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1 Step 1 – Estimation of the Marginal Implicit Price of Open Space

1 Step 1 – Estimation of the Marginal Implicit Price of Open Space

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174



S.-H. Cho



where ln pi is the natural log of the value of house i ; xi is a vector of factors

determining the value of house i ; ln oi is the natural log of open space in the vicinity

of house i ; oO i is the predicted value from (2); i is a vector of instruments that are

correlated with ln oi and uncorrelated with "i ; and ."i ; Ái / are a random disturbances

with expected values of zero and unknown variances. The instruments used in (2)

are identified in Table 1.

The GWR hedonic model with spatially autocorrelated disturbances is:

ln pi D

Ÿi



X

k



“k .ui ; vi / xO i k C ©i ;



i id 0; Â 2



âi D



Xn

j D1;j Ôi



wij âj C Ÿi ;

(3)



where xO i k is a vector of exogenous variables, including the predicted value of ln oO i ;

.ui ; vi / denotes the coordinates of the i th location in the housing market; ˇk .ui ; vi /

represents the local parameters associated with house i ; wij is an element of an

m by n spatial weighting matrix between points i and j ; and is a spatial error

autoregressive parameter.

The specification in (3) allows a continuous surface of parameter values with spatially autocorrelated disturbances, and measurements taken at certain points denote

the spatial heterogeneity of the surface (Fotheringham et al. 2002). Previous studies

have found that a log transformation of the distance and area explanatory variables

generally performs better than a simple linear functional form, as the log transformation captures the declining effects of these distance variables (Bin and Polasky 2004;

Iwata et al. 2000; Mahan et al. 2000). Thus, a natural log transformation of the

distance and area-related variables is used in this study.

Given estimation of (3), GWR residuals are tested for spatial error autocorrelation using a Lagrange Multiplier (LM) test (Anselin 1988). A row-standardized

inverse distance matrix was used to test the hypothesis of spatial error independence. Rejection of the null hypothesis suggests a GWR-spatial autoregressive error

model (GWR-SEM) as a way to address spatial heterogeneity and spatial error

autocorrelation. The GWR-SEM combines well-founded methods typically used in

conventional spatial econometric analyses, i.e., the Cochran–Orcutt method of filtering dependent and explanatory variables to address spatial error autocorrelation

(Anselin 1988), with local regression techniques in a parametric framework. The

filtering mechanism Œ.I œW/ partials out spatial error autocorrelation associated

with the explanatory and dependent variables while estimating local coefficients. It

helps to envision GWR as running n parametric regressions at n locations to control

spatial heterogeneity, and then testing whether the residuals generated by these local

regressions are spatially correlated. If the hypothesis of no spatial autocorrelation is

rejected, conventional methods are applied to filter the dependent and explanatory

variables (e.g., Anselin 1988, p. 183), and the GWR model is estimated again using

the transformed variables.

A convenient procedure to estimate is Kelejian and Prucha’s (1998) general

moments approach, based on the set of GWR residuals. Given determination of ,

the closed form solution to (3) is:



Demand for Open Space and Urban Sprawl

Table 1 Variable names, definitions, and descriptive statistics

Variable (Unit)

Definition

Dependent variable

Housing price ($)

Sale price adjusted to 2000 by

the housing price index

Variables closely associated with urban sprawl

Median household income

Incomea ($)

Finished areaa (feet2 )

Total finished square footage of

house

Total parcel square footage

Lot sizea (feet2 )

Housing density for

Housing densitya

(houses per acre)

census-block group

Area of open space within a

Open space

buffer of 1.0 mile drawn

103 feet2

around each house sale

transaction

Price of open space ($)

Marginal implicit price of

increasing additional

10,000 ft2 of open space

within 1.0-mile buffer

(assuming individual

housing price and

open-space area)

Structural variables

Agea (year)

Bricka



Poola

Garagea

Bedrooma

Storiesa

Fireplacea

Quality of constructiona



Condition of structurea



Distance variables

Distance to CBDa (feet)

Distance to greenwaya

(feet)



Year house was built subtracted

from 2006

Dummy variable for brick

siding (1 if brick, 0

otherwise)

Dummy variable for swimming

pool (1 if pool, 0 otherwise)

Dummy variable for garage (1

if garage, 0 otherwise)

Number of bedrooms in house

Height of house in number of

stories

Number of fireplaces in house

Dummy variable for quality of

construction (1 if excellent,

very good and good, 0

otherwise)

Dummy variable for condition

of structure (1 if excellent,

very good and good, 0

otherwise)

Distance to the central business

district

Distance to nearest greenway



175



Mean



Std. Dev.



129,610.227



95,460.498



51,505.871

1,929.689



20,940.122

975.633



25895.720

1.105



69956.690

0.927



53,822.711



15,490.449



47.618



38.610



29.207



21.733



0.254



0.435



0.055



0.229



0.635



0.481



3.068

1.340



0.647

0.474



0.729

0.352



0.575

0.478



0.734



0.442



44,552.592



20,713.081



7,886.866



5,573.062

(continued)



176

Table 1 (continued)

Variable (Unit)

Distance to railroada

(feet)

Distance to sidewalka

(feet)

Distance to parka (feet)

Park sizea (feet2 )

Distance to golf coursea

(feet)

Distance to water bodya

(feet)

Size of water bodya

(1,000 feet2 )



S.-H. Cho



Definition

Distance to nearest railroad



Mean

6,978.618



Std. Dev.

5,463.655



Distance to nearest sidewalk



3,060.270



4,229.282



Distance to nearest park

Size of nearest park

Distance to nearest golf course



8,652.930

1,454.759

10,680.078



5,556.530

5,094.984

4,942.615



Dist. to nearest stream, lake,

river, or other water body

Size of nearest water body



8,440.579



5,884.047



19,632.026



39,026.745



0.077



0.266



0.157



0.363



0.027



0.161



0.092



0.290



0.053



0.224



0.055



0.228



0.057



0.231



0.147



0.354



0.065



0.247



0.148



0.355



0.014



0.116



0.063



0.031



0.037



0.029



22.519



3.314



0.343



0.475



High school district dummy variables (1 if in School District)

Dummy variable for Doyle

Doylea

High School District

Dummy variable for Bearden

Beardena

High School District

Dummy variable for Carter

Cartera

High School District

Dummy variable for Central

Centrala

High School District

Dummy variable for Fulton

Fultona

High School District

Dummy variable for Gibbs

Gibbsa

High School District

Dummy variable for Halls

Hallsa

High School District

Dummy variable for Karns

Karnsa

High School District

Dummy variable for Powell

Powella

High School District

Dummy variable for Farragut

Farraguta

High School District

Dummy variable for Austin

Austina

High School District

Census block-group variables

Vacancy rate for census-block

Vacancy ratea (ratio)

group (2000)

Unemployment ratea

Unemployment rate for

(ratio)

census-block group (2000)

Average travel time to work for

Travel time to worka

(min)

census-block group (2000)

Other variables

Dummy variable for City of

Knoxvillea

Knoxville (1 if Knoxville, 0

otherwise)



(continued)



Demand for Open Space and Urban Sprawl



177



Table 1 (continued)

Variable (Unit)

Flooda



Definition

Dummy variable for 500-year

floodplain (1 if in stream

protection area, 0 otherwise)

Interfacea

Dummy variable for

rural–urban interface (1 if in

census block of mixed

rural–urban housing, 0

otherwise)

Urban growth areaa

Dummy variable for urban

growth area (1 if in urban

growth area, 0 otherwise)

Dummy variable for planned

Planned growth areaa

growth area (1 if in planned

growth area, 0 otherwise)

Seasona

Dummy variable for season of

sale (1 if April through

September, 0 otherwise)

Average prime interest rate less

Prime interest ratea

average inflation rate

a

Indicates instrumental variables used in the first step estimation



ˇO .ui ; vi / D .X0 .I



W/0 A.I



W/ X/ 1 X0 .I



Mean

0.010



Std. Dev.

0.097



0.223



0.417



0.083



0.276



0.431



0.495



0.559



0.497



4.267



2.104



W/0 A .I



W/P

(4)



which is analogous to the GLS estimator in the spatial econometric literature,

ˇ SEM D .X0 .I– W/0 .I– W/X/ 1 X0 .I– W/0 .I– W/y, where is a an n by n

diagonal matrix with a set of weights corresponding with each observation, except

that it generates i sets of local parameters. The n by n matrix A (which is a function of ui and vi ) addresses spatial heterogeneity, with diagonal elements identifying

the location of other houses relative to house i and zeros in off-diagonal positions

(Fotheringham et al. 2002). Houses near house i have more influence in the estimation of the parameters associated with house i than other houses located farther

away.

When D 0, (4) generates the usual GWR estimates. Pseudo-standard errors for

the i sets of regression parameters are based on the covariance matrix (cov):

O i ; vi // D ¢ 2 .X0 .I

cov .ˇ.u

i



œW/0 A .I



œW/X/



1



(5)



W/0 A.I

W/e=.q k/ is the variance associated with the i th

where i2 D e 0 .I

regression point (Fotheringham et al. 2002).4 Statistical significance of the estimates

from the GWR-SEM at the i th regression point is evaluated with the Pseudo-t tests



4

Those standard errors do not take into consideration the first stage estimation. Further studies will

consider a covariance matrix adjusted for the first stage regression.



178



S.-H. Cho



derived from the Pseudo-standard errors of the location-specific covariance matrices. Based on the GWR-SEM, the marginal implicit price of an additional 10; 000 ft2

of open space is estimated.



2.2 Step 2 – Open-Space Demand Estimation

The demand for open space is estimated using the marginal implicit price of open

space estimated in the first step as a proxy for the price of open space. The demand

equation for open space in the GWR framework is:

ln oi D —.ui ; vi / ln pOi C



X

k



’k .ui ; vi /xi k C Ôi ; k D 1; : : : ; m5



(6)



where ln pOi is the natural log of the estimated marginal implicit price of open space

for house i , and xi k is the kth of m variables determining the demand of open space

for house i . The xi k includes variables closely associated with urban sprawl (e.g.,

income, house and lot size, and housing density), structural attributes of the house,

census-block group variables (e.g., vacancy rate, unemployment rate, and travel time

to work), distance measures to amenities (e.g., lakes, parks) or disamenities (e.g.,

railroads), school districts, and other spatial dummy variables (e.g., urban growth

area and planned growth area) (see Table 1 for the complete list). The statistical

significance of the local estimates at the i th regression point is evaluated with t-tests

derived from the standard errors of the location-specific covariance matrices.

Another concern in regression models with many explanatory variables is multicollinearity, which occurs when two (or more) independent variables are linearly

related. Multicollinearity may inflate estimates of standard errors, rendering hypothesis testing inconclusive. Multicollinearity can be detected by variance inflation

factors (VIF) (Maddala 1992). VIFs are a scaled version of the multiple correlation

coefficients between a variable and the rest of the independent variables (Maddala 1983). There is no clear guideline for how large the VIF must be to reflect

serious multicollinearity, but a rule of thumb is that multicollinearity may be a

problem if the VIF for an independent variable is greater than ten (Gujarati 1995).

The VIFs were lower than ten for all but three variables, namely dummy variables

differentiating the rural–urban interface (22), the City of Knoxville (12), and the

Bearden high school district (11) in the demand for open space equation. In general,

multicollinearity does not appear to be too great a concern because many of the

location-specific coefficients were significantly different from zero at the 5% level.6



Covariance of pOi is not adjusted for first stage regression.

If the VIF is large but the coefficient is significant, multicollinearity is not a problem with respect

to the estimation of the standard errors. If a coefficient is significant using a weak t-test caused

by collinearity (inflated standard error), it would be significant using the stronger t-test associated

with the lack of collinearity (inflated standard error).



5



6



Demand for Open Space and Urban Sprawl



179



Nevertheless, those three variables with high VIFs were not excluded for lack of

sufficient justification.



3 Study Area and Data

Knox County, Tennessee was chosen as a case study for this research because (1)

Knoxville is the eighth most sprawling U.S. metropolitan region (Ewing et al. 2002),

and (2) the area consists of both rapid and slow regions of housing growth. Knox

County is located in East Tennessee, one of the three “Grand Divisions” in the state.

The City of Knoxville is the county seat of Knox County. Knoxville comprises

101 miles2 of the 526 miles2 within Knox County. Total populations of Knoxville

and the Knoxville Metropolitan Area were 173,890 and 655,400 in 2000, respectively (US Census Bureau 2002). The University of Tennessee and the headquarters

of Tennessee Valley Authority (TVA) are near downtown Knoxville, and the US

Department of Energy’s Oak Ridge National Laboratory is 15 miles northwest of

Knoxville. These institutions are the major employers of the area. Maryville is

located approximately 14 miles southwest of Knoxville and it is home to ALCOA,

the largest producer of aluminum in the United States. Farragut, a bedroom community, is located along the edge of the western end of Knox County (see Fig. 1). The

Smoky Mountains, the most-visited National Park in the United States, and a large

quantity of lake acreage (17 miles2 of water bodies) developed by the TVA are on

Knoxville’s doorstep.

It is important to note that push/pull factors of the geography surrounding

the study area were not modeled because data were not available. However, to

our knowledge, no other hedonic studies have successfully addressed this issue.

Admittedly, these omitted factors may cause some estimates to be biased. But understanding this context beforehand aids in the interpretation of patterns generated by

mapped coefficients. It is also important to note that the results of this study may not

be representative of other urban areas. The data set does not represent most typical

urban areas, and because of the local amenities and job opportunities, Knox County

may be more of an outlier case compared to other rapidly growing metropolitan

areas. Nevertheless, the methods used in this case study can be applied to other

urban areas where similar data exist.

This research used five GIS data sets: individual parcel data, satellite imagery

data, census-block group data, boundary data, and environmental feature data. The

individual parcel data, i.e., sales price, lot size, and structural information, were

obtained from the Knoxville, Knox County, Knoxville Utilities Board Geographic

Information System (KGIS 2009), and the Knox County Tax Assessor’s Office. Data

were used for single-family home sales transactions between 1998 and 2002 in Knox

County, Tennessee. A total of 22,704 single-family home sales were recorded during this period. Of the 22,704 houses sold, 15,500 were randomly selected for this

analysis. County officials suggested that sales prices below $40,000 were probably

gifts, donations, or inheritances, and would therefore not reflect true market value.



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