# Estimation of Housing Price Variations Using Spatio-Temporal Data

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Regression-Cokriging Predictor

#### 2.2. Direct and Cross-Correlation of Residuals

## 3. Data and Case Study

#### 3.1. Database

^{2}). The mean distance to the CBD, in which the sample dwellings are located, ranges from 1300 to 1600 m in the different years. The floor area remains fairly stable at 108–119 m across the four years. The table shows the statistics for the distance between each dwelling and its nearest neighbor, which is less than 100 m on average in all years. Moreover, as can be observed in the table, information is not available for all the explanatory variables and years, thus indicating the heterogeneity of the data.

#### 3.2. Results and Discussion

^{2}

_{cv}and the lowest mean absolute error (MAE) and mean squared error (MSE) values.

#### 3.3. Estimation of Spatial Price Variation in Multi-Years

## 4. Discussions and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Housing prices from 1987 to 2005 for Spain, Andalusia, and Granada (standard dwelling of 100m

^{2}). Source: Sociedad de Tasación.

**Figure 4.**Experimental variograms and fitted models of the residuals (${\widehat{\mathrm{u}}}_{\mathrm{j}}$) of the multi-equation model. Note: Direct variograms: ${\widehat{\mathrm{u}}}_{1}$ (2005) to ${\widehat{\mathrm{u}}}_{4}$ (1988); and cross-variograms: ${\widehat{\mathrm{u}}}_{1}/{\widehat{\mathrm{u}}}_{2}$, ${\widehat{\mathrm{u}}}_{1}/{\widehat{\mathrm{u}}}_{3}$, etc.

**Figure 5.**Regression between predicted and observed data. Note: The solid straight line represents the 1:1 line and the dashed line represents the regression line.

**Table 1.**Descriptive statistics of the variables of samples (PRICE in 1000 euros) and nearest neighbor statistics of distance (Nearest, in meters).

Minimum | Maximum | Mean | Standard Deviation | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1988 | 1991 | 1995 | 2005 | 1988 | 1991 | 1995 | 2005 | 1988 | 1991 | 1995 | 2005 | 1988 | 1991 | 1995 | 2005 | |

PRICE | 10.22 | 15.03 | 15.03 | 22.83 | 240.407 | 300.51 | 330.55 | 751.26 | 45.89 | 75.56 | 90.51 | 162.40 | 30.60 | 36.65 | 50.03 | 85.55 |

AGE | 1 | 1 | 1 | 2 | 40 | 81 | 84 | 40 | 13.58 | 11.51 | 16.96 | 23.44 | 7.56 | 9.72 | 11.69 | 8.93 |

BATH | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 4 | 1.23 | 1.51 | 1.60 | 1.45 | 0.44 | 0.56 | 0.54 | 0.58 |

DIST | 166.30 | 294.98 | 76.39 | 81.27 | 3511.29 | 3723.63 | 3695.92 | 3940.96 | 1492.94 | 1577.95 | 1326.28 | 1557.46 | 851.91 | 963.64 | 812.76 | 823.68 |

AREA | 65.00 | 49.00 | 40.00 | 40.00 | 340.00 | 320.00 | 325.00 | 390.00 | 109.86 | 112.85 | 118.13 | 108.80 | 33.00 | 35.27 | 42.48 | 38.64 |

FLOOR | - | - | - | 0 | - | - | - | 1 | - | - | - | 0.039 | - | - | - | 0.19 |

ELEV | - | - | 0 | 0 | - | - | 1 | 1 | - | - | 0.88 | 0.78 | - | - | 0.31 | 0.41 |

HEAT | - | - | 0 | 0 | - | - | 1 | 1 | - | - | 0.55 | 0.54 | - | - | 0.50 | 0.50 |

SPORT | - | - | - | 0 | - | - | - | 1 | - | - | - | 0.06 | - | - | - | 0.24 |

REHAB | - | - | - | 0 | - | - | - | 1 | - | - | - | 0.35 | - | - | - | 0.48 |

Nearest | 8.10 | 10.34 | 6.00 | 4.20 | 390.77 | 385.38 | 666.40 | 646.87 | 74.67 | 55.75 | 86.81 | 98.22 | 55.45 | 63.06 | 88.60 | 100.24 |

**Table 2.**Parameters of fitted direct variograms and cross-variograms of residuals from the multi-equation model.

Residuals | Nugget | Partial Sill | Practical Range |
---|---|---|---|

${\widehat{\mathrm{u}}}_{1}$ (2005) | 0.023 | 0.015 | 465.00 |

${\widehat{\mathrm{u}}}_{2}$ (1995) | 0.025 | 0.011 | 465.00 |

${\widehat{\mathrm{u}}}_{3}$ (1991) | 0.026 | 0.014 | 465.00 |

${\widehat{\mathrm{u}}}_{4}$ (1988) | 0.044 | 0.024 | 465.00 |

${\widehat{\mathrm{u}}}_{1}/{\widehat{\mathrm{u}}}_{2}$ | 0.024 | 0.012 | 465.00 |

${\widehat{\mathrm{u}}}_{1}/{\widehat{\mathrm{u}}}_{3}$ | 0.024 | 0.014 | 465.00 |

${\widehat{\mathrm{u}}}_{1}/{\widehat{\mathrm{u}}}_{4}$ | 0.032 | 0.019 | 465.00 |

${\widehat{\mathrm{u}}}_{2}/{\widehat{\mathrm{u}}}_{3}$ | 0.025 | 0.011 | 465.00 |

${\widehat{\mathrm{u}}}_{2}/{\widehat{\mathrm{u}}}_{4}$ | 0.034 | 0.016 | 465.00 |

${\widehat{\mathrm{u}}}_{3}/{\widehat{\mathrm{u}}}_{4}$ | 0.033 | 0.018 | 465.00 |

R^{2}_{cv} | MAE | MSE | |
---|---|---|---|

1988 | |||

Spherical | 0.8349 | 0.1862 | 0.0557 |

Gaussian | 0.8461 | 0.1806 | 0.0519 |

Exponential | 0.8507 | 0.1793 | 0.0503 |

1991 | |||

Spherical | 0.8639 | 0.1406 | 0.0340 |

Gaussian | 0.8719 | 0.1356 | 0.0320 |

Exponential | 0.8727 | 0.1371 | 0.0318 |

1995 | |||

Spherical | 0.9285 | 0.1424 | 0.0359 |

Gaussian | 0.9296 | 0.1396 | 0.0353 |

Exponential | 0.9332 | 0.1362 | 0.0335 |

2005 | |||

Spherical | 0.7946 | 0.1564 | 0.0428 |

Gaussian | 0.8075 | 0.1497 | 0.0401 |

Exponential | 0.8001 | 0.1545 | 0.0416 |

1988 | 1991 | 1995 | 2005 | |
---|---|---|---|---|

Intercept | 1.042 × 10^{1} | 1.084 × 10^{1} | 1.041 × 10^{1} | 1.148 × 10^{1} |

(0.000) | (0.000) | (0.000) | (0.000) | |

AGE | −2.119 × 10^{−2} | −7.828 × 10^{−3} | −8.497 × 10^{−3} | −6.176 × 10^{−3} |

(0.000) | (0.000) | (0.000) | (0.000) | |

BATH | 8.929 × 10^{−2} | 1.176 × 10^{−1} | 6.929 × 10^{−2} | 9.256 × 10^{−2} |

(0.020) | (0.000) | (0.000) | (0.000) | |

AREA | 7.949 × 10^{−3} | 5.782 × 10^{−3} | 8.361 × 10^{−3} | 5.951 × 10^{−3} |

(0.000) | (0.000) | (0.000) | (0.000) | |

DIST | −3.853 × 10^{−4} | −2.532 × 10^{−4} | −2.111 × 10^{−4} | −2.257 × 10^{−4} |

(0.000) | (0.000) | (0.000) | (0.000) | |

REHAB | -- | −1.517 × 10^{−1} | -- | −6.733 × 10^{−2} |

(0.000) | (0.008) | |||

ELEV | -- | -- | 1.552 × 10^{−1} | 1.187 × 10^{−1} |

(0.000) | (0.000) | |||

HEAT | -- | -- | 7.528 × 10^{−2} | 4.679 × 10^{−2} |

(0.000) | (0.060) | |||

FLOOR | -- | -- | -- | −1.056 × 10^{−1} |

(0.059) | ||||

SPORT | -- | -- | -- | 1.278 × 10^{−1} |

(0.011) | ||||

${R}_{cv}^{2}$ | 0.8507 | 0.8727 | 0.9332 | 0.8001 |

MAE | 0.1793 | 0.1371 | 0.1362 | 0.1545 |

MSE | 0.0503 | 0.0318 | 0.0335 | 0.0416 |

n | 260 | 247 | 293 | 207 |

**Table 5.**Cross-validation statistics of CKED and RCK (R-squared of cross-validation, ${R}_{cv}^{2}$; mean absolute error, MAE; mean squared error, MSE and sample size, n).

CKED | RCK | |
---|---|---|

${R}_{cv}^{2}$ | 0.9277 | 0.9331 |

MAE | 0.1400 | 0.1355 |

MSE | 0.0343 | 0.0317 |

n | 1007 | 1007 |

1988 | 1991 | 1995 | 2005 | |
---|---|---|---|---|

Intercept | 1.027 × 10^{1} e+01(0.000) | 1.074 × 10^{1}(0.000) | 1.446 × 10^{1}(0.000) | 1.650 × 10^{1}(0.000) |

AGE | −2.033 × 10^{−2}(0.000) | −6.484 × 10^{−3}(0.000) | −1.035 × 10^{−2}(0.000) | −6.583 × 10^{−3}(0.001) |

BATH | 1.269 × 10^{−1}(0.004) | 1.788 × 10^{−1}(0.000) | 9.536 × 10^{−2}(0.002) | 1.107 × 10^{−1}(0.002) |

AREA | 8.506 × 10^{−3}(0.000) | 5.766 × 10^{−3}(0.000) | 1.188 × 10^{−2}(0.000) | 6.125 × 10^{−3}(0.000) |

DIST | −3.521 × 10^{−4}(0.000) | −2.510 × 10^{−4}(0.000) | −2.685 × 10^{−4}(0.000) | −2.139 × 10^{−4}(0.000) |

REHAB | -- | −1.981 × 10^{−1}(0.000) | -- | −6.925 × 10^{−2}(0.048) |

ELEV | -- | -- | 2.540 × 10^{−1}(0.000) | 1.875 × 10^{−1}(0.000) |

HEAT | -- | -- | 1.550 × 10^{−1}(0.000) | 7.201 × 10^{−2}(0.053) |

FLOOR | -- | -- | -- | −1.482 × 10^{−1}(0.079) |

SPORT | -- | -- | -- | 1.376 × 10^{−1}(0.041) |

R-squared | 0.8176 | 0.8278 | 0.9234 | 0.7745 |

${R}_{cv}^{2}$ | 0.8060 | 0.8173 | 0.9180 | 0.7442 |

MAE | 0.2056 | 0.1664 | 0.1567 | 0.1767 |

MSE | 0.0655 | 0.0457 | 0.0411 | 0.0556 |

n | 260 | 247 | 293 | 207 |

1988 | 1991 | 1995 | 2005 | |
---|---|---|---|---|

${R}_{cv}^{2}$ | 5.5459 | 6.7784 | 1.6557 | 7.5114 |

MAE | 12.7918 | 17.6082 | 13.0823 | 12.5637 |

MSE | 23.2061 | 30.4157 | 18.4915 | 25.1798 |

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**MDPI and ACS Style**

Chica-Olmo, J.; Cano-Guervos, R.; Chica-Rivas, M. Estimation of Housing Price Variations Using Spatio-Temporal Data. *Sustainability* **2019**, *11*, 1551.
https://doi.org/10.3390/su11061551

**AMA Style**

Chica-Olmo J, Cano-Guervos R, Chica-Rivas M. Estimation of Housing Price Variations Using Spatio-Temporal Data. *Sustainability*. 2019; 11(6):1551.
https://doi.org/10.3390/su11061551

**Chicago/Turabian Style**

Chica-Olmo, Jorge, Rafael Cano-Guervos, and Mario Chica-Rivas. 2019. "Estimation of Housing Price Variations Using Spatio-Temporal Data" *Sustainability* 11, no. 6: 1551.
https://doi.org/10.3390/su11061551