Chapter 5 MEASUREMENTS OF SPATIAL AUTOCORRELATION AND SPATIAL MODELS
5.3 Spatial Error Model (SEM)
expectation and variance of the Index value are calculated. Then, the zI score and pvalue indicate whether this spatial autocorrelation is statistically significant or not (Mitchell, 2005).
5.2 Spatial Lag Model (SLM or SAR)
This section illustrates the estimation by means of maximum likelihood of spatial hedonic price model (or spatial autoregressive model) that contains a spatially lagged dependent variable. Unlike the traditional approach, which uses eigenvalues of the weighting matrix, this method is well suited to the estimation in situations with very large data sets.
Formally, spatial lag model is expressed as (Anselin, 2003):
2 coefficient on the spatially lagged dependent variable, Wy. Then, X is a matrix of explanatory variables, is estimated parameter, and is a vector of error terms (i.i.d.).
5.3 Spatial Error Model (SEM)
Next, this part shows the estimation by means of maximum likelihood of spatial hedonic price model (or spatial autoregressive model) that contains a spatial autoregressive error term.
Formally, spatial error model is expressed as (Anselin, 2003):
2
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where y is a vector of dependent variable, X is a matrix of the explanatory variables, Wis the spatial weight matrix, and is a coefficient on the spatially correlated errors. Then, is a vector of spatially autocorrelated error terms, u is a vector of error terms (i.i.d.), and is estimated parameters.
5.4 Empirical Analysis
The case study is concerned with the analysis of the spatial autocorrelation using the data described in chapter 3. The GeoDa software and GIS software package (ArcGIS) are integrated for model parameter estimation and mapping. The hedonic price model is considered over the study area. However, hedonic price parameters (OLS model) are usually estimated using procedures that assume independent observations. If hedonic residuals are spatially autocorrelated, the resulting parameter estimation will be inefficient and will be biased (Basu and Thibodeau, 1998). The treatment which considers and expresses such a correlation in set of market information is spatial hedonic price models which are generalizations of OLS model with reference to spatial issues.
In our empirical analysis, with the SLM and SEM model, the parameter of a hedonic price model may be made functions of other attributes including Age, Floor, Lot, DTTS and DCTSP in parameter estimation. LN_P is used as the dependent variable in the hedonic regression. For example, the analytical dataset consists of 251 duplex residence transactions in Taichung city during 2011. In addition, in order to identify which model is suitable, some tests are used on model specification to identify by Lagrange multiplier test:LM(lag), Robust LM(lag), LM(error) and Robust
(error)
LM .
According to spatial autocorrelation coefficient (Moran’s I) that is a measurement
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of cluster, the values of distinguished characteristic are determined by other surrounded values of the characteristics. On the basis of Moran’s I statistics, the significance tests of spatial autocorrelation may be created for the null hypothesis H 0
is no significant spatial autocorrelation and alternative hypothesis H is spatial 1 autocorrelation occurrence. Table 5.1 presents values of Moran’s I statistics based on spatial weight matrix constructed on the basis of inverse distance based on criterion and the significance level of the statistics, for different distance thresholds. Moran’s I correlation coefficient may also be used to graphical appearance of changes in spatial autocorrelation along with the distance (Figure 5.1, on the basis of statistically significant value from Table 5.1).
Table 5. 1 Values of Moran’s I Statistics for Different Threshold Distance in 2011
Moran's Index zIscore p-value
ID_0.02 0.0701 7.1671 0.0000
ID_0.025 0.0410 5.7217 0.0000
ID_0.03 0.0235 4.4404 0.0000
ID_0.035 0.0248 5.8145 0.0000
ID_0.04 0.0173 5.2330 0.0000
ID_0.045 0.0083 3.6244 0.0003
ID_0.05 0.0001 1.4233 0.1546
Note: LN_P is used as observations.
Note: Graphical appearances are based on Moran’s I (zIscore) that are statistically significant at least 1% level (Table 5.1).
Figure 5. 1 Changes in Spatial Autocorrelation Along with the Distance
0.0000
ID_0.02 ID_0.025 ID_0.03 ID_0.035 ID_0.04 ID_0.045
Moran's I
Inverse Distance
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The Spatial Autocorrelation (Moran’s I) tool returns five values: the Moran's Index, Expected Index, Variance,zI score, and p-value. The results of graphical summary are shown as a figure (Fig.5.2).
Figure 5. 2 Graphical Summary of Spatial Autocorrelation Reports in 2011 (ID_0.04) Table 5.2 shows the results from the estimation conventional OLS, SLM and SEM models. The OLS regression diagnostics reveal considerable and heteroskedasticity (Koenker-Bassett statistic)4 and spatial autocorrelation (Moran’s I¸ Table 5.1).
By comparing the values for SLM to those for OLS, an increase in the Log-Likelihood is noticed from -30.9484 to -29.1570. The improved fitting for the added variable (the spatially lagged dependent variable), the AIC (from 73.8967 to 72.3139) decreases relative to OLS, suggesting an improvement for the spatial lag
4 The Koenker-Bassett statistic (Koenker's studentized Bruesch-Pagan statistic) is a test to determine whether the explanatory variables in the model have a consistent relationship to the dependent variable both in geographic space and in data space.
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specification. There are some minor differences in the significance of the other regression coefficients between the SLM model and the OLS model: more importantly, the significance of DTTS changes from p < 0.05 to p < 0.01. The magnitude of all the estimated coefficients is also affected. To some extent, the explanatory power of these variables that is attributed to their neighboring locations. This is picked up by the coefficient of the spatially lagged dependent variable.
By comparing the values of SEM to those for OLS, an increase in the Log-Likelihood is noticed from -30.9484 (for OLS) to -27.1359. The improved fit for the added variable (the spatially correlated error variable), the AIC (from 73.8967 to 66.2718) decreases relative to OLS, suggesting an improvement of fit for the spatial error specification.
Next is the comparison between Spatial Error and Spatial Lag Models. Following the comparison steps, both LM(lag) and LM(error) are significant, but of the robust forms, the Robust LM(error) statistic is highly significant (p < 0.01), while the Robust
(lag)
LM statistic is not (p>0.1). In 2011, the SEM appeared to be more appropriate model (residuals from the OLS model highly correlated), and it gets better fit to the empirical data measured by Log Likelihood and AIC.
From Moran Scatter Plot (Fig. 5.3 to Fig. 5.5), the Moran’s I are 0.0422, 0.0098 and -0.0037, respectively. This indicates that including the spatially autoregressive error term in the model has eliminated mostly spatial autocorrelation.
In other annual results, all of spatial models show better fit. There are some minor differences in the significance of the other regression coefficients. Such as in 2006, DTTS changes from p < 0.05 (OLS) to p > 0.1(SEM); DCTSP changes from p <
0.01(OLS) to p < 0.1(SEM). Comparison between SLM and SEM, except for 2002, 2005 and 2011, the SLM provides better fit. The annual results from the estimation
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OLS, SLM and SEM models are shown in appendix3.
The SLM and SEM models are closely related to each other mathematically, but their economic interpretations are slightly different. The relevance of one model versus another depends on the particular application at hand. For example, a SEM is preferred over a SLM if the spatial pattern of residuals is considered as potentially valuable information (Yoo and Kyriakidis, 2009). The SLM differs from the spatial SEM. The spatial SEM is based on the assumption that there are omitted variables in the hedonic price equation and the spatial dependence of the error term is due to those spatially varying omitted variable(s) (Anselin, 1998).
Table 5. 2 The Results form Estimation OLS, SLM and SEM Models in 2011
OLS SLM SEM
Variable Coefficient Std .Error Coefficient Std. Error Coefficient Std. Error
0.3325 0.2537
Intercept 6.6690 0.1529*** 4.4845 1.6764*** 6.6648 0.1853***
Age -0.0123 0.0018*** -0.0125 0.0017*** -0.0133 0.0017***
Floor 0.0023 0.0002*** 0.0023 0.0002*** 0.0022 0.0002***
Lot 0.0025 0.0005*** 0.0026 0.0005*** 0.0029 0.0005***
DTTS -0.0311 0.0124** -0.0411 0.0137*** -0.0340 0.0147**
DCTSP -0.0409 0.0089*** -0.0381 0.0088*** -0.0353 0.0119***
0.6438 0.2160***
R-squared 0.7205 0.7248 0.7303
Log Likelihood -30.9484 -29.1570 -27.1359
AIC 73.8967 72.3139 66.2718
TEST VALUE
Note: Number of included observations=251. The dependent variable is the natural logarithm total house price (LN_P). Spatial weight matrix with threshold distance is 0.04 m. * Denotes 10% statistical significance; ** Denotes 5% statistical significance; *** Denotes 1% statistical significance
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Figure 5. 3Moran Scatter Plot for OLS_Residue
Figure 5. 4Moran Scatter Plot for Lag_Residue
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Figure 5. 5Moran Scatter Plot for Error_Residue 5.5 Summary
In this chapter, two real estate questions are explored (i) whether our analysis data are spatial autocorrelation and (ii) whether spatial hedonic price models increase traditional hedonic (OLS) price model prediction precision. After empirical analysis, the datasets are found to be spatially autocorrelated. Compared to OLS model, spatial hedonic price models (SLM and SEM) get better fit to the empirical data.
Results from classical regression are only reliable when the model and data meet the assumption (i.i.d.). If spatial autocorrelation is statistically significant, it means the model is not specific. This may be missing an important explanatory variable. If the key missing variables can’t be identified, regression results are inefficient. However, the spatial autoregression approach can be used to deal with spatial autocorrelation problems and offers improved predictions.
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