GWR and the spatial heterogeneity of hedonic prices: Most promising a method, not a Panacea

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GWR and the Spatial Heterogeneity

of Hedonic Prices:

Most Promising a Method, Not a Panacea

François Des Rosiers and Marius Thériault Laval University, Canada

2007 Seminar on Automated Methods of Mass Appraisal and Market Analysis

Delft, The Netherlands, November 1-2, 2007

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Purpose and Context of Study

In Chapter 6 of «Advances in Mass Appraisal Methods», focus is put on the measurement of urban externalities using hedonic modelling

Primary purpose of this paper is to handle the spatial

non-stationarity of hedonic house prices, using access to primary

school facilities as a case study

While access to primary schools has been shown to influence family household preferences, willingness-to-pay may vary over space

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Literature Review – Proximity to Schools

Influence of school quality on house prices (Hendon 1973,

Jud and Watts 1981, Jud 1985, Haurin and Brasington 1996, Bogart and Cromwell 1997, Black 1999, Downes and Zabel 2002, Reback 2005, Hayes and Taylor 1996, Kane et al. 2005 )

Impact of proximity to schools and of school size on

property values (Guntermann and Colwell 1983, Des Rosiers et al. 2001, Thériault et al. 2005)

Optimal school size (365 pupils); optimal distance to

nearest school (400 metres) (Des Rosiers et al. 2001)

Impact of urban environment (type of road, urban form

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Literature Review – Regression methods

OLS method addresses neither the issue of the non-stationarity of shadow prices nor the presence of spatial autocorrelation among residuals (Getis and Ord 1992, Dubin 1998)

Improvement to OLS: Casetti’s (1972) expansion method (Can 1990, Thériault et al. 2003 ); Spatial Autoregressive

Regression – SAR (Anselin’s 1995); Geographically Weighted Regression – GWR (Brunsdon et al. 1996)

GWR outperforms most other global and local approaches (Farber & Yates 2006 )

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Database Organization

Study based on some 8,221 single-family detached house sales in Quebec City (former- QUC territory) between January 1993 and December 1996

Median price : $89,000CAN (50,000-250,000)

Information on major property attributes, local services, time trend, socioeconomic and household structure profiles (1996), regional accessibility index, land use density

Four school-related control variables : school board

dummy; nearest primary school is English speaking; is also a high school; school size

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Analytical Approach

The GWR procedure consists in performing a series of

local regressions, thereby calibrating a specific hedonic model for every property in the database. Each local regression is calibrated using all data within a given kernel around this point –although with a weight that decreases with distance -, with the spatial kernel being either “fixed” (constant radius) or “adaptive” to local land use densities (constant number of neighbours).

Y_i= β₀ (x_i ,y_i) + β₁ (x_i ,y_i) X_i1 + β₂ (x_i ,y_i) X_i2 +…+ β_p (x_i ,y_i) X_ip + ε_i (x_i ,y_i) = coordinates of the ith point in space, with i=1 to N

β_p (x_i ,y_i) X_ip = estimated parameters for variable p at the regression point

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Research Hypotheses

Hypothesis H1: Access to, and proximity of, a primary

school is a positive externality which should enhance the value of a nearby residence (neg. signed contin. var.)

Hypothesis H2: distance dummy parameters are expected

to be statistically significant and positively signed, safe for the nearest buffer (negative proximity effects)

Hypothesis H3: location rent generated by access to, and

proximity of, a primary school is expected to decrease with

distance

Hypothesis H4: Space-related attributes are expected to

display significant spatial drifts

Hypothesis H5: GWR substantially reduces spatial

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Main Regression Findings - OLS

N= 8,221 K= 29 Standzd Coeff. Collinearity statistics

B Std. Error Beta t Sig. VIF

(Constant) 10.642 0.042 253.030 0.000 LivArea_m2 0.004 0.000 0.462 52.665 0.000 2.93 lnLotSize 0.063 0.006 0.071 11.346 0.000 1.47 AppAge -0.010 0.000 -0.418 -43.643 0.000 3.48 Cottage -0.039 0.005 -0.059 -7.238 0.000 2.50 Quality 0.073 0.007 0.058 10.446 0.000 1.19 Finbasment 0.057 0.003 0.095 16.945 0.000 1.20 SoftFacing51 -0.032 0.004 -0.052 -9.177 0.000 1.20 Fireplace 0.044 0.004 0.069 11.887 0.000 1.30 HiQualFloor 0.032 0.003 0.053 9.395 0.000 1.22 HrdWoodStair 0.036 0.005 0.047 7.569 0.000 1.49 SplAttGar 0.101 0.008 0.072 13.316 0.000 1.13 DblAttGar 0.057 0.010 0.030 5.663 0.000 1.09 SplDetGar 0.057 0.006 0.056 10.359 0.000 1.10 DblDetGar 0.071 0.007 0.052 9.660 0.000 1.08 WaterSewer 0.154 0.015 0.055 10.210 0.000 1.09 NbMonthJan_9396 -0.001 0.000 -0.023 -4.424 0.000 1.05 AvgIncome_1K 0.004 0.000 0.166 21.503 0.000 2.26 PercChild0_9years -0.004 0.001 -0.071 -8.216 0.000 2.84 PercChildFam -0.002 0.000 -0.058 -7.433 0.000 2.33 RegAcc 0.076 0.003 0.233 23.175 0.000 3.83 OldSuburbs -0.034 0.006 -0.056 -5.543 0.000 3.91 NewSuburbs -0.034 0.008 -0.036 -4.217 0.000 2.79 ScatrdSettlmt -0.007 0.008 -0.012 -0.941 0.347 6.17 Decouvreurs 0.083 0.006 0.126 14.542 0.000 2.85 PremSeigneuries 0.020 0.004 0.032 5.096 0.000 1.52 EnglishSchool 0.054 0.010 0.030 5.121 0.000 1.27 HighSchool 0.019 0.007 0.017 2.972 0.003 1.22 SchoolSize -0.004 0.001 -0.022 -3.389 0.001 1.67 Unstandardized Coefficients Coefficientsa

a. Dependent Variable: LN_SALEPRICE

Sig.= 0.000

Variable

Adj. R-sq = 0.784 F= 1,064.044 SEE= 0.1380

• School board descriptors act as proxies for housing submarkets

• Negative proximity effects tend to increase with school size

• Nearest English school raises house values by some 5.4%

• If high school education provided, market premium 1.9%

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Main Regression Findings - OLS

• Driving time coefficient displays a much stronger magnitude (-0.017) than the walking time one does (-0.001) due to greater distance covered by car

N= 8,221 K= 30

Standzd Coeff.

Collinearity statistics B Std. Error Beta t Sig. VIF

(Constant) 10.669 0.042 254.238 0.000 OldSuburbs -0.026 0.006 -0.044 -4.280 0.000 3.98 NewSuburbs -0.028 0.008 -0.030 -3.449 0.001 2.81 ScatrdSettlmt 0.002 0.008 0.003 0.259 0.796 6.28 Decouvreurs 0.079 0.006 0.120 13.863 0.000 2.87 PremSeigneuries 0.021 0.004 0.034 5.399 0.000 1.52 EnglishSchool 0.055 0.010 0.030 5.299 0.000 1.27 HighSchool 0.019 0.006 0.016 2.900 0.004 1.22 SchoolSize -0.002 0.001 -0.015 -2.256 0.024 1.70 Walking_Time -0.001 0.000 -0.051 -8.971 0.000 1.26 Unstandardized Coefficients Adj. R-sq = 0.786 F= 1,040.094 Sig.= 0.000 SEE= 0.1373 Variable Coefficientsa

N= 8,221 K= 30

Collinearity statistics

B Std. Error Beta t Sig. VIF

(Constant) 10.667 0.042 253.84 0.000 OldSuburbs -0.028 0.006 -0.046 -4.49 0.000 3.97 NewSuburbs -0.029 0.008 -0.030 -3.54 0.000 2.81 ScatrdSettlmt 0.001 0.008 0.001 0.07 0.942 6.27 Decouvreurs 0.081 0.006 0.123 14.18 0.000 2.86 PremSeigneuries 0.022 0.004 0.036 5.67 0.000 1.53 EnglishSchool 0.056 0.010 0.031 5.36 0.000 1.27 HighSchool 0.017 0.006 0.015 2.66 0.008 1.22 SchoolSize -0.003 0.001 -0.017 -2.48 0.013 1.69 Driving_Time -0.017 0.002 -0.045 -7.93 0.000 1.21 Variable Coefficientsa Adj. R-sq = 0.785 F= 1,037.280 Sig.= 0.000 SEE= 0.1375 Unstandardized Coefficients

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Main Regression Findings - OLS

• Interactive time-distance coefficients yield results identical to model 3.

Model 5 (BUF) - Includes Distance Buffers

Model 4 (ITD) - Includes Interactive Time and Distance Dummies

N= 8,221 K= 31

Collinearity statistics B Std. Error Beta t Sig. VIF (Constant) 10.663 0.042 253.974 0.000 OldSuburbs -0.027 0.006 -0.045 -4.394 0.000 3.980 NewSuburbs -0.029 0.008 -0.031 -3.554 0.000 2.820 ScatrdSettlmt 0.002 0.008 0.003 0.216 0.829 6.292 Decouvreurs 0.079 0.006 0.120 13.831 0.000 2.888 PremSeigneuries 0.021 0.004 0.035 5.475 0.000 1.525 EnglishSchool 0.055 0.010 0.030 5.266 0.000 1.272 HighSchool 0.019 0.006 0.016 2.861 0.004 1.221 SchoolSize -0.002 0.001 -0.016 -2.366 0.018 1.698 WalkTime_Less1600m -0.001 0.000 -0.024 -3.565 0.000 1.804 DrivTime_Beyond1600m -0.017 0.002 -0.059 -8.015 0.000 2.069 a. Dependent Variable: LN_SALEPRICE

Adj. R-sq = 0.785 F= 1,003.448 Sig.= 0.000 SEE= 0.1375 Variable Unstandardized Coefficients Coefficientsa N= 8,221 K= 35 Standzd Coeff. Collinearity statistics B Std. Error Beta t Sig. VIF (Constant) 10.582 0.043 247.296 0.000 0.000 OldSuburbs -0.027 0.006 -0.045 -4.405 0.000 3.985 NewSuburbs -0.029 0.008 -0.030 -3.510 0.000 2.817 ScatrdSettlmt 0.002 0.008 0.002 0.190 0.849 6.279 Decouvreurs 0.083 0.006 0.127 14.684 0.000 2.868 PremSeigneuries 0.022 0.004 0.036 5.678 0.000 1.531 EnglishSchool 0.059 0.010 0.033 5.630 0.000 1.277 HighSchool 0.017 0.006 0.015 2.696 0.007 1.222 SchoolSize -0.002 0.001 -0.014 -2.095 0.036 1.717 Buf_0_250 0.068 0.010 0.066 6.896 0.000 3.467 Buf_251_500 0.060 0.009 0.088 6.681 0.000 6.596 Buf_501_750 0.053 0.009 0.079 6.048 0.000 6.490 Buf_751_1000 0.038 0.009 0.049 4.236 0.000 5.140 Buf_1001_1300 0.040 0.009 0.043 4.252 0.000 3.954 Buf_1301_1600 0.027 0.010 0.022 2.663 0.008 2.698 SEE= 0.1374 Coefficientsa Unstandardized Coefficients

Adj. R-sq = 0.785 F= 886.338 Sig.= 0.000

Variable

• As expected, the magnitude of the buffer coefficients

decreases with distance from 0.068 to 0.027 (with an

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Spatial Autocorrelation Diagnostics for OLS

Models (GEODA software)

• Spatial dependence is still present in the models’

residuals (between 0.0613 and 0.0628) and significant at the 0.001 level.

Variable Global Moran's I* Sig.

lnSaleprice 0.3106 < 0.001 Pre_Base (1) 0.3558 < 0.001 Res_Base (1) 0.0628 < 0.001 Pre_WT (2) 0.3567 < 0.001 Res_WT (2) 0.0621 < 0.001 Pre_DT (3) 0.3558 < 0.001 Res_DT (3) 0.0628 < 0.001 Pre_ITD (4) 0.3563 < 0.001 Res_ITD (4) 0.0625 < 0.001 Pre_BUF (5) 0.3565 < 0.001 Res_BUF (5) 0.0613 < 0.001

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Main Regression Findings - GWR

• GWR outperforms the OLS method, both in terms of adjusted R-square (0.810) and of AIC (Akaike Information Criterion) value (-10,191)

• Main differences in the parameter estimates are observed with space-related descriptors

• Interactive walking and driving time

median estimate values (-0.001 and

-0.013) corroborate OLS results • Findings suggest that, as

hypothesized, access to primary schools is not stationary over space

Number of nearest neighbours = 4,668 Adj. R-sq = 0.810

Akaike Information Criterion (AIC) = -10,190.89 AIC for OLS = -9,259.29

Source SS DF MS F OLS Residuals 154.8 31

GWR Residuals 135.4 8104.76 0.0167 13.6826 GWR Improvement 19.5 85.24 0.2285 (Crit.Val.: 1.4526)

ANOVA

Overall Model Performances

Variable Minimum Lwr Quartile Median Upr Quartile Maximum

Intrcept 10.4170 10.5370 10.5865 10.6339 10.7623 LivArea_m2 0.0036 0.0038 0.0040 0.0040 0.0042 lnLotSize 0.0492 0.0695 0.0785 0.0897 0.1151 AppAge -0.0121 -0.0115 -0.0113 -0.0100 -0.0086 Cottage -0.0530 -0.0341 -0.0294 -0.0267 0.0061 Quality 0.0555 0.0616 0.0763 0.0808 0.0971 Finbasment 0.0436 0.0475 0.0546 0.0602 0.0732 SoftFacing51 -0.0540 -0.0378 -0.0326 -0.0300 -0.0276 Fireplace 0.0317 0.0377 0.0412 0.0446 0.0605 HiQualFloor 0.0200 0.0254 0.0293 0.0322 0.0362 HrdWoodStair 0.0299 0.0318 0.0401 0.0440 0.0506 SplAttGar 0.0462 0.0879 0.0931 0.0989 0.1180 DblAttGar 0.0237 0.0505 0.0626 0.0825 0.0981 SplDetGar 0.0298 0.0491 0.0528 0.0549 0.0649 DblDetGar 0.0366 0.0670 0.0803 0.0868 0.0983 WaterSewer 0.0379 0.1228 0.1554 0.1750 0.2352 NbMonthJan_9396 -0.0011 -0.0007 -0.0005 -0.0004 -0.0002 AvgIncome_1K 0.0027 0.0033 0.0038 0.0043 0.0047 PercChild0_9years -0.0082 -0.0061 -0.0051 -0.0030 0.0000 PercChildFam -0.0047 -0.0038 -0.0024 -0.0015 0.0005 RegAcc 0.0343 0.0487 0.0554 0.0785 0.1480 OldSuburbs -0.0344 -0.0212 -0.0100 0.0008 0.0064 NewSuburbs -0.0790 -0.0364 -0.0177 0.0128 0.0437 ScatrdSettlmt -0.0362 -0.0041 0.0103 0.0466 0.1059 Decouvreurs -0.0672 0.0488 0.0975 0.1855 0.3535 PremSeigneuries -0.1970 -0.0226 -0.0039 0.0066 0.0429 EnglishSchool -0.0488 0.0000 0.0119 0.2648 0.3583 HighSchool -0.0147 0.0254 0.0376 0.0531 0.1177 SchoolSize -0.0114 -0.0030 0.0023 0.0047 0.0087 WalkTime_Less1600m -0.0028 -0.0012 -0.0009 -0.0007 -0.0004 DrivTime_Beyond1600m -0.0440 -0.0209 -0.0131 -0.0088 0.0009

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Main Regression Findings - GWR

• Median parameter estimates are positively signed and uniformly decrease with distance.

• However they display substantially lower magnitudes than OLS-derived coefficients (Model 5)

• In summary, H1, H2 and H3 (access to, and proximity of, a primary school

positively affect residential values) are being validated using GWR

• GWR method strongly suggests that most non-building attributes, including school-related and access-to-school variables, are not homogeneously valued by households over space (H4)

Model 7 – GWR-BUF Overall Performance & Regression Coefficients

Number of nearest neighbours = 4,668 Adj. R-sq = 0.810

Akaike Information Criterion (AIC) = -10,191.95 AIC for OLS = -9,261.26

Source SS DF MS F OLS Residuals 154.7 35

GWR Residuals 134.8 8088.15 0.0167 12.1852 GWR Improvement 19.9 97.85 0.2031 (Crit.Val.: 1.4244)

Overall Model Performances

ANOVA

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GWR local R-Squares

Map 1 : ITD Model (computed for each case)

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Monte-Carlo Spatial Stationarity

Test for GWR Coefficients

(non stationary where significant)

Parameter P-value Level of Significance Intercept 0.630 n/s LivArea_m2 0.450 n/s lnLotSize 0.100 n/s AppAge 0.000 *** Cottage 0.390 n/s Quality 0.570 n/s Finbasment 0.070 n/s SoftFacing51 0.060 n/s Fireplace 0.420 n/s HiQualFloor 0.530 n/s HrdWoodStair 0.360 n/s SplAttGar 0.560 n/s DblAttGar 0.290 n/s SplDetGar 0.880 n/s DblDetGar 0.230 n/s WaterSewer 0.240 n/s NbMonthJan_9396 0.060 n/s AvgIncome_1K 0.010 ** PercChild0_9years 0.000 *** PercChildFam 0.000 *** RegAcc 0.000 *** OldSuburbs 0.240 n/s NewSuburbs 0.000 *** ScatrdSettlmt 0.000 *** Decouvreurs 0.000 *** PremSeigneuries 0.000 *** EnglishSchool 0.000 *** HighSchool 0.000 *** SchoolSize 0.000 *** WalkTime_Less1600m 0.020 * DrivTime_Beyond1600m 0.000 ***

Parameter P-value Level of Significance Intercept 0.610 n/s LivArea_m2 0.400 n/s lnLotSize 0.090 n/s AppAge 0.000 *** Cottage 0.430 n/s Quality 0.470 n/s Finbasment 0.090 n/s SoftFacing51 0.070 n/s Fireplace 0.480 n/s HiQualFloor 0.470 n/s HrdWoodStair 0.380 n/s SplAttGar 0.540 n/s DblAttGar 0.280 n/s SplDetGar 0.840 n/s DblDetGar 0.340 n/s WaterSewer 0.130 n/s NbMonthJan_9396 0.060 n/s AvgIncome_1K 0.000 *** PercChild0_9years 0.000 *** PercChildFam 0.000 *** RegAcc 0.000 *** OldSuburbs 0.200 n/s NewSuburbs 0.000 *** ScatrdSettlmt 0.000 *** Decouvreurs 0.000 *** PremSeigneuries 0.000 *** EnglishSchool 0.000 *** HighSchool 0.000 *** SchoolSize 0.000 *** Buf_0_250 0.020 * Buf_251_500 0.060 n/s Buf_501_750 0.050 * Buf_751_1000 0.010 ** Buf_1001_1300 0.000 *** Buf_1301_1600 0.060 n/s *** signifiant at 0.10% level ** signifiant at 1% level * signifiant at 5% level

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Spatial Distribution of Walking

and Driving Time Coefficients

Map 3 : WalkTime_Less1600m Significant Coefficients

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GWR & Spatial Dependence

Variable Global Moran's I* Sig. Pre_ITD (6) 0.3628 < 0.001

Res_ITD (6) 0.0101 < 0.001

Pre_BUF (7) 0.3634 < 0.001

Res_BUF (7) 0.0097 < 0.001 * Computed for a 3,000 metres kernel

Figure 1 : Local Spatial Autocorrelation for GWR-ITD Model (MapStat software)

Figure 2 : Local Spatial Autocorrelation for GWR-BUF Model (MapStat software)

• Global Moran’s I is much lower than the index obtained for OLS models (0.06), although it remains statistically significant at the 0.001 level

• LISA graphs suggest that local SA is no more significant at roughly 750 metres from any property

• Research hypothesis

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Discussion on GWR Limitations

The presence of space-related dummy variables in the regression equation and of continuous variables that entail some spatial discontinuity often results in the distribution of GWR regression coefficients differing from the Gaussian distribution required in order to generate reliable spatial stationarity tests

Moreover, local regressions are calibrated with dummy descriptors that, over large areas, are characterized by a

null variance (such as school boards)

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Discussion on GWR Limitations

Map 5 : Significant GWR-ITD Coefficients for the EnglishSchool Variable (Green-shaded areas indicate English school concentration in the region)

Figure 3 : Distribution of GWR-ITD Regression Coefficients 0,400000000 0,300000000 0,200000000 0,100000000 0,000000000 -0,100000000 3 000 2 000 1 000 0 Frequency Histogram Mean = 0,10260677053 Std. Dev. = 0,127720228264 N = 8 221

• Significant regression coefficients for the EnglishSchool descriptor are

found in the north-eastern portion of the territory whereas the only English

schools in Quebec City are concentrated in the green-shaded areas,

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Discussion on GWR Limitations

Map 6 : Walking Times for Homes Located Within 1,600 m of Nearest School

Figure 4 : Distribution of GWR-ITD Regression Coefficients

• Positive and significant coefficients are found in the western part of the agglomeration (north of Saint-Augustin-de-Desmaures) although no

effective walking time is computed for that area (Map 3)

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Conclusion and Suggestions for

Further Research

While GWR is a most promising regression method, it may not be applied without discrimination

The results it yields are heavily dependent upon the

specification of the hedonic model, which must therefore be

adapted consequently

For instance, accessibility to primary schools could be measured using the number of available schools from any property, based on some density-adaptive kernel

Alternately, other approaches may be best suited to deal with the above mentioned issues, such as Casetti’s expansion method or some multi-level modelling procedure.