Empir Econ (2010) 38:23–45
DOI 10.1007/s00181-009-0254-1
ORIGINAL PAPER
Analysis of county employment and income growth
in Appalachia: a spatial simultaneous-equations
approach
Gebremeskel H. Gebremariam ·
Tesfa G. Gebremedhin · Peter V. Schaeffer
Received: 15 April 2006 / Accepted: 15 September 2008 / Published online: 31 January 2009
© Springer-Verlag 2009
Abstract County median household income and employment growth rates tend
to be characterized by spatial interaction. A spatial simultaneous-equations growth
equilibrium model was estimated using GS2SLS and GS3SLS. The results indicate
strong feedback simultaneity between employment and median household income
growth rates. They also show spatial autoregressive lag simultaneity and spatial
cross-regressive lag simultaneity with respect to employment and median household
income growth rates, as well as spatial correlation in the error terms. Estimates of
structural parameters show strong agglomerative effects and significant conditional
convergence with respect to employment growth and median household income growth
in Appalachia in the 1990 s.
Keywords Employment · Income · Spatial analysis · Appalachia
JEL Classification C3 · R1 · R5
1 Introduction
State policy makers and local leaders have long placed a high priority on local
economic development (Isserman 1993; Pulver 1989; Ekstrom and Leistritz 1988).
The changing structure of traditional industries and the impact of those changes on
local communities have challenged the efficacy of established policies and strategies.
G. H. Gebremariam
Department of Economics, Virginia Polytechnic Institute, State University, Blacksburg, USA
T. G. Gebremedhin · P. V. Schaeffer (B)
Division of Resource Management, West Virginia University, Morgantown, USA
e-mail: peter.schaeffer@mail.wvu.edu
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24 G. H. Gebremariam et al.
A better understanding of factors that influence local employment, earning capacity,
and quality of life issues has therefore become important for state, regional, and local
agencies in charge of rural development policies. One of the policy challenges at the
local and county level is spatial interdependencies. Outside of the far west, counties
are small and in most instances cannot be thought of as even rough approximations
of labor markets, and often as market areas for many consumer goods, either. This
is expressed by the mean travel time to work in a rural state such as West Virginia,
which in 2005 ranked fourteenth in the nation, with 24.9min just below the national
average of 25.1min, and ahead of more urbanized states such as Ohio and Michigan
(US Census Bureau 2005).We single outWest Virginia because of its rural nature and
because it is the only state completely contained within Appalachia. The twelve states
with counties in Appalachia have commuting times (one way) of 22.4min or higher.
Dealing with spatial interdependencies is therefore one of the major objectives of this
research.
Many of the forces responsible for past economic and social changes continue to
have an impact. One of these changes was the emergence of computer-based technology
in production, administration and information, which has reduced the role of
economies of scale in many sectors. Studies by Loveman and Sengenberger (1991)
and Acs and Audretsch (1993), for example, have shown a shift in industry structure
toward decentralization and an increased role for small firms. This was mainly due
to changes in production technology, consumer demand, labor supply, and the pursuit
of flexibility and efficiency. These factors led to the restructuring and downsizing of
large enterprises and the entry of newfirms. Brock and Evans (1989) provide extensive
documentation of the changing role of small businesses in the US economy, which are
likely the result of responses to structural adjustments.
Parallel with technical changes leading to new industrial structures, new patterns
of consumer expenditures and demand resulting from rising living standards contributed
to the emergence of fragmented consumer markets, which also favored small
consumer-oriented firms over high volume, production-oriented firms. Thus, new
business opportunities in small and medium size enterprises resulted as large firms
downsized in response to a changing environment. The emerging view among policy
makers is that small business is a key element and driving force in generating employment
and realizing economic development. This paradigm shift has brought about a
revival in small businesses promotion and entrepreneurial initiatives at local, national
and international levels.
Most new businesses start small and small businesses create the majority of new
jobs (Acs and Audretsch 2001; Audretsch et al. 2000; Carree and Thurik 1998, 1999;
Fritsch and Falck 2003; Reynolds 1994; Wennekers and Thurik 1999). A growing
literature has explored the determinants of the regional variations in new business formation
(Acs and Armington 2003; Audretsch and Fritsch 1994; Callejon and Segarra
2001; Davidson et al. 1994; Fotopoulos and Spencer 1999; Fritsch 1992; Garofoli
1994; Guesnier 1994; Hart and Gudgin 1994; Johnson and Parker 1996; Kangasharju
2000; Keeble and Walker 1994; Reynolds 1994).
The geographic impact of the change from large production-oriented plants to
smaller consumer-oriented firms and plants is uncertain.While smaller unitswould tend
to make rural production sites relatively more competitive, the consumer-orientation,
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Analysis of county employment and income growth in Appalachia 25
which tends to favor locations close to markets, is more likely to have the opposite
effect. Hence, it is not possible to predict the impact of the changes discussed above
on the geographic distribution of economic activity a priori.
The literature on economic growth at the regional level has focused attention on
the so-called convergence hypothesis of neoclassical growth theory which predicts
that poorer regions tend to catch up with the richer regions in per capita income as
time passes, through the process of factor mobility. Because of the spatial structure of
our model, we can test for convergence. Previous studies by Barro and Sala-i-Martin
(1992, 2004) for US states, Japanese prefectures and between European countries,
and by Persson (1997) and Aronsson et al. (2001) across Swedish counties, found
income evidence of convergence. Similar studies by Arbia et al. (2005) of 92 Italian
provinces (1970–2000), Ertur et al. (2006) of 138 European regions (1980–1995), and
Rappaport (1999) ofUS counties (1970–1990), also found income convergence. However,
a study by Glaeser et al. (1995) did not discover significant evidence of income
convergence between US cities. Of particular interest are two papers by Higgins et al.
(2006) and Young et al. (2008) that looked at per capita income net of government
transfers in US counties in all fifty states from 1970 to 1998. They found a speed of
convergence of between 6 to 8%, considerable faster than the approximately 2% typically
reported. Higgins et al. (2006) also found a much a faster speed of convergence
in counties located in southern than in northeastern states.
The relationship between economic growth and its determinants has been studied
extensively. One issue is whether population is driving employment changes or
employment is driving population changes (do ‘jobs follow people’ or ‘people follow
jobs’?). Empirical studies on identification of the direction of causality have resulted
in empirical models of regional development that often reflect the interdependence
between household residential choices and firm location choices (Steinnes and Fisher
1974). To account for this causation and interdependency, Carlino and Mills (1987)
constructed a simultaneous system model with two partial location equations as its
components. They used data for counties in the contiguous United States. The empirical
result from their study of greatest interest to us is the finding that in the 1970 s family
income had a strong impact on the growth of population density as well as employment
density. Recently, Deller et al. (2001) expanded the original Carlino-Mills model and
presented a three-dimensional model (jobs-people-income) that explicitly traces job
quality and the role of income in the regional growth process. They also used county
data, but restricted themselves to non-metropolitan counties; the time period studied
was 1985–1995. Their empirical results indicate that initial conditions co-determine
the eventual outcome and that counties with higher initial population levels tended to
have higher employment growth. However, counties that had higher levels of population,
employment, and per capita income in 1985 tended to have lower rates of overall
growth.
There have also been efforts to model the interactions between employment growth
and human migration (Clark and Murphy 1996; MacDonald 1992), per capita personal
income and public expenditures (Duffy-Deno and Eberts 1991), and net migration,
employment growth, and average per capita income (Greenwood and Hunt
1984; Greenwood et al. 1986; Lewis et al. 2002) in simultaneous-equations models.
Among these contributions, Clark and Murphy’s (1996) findings have been particularly
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26 G. H. Gebremariam et al.
influential. Their empirical analysis covered the period 1981–1989 and was conducted
at the county level. They expanded the Carlino-Mills model by including amenity measures
beyond climate (temperature), neighborhood poverty, and fiscal variables. Their
results are consistent with those of Carlino and Mills (1987) and, specifically, they
find simultaneity between employment density and population density.
The focus of this empirical analysis is Appalachia, a region that is for many a symbol
of poverty and underdevelopment in the midst of prosperity (Pollard 2003). It is a
region of about 23 million people. Forty-two percent of the population is rural, compared
to 20% for the nation as a whole. Many parts of the region can also be considered
remote because of topography and a comparatively poor transportation infrastructure.
Appalachia also constitutes a separate policy region, with programs administered by
the Appalachian Regional Commission. The unit of analysis is the county, so that we
can trace local economic development in terms of employment and income growth
data, respectively. The time period considered is 1990–2000. This was a decade of
economic growth and expansion in most of the United States. It is of interest to study
if and/or how the boom of the 1990 s impacted Appalachian counties.
Like the studies mentioned above, this article examines the determinants of regional
variations in employment and household income growth rates using county data.
Its novel contribution lies in a methodological innovation. Specifically, the model
introduces both spatial lag and spatial error dependence into a simultaneous equation
model and obtains estimation results using Generalized Spatial 3 Stage Least Squares
(GS3SLS). This has not been previously done and yields more efficient and consistent
estimates. The estimation strategy is discussed in the estimation issue section.
2 Method of analysis
Interdependence between employment and income exists because both households and
firms are mobile and locate to maximize utility and profits, respectively. Households
migrate if they can capture better income opportunities than those available at their
current location and firms move to be near growing markets. The location decisions
of firms are also expected to be influenced by factors such as local business climate,
labor costs, tax rates, local public services and the supply of inputs. In addition,
government-provided incentives may influence where firms locate. Such regional
factors that affect households’ and firms’ decision making are also likely to exhibit
spatial autocorrelation (Anselin 1988, 2003). These assumptions are expressed as
three hypotheses to be tested: (1) Employment growth and median household income
growth are interdependent and jointly determined by regional variables; (2) Employment
growth and median household income growth in a county are conditional upon
initial conditions of that county; and (3) Employment growth and median household
income growth in a county are conditional upon business and median household
income growth in neighboring counties. Emphasis is put on determining the linkages
between employment growth and household median income, as well as on examining
the elasticity of these variables with respect to each of the regional variables.
To test the three hypotheses, a spatial simultaneous equations model of business
growth and householdmedian income is used. Following Carlino and Mills (1987) and
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Analysis of county employment and income growth in Appalachia 27
building on Boarnet (1994), a model that incorporates own-county and neighboring
counties effects is specified as follows in matrix notation:
EMP∗
i t
= f1 MHY∗
t ,WMHY∗
t ,WEMP∗
t ,
Xem
t−1 (1a)
MHY∗
i t
= f2 EMP∗
t ,WEMP∗
t ,WMHY∗
t ,
Xmh
t−1 (1b)
EMP∗
t and MHY∗
t are the equilibrium levels of private non-farm employment and
median household income, respectively, and t denotes time. W is a row standardized
spatial weights matrix with typical element wi j . Each element wi j represents a measure
of proximity between location i and location j . We define the adjacency criteria
such that wi j equals 1/ni ; ni is the number of nonzero elements in the ith row of W.
The row element is nonzero if location i and j are adjacent and 0 otherwise.WEMP∗
t
and WMHY∗
t represent the equilibrium values of neighboring counties’ effects for
private non-farm employment and median household income, respectively. They are
obtained by multiplying EMP∗
t and MHY∗
t , respectively, with W. The matrices of
additional exogenous variables in the respective equations of the system of spatial
simultaneous equations are given by Xem
t−1 and Xmh
t−1, respectively. The descriptions
of these variables are given in the data section below. Note that equilibrium levels
of private non-farm employment and median household income are assumed to be
functions of the equilibrium values of the respective right-hand endogenous variables,
their spatial lags and the vectors of the additional exogenous variables.
The system of equations in (1a, b) captures the simultaneous nature of the interactions
between employment growth and median household income at equilibrium. The
nature of interaction among the endogenous variables depends on the initial conditions
in a county.
Based on the result of a generalized PE-test, a multiplicative log-linear form of the
model was used. The model specification is discussed in greater detail in the section
“Estimation Issues.” The chosen functional form implies constant elasticity for the
equilibrium conditions given in (1a,b). A log-linear (i.e., log-log) representation of
the equilibrium conditions can thus be expressed as:
EMP∗
t
= MHY∗
t a1 × WEMP∗
t b1 × WMHY∗
t c1 ×
K1

k=1
Xem
kt−1 x1k (2a)
MHY∗
t
= EMP∗
t a2 × WMHY∗
t b2 × WEMP∗
t c1 ×
K2

k=1
Xem
kt−1 x2k (2b)
where ai , bi and ci i = 1, 2 are the exponents on the endogenous variables and their
spatial lags, xikq for i, q = 1, 2 are vectors of exponents on the exogenous variables,
is the product operator, and Ki for i = 1, 2 is the number of exogenous
variables in the private non-farm employment and median household income
equations, respectively. The log-linear specification has the advantage of yielding a
log-linear reduced form for estimation, where the estimated coefficients represent elasticities.
Duffy-Deno (1998) and Mackinnon et al. (1983) also show that, compared to a
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28 G. H. Gebremariam et al.
linear specification, a log-linear specification ismore appropriate formodels involving
population and employment densities.
Previous empirical studies suggest that employment and median household income
likely adjust to their equilibrium levels with a substantial lag (Aronsson et al. 2001;
Barkley et al. 1998; Boarnet 1994; Carlino and Mills 1987; Deller et al. 2001; Duffy
1994; Duffy-Deno 1998; Edmiston 2004; Hamalainen and Bockerman 2004; Henry
et al. 1999, 1997; Mills and Price 1984). Therefore, based on these studies, a distributed
lag adjustment is introduced and the corresponding partial-adjustment process
for Eqs. (1a,b) takes the form:
EMPt
EMPt−1
=
EMP∗
t
EMPt−1
ηem
→ ln(EMPt )
− ln (EMPt−1) = ηem ln EMP∗
t − ηem (EMPt−1) (3a)
MHYt
MHYt−1
=
MHY∗
t
MHYt−1
ηmh
→ ln(MHYt )
− ln(MHYt−1) = ηmh ln MHY∗
t − ηmh ln(MHYt−1) (3b)
The subscript t − 1 refers to the variable lagged one period, one decade in this study,
and ηem and ηmh are parameters representing the speed of adjustment of employment
and median household income to their respective equilibrium levels. They are interpreted
as the proportions of the respective equilibrium rate of growth that were realized
in each period. If both ηem and ηmh are less than one, then the system is stable and
guaranteed to converge.
The existence of spatial autocorrelation in the errors is tested by means of a Global
Moran’s I test statistic, as suggested by Anselin and Kelejian (1997) for models with
endogenous regressors. A more general version of Moran’s I test statistic and its
asymptotic distribution is given by Kelejian and Prucha (2001). The results of the test
(Table 2) indicate the existence of spatial autocorrelation in the errors of all equations
in (3a, b). Therefore, we need a model that accounts for this spatial effect.We achieve
this by substituting Eqs. (2a, b) into Eqs. (3a, b). Eliminating the unknown equilibrium
values and simplifying the model yields the following system:
EMPRt = α1 + ηema1
ηmh
MHYRt + ηemb1
ηem
WEMPRt + ηemc1
ηmh
WMHYRt
+ηema1 ln(MHYt−1)+ηemb1 ln(WEMPt−1)
+ηemc1 ln(WMHYt−1)
+
K1

k=1
ηem x1kln Xem
kt−1 − ηem ln(EMPt−1) + uem
t (4a)
MHYRt = α2 + ηmha2
ηem
EMPRt + ηmhb2
ηmh
WMHYRt + ηmhc2
ηem
WEMPRt
+ ηmha2 ln(EMPt−1)+ηmhc2 ln(WEMPt−1)
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Analysis of county employment and income growth in Appalachia 29
+ηmhb2 ln(WMHYt−1)
+
K2

k=1
ηmhx2kln Xge
kt−1 − ηmh ln(MHYt−1) + umh
t (4b)
EMPRt and MHYRt are the log differences between the end and beginning period
values of private non-farm employment and median household income, respectively,
and denote the growth rates of the respective variables. αr and ρr, for r = 1, 2, are
unobserved parameters. uem
t and umh
t are n ×1 vectors of disturbances? Note that the
disturbance vector in the r th equation is generated as:
ut,r = ρrWut,r + εt,r , r = 1, 2
This specification relates the disturbance vector in the r th equation to its own spatial
lag. The vectors of innovations (εi t,r , r = 1,2 or εem
t and εmh
t ) are distributed identically
and independently with zero mean and variance-covariance σ2
r , for r = 1, 2.
Hence, they are not spatially correlated. The specification of the mode, however, allows
for innovations that correspond to the same cross sectional unit to be correlated across
equations. As a result, the vectors of disturbances are spatially correlated across units
and across equations.
Equations (4a, b) constitute a system of simultaneous equations with feedback
simultaneity, spatial autoregressive lag simultaneity, spatial cross-regressive lag simultaneity,
and spatial autoregressive disturbances.The endogenous variables of themodel
are EMPRt and MHYRt . If each equation is investigated separately, we notice that
each of these variables is expressed in terms of the right hand endogenous variables
and their spatial lags, the logs of the lagged endogenous variables and their spatial
lags, and the logs of other exogenous variables. By structure, the spatial lags of the
lagged endogenous variables are, however, included in the spatial lags of the respective
endogenous variables. Hence, in order to avoid multicollinearity, the model is
estimated by excluding all the spatial lags of the lagged endogenous variables.
3 Data types and sources
The data for the 417 Appalachian counties used for the empirical analysis were collected
and compiled from County Business Patterns, Bureau of Economic Analysis,
Bureau of Labor Statistics, Current Population Survey Reports, County and City Data
Book, US Census of Population and Housing, US Small Business Administration,
and Department of Employment Security. Data for county employment and county
median household income were collected for 1990 and 2000.
3.1 Dependent variables
The dependent variables used in the empirical analysis include the growth rate of
employment and the growth rate of median household income.
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30 G. H. Gebremariam et al.
3.1.1 Growth rate of employment (EMPR)
The growth rate of employment is measured by the log-difference between the 2000
and the 1990 levels of private non-farm employment, exclusive of self-employment.
Empirical research indicates that in the study period most new jobs were generated
by new small businesses (Acs and Audretsch 2001; Audretsch et al. 2000; Carree and
Thurik 1998, 1999; Wennekers and Thurik 1999; Fritsch and Falck 2003). Research
by the US Small Business Administration also shows that job creation capacity in the
US is inversely related to the size of the business. Between 1991 and 1995, for example,
enterprises employing fewer than 500 people created new jobs as follows (size
of enterprise in parenthesis): 3.843 million (1–4), 3.446 million (5–19), 2.546 million
(20–99), and 1.011 million (100–499). During the same period, enterprises employing
500 or more people lost 3.182 million net jobs (US Small Business Administration
(SBA) 1999).
3.1.2 Growth rate of median household income (MHYR)
The log-difference between the 2000 and 1990 levels ofmedian household income in a
given county is used to measure the growth rate of median household income. Median
household income is used as an average overall measure of county-level income.
Median household income is preferable to using the mean household income because
unlike the mean, the median is not influenced by the presence of a few extreme values.
The spatial lags of the Growth Rate of Employment (WEMPR) and Growth Rate
of Median Household Income (WMHYR) are included on the right hand side of each
equation of (4a, b). These spatially lagged endogenous variables are created by multiplying
each of the dependent variables by a row standardized queen-based contiguity
spatial weights matrix W.
3.2 Independent variables
The independent variables include demographic, human capital, labor market, housing,
industry structure, and amenity and policy variables. In line with the literature,
unless otherwise indicated, the initial values of the independent variable are used in
the analysis. This type of formulation also reduces the problem of endogeneity. All
the independent variables are in log form except those that can take negative or zero
values. The descriptions of each of the independent variables of the models are given
below.
Equation (4a) includes a vector of control variables (Xem
kt−1) for k = 1, . . . , K1,
which includes human capital, agglomeration effects, unemployment, and other
regional socio-economic variables that are assumed to influence county employment
growth (business growth) rate. Human capital is measured as the percentage of adults
(over 25 years old) with college degrees and above (POPCD), and the percentage of
adults (over 25 years old) with high school diploma (POPHD). It is expected that educational
attainment is positively associated with employment growth. To control for
agglomeration effects from both the supply and demand sides, county population size
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Analysis of county employment and income growth in Appalachia 31
(POPs) and the percentage of the population between 25 and 44 of age (POP25-44)
are included and it is expected that agglomeration effects to have a positive impact
on employment growth. The county unemployment rate (UNEMP) is included as a
measure of local economic distress. Although a high county unemployment rate is
normally associated with a poor economic environment, it may provide an incentive
for individuals to form new businesses that can employ not only the owners, but also
others. Thus, we do not know a priori whether the impact of UNEMP on employment
growth is positive or negative. Establishment density (ESBd), which is the total number
of private sector establishments in the county, divided by the county’s population,
is included to capture the degree of competition among firms and crowding of businesses
relative to the population. The coefficient of ESBd is expected to be negative.
Vector Xem
kt−1 also includes OWHU (owner occupied housing) to capture the effects
of the availability of resources to finance businesses and create jobs on employment
growth in the county. The percentage of owner-occupied dwellings is expected to be
positively associated with employment growth in the county. Also included in Xem
kit
are property tax per capita (PCPTAX), percentage of private employment in manufacturing
(MANU), percentage of private employment in wholesale and retail trade
(WHRT), natural amenities index (NAIX), highway density (HWD), gross in-migration
(INM), gross out-migration (OTM), median household income (MHY), and direct local
government expenditures per capita (GEX). Since the percentage of the populations
between 5 and 17 years of age (POP5-17) and above 65 years of age (POP > 65)
do not constitute the prime working age of the population, they are not included in
Eq. (4a). Direct federal expenditures and grants per capita (DFEG) in Appalachia have
been mainly income support in the form of Food Stamps, Social Security Disability
Insurance (SSDI), Temporary Assistance for Needy Families (TANF), and Supplemental
Security Income (SSI) and hence not directly related to employment creation
(Black and Sanders 2004). Homeownership (OWHU) and the social capital index
(SCIX) are highly correlated. In order to avoid the problem of multicolinearity, SCIX
is not included in Eq. (4a). SCIX is a county-level index that incorporates associational
density of associations such as civic groups, religious organizations, sport clubs, labor
unions, political and business organizations, percentage of voters who vote for presidential
elections, county-level response rate to the Census Bureau’s decennial census,
and the number of tax-exempt non-profit organizations (Rupasingha et al. 2006).
We also use the natural amenity index created by McGranahan (1999) from standardized
mean values of climate measures (January temperature, January days of sun,
July temperature, and July humidity), topographic variation and water area as proportion
of county area (see http://www.ers.usda.gov/Data/NaturalAmenities/natamenf.
xls). Note that since both SCIA and NAIX are indices of many exogenous variables,
they will constitute important parts of the instrument matrix that will be used to identify
the endogenous variables of the system.
Equation (4b) contains a vector of exogenous variables (Xmh
kt−1, k = 1, . . . , K2),
which includes, among others, POPs, POPd, FHHF, POPHD, UNEMP, MANU,
WHRT, and Social Capital Index (SCIX).
The initial levels of employment (EMPt −1) and median household income
(MHYt−1) are also included in the respective equations of (4a, b). These variables
are treated as predetermined variables because their values are given at the
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32 G. H. Gebremariam et al.
beginning of each period and hence are not affected by the endogenous variables.
Table 1 provides the full list of the endogenous, and of the spatial lag and control
variables, their descriptions and the sources of the data.
4 Estimation issues
Equations (4a, b) constitute amodel with feedback simultaneity, spatial autoregressive
lag simultaneity, and spatial cross-regressive lag simultaneity with spatially autoregressive
disturbances. This creates complications, ofwhich the choice of the functional
form of each equation, whether or not each equation is identified, and the choice of
the estimator and instruments are the most important ones.
Concerning the functional form, a generalized PE test was performed (Kmenta
1986, pp. 521–522; Mackinnon et al. 1983) to determine whether a linear or log-linear
specification is most appropriate. The test indicates that the log-linear specification is
preferred to the linear form for all equations. Thus, the model is specified in log-linear
form with two modifications involving the measurement of the explanatory variables.
First, the natural log formulation is dropped for explanatory variables that can assume
negative or zero values. Second, lagged 1990 values are used for all explanatory variables
to avoid simultaneity bias.
Concerning identification, first, for each equation, the number of basic endogenous
variables that appear on the right hand side is smaller than the number of control variables
that appear in the model but not in that equation. Second, in those cases where
there are more instruments than needed to identify an equation, a test statistic1 was
computed (Hausman 1983) to investigate whether the additional instruments are valid
in the sense that they are uncorrelatedwith the error term. That is E(Q ur ) = 0,where
Q is an instrument matrix as defined below. Fulfillment of this condition ensures that
the instrument Q allows us to identify the regression parameters [α

, β

, λ

, γ
] of
Eqs. (4a, b), where α
is a vector of slope coefficients and β

, λ

, γ
are vectors of
coefficients of the right-hand dependent variables, the spatial lag variables, and the
predetermined variables, respectively.
As to the choice of estimator, the Method of Moments is preferred over the
Maximum Likelihood approach because the latterwould involve significant additional
computational complexity.2 The conventional three-stage least squares estimation to
1 This test statistic is nR2
u , where n is the sample size and R2
u is the usual R-squared of the regression of
residuals from the second-stage equation on all included and excluded instruments. In other words, estimate
Eqs. (4a, b) by GS2SLS or any efficient limited-information estimator and obtain the resulting residuals,
ˆ ur . Then, regress these on all instruments and calculate nR2
u . The statistic has a limiting chi-squared distribution
with degree of freedom equal to the number of over-identifying restrictions, under the assumed
specification of the model.
2 In theMaximum Likelihood approach, the probability of the joint distribution of all observations is maximized
with respect to a number of parameters. This involves the calculation of the Jacobian that appears in
the log-likelihood function, which is computationally challenging. The complexity becomes overwhelming
if the sample size is large, which applies in our case, and if the spatial weights matrices are not symmetric,
which also applies in our case, even if the sample size is moderate (Kelejian and Prucha 1999, 1998).
We also do not expect the error terms in our model to be normally distributed, which is required for the
Maximum Likelihood procedure.
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Analysis of county employment and income growth in Appalachia 33
Table 1 Descriptive statistics
Variable code Variable description Mean SD Minimum Maximum
Constant 1.00 0.00 1.00 1.00
EMPR Employment Growth Rate
1990–2000
0.17 0.25 −0.69 1.79
MHYR Median Household Income Growth
Rate 1990–2000
0.48 0.31 −0.49 1.40
WEMPR Spatial Lag of EMPR 0.18 0.14 −0.18 0.81
WMHYR Spatial Lag of MHYR 0.47 0.19 −0.11 1.02
POPs Population, 1990 10.30 0.94 7.88 14.11
POPd Population Density, 1990 4.28 0.90 1.85 7.75
POP5-17 Percent of Population between
5–17Years, 1990
2.92 0.12 2.17 3.22
POP25-44 Percent of Population between
25–44Years Old, 1990
3.38 0.08 2.79 3.74
POP > 65 Percent of Population above
65Years Old, 1990
2.60 0.20 1.55 3.20
FHHF Percent of Female Householder,
Family Householder, 1990
2.32 0.20 1.81 3.19
POPHD Persons 25Years and over, % High
School only, 1990
4.10 0.17 3.57 4.47
POPCD Persons 25Years and over, %
Bachelor’s Degree or above, 1990
2.27 0.41 1.31 3.73
OWHU Owner-Occupied Housing Unit in
Percent, 1990
4.33 0.08 3.87 4.47
MHV Median Value of Owner Occupied
Housing 1990
10.74 0.26 9.67 11.68
UNEMP Unemployment Rate 1990 2.15 0.35 1.22 3.25
AGFF % Employed in Agriculture,
Forestry and Fisheries 1990
3.62 2.66 0.00 17.10
MANU % Employed in Manufacturing
1990
3.14 0.57 0.79 3.98
WHRT % Employed in Wholesale and
Retail Trade 1990
2.92 0.19 2.16 3.32
FIRE % Employed Finance, Insurance
and Real Estate 1990
1.23 0.33 0.00 2.23
HLTH % Employed Health Service 1990 1.95 0.34 0.74 3.44
NAIX Natural Amenities Index 1990 0.14 1.16 −3.72 3.55
ESBd Establishment Density 1990 2.93 0.34 1.87 4.09
EFIR Earnings in Finance Insurance and
Real Estate 1990
21075.08 96011.09 0.00 1638807.0
CSBD Commercial and Saving Banks
Deposits 1990
12.21 1.07 8.83 16.95
DFEG Direct Federal Expenditure and
Grants per Capita 1990
7.99 0.38 6.98 10.18
FGCE Federal Government Civilian
Employment per 10,000 Pop.
1990
60.48 101.03 0.00 1295.00
PCTAX Per Capital Local Tax 1990 5.91 0.53 4.51 7.42
PCPTAX Property Tax Per Capita 1990 5.52 0.62 3.91 7.36
SCIX Social Capital Index 1987 −0.60 0.94 −2.53 5.64
HWD Highway Density 1990 0.69 0.40 −0.34 2.63
123
34 G. H. Gebremariam et al.
Table 1 continued
Variable code Variable description Mean SD Minimum Maximum
ESBs Establishment Size 1990 2.53 0.30 1.49 3.60
AWSR Average Annual Wage and Salary Rate 1990 9.75 0.19 9.31 10.35
EMP Employment 1990 8.83 1.25 5.42 13.38
INM In-Migration 1990 7.09 1.00 4.54 10.52
OTM Out-Migration 1990 7.04 0.97 4.50 10.55
MHY Median Household Income 1989 9.94 0.23 9.06 10.68
GEX Direct General Expenditures per Capita 1992 7.23 0.28 6.49 8.11
All variables are expressed in logs except AGFF, EFIR, FGCE, SCIX, and NAIX
handle the feedback simultaneity is inappropriate, because of the spatial autoregressive
lag and spatial cross-regressive lag simultaneities terms. The Spatial Generalized
Methods of Moments approach used by Rey and Boarnet (2004) in a Monte Carlo
analysis of alternative approaches to modeling spatial simultaneity is also inappropriate,
because the model includes spatially autoregressive disturbances. Therefore,
we use the Generalized Spatial Two-Stage Least Squares (GS2SLS) as suggested by
Kelejian and Prucha (1998, 1999), and the Generalized Spatial Three-Stage Least
Squares (GS3SLS) approach as outlined by Kelejian and Prucha (2004).
TheGS2SLS and GS3SLS procedures are carried out in three and four step routines,
respectively. The first three steps are common to both routines. In the first step, the
parameter vector α

, β

, λ

, γ
is estimated by two stage least squares (2SLS), using
an instrument matrix Q that consists of a subset of linearly independent columns
X,WX,W2X, where X is the matrix that includes the control variables in the model.
W is a weights matrix. The disturbances for each equation in the model are computed
using the estimates of α

, β

, λ

, γ
from the first step. In the second step, the estimates
of the disturbances are used to estimate the autoregressive parameter ρ for each
equation, using Kelejian and Prucha (2004) generalized moments procedure. In the
third step, a Cochran–Orcutt-type transformation is performed, using the estimates for
ρ from the second step to account for the spatial autocorrelation in the disturbances.
The GS2SLS estimates of [β

, λ

, γ
] are then obtained by estimating the transformed
model using a subset of the linearly independent columns of [X,WX,W2X] as the
instrument matrix.
Although the GS2SLS takes the potential spatial correlation into account, it does
not utilize the information available across equations because it does not account for
the potential cross equation correlation in the innovation vectors (εem
i t , εmh
i t ). The correlation
coefficient between the residuals of the GS2SLS (εem
i t and εmh
i t ) is given in
Table 2. The full system information is utilized by stacking the Cochran–Orcutt-type
transformed equations (from the second step) in order to jointly estimate them. Thus,
in the fourth step, theGS3SLS estimates of the betas, lambdas, and gammas [β

, λ

, γ
]
are obtained by estimating this stacked model. The GS3SLS estimator is more efficient
than theGS2SLS estimator. Further, consistent estimates of the covariance matrix
are used to obtain the Feasible Generalized Three-Stage Least Squares (FGS3SLS)
estimators of α

, β

, λ

, γ
.
123
Analysis of county employment and income growth in Appalachia 35
Table 2 Correlation matrix of
the residuals from generalized
spatial two-stage least squares
(GS2SLS) estimation of the
model
Equation 1 Equation 2
Equation 1 1.0000
Equation 2 −0.3974 1.0000
5 Discussion and analysis of results
The GS2SLS and GS3SLS parameter estimates of the system represented by
Eqs. (4a, b) are reported in Table 3. These values are consistent with theoretical
expectations and with the results of many other cross-sectional empirical studies (Boarnet
1994; Deller et al. 2001; Henry et al. 1997). The coefficients of the endogenous
variables (EMPR and MHYR) are positive and statistically significant, indicating
strong interdependence between employment and median household income growth
rates. This interdependence is consistent with economic theory and empirical results.
Increases in the demand for goods and services that result from increases in family
median or per capita income are associated with increases in employment (Armington
and Acs 2002), which create opportunities for even more people to work and earn
income. However, the effect of median household income growth on employment
growth is stronger than that of employment growth on median household income
growth.
In the business employment (EMPR) equation, fifteen of the coefficient estimates
are significantly different from zero at the 10% level or better. The results suggest a
positive and significant parameter estimate for the spatial autoregressive lag variable
(WEMPR). This indicates that employment growth tends to spill over to neighboring
counties. The results also show a negative coefficient for (WEMPR) in the (MHYR)
equation, indicating that employment growth rates in neighboring counties tend to
unfavorably affect median household income growth rates (MHYR) in a given county.
These estimates are important for policy because they indicate that employment growth
in neighboring counties has positive and negative spillover effects on a given county’s
EMPR and MHYR, respectively. Furthermore, the significant spatial lag effects indicate
that EMPR not only depends on characteristics within the county, but also on
those of its neighbors. Hence, spatial effects should be tested empirically involving
employment growth rates and household income growth rates. Our model specification
incorporates a spatially autoregressive spatial process besides the spatial lag in the
dependent variables. The negative estimate for ρ1 (see Table 3) indicates that random
shocks to EMPR do not only affect the county where the shocks originated and its
neighbors, but also create negative shock waves across Appalachia.
To control for agglomeration effects, the model includes population statistics, such
as the initial county population size (POPs) and the percentage of population between
25 and 44 years old (POP25_44). The result shows that both POPs and POP25_44
have positive and significant effects on EMPR, even after accounting for potential spatial
spillover effects. This result is consistent with the literature (Acs and Armington
2004) which indicates that a growing population increases the demand for consumer
goods and services as well as the pool of potential entrepreneurs which encourage
business formation. This result is important from a policy perspective. It indicates
123
36 G. H. Gebremariam et al.
Table 3 Generalized spatial 2SLS (GS2SLS) and full information generalized spatial 3SLS (GS3SLS)
estimation results
Variables GS2SLS GS3SLS
EMPR Equation MHYR Equation EMPR Equation MHYR Equation
Coefficient t-statistic Coefficient t-statistic Coefficient t-statistic Coefficient t-statistic
Constant −7.5180∗∗∗ −4.07 7.7602∗∗∗ 3.95 −8.53228∗∗∗ −5.01698 8.6547∗∗∗ 4.714
EMPR 0.2825 1.66 0.6156∗∗∗ 4.0457
MHYR 0.1685 1.59 0.3735∗∗∗ 3.8956
WEMPR 0.2492∗ 1.94 −0.1423 −0.98 0.2792∗∗ 2.2949 −0.2694∗ −1.7
WMHYR 0.1657 1.44 −0.0559 −0.43 0.1147 1.1999 −0.1063 −0.8495
POPs 0.8367∗∗∗ 4.32 0.0877 0.78 0.7724∗∗∗ 4.3572 −0.0299 −0.2807
POPd −0.0101 −0.3 −0.0123 −0.4054
POP5-17 −0.1566 −0.9 −0.1072 −0.6642
POP25-44 0.2806 1.48 0.3093∗ 1.807
POP > 65 0.1046 0.98 0.1576 1.6024
FHHF −0.0031 −0.03 −0.0034 −0.3856
POPHD −0.1589 −1.03 −0.2439 −1.15 −0.1487 −1.0167 −0.1556 −0.7667
POPCD 0.0561 1 −0.0989 −1.35 0.0789 1.4827 −0.1147 −1.6361
OWHU −0.4079∗ −1.77 −0.368∗ −1.76
MHV −0.0309 −0.32 0.0955 0.76 −0.0483 −0.5198 0.0763 0.6308
UNEMP −0.0825∗∗ −2.05 0.0442 0.79 −0.079∗∗ −2.0599 0.0706 1.3197
AGFF −0.0055 −1.11 0.0025 0.38 −0.006 −1.2612 0.0032 0.5017
MANU 0.0856∗∗ 2.65 −0.0008 −0.02 0.0772∗∗ 2.5484 −0.0324 −0.8124
WHRT 0.3734∗∗∗ 4.5 −0.0727 −0.65 0.3719∗∗∗ 4.7178 −0.1916∗ −1.8012
FIRE 0.0177 0.39 −0.0471 −0.86 0.0282 0.6542 −0.0616 −1.168
HLTH −0.0079 −0.2 0.0297 0.56 −0.0157 −0.4067 0.0277 0.5475
NAIX 0.0072 0.72 −0.0063 −0.47 0.0062 0.645 −0.0064 −0.4944
ESBd 0.7049∗∗∗ 3.82 0.0242 0.27 0.6574∗∗∗ 3.9138 −0.0495 −0.5689
EFIR −1.05216D-08 −0.09 −1.16242D-08 −0.1113
CSBD 0.0406 1.14 0.0304 0.9565
DFEG 0.0002 0.01 −0.0071 −0.1973
FGCE 0.0001 0.6 4.78E-05 0.5158
PCTAX −0.0706 −1.25 −0.062 −1.2314
PCPTAX 0.0108 0.26 0.01095 0.2924
SCIX 0.0439∗ 1.7 0.046∗ 1.974
HWD −0.002 −0.04 −0.0062 −0.1303
ESBs 0.5536∗∗ 2.87 0.5345∗∗∗ 3.0658
AWSR 0.0912 0.94 0.0822 0.9521
EMP −0.8647∗∗∗ −4.7 −0.0223 −0.28 −0.8151∗∗∗ −4.8863 0.0941 1.2818
INM 0.1122 1.38 −0.1245 −1.25 0.1424∗ 1.8427 −0.1792∗ −1.8725
OTM −0.1382 −1.65 0.0693 0.65 −0.1401∗ −1.7571 0.1248 1.215
MHY 0.2334 1.32 −0.7671∗∗∗ −4.35 0.3636∗∗ 2.2161 −0.7976∗∗∗ −4.7331
GEX 0.0608 1.33 0.0684 1.24 0.04105 0.9472 0.0477 0.8971
Rho (ρ) −0.0428 0.1913 −0.0428 0.1913
123
Analysis of county employment and income growth in Appalachia 37
Table 3 continued
Variables GS2SLS GS3SLS
EMPR Equation MHYR Equation EMPR Equation MHYR Equation
Coefficient t-statistic Coefficient t-statistic Coefficient t-statistic Coefficient t-statistic
nR2∼χ2
(30,36)a 46.4608 0.02807b 39.1464 0.3305b 46.4608 0.02807b 39.1464 0.3305b
Moran I −0.2058 −5.0284c 0.1336 3.0753c −0.2058 −5.0284c 0.1336 3.0753c
Eta (η) 0.8647 0.7671 0.8151 0.7976
Half-Life (years) 8.47 8.65 8.47 8.65
PE test log log log log
n 417 417 417 417
*, **, and *** denote statistical significance level at the 10, 5, and 1%, respectively
a 30, 36 represent the degree of freedoms which are equal to the over-identifying restrictions in the EMPR, MHYR
equations, respectively
b p-values
c Z-values for Moran I
that counties with high population concentration are benefiting from the resulting
agglomerative and spillover effects that lead to localization of economic activities, in
line with Krugman (1991a,b) argument on regional spillover effects.
The county unemployment rate (UNEMP) is included among the exogenous variables
to measure local economic distress. The results suggest that a high unemployment
rate is associated with low business growth. This indicates that the poor economic
environment in Appalachia did not provide incentives for individuals to form new
businesses that employ not only the owner, but others. Unemployed individuals may
not have the capital to start a business. Furthermore, a high level of unemployment is
indicative of a relatively low aggregate demand, which also discourages new firm formation.
This result is consistent with the findings of Acs and Armington (2004). They
found that unemployment is negatively associated with new firm formation during
economic growth periods and positively during economic recession periods.
The coefficient of the variable representing the percentage of homes that are owned
by their own occupants (OWHU) is negative and statistically significant at the 10%
level. This result indicates that high home ownership is negatively associated with
business formation in Appalachia. This is contrary to the expectation that high home
ownership signals the availability of household assets and is therefore an indicator of
the capacity to finance new businesses by potential entrepreneurs, either by using the
house as collateral for loan or as indication of availability of other personal financial
resources. The result, however, shows that in Appalachia during the study period home
ownership was positively correlated with level of economic distress (Pollard 2003),
and home ownershipwas higher in distressed counties (76%), and lowest in attainment
counties (69%). Homeownership was also higher in central Appalachia (76%) than
in the more developed northern or southern sub-regions; and Appalachia non-metro
areas had higher ownership rates (76%) than its metro areas (72%). Thus, the result
indicates that home ownership is not a good indicator of the availability of resources
to start new business, at least in Appalachia.
123
38 G. H. Gebremariam et al.
The coefficients for MANU and WHRT are positive and significant at the 5 and 1%
levels, respectively. These results indicate that counties with a higher initial percentage
of their labor force employed in manufacturing and the wholesale and retail trade
showed higher growth rates in business than other counties.
The percentage of people employed in manufacturing (MANU) and the percentage
of people employed in wholesale and retail trade (WHRT) are included in the
EMPR equation to control for the influence of sectoral employment concentration on
the overall employment growth rate. The coefficient on MANU is positive and statistically
significant at the 5% level, indicating a direct relationship between growths
in overall employment and manufacturing employment at the beginning of the
periods. The coefficient on WHRT is also positive and significant at the 1% level,
indicating the positive role played by the service sector in expanding employment in
Appalachia during the study period. Thus, these results tend to suggest that
Appalachian counties that had a higher proportion of their labor force employed in
manufacturing and whole sale and retail trade at the beginning the periods experienced
higher growth rates in overall employment. This seems realistic since Appalachia has
experienced a shift from resource-based economic activities to manufacturing and,
particularly, to services. The coefficient on WHRT is higher and even more significant
than the coefficient on MANU in the EMPR equation, indicating that the contribution
of WHRT to overall employment growth was higher and more sustained than that of
MANU.
Establishment density (ESBd), defined as the total number of private sector establishments
in the county divided by the county’s population, is included in the model
to capture the degree of competition among firms and the concentration of businesses
relative to the population density. The average size of establishment (ESBs), defined
as total private sector employment divided by the number of private establishments
in the county, is also included to capture the effects of barriers to entry of new small
firms on employment growth. The coefficient for ESBd is positive and statistically
significant at the 1% level, indicating that the Appalachian region is far below the
threshold where competition among firms for consumer demands crowds businesses.
According to the results, a high ESBd is associated with growth in employment (business
growth), indicating that firms tend to locate near each other, possibly due to
localization and agglomeration economies of scale. The coefficient for ESBs is also
positive and significant indicating the existence of low barriers to new firm formation
and employment generation in Appalachia during the study period.
The results indicate that the county employment level is dependent on gross
in-migration, gross out-migration, and median household income. The coefficient for
INM, for example, is positive and significant at the 5% level. The coefficient for OTM
is negative and statistically significant at the 1% level. These are consistent with theoretical
expectations and empirical findings (Borts and Stein 1964). In-migration tends
to shift both the labor supply and labor demand curve right-wards, and out-migration
tends to lead to leftward shift of the curves. Thus, in-migration leads to increases in
employment, whereas out-migration leads to decreases in employment. A growing
population increases the demand for consumer goods and services and is positively
related to business formation (Acs and Armington 2004).
123
Analysis of county employment and income growth in Appalachia 39
Consistent with theoretical expectations and empirical findings, the coefficient for
MHY is positive and statistically significant at the 5% level. Increases in the demand
for goods and services that result from increases in family median or per capita income
are associated with increases in employment (Armington and Acs 2002).
An interesting observation from the empirical results pertains to the role of local
government in employment growth. The model predicts that local governments,
through their spending and taxation functions, play critical roles in creating and
enabling economic environments for businesses to prosper. The empirical results, however,
indicate that local governments have not played significant roles in employment
growth in Appalachia. Given the economic hardship and high level of underdevelopment
in Appalachia, these results are indications that local governments may need to
reassess or step up their efforts to create incentives for employment growth in this
region.
The elasticity of EMPR with respect to the initial employment level (EMP) is negative
and statistically significant, indicating convergence in the sense that counties with
low levels of employment at the beginning of the period (1990) tend to show a higher
rate of business growth than counties with high initial levels of employment, conditional
on the other explanatory variables. This result is consistent with prior studies on
rural renaissance (Deller et al. 2001; Lundberg 2003). The speed of adjustment, ηem,
is calculated at 0.8151, which indicates that just over 81% of the equilibrium rate of
growth in the employment rate of growth was realized during the period 1990–2000.
That is 8.151% annually, giving a half-life time of 8.47 years.
The parameter estimates for the MHYR equation also shows a positive estimate
for ρ2. This indicates that random shocks into the system with respect to MHYR not
only affect the county where the shocks originate and its neighbors, but create positive
spillover effects across Appalachia. The elasticity of EMPR with respect to the initial
median household income (MHY) is negative and statistically significant, indicating
convergence in the sense that counties with low median household incomes at the
beginning of the period (1990) tend to show higher rates of growth of median household
incomes than counties with high initial median household incomes, everything
else being equal. The speed of adjustment, ηmh, is calculated at 0.7976, which indicates
that about 80% of the equilibrium rate of growth in themedian household income
growth ratewas realized during the period 1990–2000. That is 7.976% annually, giving
a half-life time of 8.65 years. This result is comparable to the speed of convergence
estimates obtained by Higgins et al. (2006) and Young et al. (2008).
The effect of out-migration on the growth rate of median household income is negative
and statistically significant. If migrants’ endowments of human capital in the
form of education, accumulated skills, or entrepreneurial talents are higher compared
to the sending population, then the loss of their skills, inventiveness and innovativeness
would contribute to a decline in local productivity. Migrants may also own physical
and financial capital that they may take with them leading to a loss in investment in the
sending county. Moreover, out-migrants may contribute to a decline in the growth of
markets and scale and agglomerations economies in the sending county. Such demand
effects are the sources of loss in the growth of per capita personal incomes.
The coefficient for the index of social capital (SCIX) is positive and significant,
suggesting that high levels of social capital increase the wellbeing of a county. The
123
40 G. H. Gebremariam et al.
coefficients for the proportion of school age population (POP5-17), the proportion of
the population above 65 years old (POP > 65), and the proportion of female headed
households (FHHF) are negative, positive, and negative, respectively, as expected.
Counties with high proportions of POP5-17 and FHHF tend to have low levels of
median household incomes, whereas counties with a high proportion of POP > 65
tend to have high levels of MHY. These results are consistent with empirical results
of previous studies.
6 Conclusions
Themain objective of this studywas to test the hypotheses that (1) employment growth
and median household income growth are interdependent and jointly determined by
regional variables; (2) employment and median household income growth in a county
are conditional upon initial conditions of the county; and (3) employment and median
household income growth in a county are conditional upon employment and median
household income growth in neighboring counties. To test these hypotheses, a spatial
simultaneous equations model was developed. GS2SLS and GS3SLS coefficients of
the parameters were obtained by estimating the model using data covering the 417
Appalachian counties for the 1990–2000 period. The empirical results of the study
support the three hypotheses. In particular, the employment growth rate in one county
is positively affected by the employment growth rate and themedian household income
growth rate in neighboring counties, and the median household income growth rate in
one county is negatively affected by employment growth rate and median household
income growth rate in neighboring counties.
A policy implication of the finding is that counties may be more successful in
creating environments (business climate) to make themselves attractive to firms if
several neighboring counties pool their resources. The results also indicate the presence
of spatial correlation in the error terms, which implies that a random shock into
the system spreads across the region. The results further indicate convergence across
counties in Appalachia with respect to employment growth and median household
income growth rates, conditional upon the initial conditions of the explanatory variables
in the model. This information indicates that the divergence in the economic
status among Appalachian counties is narrowing and could mean that the efforts of
the Appalachian Regional Commission are showing results.
The empirical results indicate the presence of significant agglomerative effects:
counties with higher population concentrations showed significant business growth.
Combined with the findings of spillover effects, this might justify favoring focusing
investments in areas capable of generating agglomeration effects.
The study also produces useful information concerning the creation of new or the
expansion of existing businesses in Appalachia. Establishment density, which captures
the degree of competition among firms and crowding of businesses relative
to the population, indicates that Appalachia is below the threshold where competition
among firms for consumer demands crowds businesses. In addition, the results
indicate low barriers to new firm formation and employment generation during the
study period.
123
Analysis of county employment and income growth in Appalachia 41
While incorporating spatial interdependencies adds to the model’s computational
complexities, the returns are not only improved estimates, but the analysis also yields
information about spatial relationships that would not otherwise be available. For the
study period, this research suggests that a growth pole approach that spatially concentrates
scarce policy investments could benefit the region. Such insight requires
a spatially explicit model otherwise they are based on guesswork and intuition. Of
course, given the short time period of our analysis, additional research is needed to
determine if this result is stable over time or changes with the business cycle.
In general, this study confirms the importance of spatial effects in regional development.
The empirical results indicate the presence of spatial correlation in the error
terms and of spatial autoregressive lag. Failure to account for spatial interaction effects
results in less efficient and consistent estimates, as well as loss of insight.
Acknowledgments This research was partially funded by the West Virginia Agricultural and Forestry
Experiment Station. We acknowledge helpful comments by Dale Colyer and two referees. We thank
Anil Rupasingha, Stephan Goetz and David Freshwater for allowing the use of their Social Capital Index
data set for US counties. The usual caveat applies.
Appendix A: Derivation of the reduced form of the model
Let the system given in (4a, b) be written as:
Y = YB + XΓ + WYΛ + U. (I)
U = WUC + E and
Y = ( y1, . . . , yG) X = (x1, . . . , xK ) U = (u1, . . . , uG)
WU = (Wu1, . . . ,WuG) , C = diagGj
=1 ρj , E = (ε1,…, εG)
where y j is the n by 1 vector of cross sectional observations on the dependent variable
in the j th equation, xl is an n by 1 vector of cross sectional observations on the j th
exogenous variable, u j is an n by 1 vector of error terms in the j th equation, and B
and Γ are correspondingly defined parameter matrices of dimension G by G and K
by G, respectively. B is a diagonal matrix. Λ is G by G matrix of parameter estimates
of the spatial lag variables. It not diagonal and hence each equation includes spatial
cross-regressive lag variable in addition to its own spatial lag. Hence the model has
the same structure as that in Kelejian and Prucha (2004).
Note that ρj denotes the spatial autoregressive parameter in the j th equation and
since C is taken to be diagonal, the specification relates the disturbance vector in
the j th equation only to its own spatial lag. Since it is assumed that E(ε) = 0 and
E(εε

) = Σ ⊗ In, the disturbances, however, will be spatially correlated across units
and across equations.
The system in Eq. (I) can be expressed in a form where its solution for the endogenous
variables is clearly revealed. But, first consider the following vector transformations:
123
42 G. H. Gebremariam et al.
vec(Y) = vec(YB) + vec(XΓ ) + vec(WYΛ) + vec(U)
vec(Y) = vec(YB) + vec(XΓ ) + vec(WYΛ) + vec(UWC + E)
= B ⊗ I vec(Y) + Γ
⊗ I vec(X) + Λ
⊗ W vec(Y)
+ C ⊗ W vecU + vecE
Letting y = vec(Y), x = vec(X), u = vec(U), and ε = vec(E), it follows from
Eq. (I) that:
y = B ⊗ I y + Γ
⊗ I x + C ⊗ W u + ε
or
y = B ⊗ I y + Γ
⊗ I x + u,
u = C ⊗ W u + ε
(II)
Let B∗ = [(B ⊗ I) + (Λ
⊗ W)], Γ
∗ = (Γ
⊗ I ) and C∗ = C ⊗ W = diagGj
=1
(ρ jW), then Eq. (II) can be written in more compact form as:
y = B∗ y + Γ
∗x + u,
u = C∗u + ε
(III)
Assuming that InG − B∗ and InG − C∗ are nonsingular matrices with |ρj | < 1, j =
1, . . . , G, the system in Eq. (III) can be expressed in its reduced form as:
y = InG − B∗
−1 Γ
∗x + u ,
u = InG − C∗
−1
ε
(IV)
Based on the results of our estimation, we found that InG − B∗ and InG − C∗
have full column ranks and |ρj | < 1, j = 1, 2. From this we can conclude that
the reduced form of the system [Eq. (IV)] is properly defined and there also exists
spatial multiplier working in the system.
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