Anisotropic Autocorrelation in House Prices - Springer Link

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Cox School of Business, Southern Methodist University, Dallas, TX 75275-0333 .... trade off housing and commuting costs when selecting a residence.
Journal of Real Estate Finance and Economics, 23:1, 5±30, 2001 # 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.

Anisotropic Autocorrelation in House Prices KEVIN GILLEN Real Estate Department, The Wharton School, University of Pennsylvania, 314 Lauder-Fischer Hall, 256 South 37th Street, Philadelphia, PA 19104.6330 Email: [email protected] THOMAS THIBODEAU* Cox School of Business, Southern Methodist University, Dallas, TX 75275-0333 Email: [email protected] SUSAN WACHTER Policy Development and Research, U.S. Department of Housing and Urban Development, 451 Seventh Street SW, Washington, DC 20410

Abstract This article examines anisotropic spatial autocorrelation in single-family house prices and in hedonic house-price equation residuals using a spherical semivariogram and transactions data for one county in the Philadelphia, Pennsylvania, MSA. Isotropic semivariograms model spatial relationships as a function of the distance separating properties in space. Anisotropic semivariograms model spatial relationships as a function of both the distance and the direction separating observations in space. The goals of this article are (1) to determine whether there is spatial autocorrelation in hedonic house-price equation residuals and (2) to empirically examine the validity of the isotropy assumption. We estimate the parameters of spherical semivariograms for house prices and for hedonic house-price equation residuals for 21 housing submarkets within Montgomery County, Pennsylvania. These housing submarkets are constructed by dividing the county into 21 groupings of economically similar adjacent census tracts. Census tracts are grouped according to 1990 census tract median house prices and according to characteristics of the housing stock. We ®t the residuals of each submarket hedonic house price equation to both isotropic and anisotropic spherical semivariograms. We ®nd evidence of spatial autocorrelation in the hedonic residuals in spite of a very elaborate hedonic speci®cation. Additionally, we have determined that, in some submarkets, the spatial autocorrelation in the hedonic residuals is anisotropic rather than isotropic. The empirical results suggest that the spatial autocorrelation in Montgomery County single-family house-price equation residuals is anisotropic in submarkets where residents typically commute to a regional or local central business district. Key Words: house prices, spatial autocorrelation

1. Introduction House prices are spatially autocorrelated because properties in close proximity tend to have similar structural characteristics (such as square feet of building, or living area, dwelling age, and design features). This is a natural consequence of the fact that spatially *

Corresponding author.

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proximate properties tend to be developed at about the same time. In addition, properties within the same neighborhood share important neighborhood amenities (for example, neighborhood properties have access to the same public schools and are served by the same municipal police and ®re departments). Finally, house prices are likely to be spatially autocorrelated in neighborhoods where residents follow similar commuting patterns. House-price models attempt to explain spatial or temporal variation in house prices. These models are frequently used to mark residential property values to market. Hedonic models relate house prices to characteristics of the lot, the structure, and the neighborhood (Gillingham, 1975; Goodman, 1978; Thibodeau, 1989, 1992, 1996; and many others). Repeat-sales models measure changes in house prices as the average rate of appreciation for properties that have sold at least twice and have not undergone major structural changes between sales dates (Bailey et al., 1963; Case and Shiller, 1987, 1989). Hybrid models combine hedonic and repeat-sales speci®cations to obtain more ef®cient parameter estimates (Case and Quigley, 1991; Quigley, 1995; Hill et al., 1997). Assessed-value models estimate price indices using information obtained from property tax assessment departments (Clapp and Giaccotto, 1992). The parameters of hedonic house-price equations are typically estimated using ordinary least squares (OLS). This estimation procedure assumes the residuals are independently and identically distributed with zero mean, a constant variance, and zero covariance. When the residuals are spatially autocorrelated, the assumption of a zero covariance is violated, and OLS yields inef®cient parameter estimates. More accurate parameter estimates can be obtained by explicitly modeling the spatial autocorrelation. Modeling spatial relationships in hedonic house-price equations can also signi®cantly improve the accuracy of market-value predictions. The residuals produced by house-price models may be spatially autocorrelated for three reasons. First, proximity externalities in¯uence the market values of nearby properties in similar ways. Thibodeau (1990) demonstrated that high-rise of®ce buildings reduce the market value of nearby homes by as much as 15 percent. Information on the determinants of proximity externalities in the single-family market is dif®cult and costly to obtain. Consequently, these variables are frequently omitted from empirical house-price speci®cations. Second, other information on important structural and neighborhood characteristics are not readily available and are excluded from empirical house-price speci®cations. For example, data on the quality of local public schools and on area crime rates are dif®cult and costly to obtain, particularly on a large (such as national) scale. In addition, information on neighborhood socioeconomic and demographic characteristics are typically obtained from the U.S. Bureau of the Census. This information is collected only once every 10 years. When hedonic equations are used to model house prices, the residuals will contain information on these unobserved housing characteristics. Third, even in ideal situations where all housing characteristic information is available, it is dif®cult to select the ``correct'' model speci®cation. For example, it is dif®cult to model how public-school quality gets capitalized into the price of single-family properties. Model misspeci®cation may also contribute to spatially autocorrelated hedonic houseprice equation residuals.

ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

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Several researchers have developed hedonic house-price models that examine spatial autocorrelation in house prices and in hedonic house-price model residuals. Dubin (1988) assumed the residual correlation between properties is a negative exponential function of the distance between them and estimated hedonic parameters using a maximum likelihood procedure suggested by Mardia and Marshall (1984). Can (1992) included spatially lagged values of house prices as explanatory variables in the hedonic speci®cation. Pace and Gilley (1997) model spatial dependence in house prices using a simultaneous autoregressive (SAR) model. SAR models explain variation in house prices as a function of property characteristics and spatially weighted hedonic residuals for comparable properties. Basu and Thibodeau (1998) examine spatial autocorrelation in Dallas house prices using a semilog hedonic houseprice equation and a spherical autocorrelation function with data for over 5,000 transactions of homes sold between April, 1991 and January, 1993. Properties are geocoded and assigned to separate housing submarkets within metropolitan Dallas. Hedonic and spherical autocorrelation parameters are estimated separately for each submarket using estimated generalized least squares (EGLS). They conclude that house prices are spatially autocorrelated throughout metropolitan Dallas but that hedonic house-price equation residuals are spatially autocorrelated in about half of the submarkets examined. Most research on spatial autocorrelation in house prices has assumed that the correlation structure is isotropicÐa function of only the distance between properties. The direction separating properties is ignored. Spatial data is anisotropic when spatial autocorrelation is a function of both the distance and the direction separating points in space. Anisotropic semivariograms have been examined by Journel and Huijbregts (1978), Oden and Sokal (1986), Isaaks and Srivastava (1989), and Simon (1997). Simon (1997), for example, provides an exact test for anisotropic spatial autocorrelation. The exact tests are obtained by projecting the two-dimensional spatial observations onto a single axis making an angle y with the east-west axis. Spatial autocorrelations are then replaced by the conventional product-moment correlation coef®cient. House prices and hedonic house-price equation residuals may exhibit anisotropic spatial autocorrelation. Residential location theory suggests that housing consumers trade off housing and commuting costs when selecting a residence. To reduce commuting costs, residential properties are developed initially around major transportation arteries. This pattern of residential development may result in stronger spatial autocorrelation in house prices (and in hedonic house-price equation residuals) along major transportation arteries, and in the direction of the central business district (CBD). This article examines anisotropic spatial autocorrelation for Montgomery County single-family transactions. We ®t the parameters of a spherical autocorrelation function to the empirical (or sample) semivariogram for two directions in each of 21 housing submarkets in suburban Philadelphia. Our results suggest that the spatial autocorrelation in transaction prices and in hedonic house price equation residuals is anisotropic for some housing submarkets.

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2. Speci®cation 2.1. The hedonic house-price speci®cation We model the relationship between house prices and housing characteristics using a semilogarithmic functional form. This speci®cation regresses the log of transaction prices on a linear combination of ( possibly transformed) housing characteristics. The semi-log functional form is given by V ˆ eXb ‡ e ;

…1†

where V is property value, X is a vector of (possibly transformed) housing characteristics, b is a vector of unknown hedonic coef®cients, and e is the residual. When the residual variance is constant and the residuals are uncorrelated, ordinary least squares (OLS) yields best, linear, unbiased estimators of the parameters in the transformed equation Z ˆ log V ˆ Xb ‡ e;

…2†

where e*N…0; s2 I† so that Z*N…Xb; s2 I†. OLS yields estimated coef®cients b ˆ …XT X†

1 T

X Z;

where b*N…b; s2 …XT X†

1

†:

…3†

2.2. Modeling spatial autocorrelation When the residuals are spatially autocorrelated, E{ee0 } ˆ O, a matrix with nonzero offdiagonal elements. In this situation, b can be estimated with the generalized least squares (GLS) estimator B ˆ …XT O 1 X† 1 XT O 1 Z. The empirical challenge is to estimate the elements of O. Let si ˆ …ai ; bi † denote the location of property i ( for example, ai denotes the longitude and bi the latitude for property i). Let x …si † denote the hedonic price equation residual for a property located at si . If the stochastic process is weakly stationary, the covariogram for the distribution of residuals is C…si sj † ˆ Covfx…si †; x…sj †g for all …si ; sj †. Note that C…0† is the (assumed constant) variance for the residual distribution. The semivariogram of the process is g…si

sj † ˆ 0:5 Varfx…si †

x…sj †g ˆ C…0†

C…si

sj †:

…4†

Let h denote the (Euclidean) distance separating locations si and sj . Clearly, g… h† ˆ g…h† and theoretically g…0†. Empirically, however, g…h† is sometimes discontinuous near the origin and g…h†?y0 > 0, as h ? 0. The discontinuity, y0 , is labeled the nugget.

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Observations may eventually become spatially uncorrelated as the distance between them increases. When this happens, the semivariogram stops increasing beyond some threshold and becomes constant. That is, g…h† ? C* as h ? ?. This limiting value, C* , is called the sill of the semivariogram. The range of the semivariogram is the value h0 such that g…h0 † ˆ C* . So the range of a semivariogram is the distance beyond which observations are spatially uncorrelated. Finally, a semivariogram is isotropic if g…si sj † is a function of only the distance between si and sj , ksi sj k, and not the direction separating si and sj . Spatial data is anisotropic when spatial autocorrelation is a function of both the distance and the direction separating points in space. Geometric anisotropy occurs when the range varies with direction but the sill is constant. Zonal anisotropy occurs when the sill varies with direction but the range is constant (Isaaks and Srivastava, 1989). Mixed models are models where both the sill and the range vary with the direction separating spatial observations. The empirical (or sample) semivariogram examines how the spatial autocorrelation between observations changes as the distance between observations increases. The method of moments estimator for an empirical semivariogram (Matheron, 1963) is

g…h† ˆ

X

x…si †

2 x…sj † =2jN…h†j;

…5†

N…h†

where the average is taken over N…h† ˆ f…si ; sj †: si sj ˆ hg and N…h† is the distinct number of pairs in N…h†. For irregularly spaced data (such as single-family properties), N…h† is modi®ed so that N…h† ˆ f…si ; sj †: si sj [ T…h†g, where T…h† is a tolerance region around h. Figure 1 plots the points of the isotropic empirical (or sample) semivariogram for the log of transaction prices in the Ambler submarket, an area in the north-central region of Montgomery County. A point in the empirical semivariogram (such as a dot in Figure 1) is the difference between the variance and covariance in (the log of ) house prices computed for properties within a given tolerance region. The ®rst point to the right of the vertical axis measures the spatial autocorrelation for properties within + 200 meters of the separation distance h ˆ 250 meters. The statistic is computed without regard for the direction separating properties. The points for the empirical semivariogram are computed for 40 values of h ranging from h ˆ 250 meters to h ˆ 6,500 meters. The tolerance ranges or bins are constructed so that each bin has approximately the same number of property pairs. The next challenge is to ®t a functional form to the points of the empirical semivariogram. Three popular isotropic semivariograms that are used to empirically examine spatial relationships are the spherical, exponential, and Gaussian semivariograms. Cressie (1993) provides the functional forms for these semivariograms. Additional references on modeling spatial autocorrelation include Anselin (1978), Bailey and Gatrell (1995), Cliff and Ord (1973), Isaaks and Srivastava (1989), Ripley (1981), and Dubin et al., (1998). The spherical semivariogram model has a ®nite range while the exponential

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GILLEN, THIBODEAU AND WACHTER

Figure 1. Isotropic variogram of log(price) for Ambler.

and Gaussian semivariograms asymptotically approach a limiting value. The functional form for the spherical model is 8 0; > (  >  > < khk y ‡ y 1:5 g…h; y† ˆ 0 1 > y2 > > : y0 ‡ y1 ;

0:5

 khk y2

3 )

hˆ0 ;

05khk  y2 khk  y2

9 > > > = > > > ;

:

…6†

The nugget for the spherical semivariogram is y0 the sill is y0 ‡ y1 , and the range is y2 . The parameters of the spherical semivariogram are estimated using nonlinear least squares. The three parameters of the spherical semivariogram model are ®t to the empirical semivariogram, g…h†, by minimizing the nonlinear function S…y† ˆ

K X

‰g…h…k††

g…h…k†; y†Š

2

…7†

kˆ1

with respect to the semivariogram parameters h. The sequence h…1†, . . . , h…K† denotes the separation distances for which the sample semivariogram g…h† are computed. Figure 1 also plots a spherical semivariogram ®tted to the points of the empirical semivariogram for the

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ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

Ambler submarket. The ®tted spherical semivariogram is discontinuous at the origin (with an estimated nugget ˆ 0.0747) and increases with separation distance h. The ®tted spherical semivariogram becomes horizontal for h44:38 km, so house prices for properties separated by more than 4.38 km are spatially uncorrelated. If Ambler house prices were spatially uncorrelated, then the ®tted semivariogram would be horizontal at all separation distances. To estimate the standard errors of the semivariogram parameter estimates, we assume the semivariogram residuals are independently and identically distributed with mean zero and variance s2 , then K P

^2 ˆ s

…Gi

iˆ1

gi †

2

;  K  Gi ˆ value of the empirical semivariogram at Di where gi ˆ value of the spherical semivariogram at Di

…8†

and the nonlinear least squares estimator ^y is approximately normally distributed with mean y and covariance matrix (Maddala, 1977) ^2 ‰F…^ y†T F…^ y†Š s

1

:

…9†

F…y† is an N63 gradient matrix created by evaluating the partial derivatives of (6) with respect to y0 , y1 , and y2 at each of the K categorical distances Di :   qg qg qg F…y† ˆ qy qy qy " (0 1 2  3 ) 3 Di 1 Di ˆ 1 2 y2 2 y2

 y1

   # 3 Di 3 D3i ‡ 2 y22 2 y42

for i ˆ 1; 2; . . . ; K: …10†

3. The data This article examines spatial autocorrelation in house prices and in hedonic house-price equation residuals using data for 21,562 transactions of single-family homes sold between January, 1995 and March, 1998 in Montgomery County, Pennsylvania. Montgomery County is located to the northwest of Philadelphia. The county contains many homes that predate the American revolution. The eastern half of the county is a collection of various bedroom communities of educated professionals. It includes portions of Philadelphia's famous Main Line, a sequence of af¯uent communities to the west of Philadelphia dating

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Figure 2. Philadelphia metropolitan area and Montgomery County.

back to the nineteenth century. The region includes Bryn Mawr, Villanova, and Swarthmore. The western half of the county is predominately rural, containing several farming communities that date to colonial times, including a strong Amish presence. Figure 2 illustrates the location of Montgomery County, PA. Realist, a private data vendor, provided 21,562 transactions of Montgomery County single-family properties. The typical Montgomery County single-family transaction had about 2,030 square feet of building area. The mean transaction price over the 1995 to 1998 period was $179,823, or $88.61 per square foot. While a few homes were built prior to the American revolution, the average age for a Montgomery County property sold during the 1995 to 1998 period is 38.2 years.

3.1. The submarkets Twenty-one submarkets were de®ned by clustering contiguous census tracts with similar housing characteristics. Within each submarket, we would ideally prefer a distribution of house prices with a constant average unit price and a small variance, but the nonuniform spatial distribution of properties constrained this possibility. That is, if we de®ned the

ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

13

geographic submarkets such that each submarket could be classi®ed as relatively highpriced, midpriced, or low-priced, there would be several submarkets with very few observations in them. So, for example, we were forced to occasionally include several low-priced tracts in an otherwise midpriced submarket. The result was 21 distinct geographic housing submarkets that generally conformed with known municipal and geographic boundaries. Figure 3 provides the submarket boundaries for the 21 Montgomery County submarkets. Table 1 contains descriptive statistics for transactions in the 21 submarkets. The median submarket transaction price ranged from $108,000 for Pottstown to $325,000 for the Main Line (West) submarket. The mean per square foot price ranged from $73.33 for Pottstown to $126.80 for the Main Line (West) submarket. The oldest properties are located in the eastern part of the county, and the average age declines for submarkets in the western portion of the county. For example, the oldest properties are located in the Main Line (East) submarket (with an average age of 60 years) while the newest stock is in the Salford and Franconia submarkets (with an average age of 20 years). Later in the article we provide semivariogram parameter estimates for both isotropic and anisotropic spherical semivariograms for each of the 21 submarkets. In addition, we provide detailed hedonic results for three representative submarkets: the Main Line (East)

Figure 3. Boundaries of Montgomery County submarkets.

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GILLEN, THIBODEAU AND WACHTER

Table 1. Descriptive statistics for all 21 submarkets.

Submarket

Number of Transactions

Mean Price

Median Price

Pottstown New Hanover Pennsbury Upper Providence Frederick Salford Franconia Souderton Lower Providence Main Line (East) King of Prussia Norristown Conshohocken Worcester Lansdale Ambler Whitemarsh Cheltenham Abington Bryn Athyn Main Line (West)

915 418 277 1,148 407 751 288 266 664 1,399 434 997 443 866 2,635 1,773 988 1,666 3,226 1,361 640

$113,518 $147,220 $112,027 $168,668 $138,579 $168,325 $157,958 $124,364 $180,744 $277,604 $158,585 $128,247 $138,670 $228,226 $169,242 $232,301 $214,429 $158,184 $132,461 $191,243 $376,451

$108,000 $141,310 $112,025 $160,000 $135,000 $155,000 $152,000 $124,900 $165,000 $255,000 $150,000 $120,000 $128,000 $209,400 $159,450 $215,000 $190,000 $149,000 $127,000 $174,000 $325,000

Mean Price per Square Foot $73.33 $79.42 $74.59 $80.41 $81.80 $85.61 $86.01 $81.06 $88.04 $111.66 $86.32 $81.67 $89.81 $89.81 $85.41 $94.92 $96.87 $81.57 $85.64 $88.20 $126.80

Mean Square Foot

Mean Age

Mean Household Income

1,604 1,914 1,594 2,127 1,761 1,992 1,884 1,589 2,096 2,498 1,883 1,639 1,587 2,536 2,008 2,446 2,213 2,013 1,596 2,220 2,867

41 21 52 24 23 20 20 51 27 60 34 46 46 21 25 28 36 55 48 45 47

$53,062 $67,113 $53,691 $66,795 $60,801 $70,836 $64,299 $53,471 $66,337 $135,256 $82,731 $63,391 $60,254 $91,782 $67,936 $98,255 $107,600 $91,673 $66,690 $122,598 $193,401

submarket, the Norristown submarket, and the Ambler submarket. A brief characterization of these three submarkets follows. The Main Line is one of the oldest and wealthiest suburbs of the Philadelphia MSA. Named for a still-active commuter train line that dates back to the nineteenth century, the Main Line is a contiguous sequence of af¯uent suburbs that spans three counties, beginning at Philadelphia's westernmost city line. To normalize the relative size of the submarkets, we divided the segment of the Main Line that is within Montgomery County into two distinct submarkets: Main Line (East) and Main Line (West). The Main Line's housing stock can be characterized by global homogeneity, but local heterogeneity: it is uniformly high-priced, but the value of any particular property can still differ from its neighbor substantially (for example, $600,000 versus $400,000). The 1996 estimated average median census tract household income for the Main Line (East) submarket is $136,256, and 40.7 percent of the submarket population has a college degree. The submarket is approximately 7.5 km by 9.0 km and has an area of 27,400 square meters. A typical property in the Main Line (East) submarket has 2,498 square feet of building area and sold for $277,604, or $111.66 per square foot. This area has some of the oldest homes in the Philadelphia area, with 25 percent of the transactions built prior to 1925. There are 1,399 transactions for the Main Line (East) submarket. The Norristown submarket includes the edge city of Norristown and surrounding tracts. Norristown is an old industrial mill town dating back to the American revolution that was

ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

15

built on the banks of the Schuylkill River for access to waterpower. The housing stock in the city proper contains many traditionally working-class row homes and cottages in highdensity urbanlike settings that are typical of neighborhoods in larger northeastern cities. Although Norristown's industrial base is no longer extant, and the city has become steadily enveloped in Philadelphia's spreading suburban sprawl, Norristown persists as the area of Montgomery County with the lowest median income and highest percentage of minority residents. The 1996 estimated average median census tract household income for this submarket is $63,391 and 16.2 percent of the population has a college degree. The housing stock of Norristown can be characterized by both global and local homogeneity: uniformly low-priced neighborhoods of similar rowhomes and small homes. The Norristown submarket is approximately 9.8 km by 9.1 km and has an area of 35,139 square meters. The area residents are primarily blue collar. The average transaction price in Norristown was $128,247, or $81.67 per square foot. Transactions had an average of 1,639 square feet of building area and were 46 years old at the time of sale. There are 997 Norristown transactions. The Ambler submarket is the stereotypical postwar middle-class bedroom community of the American suburban landscape. It is also the housing submarket that is most similar in characteristics to the overall Montgomery County housing market. Geographically, Ambler is the largest of the three submarkets pro®led here, and unlike the other two submarkets, it contains no real geographically de®ned center. It is best described as a collection of similar suburban housing developments near, but without any real relationship to, the township of Ambler. The 1996 estimated average median census tract household income for this submarket is $98,255, and 29.8 percent of the population have a college degree. The housing stock of Ambler is best characterized as globally heterogeneous but locally homogenous: clusters of architecturally similar and like-priced homes, but some relative variance between clusters of development due to the timing and nature of general suburban growth. The Ambler submarket is approximately 13.1 km by 11.7 km and has an area of 55,490 square meters. The average transaction price in Ambler was $232,301, or $94.92 per square foot. Transactions had an average of 2,446 square feet of building area and were 28 years old at the time of sale. There are 1,773 transactions for Ambler. 4. The empirical hedonic house-price speci®cation With a few exceptions, the empirical speci®cation is standard. The house-price speci®cation relates the log of transaction price to various structural (dwelling size, dwelling age, number of stories, etc.) and location (distance to CBD) characteristics and includes dummy variables for sales date. The continuous variables in the model are common to most traditional hedonic model speci®cations: log of building square footage, frontage, and log of number of stories. We also take the ratio of building square footage to lot square footage to measure the pricing of aesthetic proportionality of the property. As this variable becomes larger, it increasingly denotes a large house on a small lot. Hence the negative sign on this variable's coef®cient

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GILLEN, THIBODEAU AND WACHTER

denotes a discounting for the lack of yard space or privacy. Additionally, we take the ratio of total number of rooms to building square footage to measure the effects of average room size on property value. In accordance with the Victorian aesthetic of the time, many homes in Montgomery County built during the nineteenth century contain a very tight partitioning of the building into a sequence of many rooms that would be considered awkwardly small by today's standards. The positive sign on its coef®cient indicates that, for a ®xed house size, today's consumers generally prefer fewer large rooms to numerous small rooms. A set of dummy variables measure the effects of categorical housing characteristics. For example, a dummy variable is created for each unique value of qualitative variables, like exterior material and type of heating fuel. Additionally, dummy variables are created for discrete variables, like number of ®replaces or number of bathrooms to allow for nonlinear relationships between these structural characteristics and house prices. As a general rule, the category with the greatest number of observations associated with it serves as the omitted variable in the empirical speci®cation. There are eight possible categories for exterior material. The dummy variable for aluminum exterior is omitted from the speci®cation since it had the largest percentage of observations associated with it. Table 2 provides the omitted categorical variables. The empirical speci®cation includes variables designed to measure the in¯uence that housing vintage has on house price. Figure 4 illustrates the distribution for year of construction. Although the volume of construction is certainly increasing over time as the population of the region increased, the historical pattern of housing construction exhibits several development cycles. Measuring from trough-to-trough, six distinct cycles of construction emerge, which are labeled pre-1865, 1866 to 1887, 1888 to 1918, 1919 to 1945, 1946 to 1975, and 1976 to 1998. Each wave of new construction after 1865 is approximately 25 years in length, and all properties built during a particular cycle are characterized by a common set of hedonic characteristics: architectural style, exterior material, type of heating fuel, and so on. To capture the unique and nonlinear effects of each cycle, we take the interaction of a property's age with a dummy variable that equals 1 if the property was constructed during that cycle. We do the same with age-squared and age-cubed. The general speci®cation is Table 2. Omitted categories for dummy variables in hedonic speci®cation. Variable

Omitted Category

Development cycle dummy Type of exterior Type of space heating system Type of heating fuel Type of basement Number of ®replaces Number of bathrooms Source of water supply Sewer system Time period sold

Built post-1975 Aluminum Air Gas Full-sized One ®replace 2.5 bathrooms Public (as apposed to private well) Public (as opposed to private septic tank) 3Q1998

ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

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Figure 4. Montgomery County construction cycles.

AGEi; j ˆ age of the ith house* (dummy ˆ 1 if built during jth cycle, ˆ 0 otherwise) AGESQi; j ˆ age-squared of the ith house* (dummy ˆ 1 if built during jth cycle, ˆ 0 otherwise) AGECUBEi; j ˆ age-cubed of the ith house* (dummy ˆ 1 if built during jth cycle, ˆ 0 otherwise), for j ˆ 1; . . . ; 6; i ˆ 1; . . . ; 21; 562: The speci®cation also includes variables that measure the in¯uence that distance to three centers of economic activity have on house price. The three centers of economic activity are the City of Philadelphia, the Montgomery County CBD (measured at the King of Prussia) and the distance to the submarket center of economic activity. Finally, the speci®cation includes dummy variables for sales quarter beginning with the ®rst quarter of 1995 and ending with the second quarter of 1998. The omitted category is for properties sold at the end of the period (March, 1998).

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GILLEN, THIBODEAU AND WACHTER

By imposing the same expanded hedonic speci®cation on all submarkets, we reduce the likelihood of there being any spatial effects remaining in the submarket residuals. A more parsimonious speci®cation designed to ``®t the data'' in each submarket increases the likelihood that the residuals will be correlated with omitted variables. The empirical hedonic house-price speci®cation is Ln(TRANSACTION PRICEi † ˆ b0 ‡ b1* DWELLING AGE ‡ b2* AGESQ ‡ b3* AGECUBE ‡ b4* (DWELLING AGE* DEVELOPMENT CYCLE) ‡ b5* (DWELLING AGE* DEVELOPMENT CYCLE)2 ‡ b6* (DWELLING AGE* DEVELOPMENT CYCLE)3 ‡ b7* LN(BUILDING SQUARE FOOTAGE) ‡ b8* (BUILDING SF/LOT SQUARE FOOTAGE) ‡ b9* (TOTAL ROOMS/BUILDING SF) * LN(NUMBER OF STORIES) ‡ b10 * TYPE OF EXTERIOR ‡ b11 * TYPE OF SPACE HEATING SYSTEM ‡ b12 * TYPE OF HEATING FUEL ‡ b13 * TYPE OF BASEMENT ‡ b14 * NUMBER OF FIREPLACES ‡ b15 * NUMBER OF BATHROOMS ‡ b16 * GARAGE CAPACITY ‡ b17 * FRONTAGE ‡ b18 * CORNER LOT ‡ b19 * SOURCE OF WATER SUPPLY ‡ b20 * SEWER SYSTEM ‡ b21 ‡

3 X iˆ1

‡

T X jˆ1

f*i DISTANCE TO CBDi d*j SOLDj ‡ xi ;

…11†

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ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

where DISTANCE1i ˆ Distance of the ith house to the Philadelphia CBD DISTANCE2i ˆ Distance of the ith house to the Montgomery County CBD …King of Prussia† DISTANCE3i ˆ Distance of the ith house to its submarket' s CBD and SOLDj ˆ 1 if property sold in quarter j and is zero otherwise; j ˆ 1995:1, 1995:2, . . ., 1998:2. The descriptive statistics for the housing characteristics included in the hedonic speci®cation for the three featured submarkets (the Main Line (East), Norristown, and Ambler) and for all of Montgomery County are provided in Table 3. Table 3. Descriptive statistics: Means and (standard deviations). Property Characteristic Transaction price Per square foot price Age of property (years) Built 1700±1865 Built 1866±1887 Built 1888±1918 Built 1919±1945 Built 1946±1975 Built 1976±1998 Aluminum exterior Asbestos exterior Block or concrete exterior Frame exterior Masonry exterior Masonry and other exterior Stone exterior Stucco exterior Oil heating fuel Gas heating fuel Electricity heating fuel Coal heating fuel Hot air heating system Hot water heating system Electric heating system Heat pump heating system Crawlspace or no basement Partial basement

Submarket 1: Main Line (East)

Submarket 2: Norristown

Submarket 3: Ambler

Montgomery County

$277,604 ($118,498) $111.66 $27 60 (23) 0.8% 0.6% 14.1% 44.8% 34.2% 5.43% 5.3% 2.0% 0.2% 4.6% 22.1% 29.4% 20.8% 15.7% 51.9% 45.6% 2.5% 0.0% 45.2% 51.3% 0.8% 0.4% 2.4% 9.6%

$128,247 ($37,455) $81.67 ($19) 46 (24) 0.6% 0.5% 5.1% 18.4% 58.9% 16.55% 22.8% 6.0% 0.9% 2.8% 30.9% 11.7% 5.6% 19.3% 42.4% 48.4% 8.9% 0.2% 44.8% 45.6% 3.5% 1.3% 21.2% 6.0%

$232,301 ($93,250) $94.92 ($18) 28 (26) 0.3% 0.7% 3.8% 7.3% 37.3% 50.59% 42.6% 12.2% 0.1% 6.8% 4.0% 10.2% 2.1% 21.9% 31.5% 61.0% 7.1% 0.5% 75.2% 20.5% 1.6% 2.5% 3.9% 11.3%

$179,823 ($91,902) $88.61 ($22.27) 38.2 (28.6) 0.7% 0.8% 5.5% 15.2% 44.9% 32.8% 41.0% 9.9% 0.4% 5.1% 14.4% 10.1% 4.2% 14.9% 40.1% 49.6% 10.2% 0.1% 61.7% 31.0% 3.8% 2.8% 8.4% 10.8%

20

GILLEN, THIBODEAU AND WACHTER

Table 3. (continued ) Property Characteristic Full basement Number of ®replaces

Submarket 1: Submarket 2: Submarket 3: Montgomery Main Line (East) Norristown Ambler County

88.0% 0.9 (0.4) Number of bathrooms 2.7 (0.8) Garage capacity (number of automobiles) 1.3 (0.8) Building square footage 2,498.0 (841.3) Building square footage/lot square footage 23.6% (10.5%) Frontage 88.9 (52.0) Number of stories 1.8 (0.4) Total number of rooms/building square footage 0.3% (0.1%) Property on street corner 8.3% Property in middle of block 91.7% Well water supply 0.0% Public water supply 100.0% Septic tank waste drainage 2.9% Public sewer waste drainage 97.1% Number of transactions 1,399

72.8% 0.5 (0.5) 1.9 (0.5) 0.9 (0.8) 1,638.7 (545.1) 17.1% (10.4%) 77.5 (43.4) 1.5 (0.5) 0.4% (0.1%) 7.2% 92.8% 1.8% 98.2% 1.3% 98.7% 997

84.8% 0.8 (0.4) 2.4 (0.6) 1.7 (0.8) 2,446.3 (821.7) 15.4% (10.0%) 106.2 (62.2) 1.7 (0.4) 0.3% (0.1%) 5.3% 94.7% 4.6% 95.4% 6.9% 93.1% 1,773

80.7% 0.6 (0.5) 2.2 (0.6) 1.3 (0.8) 2,028.5 (744.1) 0.17 (0.09) 92.4 (53.9) 1.6 (0.5) 0.004 (0.001) 5.5% 94.5% 6.5% 93.5% 6.2% 93.8% 21,562

5. Estimation results 5.1. Hedonic parameters The parameters of the hedonic equation and the spherical semivariogram are estimated separately for the 21 submarkets. Table 4 provides the OLS statistics for the Main Line (East), Norristown, and Ambler submarkets as well as for a Montgomery County hedonic. (Results for the remaining submarkets are available from the authors on request.) In general, dwelling size and age explain most of the variation in the log of transaction prices. t-statistics for the logarithm of square feet of building area coef®cients range from 15.7 for Norristown to 23.9 for Ambler. (The estimated coef®cient for the log of building area for the Montgomery County hedonic has a t-statistic of 78.6.) Estimated coef®cients for dwelling age polynomials are jointly statistically signi®cant for each submarket and document different depreciation/vintage patterns. The omitted type of exterior is aluminum, so the estimated coef®cients measure differences from this type of exterior. The frame, masonry, stone, and stucco exteriors command premiums in all submarkets. Properties with electric heat and heat pumps receive discounts relative to properties with gas heat. A

21

ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

Table 4. OLS regression statistics: Parameter estimates and (t-scores). Property Characteristic

Submarket 1: Main Line

Intercept Age of property Age of property squared Age of property cubed Age* (dummy if built 1700± 1865) Age* (dummy if built 1866± 1887) Age* (dummy if built 1888± 1918) Age* (dummy if built 1919± 1945) Age* (dummy if built 1946± 1975) Age squared* (dummy if built 1700±1865) Age squared* (dummy if built 1866±1887) Age squared* (dummy if built 1888±1918) Age squared* (dummy if built 1919±1945) Age squared* (dummy if built 1946±1975) Age cubed* (dummy if built 1700±1865) Age cubed* (dummy if built 1866±1887) Age cubed* (dummy if built 1888±1918) Age cubed* (dummy if built 1919±1945) Age cubed* (dummy if built 1946±1975) Asbestos exterior Block or concrete exterior Frame exterior Masonry exterior Masonry and other exterior

(

( (

(

( ( ( ( (

5.436827 (3.034) 0.023228 (0.377) 0.002816 0.49) 6.9941E-05 (0.454) 0.004298 (0.066) 0.148325 (0.477) 0.055984 (0.865) 0.019608 0.354) 0.034308 0.687) 0.002436 (0.424) 0.000184 0.024) 0.001039 (0.181) 0.002546 (0.45) 0.003024 (0.553) 6.8749E-05 0.447) 5.7084E-05 0.368) 6.0437E-05 0.393) 6.7563E-05 0.44) 7.1635E-05 0.473) 0.07153 (1.587) 0.274149 (2.323) 0.048502 (1.396) 0.085875 (3.174) 0.093566 (3.585)

Submarket 2: Norristown

(

(

( ( (

(

(

( ( ( (

16.78874 (1.3) 0.068315 (2.331) 0.007309 2.459) 0.000209 (2.44) 0.044185 1.108) 2.792119 (2.403) 0.03274 0.826) 0.081289 2.782) 0.077539 3.123) 0.007038 (2.359) 0.041623 2.414) 0.006512 (2.158) 0.007758 (2.625) 0.007924 (2.767) 0.000208 2.432) 0 Ð 0.000204 2.39) 0.000213 2.49) 0.000217 2.569) 0.021416 1.066) 0.01376 (0.293) 0.060242 (2.104) 0.002422 (0.15) 0.006049 (0.33)

Submarket 3: Ambler ( (

(

( ( ( ( (

30.116753 5.258) 0.017173 1.277) 0.000957 (0.687) 3.15E-05 0.782) 0.022891 (0.667) 1.304315 (2.486) 0.092561 (2.316) 0.00475 (0.215) 0.003009 (0.256) 0.001139 0.792) 0.021591 2.507) 0.002612 1.642) 0.000731 0.491) 0.000725 0.538) 3.225E-05 (0.801) 0.000114 (2.136) 4.013E-05 (0.992) 2.99E-05 (0.739) 3.047E-05 (0.764) 0.001842 (0.148) 0.008699 (0.054) 0.053306 (3.549) 0.105869 (5.579) 0.024954 (1.906)

Montgomery County 7.938776 (112.493) 0.001626 ( 0.36) 0.000138 ( 0.291) 0.000006083 ( 0.439) 0.00367 ( 0.784) 0.01735 ( 0.263) 0.036762 (4.192) 0.002868 (0.506) 0.005766 ( 1.454) 0.000153 (0.323) 0.000337 (0.286) 0.000688 ( 1.382) 0.00001064 ( 0.022) 0.000177 (0.386) 0.000006099 (0.441) 0.000005481 (0.377) 0.000010448 (0.754) 0.000007166 (0.517) 0.000006416 (0.468) 0.012598 ( 2.844) 0.010683 ( 0.546) 0.021714 (3.899) 0.032667 (7.844) 0.031838 (6.94)

22

GILLEN, THIBODEAU AND WACHTER

Table 4. (continued ) Property Characteristic Stone exterior Stucco exterior Oil heating fuel Electricity heating fuel Coal heating fuel Hot water heating type Electric heating type Heat pump heating type Crawlspace or no basement Partial basement No ®replace Two or more ®replaces 1.5 bathrooms 2 bathrooms 3 bathrooms 3.5 bathrooms 4 or more bathrooms Garage capacity Log (building square footage) Building square footage/lot square footage Well-water supply Frontage Property on street corner Log (number of stories)

Submarket 1: Main Line

Submarket 2: Norristown

Submarket 3: Ambler

Montgomery County

0.138478 (4.925) 0.097515 (3.52) 0.012082 (1.091) 0.009043 (0.198) Ð Ð 0.019483 ( 1.433) 0.135597 ( 1.851) 0.084008 ( 0.833) 0.039187 ( 1.091) 0.003089 ( 0.159) 0.09876 ( 5.505) 0.08478 (0.6) 0.040388 ( 2.225) 0.047246 ( 0.717) 0.016555 (0.402) 0.041605 (2.667) 0.119314 (4.848) 0.01925 (2.599) 0.729654 (19.552) 0.919653 ( 13.783) Ð Ð 5.3753E-05 (0.404) 0.05863 ( 2.966) 0.104783 (3.795)

0.054877 (2.276) 0.004607 (0.308) 0.004091 ( 0.395) 0.022733 ( 1.07) 0.084131 (0.903) 0.008659 ( 0.808) 0.029409 ( 0.993) 0.002846 ( 0.066) 0.028625 ( 1.645) 0.006948 (0.358) 0.028858 ( 2.712) 0.037553 (0.489) 0.032649 ( 2.78) 0.01721 ( 0.255) 0.04536 (0.334) 0.060844 (1.826) Ð Ð 0.028147 (4.226) 0.477578 (15.734) 0.712429 ( 9.507) 0.014271 (0.424) 0.000773 (5.457) 0.004081 ( 0.24) 0.034208 (1.732)

0.085177 (3.11) 0.029207 (2.879) 0.013219 ( 1.418) 0.000462 ( 0.024) 0.075567 ( 1.43) 0.019557 ( 1.82) 0.052233 ( 1.634) 0.007495 ( 0.266) 0.029639 ( 1.542) 0.011774 ( 0.991) 0.08111 ( 7.404) Ð Ð 0.024175 ( 1.882) 0.075392 (1.033) 0.018084 (0.535) 0.03446 (2.56) 0.137937 (2.506) 0.059498 (9.644) 0.668544 (23.89) 0.709199 ( 15.026) 0.025984 (1.211) 1.357E-05 (0.188) 0.037653 ( 2.326) 0.072262 (4.34)

0.108214 (15.586) 0.054333 (14.317) 0.009003 (3.077) 0.021513 ( 3.817) 0.032424 ( 0.893) 0.014326 ( 4.64) 0.007104 ( 0.902) 0.042416 ( 5.051) 0.069235 ( 15.387) 0.018004 ( 4.409) 0.05799 ( 19.567) 0.115142 ( 2.539) 0.016455 ( 4.86) 0.014743 ( 1.161) 0.075068 (5.289) 0.148934 (27.79) 0.409254 (36.802) 0.049232 (27.253) 0.616347 (78.577) 0.729679 ( 42.884) 0.013301 ( 2.114) 0.000399 (13.7) 0.050819 ( 9.843) 0.026814 (5.198)

23

ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

Table 4. (continued ) Property Characteristic Total rooms/building square footage Septic tank waste drainage Distance to Philadelphia CBD (meters) Distance to Montgomery County CBD (meters) Distance to submarket CBD (meters) Distance to Philadelphia CBD squared Distance to Montgomery County CBD squared Distance to submarket CBD squared Distance to Philadelphia CBD cubed Distance to Montgomery County CBD cubed Distance to submarket CBD cubed 1Q 1995 2Q 1995 3Q 1995 4Q 1995 1Q 1996 2Q 1996 3Q 1996 4Q 1996 1Q 1997 2Q 1997 3Q 1997 4Q 1997 1Q 1998

Submarket 1: Main Line 1.135041 0.095) 0.048326 ( 1.088) 0.000298 (0.619) NA NA 0.000236 (3.999) 2.0411E-08 ( 0.461) NA NA 6.8908E-08 ( 3.29) 5.0559E-13 (0.379) NA NA 5.9895E-12 (2.626) 0.09101 ( 2.677) 0.075381 ( 2.621) 0.02997 ( 1.217) 0.084785 ( 3.198) 0.06321 ( 2.175) 0.068567 ( 2.513) 0.037371 ( 1.608) 0.050609 ( 1.753) 0.090109 ( 2.982) 0.053624 ( 2.098) 0.047122 ( 2.239) 0.08163 ( 3.139) 0.058129 ( 2.134)

Submarket 2: Norristown

(

(

( ( (

(

( ( ( (

( ( ( (

2.369144 (0.286) 0.005076 (0.13) 0.001538 0.989) 0.000216 (1.86) 0.000409 (4.284) 8.1511E-08 (1.299) 2.2388E-08 1.429) 8.4332E-08 2.693) 1.4107E-12 1.664) 1.5748E-14 (0.022) 1.0363E-11 (3.025) 0.010704 (0.431) 0.018851 0.812) 0.00463 (0.215) 0.003752 0.166) 0.012778 0.493) 0.000305 0.013) 0.000655 0.031) 0.000779 (0.038) 0.061174 2.443) 0.016121 0.738) 0.003641 0.17) 0.028627 1.409) 0.012534 (0.543)

Submarket 3: Ambler

(

( ( (

( ( ( ( ( ( ( ( ( ( ( ( ( (

36.386954 (4.17) 0.049935 2.975) 0.001624 (5.251) 0.004379 (6.047) 9.905E-05 (4.267) 7.872E-08 5.722) 2.4E-07 6.348) 1.856E-09 0.285) 1.24E-12 (5.996) 4.356E-12 (6.61) 2.783E-12 4.026) 0.086168 4.183) 0.072894 4.008) 0.073246 4.451) 0.102734 5.991) 0.073774 3.686) 0.077852 4.022) 0.063646 3.99) 0.09558 5.623) 0.109072 5.541) 0.073104 3.981) 0.077135 4.578) 0.065492 3.513) 0.091353 4.429)

Montgomery County 23.60211 (9.939) 0.036448 (6.075) 0.000046475 ( 21.456) 0.000046544 (22.927) 0.000001075 ( 0.259) 1.10828E-09 (15.07) 3.10415E-09 ( 24.458) 1.43118E-09 (1.51) 9.99662E-15 ( 12.982) 5.88171E-14 (24.035) 2.88033E-13 ( 4.654) 0.048998 ( 7.202) 0.033713 ( 5.384) 0.02936 ( 5.291) 0.044764 ( 7.684) 0.045978 ( 6.964) 0.029034 ( 4.873) 0.02393 ( 4.437) 0.042517 ( 7.28) 0.056105 ( 8.263) 0.024327 ( 4.008) 0.020713 ( 3.712) 0.033719 ( 5.702) 0.037171 ( 5.643)

24

GILLEN, THIBODEAU AND WACHTER

Table 4. (continued ) Property Characteristic 2Q 1998 Regression statistics: R-squared Adjusted R-squared F-statistic Probability 4 F

Submarket 1: Main Line (

0.022311 0.913)

0.7797 0.7686 70.322 5 0.0001

Submarket 2: Norristown (

0.006263 0.302)

0.7879 0.7718 49.135 5 0.0001

Submarket 3: Ambler (

0.040137 2.291)

0.8773 0.8722 171.283 5 0.0001

Montgomery County (

0.017584 3.046)

0.8308 0.8303 1,466.0 5 0.0001

®replace is worth between 3 percent and 8 percent of the value of the property. The estimated coef®cients for the ratio of building square feet to lot size are all negative and highly signi®cant. Estimated coef®cients for square feet of garage space are statistically signi®cant with the expected in¯uence in each of the submarkets. Corner lots are capitalized into all house prices as a discount. Additional rooms per square foot of building area command premiums in Norristown and Ambler but not in the Main Line. The distance of any given property to the Philadelphia CBD is strongly signi®cant for the entire county but is generally insigni®cant within any given submarket. However, the distance to the submarket's CBD is strongly signi®cant for all submarkets, thus re¯ecting the importance of relative global and local effects. Finally, between January, 1995 and March, 1998, house prices increased 5 percent for Montgomery County properties. Similarly, prices increased over 9 percent for the Main Line properties and over 8 percent for Ambler properties, but house prices have been essentially constant for Norristown homes. 5.2. House price spherical semivariograms We examine directional spatial autocorrelation in Montgomery County house prices and in hedonic house-price equation residuals by ®tting a spherical function to the empirical semivariogram computed using properties separated by a given direction. Our analysis examines directional autocorrelation for two directions: north±south and east±west. Since few properties are located exactly north±south (or east±west) of one another, we compute the spatial autocorrelation for properties within a tolerance range of the desired direction. The tolerance region used here is + 45 degrees. Consequently, the points for the empirical semivariogram for property pairs separated by a northerly direction are computed for properties located 90 + 45. Similarly, the points for the empirical semivariogram for property pairs separated in an easterly direction are computed for properties located 0 + 45. Table 5 presents the empirically estimated thetas and (approximate) t-scores of the spherical semivariogram for log(house price) in the Main Line (East) submarket. The measurement of y2 , the range, has been rescaled to kilometers. The empirical semivariogram results clearly indicate that the spatial autocorrelation of house prices in

25

ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

Table 5. Estimated semivariogram parameters: Thetas and (t-scores). Log (House Price) Isotropic spherical semivariogram: y0 (t-score) y1 (t-score) y2 (t-score) Anisotropic spherical semivariogram: Properties north-south: y0 (t-score) y1 (t-score) y2 (t-score) Properties east-west: y0 (t-score) y1 (t-score) y2 (t-score)

Main Line (East) Submarket 0.099003 (12.801) 0.08001 (9.1672) 4.4115 (5.7187) 0.099888 (11.746) 1.4489 (0.0084357) 85.228 (0.0084211) 0.10862 (12.637) 0.044479 (4.9164) 3.3459 (3.1912)

this submarket is anisotropic. The values of the spherical semivariogram parameters for the isotropic semivariogram are statistically different from zero, indicating that house values are correlated up to a distance of approximately 4.4 km. However, the anisotropic semivariograms differ in their results: although the estimated semivariogram parameters are uniformly signi®cant for the east semivariogram, only y0 is signi®cant for the north semivariogram. This suggests that the Main Line (East) spatial autocorrelation in house prices exists solely along the east±west axis of the submarket. Figure 5 plots the isotropic and anisotropic semivariograms. First, note that the isotropic semivariogram lies exactly between the anisotropic semivariograms. This would suggest that the isotropic measurement of spatial dependence is an average of the anisotropic measurements. This isotropic semivariogram illustrates that (the log of ) the Main Line house prices are spatially autocorrelated up to a range of about 4.4 km. Beyond 4.4 km, (the log of ) the Main Line house prices are spatially uncorrelated. But the anisotropic semivariogram for east indicates that Main Line house prices are spatially autocorrelated up to a distance of only 3.3 km. At ®rst glance, the nonconverging anisotropic semivariogram for north would suggest that the spatial stochastic process of house prices is nonstationary, and the actual range of spatial autocorrelation is much greater than 4.4 km. However, the insigni®cant t-scores for its parameters suggest just the opposite. In actuality, there is no spatial autocorrelation in house prices in a north±south direction. This spatial stochastic process is just a pure nugget effect as house prices vary randomly from one

26

GILLEN, THIBODEAU AND WACHTER

Figure 5. Semivariograms of log(price) for Main Line (East).

property to the next with no covariance. All spatial autocorrelation occurs along the east±west axis. Cities generally develop outwards from their centers. Consequently, we might a priori expect that the direction of greatest autocorrelation would be in the direction of the city's CBD. This is exactly true for this submarket: Philadelphia lies directly across the eastern edge of this submarket's border.

5.3. Spherical semivariograms for hedonic residuals Table 6 provides the semivariogram parameter estimates for the residuals from the hedonic house-price equations for all 21 submarkets. The top half of the table lists semivariogram parameters for the isotropic spherical semivariogram while the bottom half of the table presents the spherical semivariogram parameters for two anisotropic spherical semivariograms. Isotropic and anisotropic estimates of the nuggest are statistically signi®cant in each of the 21 submarkets. Isotropic estimates of y1 and y2 (the range) are statistically signi®cant in nine submarkets while anisotropic estimates of these parameters are signi®cant in 11 submarkets. For ®ve of these 11 submarkets, the estimated parameters for y1 and y2 are

27

ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

statistically in only one direction. In Pottstown, New Hanover, Bryn Athyn, and the Main Line (West) submarket, hedonic house-price equation residuals are spatially correlated only for properties separated in a north±south direction. For Cheltenharn, hedonic house± price equation residuals are spatially correlated only for properties separated by an east± west direction. Table 6. Estimated semivariogram parameters: Thetas and (t-scores). Submarket 1: Pottstown

Submarket 2: New Hanover

Submarket 3: Pennsbury

Isotropic spherical semivariogram: y0 0.013509 0.007775 0.009848 (t-score) (22.326000) (16.539) (13.808) 0.000694 0.001824 0.001172 y1 (t-score) (0.953030) (2.5988) (1.1755) y2 5.067700 7.027400 4.084200 (t-score) (0.571100) (1.576000) (0.671410) Anisotropic spherical semivariogramÐProperties north: y0 0.011898 0.007588 0.010164 (t-score) (23.437) (16.321) (9.3265) y1 0.002907 0.003092 0.001702 (t-score) (4.2959) (4.8419) (1.218) y2 4.298700 6.970300 4.093200 (t-score) (2.634200) (2.879000) (0.721300) Anisotropic spherical semivariogramÐProperties east: y0 0.014094 0.008596 0.009141 (t-score) (8.4499) (10.395) (16.573) y1 0.001984 0.001007 0.000441 (t-score) (1.0822) (0.81781) (0.46532) y2 5.122200 6.139400 3.974700 (t-score) (0.676880) (0.479040) (0.243700) Submarket 8: Souderton

Submarket 9: Lower Providence

Submarket 10: Main Line (East)

Isotropic spherical semivariogram: y0 0.010266 0.011274 0.029837 (t-score) (5.8594) (13.661) (61.292) y1 0.002114 0.000837 0.007598 (t-score) (1.0865) (0.89271) (13.833) y2 1.730800 4.709700 4.411100 (t-score) (0.657440) (0.548130) (8.631600) Anisotropic spherical semivariogramÐProperties north: y0 0.009919 0.010685 0.028353 (t-score) (7.3234) (13.018) (62.699) y1 0.001311 0.000124 0.010826 (t-score) (0.88968) (0.12143) (20.804) y2 1.751200 4.685500 4.383600 (t-score) (0.525630) (0.072206) (12.662000) Anisotropic spherical semivariogramÐProperties east: y0 0.010724 0.011721 0.029623 (t-score) (4.5296) (8.5227) (51.825) y1 0.003122 0.001092 0.006407 (t-score) (1.1422) (0.75259) (10.649) y2 1.672900 3.664600 3.346500 (t-score) (0.693450) (0.475150) (6.907600)

Submarket 4: Upper Providence

Submarket 5: Frederick

Submarket 6: Salford

Submarket 7: Franconia

0.010895 (15.846) 0.000749 (0.89185) 9.171300 (0.536740)

0.007341 (13.674) 0.002948 (3.2809) 10.174000 (1.977000)

0.008872 (16.028) 0.002905 (3.7875) 7.536000 (2.069700)

0.006115 (4.3862) 0.000982 (0.68261) 5.125600 (0.465280)

0.010138 (15.062) 0.000896 (1.0644) 9.272000 (0.656990)

0.007217 (10.999) 0.002942 (3.1221) 10.191000 (1.911300)

0.008667 (15.594) 0.002705 (3.7496) 7.506300 (2.072600)

0.0062737 (3.3021) 0.001182 (0.57906) 5.1705 (0.39166)

0.010729 (13.379) 0.000311 (0.35996) 6.910600 (0.225530)

0.005972 (7.0889) 0.006538 (3.9482) 4.281800 (2.542600)

0.009032 (12.236) 0.003210 (3.3335) 6.631600 (1.761300)

0.006200 (4.4835) 0.000738 (0.52268) 4.739500 (0.339260)

Submarket 11: King of Prussia

Submarket 12: Norristown

Submarket 13: Conshohocken

Submarket 14: Worcester

0.010124 (22.327) 0.000023 (0.039333) 2.731400 (0.023442)

0.015255 (57.33) 0.000114 (0.39994) 4.372100 (0.254680)

0.011556 (22.582) 0.003695 (5.7824) 4.607800 (3.392500)

0.010638 (15.85) 0.002546 (3.4225) 5.700000 (2.074700)

0.0095929 (11.662) 0.0012116 (1.2795) 2.2615 (0.764220)

0.015260 (36.321) 0.000184 (0.4029) 3.516800 (0.254590)

0.010202 (10.045) 0.005054 (4.1442) 2.829300 (2.562800)

0.010917 (11.903) 0.003593 (3.5116) 5.294300 (2.098100)

0.010366 (11.698) 0.000719 (0.68827) 2.789000 (0.420160)

0.015528 (20.359) 0.000640 (0.81586) 4.344500 (0.526520)

0.011925 (20.507) 0.002695 (4.069000) 4.593200 (2.382500)

0.0095338 (15.255) 0.0033431 (4.8682) 5.7355 (3.002700)

28

GILLEN, THIBODEAU AND WACHTER

Table 6. (continued ) Submarket 15: Lansdale

Submarket 16: Ambler

Submarket 17: Whitemarsh

Isotropic spherical semivariogram: y0 0.012226 0.019784 0.017498 (t-score) (20.017) (24.045) (27.053) 0.00046023 0.000465 0.003331 y1 (t-score) (0.721270) (0.53701) (4.5588) y2 6.3653 6.437900 6.269600 (t-score) (0.475590) (0.340430) (2.709700) Anisotropic spherical semivariogramÐProperties north: 0.01231 0.019983 0.016957 y0 (t-score) (27.355) (15.384) (25.231) y1 0.00015051 0.000714 0.003434 (t-score) (0.31426) (0.53448) (4.818) y2 6.0585 6.422500 6.247900 (t-score) (0.20623) (0.348070) (2.960700) Anisotropic spherical semivariogramÐProperties east: y0 0.011956 0.019283 0.016213 (t-score) (18.008) (21.233) (20.297) y1 0.00034907 0.000405 0.006070 (t-score) (0.51301) (0.42103) (6.4207) y2 6.3622 6.057200 4.407900 (t-score) (0.33929) (0.262260) (3.892300)

Submarket 18: Cheltenham

Submarket 19: Abington

Submarket 20: Bryn Athyn

Submarket 21: Main Line (West)

0.020078 (37.057) 0.002152 (3.1104) 5.895600 (1.802800)

0.018771 (18.43) 0.0012617 (1.071) 5.9693 (0.64124)

0.024099 (41.484) 0.002310 (3.4852) 5.854500 (2.158100)

0.037062 (35.28) 0.0016473 (1.327) 5.7292 (0.79497)

0.019773 (25.714) 0.001121 (1.0703) 5.692500 (0.622180)

0.018292 (19.884) 0.0009111 (0.92195) 6.0464 (0.58431)

0.024058 (28.458) 0.003129 (3.3156) 5.862600 (2.025300)

0.036736 (23.995) 0.0098765 (5.3327) 5.7387 (3.1858)

0.020727 (31.663) 0.001866 (2.3766) 5.925900 (1.407200)

0.019297 (18.386) 0.001493 (1.1339) 5.1575 (0.69985)

0.024012 (29.371) 0.001826 (1.9329) 5.578300 (1.2101)

0.036814 (9.1017) 0.0047262 (1.0104) 5.7213 (0.60748)

The range of the anisotropic spatially autocorrelated residuals also varies for several submarkets that exhibit both north±south and east±west spatial autocorrelation. For example, the range of spatial autocorrelation in the Main Line (East) submarket for properties separated in a north±south direction is over 1 km further than the range for properties separated by an east±west direction. 6. Conclusion This article examines anisotropic spatial autocorrelation in house prices and in hedonic house-price equation residuals for 21 housing submarkets in suburban Philadelphia. We ®nd that both house prices and hedonic house-price equation residuals are spatially autocorrelated and the spatial autocorrelation changes with the direction separating properties for some submarkets. Since most real estate development tends to spread outward from existing city centers along major transportation arteries, it is reasonable to expect that the direction of greatest spatial autocorrelation occurs in the direction of the CBD. Our empirical results are consistent with this hypothesis. Submarkets where anisotropy occurs are typically innerring suburban bedroom communities located close to Philadelphia. For two of the submarkets (Bryn Athyn, Cheltenham), the direction of spatial autocorrelation is directly toward the Philadelphia city center. For a third submarket (Main Line West), the direction of anisotropy is toward a major transportation artery connecting the Philadelphia CBD with the County CBD, both of which are major employment centers for the residents of

ANISOTROPIC AUTOCORRELATION IN HOUSE PRICES

29

this submarket. For two other submarkets on the rural western edge of the county (Pottstown, New Hanover), the direction of spatial autocorrelation is greatest toward the Pottstown city center, which is the local CBD for this area of the county. Especially interesting is the characterization of local versus global anisotropic effects. While the global pattern of development is outward from the Philadelphia CBD, local development also spreads away from local CBDs such as the county and submarket CBDs. The direction anisotropy is in¯uenced by which CBD is the primary shopping and employment center for submarket residents. For bedroom communities of Philadelphia commuters, anisotropy obtains in the direction of the Philadelphia CBD. For submarkets where residents are equally likely to commute to either the MSA CBD or county CBD, anisotropy obtains in the direction of the transportation artery connecting these two CBDs. Finally, for the more rural and isolated submarkets where the local CBD is the primary retail and employment district, anisotropy obtains in the direction of this CBD, unaffected by the larger spread of development outward from Philadelphia. Further investigation into the estimation of the local versus global layers of anisotropy is an area for further research. The goal of this article was to empirically test the isotropy assumption for house-price models. Our eventual goal is to evaluate the gain in prediction accuracy associated with explicitly modeling anisotropic spatial dependence in hedonic house-price equation residuals. Acknowledgments We acknowledge Professor Tony Smith, John Green of REALIST, and Paul Amos for their assistance with this article. References Anselin, L. (1978). Spatial Econometrics: Methods and Models. Dordrecht: Kluwer Academic. Bailey, M., R. Muth, and H. Nourse. (1963). ``A Regression Method for Real Estate Price Index Construction,'' Journal of the American Statistical Association 58(304), 933±942. Bailey, T. C., and A. C. Gatrell. (1995). Interactive Spatial Data Analysis. Essex, England: Addison Wesley Longman. Basu, S., and T. Thibodeau. (1998). ``Analysis of Spatial Autocorrelation in House Prices,'' Journal of Real Estate Finance and Economics 17(1), 61±85. Can, A. (1992). ``Speci®cation and Estimation of Hedonic Housing Price Models,'' Regional Science and Urban Economics 22, 453±477. Case, B., and J. Quigley. (1991). ``The Dynamics of Real Estate Prices,'' Review of Economics and Statistics 83, 50±58. Case, K., and R. Shiller. (1987). ``Prices of Single-Family Homes Since 1970: New Indexes for Four Cities.'' National Bureau of Economic Research Working Paper No. 2393. Cambridge, MA. Case, K., and R. Shiller. (1989). ``The Ef®ciency of the Market for Single-Family Homes,'' American Economic Review 79(1), 125±137. Clapp, J., and C. Giaccotto. (1992). ``Estimating Price Trends for Residential Property: A Comparison of Repeat Sales and Assessed Value Methods,'' Journal of the American Statistical Association 87, 300±306. Cliff, Andrew D., and J. K. Ord. (1973). Spatial Autocorrelation. London: Pion.

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