# Application of Hierarchical Spatial Autoregressive Models to Develop Land Value Maps in Urbanized Areas

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Basic of Conducted Research

- Y—is an N × 1 vector of dependent variable,
- ρ, λ—parameters of spatial interactions,
- W—spatial weight matrix at the individual level,
- β—parameter vector,
- X—matrix of explained variables,
- Δ—matrix demonstrating the classification of entities i to objects j,
- θ—vector of random effects for absolute term,
- u—vector of random group effects,
- ε—vector of a random component,
- M– spatial weight matrix at the group level.

_{u}

^{2}, σ

_{ε}

^{2}} is proportional to the product of the data and prior distributions [34]. For the k-element vector of β parameters, we consider the multidimensional normal distribution with the expected value M

_{β}and diagonal matrix of variance-covariance matrix T

_{β}. Therefore, the posterior distribution of β parameters will be as follows [56,63]:

_{ε}can be presented in the following way:

_{ε}, b

_{ε}) stands for reverse distribution gamma with the shape parameter a

_{ε}and scale parameter b

_{ε}. Posterior distribution of random effect variance σ

_{u}

^{2}will have the following form:

## 3. Data Description

^{2}currently inhabit the city. The varied spatial structure of the city, numerous lakes and forests, as well as a quite strong planning intervention have resulted in significant spatial heterogeneity of property prices. Transaction data used for the analyses originate from the register of prices and values of real estate properties, held by the City Hall in Olsztyn. Overall, 520 data entries concerning undeveloped land property transactions, carried out in 2010–2017, were used for analyses. A similar number of transactions were assumed in many research carried out so far concerning, e.g., mass valuation (e.g., References [65,66,67]. The logarithm of the price per 1m

^{2}was assumed as an explained variable. The assumption of the logarithm was dictated by a relatively large span of prices and a distribution demonstrating strong right-skewness. Each of the sold properties was additionally described in the form of a set of eleven features forming explaining variables, as presented in Table 1.

## 4. Results and Discussion

_{resid}also proves the fact that the hierarchical model slightly better explains the price variability. In the SAR model, four variables proved statistically significant (property right, density of main roads, distance from public transport stops and utility network).

^{2}. In order to obtain the map of values, after overlaying individual layers, the obtained values in the form of logarithms were converted directly into the value in PLN. The value estimated on the basis of the LM model results from simple prediction, while the map generated on the basis of the SAR model required additional interpolation of the spatial lag (ρWy). Value maps developed on the basis of LM and SAR models are schematically presented in Figure 6.

## 5. Conclusions

_{resid}errors was also found compared to the HLM model. Hierarchical spatial models, HSAR, take into account both micro-scale spatial effects and the context resulting from the location of specific observations at subsequent levels in the spatial hierarchy. Therefore, they carry greater informational content compared to classic models.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

HLM | Hierarchical Linear Model |

HSAR | Hierarchical Spatial Autoregressive |

LM | Linear Model |

SAR | Spatial Autoregressive |

## Appendix A

lnprice | ||

date | ||

right | ||

lnarea | ||

type | ||

lnlake | ||

lnforest | ||

densdev | ||

densroad | ||

lncentr | ||

lnbus | ||

utility |

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**Figure 3.**Diagnostics of random elements of HLM (

**a**) and HSAR (

**b**) models (the chart presents random effects for individual zones, in the ascending order, specifying the confidence intervals for those effects).

**Figure 7.**Maps of land values developed based on HLM, (

**a**) without random effect, and (

**b**) with random effect.

**Figure 8.**Maps of land values developed based on HSAR, (

**a**) without random effect, and (

**b**) with random effect.

Symbol | Description |
---|---|

lnprice | logarithm of unit price of land in PLN/m^{2} |

date | sale date [number of months] |

right | property right [ownership or perpetual usufruct—dummy variable] |

lnarea | logarithm of property area |

type | prevailing type of development [compact multi-family urban development, scattered detached houses—dummy variable] |

lnlake | logarithm of distance from lake |

lnforest | logarithm of distance from forest |

densdev | density of development [based on a kernel function with a range of 1 km] |

densroad | density of main roads [based on a kernel function with a range of 1 km] |

lncentr | logarithm of distance from the center, |

lnbus | logarithm of distance from public transport stop |

utility | utilities network [continuous variable from 0—no utilities, to 1—full utilities network]. |

Variable | Min | Max | Mean | Median | Std. Dev. |
---|---|---|---|---|---|

lnprice | 4.055 | 6.783 | 5.431 | 5.420 | 0.507 |

date | 0.000 | 96.000 | 43.262 | 44.000 | 27.181 |

right | 0.000 | 1.000 | 0.879 | 1.000 | 0.327 |

lnarea | 1.431 | 11.550 | 6.443 | 6.665 | 1.570 |

type | 0.000 | 1.000 | 0.221 | 0.000 | 0.415 |

lnlake | 2.303 | 8.577 | 6.995 | 7.394 | 1.171 |

lnforest | 0.000 | 7.847 | 6.333 | 6.722 | 1.422 |

densdev | 2.429 | 1515.110 | 387.088 | 324.610 | 276.969 |

densroad | 0.000 | 7.226 | 1.455 | 0.947 | 1.538 |

lncentr | 5.646 | 8.842 | 8.187 | 8.387 | 0.559 |

lnbus | 0.000 | 7.097 | 5.391 | 5.410 | 0.757 |

utility | 0.000 | 1.000 | 0.916 | 1.000 | 0.193 |

LM | HLM | |||
---|---|---|---|---|

Variable | β | SE | β | SE |

intercept | 6.439 * | 0.526 | 7.827 * | 0.750 |

date | 2.3e-04 | 7.5e-04 | −3.9e-04 | 6.8e-04 |

right | 0.241 * | 0.065 | 0.291 * | 0.063 |

lnarea | −0.012 | 0.014 | −0.020 * | 0.014 |

type | 0.030 | 0.061 | −0.062 | 0.067 |

lnlake | 0.048 * | 0.021 | 0.062 * | 0.032 |

lnforest | −0.012 | 0.017 | 0.023 | 0.021 |

densdev | 7.7e-05 | 1.1e-04 | −1.1e-04 | 1.4e-04 |

densroad | 0.036 * | 0.016 | 0.016 | 0.021 |

lncentr | −0.160 * | 0.057 | −0.281 * | 0.079 |

lnbus | −0.125 * | 0.028 | −0.175 * | 0.031 |

utility | 0.544 * | 0.115 | 0.271 * | 0.180 |

logLik | −313.477 | −315.573 | ||

AIC | 652.953 | 631.153 | ||

SE_{resid} | 0.447 | 0.382 |

_{resid}—standard error of residuals, *—statistically significant at 95% credible level.

SAR | HSAR | |||
---|---|---|---|---|

Variable | β | SE | β | SE |

intercept | 3.139 * | 1.081 | 7.348 * | 1.805 |

date | −3.0e-04 | 7.3e-04 | −3.0e-04 | 6.8e-04 |

right | 0.238 * | 0.064 | 0.288 * | 0.064 |

lnarea | −0.012 | 0.014 | −0.020* | 0.014 |

type | 0.016 | 0.060 | −0.090 | 0.073 |

lnlake | 0.034 | 0.021 | 0.067 * | 0.035 |

lnforest | −0.009 | 0.017 | 0.019 | 0.022 |

densdev | 2.8e-05 | 0.016 | −1.3e-04 | 1.5e-04 |

densroad | 0.022 * | 0.016 | 2.5e-03 | 0.024 |

lncentr | −0.097 | 0.062 | −0.305 * | 0.105 |

lnbus | −0.136 * | 0.028 | −0.180 * | 0.031 |

utility | 0.521 * | 0.112 | 0.278 * | 0.184 |

logLik | −308.539 | −1048.294 | ||

AIC | 617.078 | 2087.616 | ||

SE_{resid} | 0.437 | 0.361 | ||

ρ | 0.551 | 0.119 | ||

λ | NA | 0.383 |

_{resid}—standard error of residuals, *—statistically significant at 95% credible level, NA—not applicable.

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**MDPI and ACS Style**

Cellmer, R.; Kobylińska, K.; Bełej, M.
Application of Hierarchical Spatial Autoregressive Models to Develop Land Value Maps in Urbanized Areas. *ISPRS Int. J. Geo-Inf.* **2019**, *8*, 195.
https://doi.org/10.3390/ijgi8040195

**AMA Style**

Cellmer R, Kobylińska K, Bełej M.
Application of Hierarchical Spatial Autoregressive Models to Develop Land Value Maps in Urbanized Areas. *ISPRS International Journal of Geo-Information*. 2019; 8(4):195.
https://doi.org/10.3390/ijgi8040195

**Chicago/Turabian Style**

Cellmer, Radosław, Katarzyna Kobylińska, and Mirosław Bełej.
2019. "Application of Hierarchical Spatial Autoregressive Models to Develop Land Value Maps in Urbanized Areas" *ISPRS International Journal of Geo-Information* 8, no. 4: 195.
https://doi.org/10.3390/ijgi8040195