *
Model 3
Type 3 (1998) 0.9535* 0.9548 * 0.9144 *
Type 4 (1998) 0.9560 * 0.9574 * 0.9202 *
Note: Asterisk (*) indicates signiﬁcance at the 95 percent conﬁdent level.

73

Table 9 reports the comparison of estimated efﬁciency in different dimensions.
Except the category of city (urban district is deﬁned as when at least half of schools are
located in city), all other categories divide 511 school districts into two even parts by
ordering from least to best and cutting right in half. We make the dummy variables for
all categories, and run OLS regression on those dummy variables in each dimension once
a time. Although the efﬁciency measure is estimated by assuming they are distributed
half-normally, truncated normally or exponentially, the OLS estimates are still the best

unbiased estimates because our sample size is large enough (511 observations).

We found that urban districts always have lower estimated efﬁciency. The
difference of estimated efﬁciency between urban districts and non-urban districts are
signiﬁcant, and with the value of 3.65 percent. Besides, the poorer district (no matter
measured by per capita income or the percentage of children in poverty) tends to have
lower efﬁciency and the difference of 2.16 percentage points is signiﬁcant. The adult
educational level is also an important factor of efﬁciency in our model. The districts with
higher-educated parents are more efﬁciency for Mathematics. The difference of
estimated efﬁciency is signiﬁcantly large, which is 3.18 percentage points. The other
factors, which caused signiﬁcant difference of efﬁciency, are K-12 enrollment,
percentage of children who are black and total expenditure. Larger districts, the districts
with higher percentage of black children, and the districts with more total expenditure

tend to be more inefﬁcient, and the decreased efﬁciency is around 1 percentage point.

74

Table 9: Comparison of Estimated Efﬁciency (HST Math)

 

Variable

 

City
K-12 Enrollment, 1998
Enrollment Density, 1998
Percentage of Children in Poverty
Percentage of Children in Black
Percentage of Children in Hispanic
Average Schooling Years

Per Capita Income, 1989
Pupil-Teacher Ratio, 1998

Total Expenditure, 1998

Estimated Efﬁciency

 

 

Urban

0.8506
Large
0.8882
High Density
0.8845
Poor
0.8725
Black
0.8782
Hispanic
0.8861
High Education
0.8992
Poor
0.8719
Large Class
0.8850
High Exp.
0.8781

Rural _. .. . __ .

0.8855
Small
0.8784
Low Density
0.8820
Rich
0.8941
Non-Black
0.8884
Non-Hispanic
0.8805
Low Education
0.8673
Rich
0.8946
Small Class
0.8816
Low Exp.

0.8885

Difference

--.tStd; Dew)--.

0.0365“
(0.0115)

0.0098“
(0.0056)

0.0025
(0.0056)

0.0216"
(0.0055)

0.0101*
(0.0056)

0.0056
(0.0056)

0.0318"
(0.0054)

0.0227"
(0.0055)

0.0034
(0.0056)

0.0103*
(0.0056)

 

Note: Number in parenthesis represents standard error.

One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (") indicates signiﬁcance at the 95 percent conﬁdence level.

75

Other than comparing the mean of efﬁciency measure for each categories, we can
also specify the efﬁciency measure as the function of some explanatory variables. That is,
instead of treating efﬁciency measure as an independent random variable, we doubt it is
correlated with some of the explanatory variables. FRONT41 provides this function, and
the result is reported in Table 10. In Table 10, we specify city, total expenditure, poverty
rate, and K-12 enrollment may explain the efﬁciency measure, and we found that none of
them have signiﬁcant impact on efﬁciency measure. Moreover, the coefﬁcients lose their

signiﬁcance when they simultaneously explain both test score and efﬁciency measure.

76

TablelO: MLE Estimates for Technical Efﬁciency Models for Math Test

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model 1
Math Score Efﬁciency
City 0.0432
” (0.0520)

Per Capita Income, 1989 -0.2074

(0.2955) "
Percentage of Children in -0.0712 0.2841
Poverty, 1989 (0.1436) (0.2572)
Percentage of Children Black, -0.3540**
1989 (0.0602) “
Percentage of Children -0.3560**
Hispanic, 1989 (0.1761) “
Educational Herf. Index, -0.1499
1989 (0.2169) "
Avg. Schooling Years, 1989 8.5617"

(0.8745) --
K-12 Enrollment, 1998 -0.0210** -0.5284

(0.0617) (0.4998)
Enrollment Density, 1998 -0.0130**

(0.0055) --
Pupil-Teacher Ratio, 1998 -1.0672**

(0.2801) "
Total Expenditure, 1998 -0.9931 -0.2425

(0.8429) (0.7367)
Average Teacher Salary, 0.6251 **
1998 (0.1399) "
Special Educational -0.8340
Expenditure, 1998 (0.9993) --
Constant -27.5924** -0.0595

(1 .0000) (1.0076)
Note: Number in parenthesis represents standard error.

 

One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

77

IV. Conclusion

In our model, we use the stochastic production frontier models to estimate
efﬁciency for public school districts in Michigan. Three different speciﬁcations for the
efficiency term including the half-normal, truncated-normal, and exponential are used.
By estimating the measure of individual efﬁciency for HST math and science
achievement, we ﬁnd that urban school districts are signiﬁcantly less efficient than non-
urban school districts. Besides, the school districts with higher percentage of children in
poverty, higher percentage of children who are black, larger enrollment (large districts),

parents who are lower-educated, and larger total expenditure are less efﬁcient.

We also found that the correlation coefﬁcient between the test score and the
efﬁciency measure for HST math result is relatively high, however, it is not so high
between the test score and some of the efficiency measures for HST science result.
According to the test score, Detroit is pretty inefﬁcient in both math and science, with
rank of 6 to 10. Nevertheless, according to the measure of efficiency, although Detroit is
relatively inefﬁcient in the production of schooling, especially for HST science result, it
is not the worst one. Moreover, the average measure of efﬁciency of Detroit is roughly

twice as large as the least efﬁcient one.

In the face of the evidence presented in this paper, it seems inappropriate to single
out the school district of Detroit for special institutional reform. If special institutional
reforms are applied to any district they should be applied to those districts who have the

lower score in our measure of efﬁciency.

78

APPENDICES

79

Appendix A: Public School Districts with Missing Data

Table A: Public School Districts with Missing Data

 

Relative Relative

Public School District Name District Size" Publlc School Distrlct Name District Size”

 

Autrain-Onota Public Schools 0.0149 Bloomﬁeld Twp S D 7f 0.0075
Burt Township School Dist 0.0236 Colfax Twp Sch Dist 1f 0.0047
Saugatuck Public Schools 0.2153 Sigel Twp School District 3 0.0062
Ganges School District No.4 0.0081 Sigel Twp School District 4 0.0062
Arvon Township School Dist 0.0084 Sigel Twp School District 6 0.0050
Hagar Township School Dist 6 0.0168 Verona Twp Sch Dist No If 0.0068
Sodus Twp Sch Dist 5 0.0245 Stockbridge Comm Schools 0.5577
Mar Lee School District 0.0811 Palo Comm School District 0.0612
Beaver Island Comm Schools 0.0283 Berlin Twp School District 3 0.0056
Whiteﬁsh Schools 0.0286 Easton TWp School District 6 0.0099
Crawford Ausable Schools 0.7046 Ionia Twp School District 2 0.0093
Breitung Twp School District 0.6940 Excelsior District #1 0.01 15
Maple Valley School District 0.5052 Grant Township Schools 0.0031
Oneida Twp School District 3 0.0059 Bois Blane Pines Sch Dist 0.0016
Loucks School-Roxand #12 S/D 0.0025 Moran Township School Dist 0.0401
Littleﬁeld Public Sch Dist 0.1491 Powell Township School Dist 0.0245
Genesee School District 0.3014 Wells Township School Dist. 0.0143
Wakeﬁeld Twp School Dist 0.1106 Holton Public Schools 0.3936
Litchﬁeld Community Schools 0.2038 Pineview School District NA52

Elm River Twp School Dist 0.0056 Big Jackson School District 0.0118
Stanton Twp School District 0.0572 Ferry Community School Dist NA”

Church School District 0.0075 Nottawa Community School 0.0522
Bloomﬁeld #1 School District 0.0000 Bangor Twp School District 8 0.0065

 

 

" Numbers are reported as K-12 enrollment in 1998 of each district divided by average K-12
enrollment in 1998.

’2 Missing Data.

80

Appendix B: MLE Estimates for Different Models for Math Test

Table B: MLE Estimates for Different Models for Math Test

Observation: 511 public school districts in Michigan
Output Variable: Math Grade 11 High School Test, 1999

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Model 1 Model 2 Model 3
Per Capita Income, 1989 -0.l613 -0.2564 -0.2760
(0.38143 (0.3970) (0.4015)
Percentage of Children in Poverty, 1989 -0. 1951 -0.3419 -0.2534
(0.0937) "”" (0.0933) ** (0.0967ﬁ"
Percentage of Children in Black, 1989 -0.3334 -- --
(0.0485) *" -- --
Percentage of Children in Hispanic, 1989 -0.3618 -- --
(0.1678 *"' -- --
Racial Herf. Index, 1989 -- 0.1911 0.1970
-- (0.0516)“ (0.0515) **
Educational Herf. Index, 1989 -0.2206 -0.1629 -0.1667
(0.2561) (0.2585) (0.2578)
Avg. Schooling Years, 1989 8.2446 8.3192 8.9857
(1.7712) *"‘ (1.8434) *‘1‘ (1.9701) **
K-12 Enrollment, 1998 0.0514 -0.0305 0.0047
(0.1832) (0.2654) (0.2499)
Enrollment Density, 1998 -0.01 16 -0.0154 -0.0176
(0.0067) "‘ (0.0060) ** (0.0059) **
Pupil-Teacher Ratio, 1998 -1.0593 -l.4308 -10.0091
(0.3296) " (0.3457) ** (2.2744) **
Pupil-Teacher ratio (Square), 1998 -- -- 0.1967
-- -- (0.0508) **
Total Expenditure, 1998 -0.9703 -l.5360 -3.4421
(0.7824) 40.9631) (4.6889)
Total Expenditure (Square), 1998 -- -- 0.0018
-- -- (0.3255)
Average Teacher Salary, 1998 0.6469 0.7695 4.9325
(0.1318) "”" (0.1330) ** (1.1341)"
Average Teacher Salary (Square), 1998 -- -- -0.043O
-- -- (0.0122) **
Special Educational Expenditure, 1998 -l.0690 -0.4224 1.8380
(3.5811) (3.5664) (10.6519)
Special Educational Expenditure -- -- - l .4590
(Square), 1998 -- -- (10.6194)
Constant -22.2901 -34.2018 -40.6176
(25.7425) (27.4516) (41.2992)
Note: Number in parenthesis represents standard error.

 

One asterisk (’) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

81

Appendix C: 20 Most Efﬁcient Public School Districts

Table Cl: 20 Most Efﬁcient Public School Districts for Math

 

 

N... 2.3:...” “3°“
Osceola Twp School District 0.9458 0.0935 0.00 1.2843
Croswell Lexington Comm Sd 0.9485 0.7876 0.00 1.1286
Holland City School District 0.9488 1.7634 0.00 1.2522
New Lothrop Area Public Sd 0.9502 0.2666 0.00 1.2667
Cadillac Area Public Schools 0.9502 1.2188 0.00 1.2442
Pickford Public Schools 0.9509 0.1563 0.00 1.4721
Brown City Comm School Dist 0.9518 0.3492 0.00 1.3228
Fowler Public Schools 0.9520 0.1553 0.00 1.3277
Alba Public Schools 0.9532 0.0584 0.00 1.1238
Ludington Area School Dist 0.9537 0.8572 0.00 1.4208
Rapid River Public Schools 0.9539 0.1637 0.00 1.2667
Charlevoix Public Schools 0.9542 0.4182 0.00 1.4095
Public Schools Of Calumet 0.9544 0.5412 0.00 1.3100
Dowagiac Union Schools 0.9544 0.9550 0.00 1.2795
Hamilton Community Schools 0.9545 0.6894 0.00 1.4079
Onekama Consolidated Schools 0.9546 0.1612 0.00 1.2618
N.I.C.E. Community Schools 0.9555 0.4384 0.00 1.4593
Saranac Community Schools 0.9575 0.3821 0.00 1.3453
Peck Community School Dist 0.9597 0.1793 0.00 1.4111
Zeeland Public Schools 0.9608 1.3160 0.00 1.3373

 

 

’3 The mean of average efﬁciency for graduation rate is 0.9001, and with 10% percentile of
0.8099, 25% percentile of 0.8627.

5" Numbers are reported as K-12 enrollment in 1998 of each district divided by average K-12
enrollment in 1998.

’5 Numbers are reported as HST Math test score in 1998 of each district divided by average HST
Math test score in 1998.

82

Table C2: 20 Most Efﬁcient Public School Districts for Science

 

 

School District Name Qggéiency“ lsllegstive District 13:22:? of Schl. ﬁgiat;:::cience
Wyoming Public Schools 0.9439 1.7637 0.00 1.3836
Kaleva Norman Dickson Schs 0.9440 0.2874 0.00 1.3463
North Central Area Schools 0.9445 0.1855 0.00 1.3855
White Pine School District 0.9447 0.0510 0.00 1.4657
Rapid River Public Schools 0.9470 0.1637 0.00 1.4507
Gladwin Community Schools 0.9473 0.6633 0.00 1.3985
Saranac Community Schools 0.9477 0.3821 0.00 1.4433
Mid Peninsula School Dist. 0.9489 0.1109 0.00 1.4433
Brown City Comm School Dist 0.9490 0.3492 0.00 1.4153
Caseville Public Schools 0.9492 0.0935 0.00 1.4918
Charlevoix Public Schools 0.9496 0.4182 0.00 1.5570
Pentwater Public School Dist 0.9501 0.1156 0.00 1.5104
Public Schools Of Calumet 0.9509 0.5412 0.00 1.4582
Watersmeet Twp School Dist 0.9510 0.0596 0.00 1.5533
Mio Au Sable Schools 0.9513 0.2936 0.00 1.4414
Marlette Community Schools 0.9550 0.4499 0.00 1.5197
Onekama Consolidated Schools 0.9551 0.1612 0.00 1.5981
Pickford Public Schools 0.9581 0.1563 0.00 1.5533
Freeland Comm School Dist 0.9592 0.4595 0.00 1.6279
Bark River Harris Sch Dist 0.9600 0.1796 0.00 1.6447

 

 

’6 The mean of average efﬁciency for science test is 0.8771, and with 10% percentile of 0.7462,
25% percentile of 0.8292.

57 Numbers are reported as K-12 enrollment in 1998 of each district divided by average K-12
enrollment in 1998.

’8 Numbers are reported as HST Science test score in 1998 of each district divided by average
HST Science test score in 1998.

83

Table C3: 20 Most Efﬁcient Public School Districts for Graduation Rate

 

 

isiissiisiis 111......» 1:11; “5:“- W
Grant Public School District 0.9763 0.7652 0.00 1.1409
Suttons Bay Public Sch Dist 0.9765 0.3330 0.00 1.1717
Glen Lake Community Sch Dist 0.9766 0.2805 0.00 1.1860
Bear Lake School District 0.9770 0.1025 0.00 1.1860
North Central Area Schools 0.9771 0.1855 0.00 1.1860
Posen Cons School District 0.9773 0.1202 0.00 1.1860
Coloma Community Schools 0.9775 0.7298 0.00 1.1634
Climax Scotts Comm Schools 0.9776 0.2203 0.00 1.1860
Atlanta Community Schools 0.9779 0.1789 0.00 1.1611
Beal City School 0.9780 0.1839 0.00 1.1860
Chippewa Valley Schools 0.9781 3.3537 0.00 1.1622
Pellston Public School Dist 0.9783 0.2308 0.00 1.1860
Sandusky Comm School Dist 0.9786 0.4579 0.00 1.1860
Adams Twp School District 0.9787 0.1656 0.00 1.1860
Godfrey Lee Public Sch Dist 0.9790 0.3871 0.00 1.1373
Akron F airgrove Schools 0.9800 0.1553 0.00 1.1860
Vanderbilt Area School 0.9801 0.1031 0.00 1.1860
Caro Community Schools 0.9802 0.7456 0.00 1.1765
Westwood Heights Sch Dist 0.9852 0.3725 100.00 1.1717
Walkerville Rural Comm Sd 0.9857 0.1274 0.00 1.1860

 

 

’9 The mean of average efﬁciency for graduation rate is 0.9135, and with 10% percentile of
0.7965, 25% percentile of 0.8555.

6° Numbers are reported as K-12 enrollment in 1998 of each district divided by average K-12
enrollment in 1998.

6' Numbers are reported as graduation rate in 1998 of each district divided by average graduation
rate in 1998.

84

Appendix D: Counts for Appearing in the 20 Least Public School Districts

Table D: Counts for Appearing in the 20 Least Public School Districts

 

Counts of Appearing in the 20 Least Efﬁcient Public School Districts for

 

 

School District Name Math Test
Math Science Grad. rate

(efﬁciency rank ‘2 (efﬁciency rank) (efficienchank)
Addison Comm Schools 8 (147, 18, 67.51) 0 (241,214, 228.44) 0 (277, 253, 265.70)
Akron Fairgrove Schools 0 (431,82, 287.58) 18 (16,10,13.17) 0 (509, 503,507.30)
AlconaCommunity Schools 0 (368, 241,299.91) 0 (475, 443,456.33) 27 (17,13,14.96)
Arenac Eastern School Dist 36 (23, 5,1380) 0 (173,145,158.50) 0 (146,129,138.33)
Ashley Community Schools 9 (147, 12, 62.67) 0 (278, 232, 258.94) 0 (203,172,188.04)
Atlanta Community Schools 0 (111,24, 64.24) 5 (25,17,21.78) 0 (505, 493, 499.07)
Bad Axe Public Schools 13 (304,17,177.13) 0 (168,142,156.56) 0 (344,318, 334.26)
Baldwin Comm Schools 40 (25,9, 14.44) 0 (46,21,29.44) 0 (401,329, 363.52)
BeecherComm Sch Dist. 19 (112,10,41.18) 0 (129, 74, 85.56) 0 (150,71,113.78)
Bellevue Comm Sch Dist 0 (46l,13l,279.29) 0 (431,400,419.83) 27 (18,16,16.78)
Benton Harbor Area Schools 45 (13, 4, 6.84) 17 (21,11,1173) 0 (116,44, 69.44)
Britton Macon Area Sd 18 (73, 8, 37.69) 0 (115, 83, 98.00) 0 (345, 305, 323.89)
Buckley Comm Sch Dist- 4 (507,16,217.07) 0 (251,202, 227.50) 0 (267,217, 236.30)
BuenaVista SchoolDist. 42 (25,2, 8.84) 18 (18, 8, 10.83) 0 (371,154, 236.44)
BurrOak Comm Sch Dist 45 (8, 2,3.82) 18 (4,1,1.61) 0 (40, 26.33.26)
Capac Comm Sch District 18 (263,16,154.11) 0 (56,45, 50.06) 0 (312, 291,300.70)
Carrollton School District 16 (185,15, 72.38) 0 (177,116,151.50) 0 (415,378, 398.56)
Muskegon Heights Sd 45 (12, 2, 4.91) 18 (7,1,3.56) 0 (265, 86, 147.70)
Coleman Community Sd l8 (294,10,169.51) 0 (64, 52,5806) 0 (268, 249, 256.85)
Coloma Community Schools 0 (348, 172, 249.96) 18 (9, 3, 7.00) 0 (508, 487, 497.37)
Colon Comm School Dist 0 (393, 126, 286.33) 3 (25, 19, 22.11) 27 (4, 4, 4.00)
Dearbom City School Dist 0 (223, 27, 128.98) 15 (23,14,19.17) 0 (336, 249, 293.63)
Dearbom HgtsSdN0-7 18 (126,4, 56.33) 0 (30,21,25.17) 0 (229,171,190.63)
Deerﬁeld Public Schools 18 (176, 2, 67.13) 0 (60, 27, 44.50) 0 (164, 111, 141.59)
Detroit City School District 1 (56,16,4l.42) 17 (21,14,1633) 0 (380, 277, 343.74)
EauClaire Public Schools 17 (150,13, 56.04) 0 (163,85,109.39) 27 (7, 5, 5.85)
Ecorse Public School Dist 27 (62, 5, 24.53) 0 (147,103,11739) 27 (3, 5, 6.15)
Engadine Consolidated Schs 45 (2, 1, 1.62) 18 (3, 1, 1.33) 0 (279, 193, 22237)

 

85

Table D (cont’d)

 

Counts of Appearing in the 20 Least Efﬁcient Public School Districts

 

 

School District Name for Math Test
Math Science Grad. rate

(efﬁciency rank) (efﬁciency rank) (efﬁciency rank)
Fairvicw Area 8d 9 (88, 14, 44.33) 1 (28, 20, 24.78) 0 (436, 396,413.93)
Fennville Public Sch 0 (464, 45, 193.51) 0 (68, 24,4272) 27 (12,10,10.70)
Fulton Schools 45 (123,849) 18 (14,8,10.78) 0 (472, 454, 463.67)
GalienTwoSd 0 (307, 54,188.04) 0 (162, 108, 134.61) 27 (3,2300)
Godfrey Lee Public Sch Dist 0 (400, 174, 271.76) 18 (6, 3, 4.83) 0 (508,495, 503.41)
GrandHavenCity so 0 (443,410,42453) 0 (319, 306,312.67) 27 (16, 14, 14.67)
Hachrca Schools 0 (63,38,51.04) 0 (196,153,176.28) 27 (15,12,13.26)
Hamtramck Public Schools 0 (300,65, 180.38) 0 (64, 49, 57.89) 5 (35, 16, 22.56)
Hanover Horton Schools 6 (76,16,41.47) 0 (139,96, 118.00) 0 (268, 232, 249.85)
Highland Park City Schools 45 (20,1,858) 13 (511,1,147.72) 0 (226,41,102.37)
Huron School District 0 (260, 49, 131.78) 0 (172,135,151.17) 26 (21,18,18.67)
InksterCity School District 12 (201,10, 69.76) 18 (6, 4, 423) 27 (22,200)
Jackson Public Schools (241,133,177.07) 0 (178,138,159.28) 27 (9, 7, 8.41)

Kingsley Area School (424, 165, 279.91) 18 (18,13,16.67) 0 (361,337, 350.52)

0
0
LakevillcComm School Dist 0 (311,116, 225.11) 0 (182,147,166.94) 27 (13, 9,1148)
Lansing Public School Dist 0 (176, 87, 127.36) 0 (824264.72) 27 (1,1,1.00)
Lincoln Park Public Schools 8 (34,15, 38.64) 0 (69,47, 57,33) 0 (32, 54, 63.00)
Mackinac lslandPub Sch 3 (511,17,315.47) 0 (79, 27,4878) 0 (233,138,17459)
MancelonaPublic Schools 0 (101,54,7156) 0 (146,130,135.22) 27 (7, 5,6.04)
Marcnisco School District 0 (447,165,313.29) 18 (9, 3,6.39) 0 (227,159,189.30)
Mason Cons School District 0 (99, 22, 58.44) 18 (17,12,14.83) 0 (47,41,4356)
Memphis Comm Schools 27 (346, 4, 126.84) 0 (346, 285,314.00) 0 (228,199,212.33)

0

0

0

Mendon Comm Sch Dist 11 (43, 17, 28.29) 0 (183, 139, 164.11) (32, 24, 29.19)

Morenci Area Schools 1 (46, 20, 33.62) 0 (90, 76, 83.83) (104, 94, 98.59)
Oak ParkCity School Dist 23 (70, 12, 24.71) 15 (29,13,1794) (491,397, 444.41)
Onaway AreaComm Sd 0 (447,176, 309.02) 0 (276,198, 235.28) 27 (17,13,1541)
Onsted Community Schools 18 (411,13,230.62) 0 (76, 54, 66.33) 0 (240,210, 225.56)
Pellston Public School Dist 3 (342,17, 147.56) 0 (453, 441,446.61) 0 (508,495, 501.59)
Pontiac City School District 12 (48, 14, 26.07) 0 (60, 34, 47.28) 0 (51,36,4144)

 

 

62 The ﬁrst number in parenthesis represents maximum rank, the second one represents minimum
rank, and the last one represents average rank.

86

Table D (cont’d)

 

Counts of Appearing in the 20 Least Efﬁcient Public School Districts

 

 

School District Name for Math Test
Math Science Grad. rate

(efﬁciency rank) (efﬁciency rank) (efficiency rank)
River Rouge City Schools 22 (35,7,18.87) 0 (45, 26. 33.22) 27 (10, 8, 8.59)
River Valley School District 22 (192,ll,76.44) 0 (338, 272, 303.22) 27 (12,10,10.85)
Saranac Community Schools 0 (506.490.500.60) 0 (501.496.497.94) 25 (21.18.18.93)
South Haven Public Schools 0 (331,4219540) 0 (133,101,119.39) 20 (2219,2015)
Southﬁcld Public Sch Dist 5 (104.13.49.20) 5 (33, 17, 23.94) 0 (339.266.300.67)
Springport Public Schools 18 (179.7, 103.22) 1 (29, 20, 25.78) 0 (33, 26, 29.78)
Tahquamenon Area Schools 0 (129.67.102.18) 16 (23,13,18.39) 0 (123,87,108.85)
Vanderbilt Area School 26 (721034.67) 0 (104.82.91.50) 0 (509.505.507.44)
Waldron Area Schools 27 (79,1,26.13) 0 (89, 64. 73.50) 0 (143.122.129.70)
Warren Woods Public Schs 7 (118.18.61.89) 0 (69, 60, 66.06) 0 (486.460.473.04)
Webbcrville Comm Schs 18 (383. 14, 230.16) 18 (13, 7, 10.44) 5 (22, 20.21.33)
Westwood Community Schs 20 (74. 14, 29.02) 0 (46.23.3094) 0 (498.472.483.59)
WillowRunCommSchs 0 (113,25,67.33) 18 (12, 8.9.50) 0 (214,170,19848)

 

87

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88

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91

CHAPTER 3

MIGRATION AND ECONOMIC GROWTH

I. Introduction

The study of internal migration in the United States has been a long and
respectable tradition in the ﬁeld of regional development, economic growth, and labor
mobility. Early studies investigated the relationship between migration and the output
level or the rate of economic growth, while recent studies emphasized more the impact of

migration on the speed of convergence of economic growth between different areas.

There is substantial evidence that internal migration does speed up the
convergence of economic growth between different areas. Barro and Sala-i-Martin (1992)
include migration in a modiﬁed neoclassical growth model to make endogenous
migration as a source of linkage between population growth and the log of per capita
income. The estimated parameter of convergence speed is much higher in the model with
migration than the model without migration (about 30~50% higher). Persson (1994) also
found that migration has a positive though small effect on the speed of convergence in

per capita income across the twenty-four Swedish counties from 1906 to 1990.

If internal migration improves the convergence across different economies, it
implies in-migration will lower the speed of growth in the rich area, and speed up the
growth in the poor area. Out-migration has opposite effects. However, what we want to
know is how will migration affect income growth when we control for the level of

income.

92

The simple Solow model merits that the capital stock in steady state is determined
by the intersection of break-even investment ( n+g+ 5) k and the actual investment sf(k).
Other things equal, the higher the growth rate of population n, the lower the level of
capital stock in steady state is. Since output is an increasing function of capital stock,
therefore, the level of output in steady state will be lower too. To be one of the positive
components of population, net migration might have a negative relationship with the
output level in steady state. However, there are many early empirical studies ﬁnding a
positive relationship between the net migration and the level of per capita income. Okun
(1968) found that internal population migration tends to increase the rate of economic
growth of a country. Olvey (1972) also found that the in-migration responds to both
employment growth and wage levels. Greenwood (1976) provided evidence that white
and nonwhite migrants have different performance, while the former is more responsive
to the growth of job opportunities, and the latter is more responsive to high income levels

and income growth.

To explore the possible reason that explains the positive impact of migration on
the economic growth, we consider that the migrants may have different characteristics
between different groups, such as between white people and nonwhite people. There
may exist self-selection behavior that people who carry more hmnan or physical capital
are more likely to move to ﬁnd a better environment. Our data also support this possible
reason. About 80 to 90 percent of migration ﬂows is constituted by white native

population in most of states.

The other issue we are concerned with is that the net migration and the level of

per capita income may have a circular relationship. That is, the observed positive

93

relationship between the net migration and the level of per capita income may come from
the attractions of higher income level for the migrants, but not vice versa. To explore this
simultaneous relationship, a simultaneous-equations model should be adopted, such as

Greenwood (1976), Greenwood (1978), Olvey (1972), and Okun (1968).

In this paper, we will use a recently developed econometric approach “Dynamic
Panel Data Model” to estimate the GMM coefﬁcients for the relationship between
migration and growth. In Section II, we review some previous studies in the migration
model, and empirical studies of the migration and economic growth issues. Section III
introduced the methodology of dynamic panel data we used in our analysis. Section IV
describe the speciﬁcation of our model, the data set used in our model is also summarized

as well. Section V reports the regression result, and Section VI concludes.

11. Literature Review

Most of the early studies on the migration and economic growth use the cross-
sectional data model. Okun (1968), Olvey (1972), and Greenwood (1976) employed a
simultaneous equation approach for their estimation since they believed there are
causality relationships between migration and economic growth. The econometric
methodology of two-stage least squares or three-stage least squares was applied to
estimate the system of simultaneous equation. In addition, they all used the data for the

United State.

Okun (1968) focused mainly the effect of interstate migration on the inequality of

per capita income for 1940 to 1950 among the states in the United States. He claimed

94

that at that time labor was generally in short supply as a result of the World War II and its
after-effects, therefore, an inﬂow of migration may be relatively more beneﬁcial to the

receiving region than in a more normal time period.

Two of the endogenous variables in Okun’s model are net migration rate and the
change of per capita service income“. In addition to the change of per capita income, the
explanatory variables for net migration rate include the level of per capita service income,
the change of agricultural employment, and the fertility rate. Turning to the growth
equation, the level of per capita service income, racial composition of migration streams,
and age-sex composition in migration as well as the net migration are used as the
explanatory variables, in which the net migration and age-sex composition in migration
are endogenous. Age-sex composition in migration relative to that in total population is
the third endogenous variables in Okun’s model which is determined by the fertility rate

and net migration in deﬁnition.

Okun found that the states with relatively high service incomes per capita tend to
attract migrants. However, the decline of agriculture share in the labor force causes out-
migration to increase, which is in conﬂict with the fact that the net migration and change
of agricultural employment have positive correlation with each other in the data set.
Okun suggested this maybe because those factors affecting net migration of labor
between the state’s farm sector and other states are of more considerable importance than
those factors affecting net migration of labor between the state’s nonfarm sector and

other states during the 1940’s. This will contribute to a net positive correlation between

 

63 Note that Okun deﬁnes service income per capita as total income per capita minus property
income per capita.

95

 

the net migration and the change of the agricultural employment. He also found a
signiﬁcant but negative effect of migration on the growth of service income per capita.
Okun tried to explain this striking result by examining the correlation coefﬁcient between
the level of per capita service income and its growth rate. Because they are so strongly
positively correlated, Okun suggested that migrants necessarily moved in the direction of
states with both high level of income and growth rate and it is difﬁcult to separate these
two effects. He concluded that it was the high level but not the large increase in service
income per capita, which attracted migrants to the receiving states since the coefﬁcient

for the level of per capita service income is signiﬁcantly positive.

Turning to Okun’s growth equation, he found that high levels of service income
per capita tended to contribute to large increases in service income per capita. Besides,
the positive net migration also tended to contribute positively to the absolute growth in

service income per capita.

Okun’s ﬁnding concerning the income level and the growth rate is not consistent
with the literature on the convergence of economic growth. However, he did ﬁnd that net
migration and income level or income growth and net migration have positive and

signiﬁcant relationship.

Unlike Okun’s use of the state as the unit of observations, Olvey (1972) did
related empirical work using data for the 56 largest Standard Metropolitan Statistical
Areas (SMSA’s) in the United States for the period of 1955 to 1960. He deﬁned
economic growth as employment growth, and also claimed that the employment growth

and migration involve a circular relationship. In his multi-equation model, he speciﬁed

96

 

two employment growth equations in manufacturing and service sectors, three migration
equations for in-migration from contiguous states, long distance in-migration, and out-
migration, as well as three identities for prospective unemployment“, net population
growth, and growth of total employment. Besides, he used population growth without
migration, industrial composition index (employment demand side), metropolitan wage,

income level, and climate (employment supply side) to be exogenous variables.

Most of his regression coefﬁcients had expected and signiﬁcant signs. He found
that manufacturing employment growth is deterred by high wage levels and stimulated by
net population growth and a mild climate. Service employment growth is closely related
to manufacturing employment growth and to population growth and income levels. In-
migration responds to both employment grth and wage levels. Short distance or
contiguous in-migration is signiﬁcantly higher for metropolitan areas located in low-
income regions. Long distance in-migration is responsive to a mild climate as well. His
ﬁndings did support the apparent strength of the connection between regional
employment growth and migration pattern. Besides, one of the most striking results in
Olvey’s model is that the responsiveness of out-migration to prospective unemployment
is positive and signiﬁcant. Thus out-migration seems to be explained largely in terms of

push factors on the wage and employment condition at the origin.

Greenwood (1976) utilized race speciﬁc data relating to the 100 largest Standard

Metropolitan Statistical Areas (SMSA’s) in the United States for the 19505 and

developed a simultaneous-equations model of migration and urban change for white and

 

6“ Ovley deﬁned the prospective unemployment as the total increase in population less the
increase in employment, assuming zero out-migration.

97

nonwhite civilian labor force members. In his model, he believed income, employment
and unemployment are important determinants of the direction and magnitude for
migration. Hence he constructed a larger system of equations than Okun’s and Ovley’s
and included 12 endogenous variables (out-migrants, in-migrants, employment change,
unemployment change, income change, and natural population change, each for whites
and nonwhites) in his model. The most interesting ﬁnding in his regression results is that
no endogenous variables in the out-migration equations are signiﬁcant, that is, there are
no important relationships between the change of income, employment, unemployment
and the migration. On the contrary, in the in-migration equation, the regressions results
show that the growth of income and employment has positive and signiﬁcant effect on
nonwhite in-migration, while the growth of income is positive and signiﬁcant related

with white in-migration and the growth of unemployment discourages white in-migration.

There is also some evidence for the signiﬁcant relationship between migration
and economic growth from the countries other than the United States. Greenwood (1978)
estimated a simultaneous-equations model of internal migration and regional economic
growth in Mexico for 32 states for the period of 1960 to 1970. By using ordinary least
square, two-stage least squares and three-stage least squares, he estimated a system of 10
equations, which contains migration equations, labor force equations and earnings and
earnings concentration equations. Eight endogenous variables, in-migration, out-
migration, the rate of employment growth in agricultural, manufacturing, and other
sectors, the rate of change of unemployment, the rate of change of earnings, and earnings

concentration, are included in his model.

98

Greenwood used both of the growth of employment and earnings to proxy the
concept of economic growth. He found that internal migrants in Mexico are quite
responsive to employment opportunities. More job opportunities (employment growth)
attract more migrants. However, the regression result showed the effect of
unemployment rate and its growth rate on migration is positive and signiﬁcant, which is
counterintuitive. For the employment growth equation, in-migration is helpful for the
employment growth, while out-migration discouraged the employment in most of sectors.
In the earnings growth equations, in-migration has positive and signiﬁcant effect on
earnings growth, which implies labor demand shift derived from in-migration dominates

the labor supply shift (in-migration has downward pressure for wage).

lII. Methodology of the Estimation

3.1 Dynamic Panel Data Model

Recently, the panel data model has become inﬂuential in growth estimation,
compared to cross-sectional estimation in most of earlier literatures. Hsiao (1986)
claimed the major advantage of using panel data is that it provides a larger information
set than conventional cross-section data or time-series data, increasing the degree of
freedom, hence improving the efﬁciency of econometric estimates. We believe a number
of unobservable individual-speciﬁc, time-invariant ﬁxed effects, which are responsible
for individual income level in steady state exist in our model. Panel data estimation

suggests that in the presence of those ﬁxed effects, the cross-sectional error term v... can

be decomposed into two terms u,, and 77., where 77, is unobserved state-speciﬁc ﬁxed

99

factors that we mentioned above, and u... is usual error term. We assume u... to be

independently normally distributed across individuals with zero mean.

The general model to be estimated in this essay is of the following form:
P

y... = 261.32,...) +13'(L) x... + 1., +11, +1... , t = q +1,...,T, ; i = 1,... N, (3.1)
k=l

where y,, is dependent variable, that is, log of per capita income in the growth
equation and net migration rate in the migration equation. We think that the grth rate
in the current period may be partly determined by its value in last period, and net
migration rate may be also closely correlated with its previous level because the area with
more migrants may attract more people moving in than other areas. Therefore, the
dependent variable should be auto-regressive of some order. We include the lags of
dependent variable in the right hand side of the structural equation. Since this kind of

models capture the dynamic property, it is also referred as “dynamic panel data model”.

Here x,, is a vector of explanatory variables except the lags of dependent variables,
which may or may not correlated with the error term, ,B(L) is a vector of associated
polynomials in the lag operator, q is the maximum lag length in the model, and it, is time
speciﬁc effect that could capture the random shocks maybe present in a particular period.
The variable T, is the number of time periods available on the ith individual, which is

small and the number of individuals N is large. In our model, the maximum value of T, is

7 (decennial data from 1930 to 1990), and N equals 486’.

 

6’ This includes 48 states in the United States. We exclude Hawaii, Alaska, and Washington DC.
because they are very different from other states.

100

We can also write the general form of our model in the following form;
yr! =mt6+vlt2 (3'2)

where W., is a data matrix containing the time series of the lagged dependent variables,
the x’s and the time dummies. The error term is v" = 11, +111. , Which is the sum of

individual effect and usual error term.

3.2 Endogeneity Problem

Since 11, is unobservable and constant over time. even W., only contains the
exogenous explanatory variables, E(W.,v,,) will not be zero. Therefore, OLS estimates
will not be consistent. In order to remove the individual ﬁxed effect, there are several

transformation methods available. One of them is the ﬁxed effect transformation:

yr: _y1 =(VVu—VT/n)5+(n1_nt)+(uu—l71)=(I/V11_W11)8+(u11_l71)

where y, is average value of y,, across time. It is the same for W, and 17,.

However, the ﬁxed-effects transformation is not well suited because the errors

after the ﬁxed-effect transformation are u” — 17,. and E, is correlated with all W.,. Instead

of the ﬁxed effect transformation, ﬁrst differencing is preferred. After ﬁrst differencing,

the equation becomes:
yu —yl,l-l = (W11 _I’Vt,t-I)8+(utt —ut,1-l) : Ayn = AWIS+Au11°

Then we have

101

E(AVVIlAut) = 1304/1111”) + E(W.:1—1“1,1-1) — E(”/l:t-lu'l ) - E(VV1'ru1,t-1)° (33)

l I

In order to get consistent OLS estimators, we need the four terms in the right hand
side of equation (3.3) to equal zero. Generally, we always assume W., has no correlation
with the error term in past periods, which means the last term of equation (3.3) is zero.
However, the other three terms in our model are not zero. Since W., includes lags of the
dependent variable, the assumption of strictly exogeneity is not satisﬁed. In this situation,
we need to allow the existing of correlation of W., with future error term. To see how this

works, suppose W., only includes y,.,_, for simplicity.

yll = BlyIJ-I +111 +utt (3'4)

Let W., = y,',_,, then W... , and u,, are necessarily correlated because

E04]: um) = E(ynun) = E(u121) : Var(uu) > O s

,t+l

where u,, is also correlated with all other future values of W.,.

E(u,, III/”,WU_,,...,W“,11,)=0 is also known as weakly exogeneity condition.

Therefore, the third term in equation (3.3) is not zero, either.

Beside the problem of lags of dependent variables, there is another endogeneity‘56

problem with our model. Since one or more of the explanatory variables, such as net

 

6" Economically endogenous and Econometrically endogenous: The former means variables that
are jointly determined within the speciﬁcation of the economic model, while the latter one
describes the variables as they are determined simultaneously with the dependent variables, that is,
correlated with the error terms.

102

migration rate in growth equation, may be determined simultaneously with the dependent

variable, E(x,, u,,)¢0, then E( W., 1.1,.) at 0. Similarly, E( W.,, a...) ¢ 0

3.3 Choices of Instrumental Variables

In our estimation, the instrumental variable method is used to deal with the
endogeneity problem. There are many choices of instrumental variables. Anderson and
Hsiao (1982) proposed two simple choices. One possibility is to use the fact that AW,” is
uncorrelated with Au“ under weak exogeneity, and we also assume AW,” is sufﬁciently

correlated with AW”. Then the model

Ayn =AWHS+AuH (3.5)

can be estimated by pooled 2SLS using instruments AWL”. Note here t=3, T since we
lose one more observation by using instrumental variables. The second possibility to be
the instrumental variable is lagged levels of W... For instance, WW can be the
instrumental variables for AW... The disadvantage with the previous two procedures is

that they do not use all available instruments, therefore, cannot be expected to be efﬁcient.

Arellano and Bond (1991) suggested not only W.,,“ W.,,2 are good instrumental
variables, but also any further lags of W., are. They also tested several speciﬁcations for
dynamic panel data by estimating employment equation. Among eight different
equations, which includes one-step GMM, two-step GMM, Anderson and Hsiao type
estimates, OLS, within-group...etc., they found that the GMM estimators offer
signiﬁcant efﬁciency gains compared to other simpler IV alternatives, and produces

estimates that are well determined in dynamic panel data models.

103

Let us deﬁne the matrix of GMM-type instruments as follows:

 

W3
0
Z, = 0
_ 0
where W0

I!

is:

8=[(ZW°‘Z.)A~(ZZIWC°)] [28"2.>A.<zz.'y:>]

0 0
W3 0
0 W3
5 0
0 0
a(W,,,W,,,...,

where AN =(—IIVZ:Z,'H,Z,)—l a

 

0
W1.7'-1_

(3.6)

W”). The general form of linear GMM estimators of 6

l

is, ﬁrst differencing in our estimation.

For one-step estimation, we use

2
-l
0

 

—l
2

—1
O

0
—l
2

O
0

 

W i and y: denote some transformation of W, and y,, that

. c o A. A“ A. .
and if u,, are heteroskedastrc, a two-step estrmator use H, = ulu, , where u, 18

transformation of one-step residual.

For instance, the GMM-type instrumental variable for equation (3.1) is

"y” x,, x,, 0 0 0 0 0 0 0 0 0 0 0 0
Z- O 0 0 y,, yd x,, x,2 x,, O O 0 0 0 O 0 0
' ' 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 y,, y,, y,“ )1” x1,

 

Note that the most distant observation used as instrument is y,, for y and x,,, x.) for
x here since we ﬁrst-difference our original model. Therefore, one more observation is

lost.

3.4 Statistical Tests

There are several statistical tests to measure the efﬁciency of the model. The ﬁrst
one is Wald test statistics, which measure the joint signiﬁcance of explanatory variables.
The second statistics is Sargan test statistics that tests the validity of instrumental
Finally, the m2 test statistics is a test for the lack of second-order serial

variables.

correlation of the errors.

105

 

IV. Specification of the Model

In this section we apply the strategies for estimation and testing outlined earlier to
a model of migration and economic growth, using decennial data for forty-eight states in
the United States from 1930 to 1990. Alaska, Hawaii and Washington DC. are not
included in our estimation because their characteristics on economic and social aspects
are very different compared to the other states. We consider the dynamic migration and

economic growth equations in the following:

The migration equation we estimated in our model is as follows:

m” =ym +91ny..+¢X/3+N,"+nf’+u,": (4.1)

1,1—1
The growth equation we estimated in our model is as follows:

1“ yu _ In yl,l—l = 61“ le—l + Bmu—l + WXJ‘i-r + A); +rlty +u1i/ (4-2)

3111))”: alnytJ—l +Bmt,t—l +wa +1)! +111), +111): (4'3)

t,t-l

Here the subscript i indexes 48 states in the United States, the subscript t = 1930,
1940, 1950, 1960, 1970, 1980 and 1990, and X ,",'.X 5 represent the exogenous

explanatory variable matrix for the migration and economic growth, respectively.

Considering the data used in our regressions, we should distinguish the ﬂow
variables from the stock variables. Note the time subscript t in our speciﬁcation has
different meanings for ﬂow variables and stock variables. For ﬂow variables, such as the

net migration, the growth rate of population. . .etc., t represents the current decade instead

106

of a speciﬁc year. Hence, the variable with subscript t represents the ﬂow amount or the
value change during the decade. On the other hand, for stock variables, such as the
average years of schooling completed, the number of farms per person. . .etc., t represents
the speciﬁc year at when the stock variables were measured. In the following we

introduce variables used in our model, and descriptive statistics is reported in Table 1.

The main focus in this paper is on migration. We use the migration rate, which is
the percentage of net inﬂow of the migrants in total population for each state, to be the
dependent variable in the migration equation. As we mentioned earlier, the migration
rate is also one of the explanatory variables in the growth equation because we suspect
the migration rate and growth rate have causality relationship with each other. When the
migration rate is positive, the state attracts people to move in. On the other hand, when
the migration rate is negative, the state is losing residents. We can see the migration rate
is pretty high in 19403 and 19505, for both moving in and moving out. North Dakota and
South Dakota always experience large out-migration with the migration rate around
15~16% before 19505, while Florida and Nevada experience large in-migration with a
migration rate of up to 30% in some decades. We can also ﬁnd that after 1950s, both of

the lowest and highest migration rate increase a lot decade by decade.

107

Table 1: Data Statistics Summary by Decades

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Variable“ 1930 I 1940 I 1950 l 1960 j 1970 I 1980 1 1990
Area 61655.98
(46819.13)
Total Population 2547.71 2729.29 3122.90 3702.06 4195.54 4678.00 5134.42
(2529.20) (2690.55) (3101.28) (3785.45) (4361.41) (4745.77) (5505.94)
Total P0pulation Growth 7.94 15.04 18.90 13.1 1 15.81 8.71 __
(7.38) (13.96) (18.94) (12.62) (14.43) (10.97)
P0pu1ation Density 0.09 0.10 0.11 0.13 0.15 0.16 0.17
(0.15) (0.15) (0.17) (0.20) (0.22) (0.23) (0.24)
Minority Race Share 10.55 10.20 9.92 10.08 10.42 13.57 14.92
(13.22) (12.63) (11.20) (10.17) (8.86) (10.02) (9.38)
Absolute Value of the 4.37 8.35 8.66 6.33 7.64 5.40
Migration Rate (4.53) (6.98) (8.15) (5.96) (7.28) (5.19) "
Number of Farms Per 62.83 57.12 45.33 28.47 19.51 14.43 13.28
Hundred Persons (34.80) (30.76) (28.40) (21.02) (17.07) (12.98) (12.31)
ﬁgiiiﬂrig 6.94 5.96 7.62 7.46 8.33 7.77 7.32
Employees (4.66) (4.22) (5.05) (3.87) (3.50) (2.98) (2.73)
Investment in __ __ 42.41 58.29 110.73 299.75 382.81
Manufacturrng (23.49) (30.48) (42.75) (129.63) (171.08)
Percentage 0i Foreign 9.83 7.20 5.46 4.07 3.31 4.10 4.62
Born Population (7.54) (5.88) (4.44) (3.26) (2.74) (3.36) (4.38)
Per Capita Income 617.63 536.83 1401.27 2058.33 3675.81 5077.06 19351.19
(224.24) (191.59) (324.71) (407.24) (544.63) (667.40) (3202.25)
Groyvth Rate of Per -12.47 173.75 48.36 80.71 38.59 280.93 __
Capita Income (7.57) (42.88) (9.46) (13.74) (7.25) (32.84)
Percentage of Urban 46.02 47.75 55.57 61.95 65.77 66.59 67.06
Population (19.92) (18.24) (16.00) (14.85) (14.37) (14.42) (21.91)
Average Years of Adults __ 8.42 9.45 10.05 10.92 11.88 12.53
Schooling (0.85) (0.77) (0.68) (0.62) (0.55) (0.42)
:iiiertiiztiesiiixium __ 24.99 34.31 41.35 52.64 66.90 75.98
13 ducaﬂon (5.76) (7.90) (7.21) (7.89) (7.33) (5.51)
Crime Rate __ 1.71 1.59 0.92 2.31 5.43 5.05
(0.69) (0.62) (0.38) (0.89) (1.40) (1.41)
Percentage of Age of 9.69 8.56 11.20 11.63 8.54 7.54 7.60
Under 5 Pepulation (1.49) (1.49) (1.11) (0.92) (0.57) (1.18) (0.64)

 

Note: Numbers in parenthesis present standard deviation.

 

67 Detailed data deﬁnition and source please see Appendix A.

108

 

The growth variable is the other one of main variables in our model. In the
growth equation, we use the logarithm of per capita income in thousand dollars in state i
at the beginning year during a speciﬁc decade t, to be the dependent variable. It is also
one of the explanatory variables in migration equation. In some regressions, we replaced
the logarithm of per capita income by the growth rate of per capita income for
comparison. Although theoretically lny .- lny M is approximately equal the growth rate,
empirically their values are somewhat different, especially when the per capita income
grows rapidly. For example, for state of Maine, lnymo= 8.09, lnymo= 8.40, lnyI990 =9.82,
however, yg1980 = 35%, yg1990 = 317%. This makes the coefﬁcients in our estimations

using different measures of economic growth change a lot.

Note that the coefﬁcient 01 in equation (4.3) equal (1 + c) in equation (4.2). From
the previous studies on the convergence of economic growth, c is expected to be negative.
The region with low-income level grows faster than the region with high-income level.
Therefore, we expect or much smaller than 1, and if 01 approaches to 1, we will have the
unit root problem. It also means that the income level has no impact on the economic

growth rate, which is contradicted by the theories.

Besides the growth variables, several geographic variables are included in the

right hand side of the migration equation.

Persky and Kain (1970) argued that social factors such as the racial composition
are as important as the economic opportunities to explain the ﬂow of migrants between
different areas. The migrants always face the problems of assimilation in the destination

area. People might like to live with others of the same ethnic group. This is why there

109

are always Chinatown, Italiantown...etc. in the metropolitan area. Persky and Kain
(1970) also prove that the racial distribution of employment is a prime determinant of the
racial composition of migration. In our estimation, we do not have the data of racial
distribution of employment, however, we believe the racial distribution of population also
has similar impacts on the migration ﬂow. Therefore, we include the percentage of
nonwhite people in total population to be one of the explanatory variables in the
migration equation. Since the internal migration ﬂow is composed mostly of white
people, we expect a negative relationship between net migration and the percentage of

nonwhite population.

Some people argued that the characteristics of states possibly affect interstate
migrants and international immigrants differently. For example, international immigrants
tend to choose the cities with major entries points such as an international airport or
harbor as their migration destination, regardless the level of per capita income, public
security, and the geographic environment. . .etc. While the internal migrants may be more
concerned with a better living environment and more job opportunities when they make
the decision to move. What we focus on in this paper is the internal migration within the
United States, however, the migration data we have were measured as the net amount of
people who move into the speciﬁc state, regardless of point of origin. In order to separate
out the impact from international migrants, we include the number of foreign-born people

in both of the migration and growth equations.

The percentage of urban resident in total population is expected to positively
relate to the migration rate. A more urbanized area might have more job opportunities.

Therefore, it maybe attracts more people moving in. When people consider moving, the

110

amenity of an area matters. For example, the mild weather is a positive factor, while
crime is a negative one. Since the weather in an area doesn’t change much over time, the
two weather variables, highest temperature in summer and lowest temperature in winter,

are constant over time.

Now let us consider the factors affecting the growth rate. Population growth is an
important one of them. In the simple Solow model, the population growth is expected to
have a negative impact on the level of income in steady state. The accumulation of
human capital and physical capital is also a positive factor to the economic growth. The
former can be measured by the education level for a region, while we use the new capital
establishment to measure the accumulation of physical capital. We have two measures of
education level in our estimation. The ﬁst one is average years of schooling completed
for persons 25 years old and over. The other education variable is percentage of persons
25 years old and over who have at least high school education. In the modern growth
literature, accumulation of human capital is expected to have positive effect on economic

growth.

Industry decomposition may also have impacts on economic growth. The number
of farms per thousand people and the percentage of people who work for manufacturing

industry in the total population are our industry variables.

V. Regression Results

We compare the regression results estimated by different econometric methods

for migration and growth equation in Table 2A and Table 2B respectively.

111

The ﬁrst column in Table 2A and Table 2B reports coefﬁcients and standard
errors estimated by Anderson and Hsiao type of instrumental variable method, while the
other three columns are estimated by GMM method. All four estimations use the
transformation of ﬁrst differencing to remove the unobserved individual effects. In
addition, they are all estimated by using variables in levels as instruments. Moreover,
robust variance-covariance matrix is also applied except in the second column. Note that
the m2 statistics for all of four estimations are small enough that we can accept the null
hypothesis of no serially correlation between error terms. Besides, the Sargan Tests in all
GMM estimations are signiﬁcant, except the last column in Table ZB, which shows we

use the right instrumental variables.

The ﬁrst two GMM regressions are estimated by one-step procedure and the last
one is estimated by two-step procedure. However, the evidence from the simulation of
Arellano and Bond (1991) showed that the standard errors are not reliable in two-step
estimation when we have ﬁnite sample size. In their simulation, the standard errors of
two-step GMM estimates are much smaller than of one-step GMM estimates. Therefore,
they suspect that most of this apparent gain in precision may reﬂect a downward ﬁnite-
sarnple bias. We can also see the same situation occurred in our estimation. The
standard errors of the estimates from GMM two-step estimation in the fourth column are
always much smaller than that in the second and third column, which uses GMM one-

step estimation.

112

Table 2A: Migration Equation in Different Estimation Methods

 

 

 

 

 

 

 

 

 

 

Estimation Method AHi GMM] GMer GMM2
Variable Transformation First Difference First Difference First Difference First Difference
Time Period 4 4 4 4
Dependent Variable Migration Rate Migration Rate Migration Rate Migration Rate
Instrumental Variable 1y: 1 lag 1y: 2 lag 1y: 2 lag 1y: 2 lag
m: l lag m: 2 lag m: 2 lag m: 2 lag
Migration Rate in Previous 0.6030 -0.0571 -0.0571 -0.0593
Decade (0.2683) ** (0.0979) (0.0665) (0.0441)
lhiiagéhgthglgggﬂihag Year 8.5653 16.6719 16.6719 14.6598
it it it
of Current Decade (6.8970) (5.8760) (6.5940) (4.4380)
{13:33: 5]: :ggm’i‘n: Year -2.8937 12.3922 12.3922 11.5140
*# 1* t
of Previous Decade (7.2080) (5.6610) (4.1660) (2.6780) "‘
Percentage of Urban -0.2894 -0.2553 -0.2553 -0.1900
Population (0.1627) * (0.1259) ** (0.1854) (0.1338)
Crime Rate -1.9685 -2.5643 -2.5643 -2.2532
(1.1890) 1' (0.9496) ** (0.9198) ** (0.5788) **
. . -0.0701 -0.0566 -0.0566 -0.l317
Mummy Race Share (0.1933) (0.1771) (0.1101) (0.0631) ..
Percentage of Foreign-Bom 1.4470 0.3261 0.3261 0.2534
Population (0.5389) *"‘ (0.3674) (0.3331) (0.2407)
Constant -4.5875 -12.7l91 -12.7191 -11.7520
(7.5690) (6.6290) * (6.9890) * (4.4090) **
Year Dumm . 1960 7.5526 -4.9299 -4.9299 -4.8424
y' (10.7100) (8.4510) (7.4490) (4.4160)
Y ar Dumm , 1970 11.0603 8.8254 8.8254 8.6508
c y' (4.5650) .. (5.0420) * (4.2220) .. (2.3530) ..
. 1.6988 4.7840 4.7840 4.3251
Yea’ Dm‘m‘y' 1980 (7.6340) (7.7170) (6.6100) (3.9440)
Wald Test" 77.30 ** 103.7 ** 120.3 ** 246.2 **
Sargan Test" -- 53.62 ** 53.62 ’1‘" 27.25“
m2 Test -0.5158 -1.852 -1.901 -1.905

 

 

Note: Number in parenthesis represents standard error.

One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

113

 

 

Table 2B: Growth Equation in Different Estimation Methods

 

 

 

 

 

 

Estimation Method AHi GMM] GMer GMM2
Variable Transformation First Difference First Difference First Difference First Difference
Time Period 4 4 4 4
Dependent Variable Logarithm of Per Logarithm of Per Logarithm of Per Logarithm of Per
Capita Income Capita Income Capita Income Cagita Income
Instrumental Variable 1y: 1 lag 1y: 2 lags 1y: 2 lags 1y: 2 lags
m: 1 lag m: 2 lags m: 2 lags m: 2 lags
Logarithm of Per Capita 0.1235 0.1298 0.1298 0.0980
Income 10 Years Ago (0.0749) * (0.0505) 1‘" (0.0742) "‘ (0.0600) *
Migration Rate in Last 10 0.0025 0.0019 0.0019 0.0021
Years (0.0018) (0.0009) ” (0.0010) "' (0.0006) "
Natural Population Growth in -0.0423 -0.0471 -0.0471 -0.0443
Last 10 Years (0.0125) " (0.0074) " (0.0107) ** (0.0080) ‘1‘
Average Years of Adults 0.0407 0.0554 0.0554 0.0673
Schooling (0.0268) (0.0269) *s (0.0262) n (0,0211) ..
Percentage of Manufacturing -0.0003 -0.0022 -0.0022 -0.0009
Employee (0.0035) (0.0033) (0.0032) (0.0026)
Percentage of Foreign-Bom 0.0039 0.0027 0.0027 0.0025
Population in Last 10 Years (0.0038) (0.0026) (0.0035) (0.0024)
. . 0.0001 0.0001 0.0001 0.0001
lnvestrnent 1n Manufacturing (0000]) 2 (0.0001) n (0.0001) n (0.0000) n
Constant 0.3601 0.3549 0.3549 0.3731
(0.0832) " (0.0522) '1'" (0.0827) " (0.0684) '1“
. 0.1588 0.1500 0.1500 0.1319
Decade Dmmy' 1970 (0.0603) .. (0.0424) *s (0.0598) as (0.0488) 1.
_ -0.3156 -0.3472 -0.3472 -0.3414
Decade 9mm" 1980 (0.0764) .. (0.0492) n (0.0743) ** (0.0580) n
. 0.8630 0.8460 0.8460 0.8314
Decade Dummy 1990 (0.0774) H (0.0466) *s (0.0743) "1 (0.0588) as
Wald Test” 173.8 1“ 362.7 "1‘ 262.6 " 753.2 "1‘
Sargan Test" -- 32.66 " 32.66 "‘"' 20.10
m2 Test -0. 1335 -0.3592 -0.3898 -0.3713

 

 

 

 

 

Note: Number in parenthesis represents standard error.

One asterisk (‘) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (") indicates signiﬁcance at the 95 percent conﬁdence level.

114

 

In the rest of this paper, we will focus on the estimation using GMM one step
without robust variance-covariance matrix and refer it as our base regression. The second
column in Table 2A represents the estimated relationship between the migration rate and
its explanatory variables. The result suggests that the previous migration rate does not
have an important impact on current migration rate. That is, the migration rate does not
have a dynamic property. However, the previous income level and the growth rate of
income do have signiﬁcant effect on migration rate. If we transform lny and lny(-l) into
1ny-lny(-1) (growth rate), and lny (income level), we will get the coefﬁcient for growth
rate with value of —12.3922, and the coefﬁcient for previous income level with the value
of 29.0641. Therefore, the migration rate is positively related with the previous level of
logarithm of per capita income. When per capita income increases one percent, the
migration rate will increase 0.29 percentage points. However, the growth rate of income
is negatively and signiﬁcantly related with the migration rate. It shows that people do not
move to the area with high growth rate, but to the area with high level of per capita
income. It may be clear that people will choose to move into the area with higher income
level conditional on the same growth rate. However, it seems surprisingly that people

prefer the area with the lower growth rate when conditional on the same level of income.

Another signiﬁcant impact of explanatory variable comes from the crime rate.
This is a negative factor to the migration ﬂow as we expected. The GM estimates
suggest that one more percentage point of the crime rate will reduce about 2.56
percentage points of the migration rate. Besides, the percentage of urban population also
has negative impact on the migration rate in our estimation. It is a surprising result

because we expect people will move to the state with large city, which can provide more

115

 

job opportunities. On the other hand, large cites may be congested, which can generate

incentives for out-migration.

Now let us consider the growth equation. The second column in Table 2B
represents the estimated relationship between current income level and its explanatory
variables. Note that the coefﬁcient of lagged per capita income is signiﬁcantly positive
and much smaller than one. Remember that in equation (4.2) and (4.3), a (the coefficient
of lagged of lny) = 1 + c. In the third column, or = 0.1298, therefore, c = -0.8702.
Therefore, the approximate growth rate of per capita income is negatively related with the
previous level of per capita income. When the previous level of per capita income
increases one percentage point, the growth rate decreases about 0.87 percentage point.
The growth rates of different states indeed converge, as expected by the previous studies

on the convergence of the economic growth.

The migration rate is also a signiﬁcantly positive factor to the growth of income,
though the effect is not large. One percentage increase of the migration rate will increase
1.9 percent of per capita income. We also include the natural population growth in our
explanatory variable set. As the Solow model predicts, we also ﬁnd that higher natural
population growth will decrease the level of per capita income. Additions to the
population which come from immigration have a positive effect on growth, increasing
population through a rise in fertility has positive impact on the level of income. This
ﬁnding can be explained as follows: Immigrants might bring with them a diverse set of
skills which can provide a stimulus for growth. The negative relationship between

fertility and growth can be explained easily by a Solow type growth model.

116

Besides, the average year of schooling completed is also a positive factor to per
capita income in our estimation. One more year of average schooling will result in a 5.54
percent increase of income. Finally, investment plays a very important role in economic

growth. In our estimation, the coefﬁcient for investment is positive and signiﬁcant.

117

Table 3A: GMM Estimations with Different IVs for Migration Equation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Egression 1 2 3 4 5 6 7
Time period 4 4 4 4 4 4 4
Dependent . .
Variable Migratron Rate
Instrumental 1y: 2 lag m: 2 lag m: 3 lags m: 3 lags m: 3 lags m: All lags 111: All lags
Variables m: 2 lag 1y: 3 lag 1y: 2 lags 1y: 3 lags 1y: All lags 1y: 3 lags 1y: All lags
Migration Rate in -0.0571 -0.0569 -0.0629 -0.0657 -0.0612 -0.0814 -0.0768
Previous Decade (0.0979) (0.0976) (0.0968) (0.0965) (0.0962) (0.0952) (0.0949)
Logarithm of Per
gaepgtzglﬁ°£ ‘" 16.6719 16.0063 16.4933 15.6699 15.4804 15.5294 15.3221
it it it it 0! it it
Year of Current (5.8760) (5.7910) (5.8540) (5.7610) (5.7550) (5.7380) (5.7310)
Decade
Logarithm of Per
Eipétiglﬁfge '" 12.3922 12.4306 12.4402 12.4845 12.4139 12.9049 12.8360
it it It #i it it ti
Year of Previous (5.6610) (5.6390) (5.6430) (5.6190) (5.6220) (5.5860) (5.5890)
Decade
Percentage of -0.2553 -0.2613 -0.2543 -0.2655 -0.2610 -0.2667 -0.2619
Urban population (01259)" (01225)" (0.1252)M (01220)" (0.1218)" (01216)" (01214)"
Crime Rate -2.5643 -2.5829 -2.6073 -2.6512 -2.6558 -2.7169 -2.7230
(0.9496)" (09464)“ (09433)" (09392)" (09399)" (09339)" (09345)"
Minority Race -0.0566 -0.0747 -0.0552 -0.0795 -0.0840 -0.0828 -0.0876
Share (0.1771) (0.1746) (0.1766) (0.1741) (0.1740) (0.1734) (0.1733)
23:31:52: 0.3261 0.3426 0.3390 0.3593 0.3633 0.3174 0.3210
population (0.3674) (0.3637) (0.3623) (0.3594) (0.3596) (0.3562) (0.3564)
Constant -12.7191 -11.9837 -12.5244 -1 1.5872 -11.4394 -11.4591 -11.2969
(6.6290)* (6.5390)‘ (6.6000)‘ (6.5030)* (6.5020)* (6.4770)‘ (6.4760)*
Year Dummy: -4.9299 -5.3911 -5. 1205 -5.7045 -5.7330 -6.2909 -6.3322
1960 (8.4510) (8.3990) (8.4290) (8.3660) (8.3730) (8.3180) (8.3240)
Year Dummy: 8.8254 8.5341 8.7873 8.4483 8.4295 8.3167 8.2940
1970 (5 .0420)‘ (5.0190) (5.0290)* (5 .0020)* (5 .0060)‘ (4.9810)"I (4.9850)‘
Year Dummy: 4.7840 4.3478 4.7722 4.3002 4.2502 4.3125 4.2586
1980 (7.7170) (7.6830) (7.6890) (7.6530) (7.6590) (7.6240) (7.6290)
Wald Test" 103.7 " 104.0 "“" 105.0 ‘1‘ 105.1" 105.1"”'I 106.1 " 106.2"
Sargan Test” 53.62 " 54.26 "”" 54.40 ** 55.48 ** 55.67 ** 57.29 " 57.54 "'*
m2 Test -l.852 -1.822 -1.860 -l.840 -1.818 -1.888 -1.865
Note: Number in parenthesis represents standard error.

One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

118

 

 

Table 3B: GMM Estimations with Different IVs for Growth Equation

 

 

 

 

 

 

 

 

Regression 1 2 3 4 5 6 7
Time Period 4 4 4 4 4 4 4
Dependent Variable Logarithm of Per Capita Income
Instrumental Variable 1y: 2 lags 1y: 3 lags 1y: 2 lags 1y: 3 lags 1y: All lags 1y: 3 lags 1y: All lags
m: 2 lags m: 2 lags m: 3 lags m: 3 lags m: 3 lags m: All lags 111: All lags
Egggmoﬁjfg 0.1298 0.1405 0.1256 0.1339 0.1332 0.1281 0.1283
p (0.0505)" (0.0495)" (0.0497)" (0.0486)" (0.0479)M (0.0483)" (0.0476)"
Years Ago
Migration Rate in 0.0019 0.0020 0.0018 0.0021 0.0022 0.0022 0.0023
Last 10 Years (0.0009)" (0.0009)" (0.0009)‘ (0.0009)" (0.0009)" (0.0009)” (0.0009)"
2:331: 3:133] -00471 -00434 -0.0458 -0.0422 -0.0421 -00415 -00417
Years (0.0074)" (0.0074)" (0.0074)" (0.0073)" (0.0072)" (0.0072)" (0.0071)"
Average Years of 0.0554 0.0506 0.0622 0.0527 0.0516 0.0511 0.0521
Adults Schooling (0.0269)" (0.0259) * (0.0264)" (0.0254)" (0.0246)" (0.0252)" (0.0244)"
iiﬁi’éiﬁigg -00022 -00007 -0.0018 -0.0005 -0.0019 -0.0004 -0.0016
Employee (0.0033) (0.0032) (0.0033) (0.0032) (0.003 1) (0.003 1) (0.003 1)
Percentage of
Foreign-Born 0.0027 0.0027 0.0023 0.0029 0.0034 0.0034 0.0034
Population in Last 10 (0.0026) (0.0026) (0.0025) (0.0025) (0.0025) (0.0025) (0.0025)
Years
Investment in 0.0001 0.0001 0.0002 0.0002 0.0002 0.0002 0.0002
Manufacturing (0.0001)" (0.0001)“ (0.0001)" (0.0001)" (0.0001)” (0.0001)" (0.0001)"
Constant 0.3549 0.3377 0.3512 0.3402 0.3415 0.3457 0.3453
(0.0522)" (0.0516)" (0.0513)" (0.0507)” (0.0501)“ (0.0503)" (0.0498)"
Decade Dummy: 0.1500 0.1643 0.1481 0.1615 0.1618 0.1591 0.1593
1970 (0.0424)" (0.0411)" (0.0417)M (0.0404)" (0.0394)" (0.0402)" (0.0392)"
Decade Dummy: -0.3472 -0.3193 -0.3433 -0.3189 -0.3241 -0.3201 -0.3241
1980 (0.0492)" (0.0481)" (0.0484)" (0.0472)" (0.0459)" (0.0470)" (0.0457)**
Decade Dummy: 0.8460 0.8679 0.8479 0.8668 0.8646 0.8648 0.8633
1990 (00466)" (00455)" (0.0458)“ (0.0446)“ (0.0435)” (0.0444)” (0.0433)"
Wald Test” 362.7 ** 360.9 "'* 363.2 *" 363.7 " 365.8 ** 369.5 ** 370.4 *1‘
Sargan Test“ 32.66 ** 49.05 ** 39.49 ** 53.84 ’1‘ 59.84 ** 56.13 ** 62.65 **
m2 Test -0.3592 -0.3696 -0.4344 -0.4049 -0.4444 -0.3760 -0.4191

 

 

 

 

 

 

 

 

Note: Number in parenthesis represents standard error.

One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

119

 

Different sets of instrumental variables are applied in the estimations both for
migration and growth equations in Table 3A and Table 3B. The ﬁrst column is our base
regression, and more lags of the potential endogenous explanatory variables are added
into the GMM instrumental variable set for the regressions in other columns. We can see
the coefﬁcients and signiﬁcance level do not change much except some insigniﬁcant
variables, and the regression results are robust across estimations. We can also see the
Sargan test improves when more lagged dependent variables and more lagged

endogenous variables are added as the instrument variables in our regressions.

Table 4A reports the results from alternative speciﬁcations for the migration
equation. The ﬁrst column is the base regression. Since we found that the migration rate
does not have a signiﬁcant dynamic property, we remove the lag of migration rate in the
second column. The result shows that most of the coefﬁcients do not change much. The
fourth column replaces the percentage of urban population by the population density.
The results shows that people like to move to the state with lower population density and
which is parallel to our previous ﬁnding, which shows the state with smaller percentage

of urban population attracts more migrants.

The last column includes 8 regional dummies in the regression. The level of
income still has positive impact on the migration rate, and the growth of income and the
crime rate are negative factors to the migration rate. However, urban population loses its

signiﬁcance now. Besides, we ﬁnd that joint signiﬁcance decreases a lot.

120

Table 4A: Alternative Speciﬁcations for Migration Equation

 

 

 

 

 

 

 

Regression 1 2 3 4 5
Time Period 4 4 4 4 4
Dependent Variable Migration Rate in This Decade
hrs ental Variables 1y: 2 lags 1y 2 lags 1y: 2 lags ly and lyg: 2 lags 1y: 2 lags
m: 2 lagsL ' m: 2 la s m: 2 lags m: 2 la 3
Migration Rate in Previous -0.0571 __ -0.0429 -0.0648 -0.0559
Decade (0.0979) (0.0972) (0.097 1) (0. 1036)
Growth of Logarithm of Per 12 5214
Capita Income in Previous -- -- -- ' ' u --
Decade (5.6320)
ﬁgf‘gmgizerfg‘fflggxt 16.6719 17.6223 14.7487 28.8798 10.9585
Decade g g (5.8760) .. (6.0830) .. (5.8220) .. (6.6630) as (6.7630) s
50331132111131: iﬁergg‘ﬁflmm 12.3922 1 1.8908 7.1925 __ 10.0878
Previous a“; (5.6610) .. (5.6120) *s (5.4110) (5.7150) s
. -0.2553 -0.2795 -0.2572 -0.2000
Percentage °f urban P°p“'a”°“ (0.1259) *1: (0.1332) is " (0.1247) .. (0.1800)
. . -35.3526
Populatron Densrty -- -- (13.0300) u -- -
Crime Rate -2.5643 -2.2l47 -l.9026 -2.5809 -2.6681
(0.9496) " (0.9794) "“" (0.9834) "' (0.9441) ” (0.9644) "
Minority Race Share -0.0566 -0.0572 -0.1006 -0.0644 -0.0146
(0.1771) (0.1849) (0.1761) (0.1764) (0.2026)
Percentage of F oreign-Bom 0.3261 0.3905 --0.1 123 0.3303 0.5491
Population (0.3674) (0.374 1) (0.3056) (0.3642) (0.4799)
Constant -12.7l9l -l3.4007 -l3.7472 -12.3759 -5.8932
(6.6290) "' (6.8190) "‘ (6.2790) " (6.5910) " (7.5520)
Year Dumm . 1 9 60 4.9299 -3.6397 0.9207 -5.2691 -5.8369
y' (8.4510) (8.4050) (8.03 80) (8.4290) (8.5400)
Year D . 1970 8.8254 8.7494 11.4339 8.6602 7.8599
“mm" (5.0420) * (5.1530) * (4.7190) ** (5.0270) * (5.2440)
Year D _ 1980 4.7840 3.9923 7.9560 4.5859 2.6379
“"‘my' (7.7170) (7.9200) (7.2100) (7.6890) (8.1 160)
Wald Test" 103.7 " 98.42 " 111.8 Mi 104.1 in 34.67 "
Sargan Test" 53.62 in 41.57 ** 57.79 *"' 55.66 ** 51.62 **
m2 Test -l.852 -l.576 -l.7l4 -l .872 -l.828

 

 

 

 

 

 

 

Note: Number in parenthesis represents standard error.

One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (1") indicates signiﬁcance at the 95 percent conﬁdence level.

121

 

Table 4B: Alternative Speciﬁcations for Growth Equation

 

 

 

 

 

 

Regression 1 2 3 4 5 6
Time Period 4 4 4 4 4 4
Dependent Variable Logarithm of Per Capita Income
Instrumental Variables 1y: 2 lags 1y: 2 lags 1y: 2 lags 1y: 2 lags 1y: 2 lags 1y: 2 lags
m: 2 lags m: 2 lags m: 2 lags m: 2 lags m: 2 lags m: 2 lags
Logarithm of Per Capita 0.1298 0.1432 0.1668 0.0047 0.0007 0.0959
Income 10 Years Ago (00505)" (00555)" (0.0495) .. (0.0588) (0.0579) (0.0494)*
Migration Rate in Last 0.0019 0.0016 0.0023 0.0011 0.0013 0.0015
10 Years (0.0009)" (0.0008)" (0.0009)" (0.0009) (0.0009) (0.0009)*
Natural population -0.0471 -0.0486 -0.0478 -0.0459 -0.0450 -0.0450
Growth in Last 10 Years (0.0074)" (0.0069)" (0.0076) ** (0.0069)" (0.0068)" (0.0072)M
Average Years of Adults 0.0554 0.0593 __ 0.0683 0.0668 0.0350
Schooling (0.0269) ** (00264)" (00254)" (00251)" (0.0288)
Percentage of Adults
with High School -- -- (83820:) -- -- --
Education '
ﬁgigﬁuegg -0.0022 -0.0012 -0.0029 -0.0047 __ 0.0086
*#
Employee (0.0033) (0.0031) (0.0034) (0.0033) (0.0039)
Number of Farms Per -- __ __ -0.0024 -0.0019 __
Hundred Persons (0.0008) ” (0.0007) *“
girrﬁlegtjgjlggﬁﬁ‘ﬂ;t 0.0027 __ 0.0035 0.0034 0.0021 0.0050
10 Year: (0.0026) (0.0027) (0.0025) (0.0022) (0.0028) *
Growth Rate of 0 0043
Percentage of Foreign- -- (0' 0056) -- -- -- --
Born Population '
Investment in 0.0001 0.0001 0.0002 0.0001 0.0001 0.0000
Manufacturing (00001)" (00001)" (00001)" (00001)" (00001)" (0.0001)
Constant 0.3549 0.3386 0.3580 0.4284 0.4384 0.4498
(00522)" (00578)” (00532)" (0.0533) .. (00521)" (00528)"
D d D mm .1970 0.1500 0.1548 0.1858 0.0957 0.0884 0.1318
“‘1 e “ y' (00424)" (0.0441)M (00439)" (0.0424) *4 (00411)" (0.0415) **
D deD mm , 1980 -0.3472 -0.3476 -0.3244 -0.3649 -0.3597 -0.3189
“3 u y' (0.0492) .. (0.0487) .. (0.0541)M (0.0467) .. (0.0465)" (0.0503)M
D de Dumm , 1990 0.8460 0.8564 0.8690 0.7991 0.7980 0.8375
“3 Y' (00466)" (0.0494)M (0.0486)" (0.0462)" (0.0456)” (0.0475) in
Wald Test" 362.7 .. 353.1 .... 337.4 1. 421.1 "1 424.5 H 203.6 ..
Sargan Test" 32.66 ** 32.99 31.82 ** 33.77 ** 33.01 .. 26.67 *
m2 Test -0.3592 -0.5245 0.0132 0.4292 0.4457 -1 .648

 

 

 

 

 

 

 

Note: Number in parenthesis represents standard error.

One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

122

 

We also try different speciﬁcations for the growth equation in Table 4B and the
ﬁrst column is our base regression again. The second column uses the measure of
foreign-born population growth instead of the percentage of foreign-born population, and
we get very similar result as the base regression. The foreign-born population still has no
important impact on per capita income. We replace the average years of adult schooling
by the percentage of adult with at least high school education, and now the educational
variable becomes insigniﬁcant. The result that the average year of schooling completed
is positive and signiﬁcant, but the percentage of high school education is insigniﬁcant
suggests the overall level of education, instead of distribution of higher education, has

important effects on the growth of income.

In the fourth column, we add the number of farms per hundred person to proxy
composition of industry in our estimation, while in the ﬁfth column, we remove the share
of employee in manufacture and leave the number of farm per hundred person to be the
only one industry variable. Both of the columns show that the number of farm per
hundred person have negative impact on per capita income. The reason may be that our
estimation is focused on the period after 1960. During this period, agriculture is no
longer the push power for economic growth. The area with less share of agriculture,

more share of manufacture and service will have higher level of per capita income.

The 8 regional dummies are included in the last column. Now the share of
manufacture employees and percentage of foreign-born population become the positive

important factors to the income level. This may imply within the same region, different

123

share of manufacturing employee and foreign-born population determine different level

of per capita income.

Up to this point, we haven’t deal with the possible endogeneity problem caused
by causality between the migration rate and the level of per capita income. Due to the
restriction of software we used, we can not do actual two stage or three stage least square.
However, we can deal with this problem to some extent by adding migration’s
explanatory variables into the instrumental variable set of the growth equation, and do the
same thing for the migration equation, therefore, get the similar effect of simultaneous

equations.

Table 5 reports the “quasi-simultaneous” equation results. The additional
instrumental variables for per capita income in the migration equation are natural
population growth, average years of schooling, and investment. The additional
instrumental variables for the migration rate in the growth equation are the percentage of

urban population and the crime rate.

Some coefﬁcients in the migration equation change the value a lot. but are still
signiﬁcant. The reason may be that we lose one more observation by using investment as
instrumental variable for per capita income. However, we can see that the m2 test

improves a lot in the migration equation.

124

Table 5: “Quasi-Simultaneous” Migration and Growth Equation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Migration Equation Growth Equation
Instrumental Variables natural population growth, the percentage of urban
average years of population, crime rate
schooling, investment
GMM-Type Instrumental Variables 1y: 2 lag 1y: 2 lag
m: 2 lag m: 2 lag
Migration Rate in Last 10 Years -0.6196 0.0020
(0.1249) ** (0.0009) **
Logarithm of Per Capita Income in the 63.5687
Beginning Year of Current Decade (12.7300) ** '-
Logarithm of Per Capita Income in the 12.4301 0.1272
Beginning Year of Previous Decade (6.1400) ** (0.0503) **
Percentage of Urban Population -0.1600
(0.1669) --
Crime Rate -4.4001
(0.9688 ** --
Natural Population Growth in Last 10 -0.0465
Years -- (0.0074) **
Average Years of Adults Schooling 0.0553
-- (0.0268) **
Percentage of Manufacturing -0.0023
Employee -- (0.0033)
Minority Race Share 0.0947
(0.1951) --
Percentage of Foreign-Bom -0.l691
Population (0.5 109) --
Percentage of Foreign-Bom 0.0029
Population in Last 10 Years -- (0.0026)
Investment in Manufacturing 0.0002
" (0.0001) **
Constant -38.6848 0.3563
(7.1060) ** (0.0520) **
Decade Dummy: 1960
Decade Dummy: 1970 9.4034 0.1494
(5.0420) * (0.0423) **
Decade Dummy: 1980 24.3865 -0.3466
(4.4550) ** (0.0491) **
Decade Dummy: 1990 0.8461
" (0.0465) **
Wald Test ** 104.6 ** 364.7 **
Sargan Test ** 45.75 ** 33.38 **
M2 Test 0.4159 -0.3559

 

Note: Number in parenthesis represents standard error.

One asterisk (*) indicates signiﬁcance at the 90 percent conﬁdence level.

Two asterisks (**) indicates signiﬁcance at the 95 percent conﬁdence level.

125

 

VI. Conclusion

In our dynamic panel data model, the migration rate and the income growth do
have signiﬁcant relationship with each other. The high level of income instead of income
growth rate attracts people to move into that state, and the inﬂow of migrants also
increases economic growth. Results also show that per capita income does have dynamic
properties themselves, however, the migration rate does not. To deal with the dynamic
problem, we use GMM estimation and we ﬁnd that the serial correlation problem can be
improve when we employ more lags of potentially endogenous variables as the

instrumental variables.

In addition, the percentage of urban population and the crime rate are important
negative factors to the migration rate. The area with lower rate of crime committed will
lead to larger net inﬂow of internal migrants. The more surprising result is that people do
not move to the area with higher percentage of urban population or higher population

density.

126

APPENDIX

127

 

Appendix: Data Deﬁnition and Source

Area: Land area of state in 1990, measured by square mile. Source: US. Bureau of the
C ensus, collected by Statistical Abstract of the United States (Annual).

Imalﬂopulatign: In thousand persons, 1930-1990. Source: US. Bureau of the Census,
collected by Historical Statistics of the States of the United States.

Naturalﬁopulationﬁmmh: Percentage of population of age under 5 in total population,
in percent. Source: Historical Statistics of the US, Colonial Times to 1970 and

Statistical Abstract of the United States (for 1980 and 1990 data).

Populafimllensity: Total population divided by land area, in thousand people per square
mile.

MinoriILRacLShare: Percentage of nonwhite population in total population, in percent.
Source: Historical Statistics of the States of the United States.

HEW: Percentage of population in urban area in total

population, in percent. Source: Historical Statistics of the States of the United
States and State and Metropolitan Area Data Book (for 1990 data).

MigratiQrLRate: Percentage of estimated net intercensal total migration in total population,
in percent. Source: Historical Statistics of the US, Colonial Times to 1970 and
Statistical Abstract of the United States (for 1980 and 1990 data).

Remntamfﬁorelgnzﬂﬂmmmmn: In percent. Source: US. Bureau of the Census.

WW: Current foreign-born population minus

previous foreign-born population then divided by previous foreign-born
population, in percent.

WWW: Total number of farms divided by total
population, in number. Source: Historical Statistics of the States of the United
States.

128

RemenmgeJLMannfamuﬁnLEmpleeﬁsz Total employees in manufacturing industry

divided by total population. Source: Historical Statistics of the States of the
United States.

InvestmenLjnMnufacmn’ng: New capital expenditure in manufacturing industry per
capita, in thousand dollars. Source: Historical Statistics of the States of the

United States.

Wm: In thousand dollars. Source: Statistical Abstract of the United States
(Annual).

WWW: Current per capita income minus previous per capita

income then divided by previous per capita income, in percent.

W: In years. Source: Statistical Abstract of the United

States.

W: In percent. Source: Statistical

Abstract of the United States.

grim: Offenses known to the police in urban communities, measured by rate per
100 inhabitants, sum of murder, non-negligent manslaughter, forcible rape,
robbery, aggravated assault, burglary-breaking or entering, larceny theft, and auto
theft. Source: Statistical Abstract of the United States.

129

REFERENCES

130

 

REFERENCES

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Models Using Panel Data,” Journal of Econometrics, 18, pp. 247-82.

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Baltagi, Badi H., (1995) “Econometric Analysis of Panel Data,” John Wiley & Sons Ltd.

Barro, Robert J ., and Xavier Sala-i-Martin, (1992) “Convergence Across States And
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Doornik, Jurgen A., Manuel Arellano, and Stephen Bond, (1999) “Panel Data Estimation
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Greenwood, Michael J ., (1976) “A Simultaneous-Equations Model of White and
Nonwhite Migration and Urban Change,” Economic Inquiry, Vol. XIV, pp.1-15.

Greenwood, Michael J., (1978) “An Econometric Model of Internal Migration and
Regional Economic Growth in Mexico,” Journal of Regional Science, Vol. 18,
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Hsiao, Cheng, (1986) “Analysis of Panel Data,” Cambridge University Press.

Okun, Bernard, (1968) “Interstate Population Migration and State Income Inequality: A

Simultaneous Equation Approach,” Economic Development and Cultural Change,
Vol. 16, No. 2, Part I, pp. 297-313.

Olvey, Lee D., (1972) “Regional Growth and Inter-Regional Migration—Their Pattern of
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131

 

Persky, Joseph J ., and John F. Kain, (1970) “Migration, Employment, and Race in the
Deep South” The Southern Economic Journal, pp. 268-276.

Persson, Joakim, (1994) “Convergence in Per Capita Income and Migration Across the
Swedish Counties 1906-1990” Institute for International Economic Studies.

Tondl, Gabriele, (1999) “What determined the Uneven Growth of Europe’s Southern
Regions? An Empirical Study with Panel Data,” Working Paper.

Wooldridge, Jeffrey M., (2000) “Introductory Econometrics,” South-Western College
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