Chapter 4 GEOGRAPHICALLY WEIGHTED REGRESSION APPROACH
4.3 Empirical Analysis
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is an appropriate measurement for the data. Furthermore, the Akaike Information Criterion (AIC) index is used to compare the fitted results from the global (OLS) model with those from the local (GWR) model. The AIC index would disclose whether the spatial viewpoint will significantly improve the model fitted effect.
4.3 Empirical Analysis
In this section, a practical example of GWR will be given. The GeoDa software and GIS software package (ArcGIS) are integrated for model parameter estimation and mapping. This will be based on data taken from Dept of Land Administration, M. O. I.
(Official Cadastre Agency) of Taiwan. The analytical dataset consists of 251 duplex residence transactions in Taichung city during 2011. The geographic area of this study is shown in Fig.3.1 (Chapter 3). The Dept of Land Administration provides information for all housing transactions that occurred during this period and it included the structural characteristics of each property such as transaction price, age of house and size of lot and floor, and road width. The detail description of the property characteristics is provided in Table 3.1.
In this case, LN_P is the natural logarithm total transaction price of the house which is used as the dependent variable in the hedonic regression estimations. “Age”
means the age of house which is one of the negative effects affecting house price.
Thus it is expected to have a negative coefficient for Age. Lot and Floor are the total areas of the lot and house, respectively, which measured in square meter. The size of Lot and Floor are expected to positively affect the Price. Rwidth is the width of the road, and it is expected to contribute to a higher house value since this would indicate more convenient of communication. DTTS and DCTSP represent the distance to Taichung Train Station (TTS) and Central Taiwan Science Park (CTSP), respectively, and estimate the effect of nearby amenities on house prices. However, the nearby
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amenities may increase or decrease housing prices. Hence the direction of price effects is not assumed.
4.3.1. Results of Global (OLS) Model
Firstly, in order to understand how the independent variables impact on price, the multicollinearity is worthy of further investigation. Therefore, the multicollinearity between these independent variables in this model is tested with the correlations matrix in Table 4.1. As the results, all absolute values of the correlation coefficient are almost less than 0.5. The largest correlation is between Floor and Lot and this is quite intuitive. The maximum negative correlation is between Floor and Age, and this might be a phenomenon of luxury as compared to the past. Then the multicollinearity diagnostics are conducted by VIF (Variance Inflation Factor) of the predictor variables for further determination. In general, VIF values above 10 suggest a mulitcollinearity problem. Table 4.2 shows the VIF value for each variable and indicates that the data had no significant problems with collinearity.
Second, the OLS regression model will be measured. Based on the principle of best fit, Rwidth is less statistically significant, and then it is excluded from the initial model. From the results shown in Table 4.2, it can be seen that, at global OLS model, every variable has a statistically significant coefficient. About 73.8967% of the variations in house prices are explained by the variations in the selected explanatory variables indicated by the R-squared statistic. As expected, Age decreases the house price by about 0.0123%. Both Lot and Floor positively affect the house price. Then, the accessibility variables (DTTS and DCTSP) are statistically significant for distances to Taichung Train Station and Central Taiwan Science Park, respectively. The coefficients for DTTS and DCTSP are both negative.
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Table 4. 1 Correlation Matrix of Variables used in Estimations
Correlation
Probability Age Floor Lot Rwidth DTTS DCTSP
Age 1.0000
---
Floor -0.5395 1.0000
(0.0000)
---Lot -0.2431 0.5353 1.0000
(0.0001) (0.0000)
---Rwidth -0.0009 -0.0041 0.0738 1.0000
(0.9889) (0.9483) (0.2442)
---DTTS -0.4801 0.1534 0.1248 0.1296 1.0000
(0.0000) (0.0150) (0.0482) (0.0402) ---
DCTSP -0.1005 0.0120 0.0204 -0.0539 -0.3206 1.0000
(0.1121) (0.8498) (0.7473) (0.3954) (0.0000) ---Note: The values are based on the total samples of 251 observations (year 2011). Numbers in brackets are p-values.
Table 4. 2 Results of the Global Regression Model
Variable Coefficient Std. Error VIF
Intercept 6.6690 0.1529***
AGE -0.0123 0.0018*** 2.0682 Adjusted R-squared 0.7148
Note: Number of included observations=251. The dependent variable is the natural logarithm total house price (LN_P). Large Variance Inflation Factor, VIF (> 7.5, for example) indicates explanatory variable redundancy. ** Denotes 5% statistical significance; *** Denotes 1% statistical significance
4.3.2. Results of Local (GWR) Model
In this section, the results of using a GWR model for the house price data are discussed. Firstly, 4-indicator summary of parameter estimation describes the degree of variability in the parameter estimation (the 4-indicator summary is based on the maximum, minimum, median, and mean local parameter estimates reported in the GWR model). Next, in order to know the heterogeneity in individual parameters, GWR technique can visualize the local parameter estimates. Each parameter can be mapped as surface illustration of the spatial variation (Matthews and Yang, 2012). In
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GWR, the regression is re-centered many times (on each observation) to generate locally GWR parameter results. These local GWR results generate a complete map of the spatial variation of the parameter estimates. The local statistics can take on different values at each location. GWR results is unlike global model results, it is mappable and given a very large number of potential parameters estimation. It is almost essential to map the potential parameters in order to make some sense of the patterns they display (Matthews and Yang, 2012).
There is a short summary of results obtained from application GWR to house price assessment (Table 4.3). The regression parameters seem not to be constant over the entire study area, and they change along with location. Furthermore, AIC index also shows slight decline (from 73.8967 to 67.5369). The R-squared (Adjusted R-squared) value increases when GWR tool are considered in the model. Comparing the GWR diagnostics to the OLS diagnostics, the GWR performs better model fitting.
Table 4. 3 Results of the Local Regression Model
GWRMAX GWRMIN GWRMEDIAN GWRMEAN
Intercept 6.7975 6.8192 6.7691 6.6757
(39.0464) (39.4201) (38.2937) (39.0262)
AGE -0.0126 -0.0131 -0.0122 -0.0120
(-6.9180) (-7.1202) (-6.6844) (-6.5163)
Floor 0.0023 0.0022 0.0023 0.0023
(10.7481) (10.2338) (10.8866) (10.7041)
Lot 0.0026 0.0026 0.0026 0.0028
(5.4094) (5.3724) (5.4700) (5.7121)
DTTS -0.0386 -0.0447 -0.0348 -0.0364
(-2.6948) (-3.1202) (-2.3916) (-2.5382)
DCTSP -0.0511 -0.0492 -0.0508 -0.0424
(-4.9624) (-4.8380) (-4.8423) (-4.1761) AIC 67.5369
R-squared 0.7347 Adjusted R-squared 0.7255
Note: In this case, adaptive kernel type (Gaussian kernel) is used. The spatial context is a function of a specified number of neighbors. Where feature distribution is dense, the spatial context is smaller;
where feature distribution is sparse, the spatial context is larger. The dependent variable is the natural logarithm total house price (LN_P). Numbers in brackets are t-values.
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Next, to give a brief display of the spatial variation, Fig.4.3 is taken into consideration. This is interesting when the coefficient takes both high and low values.
In particular, in Nantun district and Xitun district, there are strong negative relationship (the house prices decrease as the distance to the CTSP increases) whereas in other areas the relationship is reduced gradually. After statistical detection and schema description, the direct evidence indicates that the characteristic influence isn’t the same in all regions. Thus, GWR model can help to analyze the spatial heterogeneity of our model parameters.
Figure 4. 3 Spatial Variation of the DCTSP Coefficient 4.4 Summary
In this chapter, the technique for producing visual local regression coefficients is discussed. In the past, the coefficients for a global regression may fail to notice geographical features in the relationships between variables. However, the GWR technique has provided a straight and reasonable approach to deal with the problem.
GWR is a helpful tool for investigation that produces a set of location-specific parameter estimates which can be analyzed and mapped to offer information on spatial non-stationary in regression parameters.
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