# A space-time-categorical local linear smoother for predicting land/house price

Ghislain Geniaux, Davide Martinetti

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## To cite this version:

Ghislain Geniaux, Davide Martinetti. A space-time-categorical local linear smoother for predicting land/house price. 1. International Conference on Econometrics and Statistics (EcoSta 2017), Honk-Kong, Jun 2017, Honk-Kong, Hong Kong SAR China. 112 p. hal-01607264

HAL Id: hal-01607264

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Submitted on 2 Jun 2020

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# A space-time-categorical local linear smoother for predicting house prices

Ghislain Geniaux and Davide Martinetti

INRA UR 767 Ecodéveloppement

17 June 2017

EcoStat 2017, Honk Kong
Market segmentation / submarket

A housing (resp. land) submarket can be
defined, roughly, as a set of dwellings
данная land) that are reasonably close
(resp. lands) substitutes of one another, but there are
not substitute of dwellings (resp. lands)
belonging to other submarkets.
Islam and Asami (2009) → 3 approaches:

1. hedonic price models are used to cluster
**the properties that are similar with
respect to a bundle of qualitative
characteristics**, such as lot size, number of
rooms and bathrooms, garden, parking slot,
etc. (Grigsby et al., 1986; Kauko, 2002;
Leishman, 2001; Schnare and Struyk, 1976;
Tu and Goldfinch, 1996; Tu, 1997)
2. On the other hand, housing market can be analyzed with respect to **the spatial distribution of properties and other spatial features.** In this context, spatial proximity and clustering are the prime determinants of submarket's definition (Gallet, 2004; Goodman, 1978; Goodman and Thibodeau, 1998, 2003).
3. There exist mixed approaches that consider both topographic and quality segmentation, sometimes referred as hybrid-related submarkets, (O'Sullivan and Gibb, 2008).
OUR PROPOSAL

Since we postulate a strong dependence
between house quality and its location, we
cannot rely on two-stage models such as the
ones proposed by (Goodman and
Thibodeau, 2007; O'Sullivan and Gibb, 2008;
Tu, 1997).
OUR PROPOSAL

We prefer instead a smoother approach,
where the hedonic regressions coefficients
can vary across space, time and
submarkets.

Extended version of geographically-weighted
regression with spatial dependence, namely
MGWR-SAR, Geniaux and Martinetti (2017)
OUR PROPOSAL

Geniaux and Martinetti(2017) « A new method for
dealing simultaneously with spatial autocorrelation
and spatial heterogeneity in regression models »
RSUE, forthcoming hereafter GM2017

+

Li and Racine 2010 « Smooth varying-coefficient
estimation and inference for qualitative and
quantitative data ». *Econometric Theory* 26 (06)

hereafter LR2010
Local linear regression framework
(Cleveland, 1979; Hastie and Tibshirani,
1990,1993)

$$
Y_i = \beta(u_i, v_i; h)X_i + \epsilon_i,
$$

Each Local Regression for point i is based
on a local subsample
Each local subsample is defined by a kernel that produces a vector of weights based on spatial proximity between *i* and *j*:

$$w_{ij} = K(d_{ij}, h)$$

where $d_{ij}$ is a metric of proximity between *i* and *j* and *h* a bandwidth.

Various kernels $K()$ can be used, but the main issue is to choose a suitable bandwidth *h* using Cross Validation (leave-one-out) or Plug-in Methods.
Add time differences to the kernel :

$$w_{ij} = K(d_{ij}, T_{ij}; h_d, h_t)$$

Huang et al., 2010; Wrenn and Sam, 2014; Fotheringham et al., 2015

Wu et al. (2014) proposed a GWR technics with spatial autocorrelation,

Wei et al. (2017) proposed to extend GWR using spatial SUR models in order to explore spatio-temporal heterogeneity
## ADD OTHER DIMENSIONS OF ATTRIBUTE'S SPACE ?

* Why not choosing a full non-parametric framework ?

* Because convergence time precludes such option for moderate and big samples as soon as you have more than 3-5 covariates.

* To provide results easier to interpret and to share with practitioners, notably using maps/time and map/housing submarkets : space + time + market segment
## ADD OTHER DIMENSIONS OF ATTRIBUTE'S SPACE ?

* Why choosing categorical submarkets:

* Because by merging all submarkets in a global local linear regression, it allows to increase the amount of information used in each submarket for taking into account unobserved heterogeneity.

> It's what we call « shared spatial heterogeneity ».
Extending mgwrsar R package (Geniaux Martinetti 2017)
Mixed GWR + 2SLS for spatial autorcorrelation

$$
y = \beta_c X_c + \epsilon_i \quad (\text{OLS})
$$

$$
y = \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{GWR})
$$

$$
y = \beta_c X_c + \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{MGWR})
$$

$$
y = \lambda W y + \beta_c X_c + \epsilon_i \quad (\text{SAR})
$$

$$
y = \lambda W y + \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{MGWR-SAR}(0, 0, k))
$$

$$
y = \lambda W y + \beta_c X_c + \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{MGWR-SAR}(0, k_c, k_v))
$$

$$
y = \lambda(u_i, v_i)W y + \beta_c X_c + \epsilon_i \quad (\text{MGWR-SAR}(1, k, 0))
$$

$$
y = \lambda(u_i, v_i)W y + \beta_v(u_i, v_i)X_v + \epsilon_i \quad (\text{MGWR-SAR}(1, 0, k))
$$

$$
y = \lambda(u_i, v_i)W y + \beta_c X_c + \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{MGWR-SAR}(1, k_c, k_v))
$$
Extending mgwrsar R package (Geniaux Martinetti 2017)
Mixed GWR + 2SLS for spatial autorcorrelation

$$
y = \beta_c X_c + \epsilon_i \quad (\text{OLS})
$$

$$
y = \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{GWR})
$$

$$
y = \beta_c X_c + \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{MGWR})
$$

$$
y = \lambda W y + \beta_c X_c + \epsilon_i \quad (\text{SAR})
$$

$$
y = \lambda W y + \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{MGWR-SAR}(0, 0, k))
$$

$$
y = \lambda W y + \beta_c X_c + \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{MGWR-SAR}(0, k_c, k_v))
$$

$$
y = \lambda(u_i, v_i)W y + \beta_c X_c + \epsilon_i \quad (\text{MGWR-SAR}(1, k, 0))
$$

$$
y = \lambda(u_i, v_i)W y + \beta_v(u_i, v_i)X_v + \epsilon_i \quad (\text{MGWR-SAR}(1, 0, k))
$$

$$
y = \lambda(u_i, v_i)W y + \beta_c X_c + \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{MGWR-SAR}(1, k_c, k_v))
$$
$$y = \lambda Wy + \beta_c X_c + \beta_v(u_i, v_i) X_v + \epsilon_i \quad (\text{MGWR-SAR}(0, k_c, k_v))$$

+ GENERAL KERNEL PRODUCT of Li and Racine 2010
ADDING TIME AND HOUSING SUBMARKET IN THE KERNEL

Spatial, temporal and categorical kernel are
combined by means of the Generalized Kernel
Product function:

$$
GPK(i, j) = K(d_{ij}, hs) * K(T_{ij}, ht) * K(S_i, \rho)
$$
The categorical kernel (Aitchison and Aitken, 1976; Li and Racine, 2010) takes the following form:

$$K(S_i, \rho) = \begin{cases} 1 & \text{if } S_j = S_i = s \\ \rho_s & \text{if } S_j \neq S_i = s \end{cases}$$
## Planed Extensions of GM2017

$$ Y_i = \lambda WY + \beta_c X_c + \beta_v((u_i, v_i), T, S; h_d, h_t, \rho_s) X_v + \epsilon_i, $$

$$ Y_i = \sum_s \lambda_s WY + \sum_s \beta_c^s X_c + \beta_v((u_i, v_i), T, S; h_d, h_t, \rho_s) X_v + \epsilon_i, $$

$$ Y_i = \lambda((u_i, v_i), T, S; h_d, h_t, \rho_s) WY + \beta_v((u_i, v_i), T, S; h_d, h_t, \rho_s) X_v + \epsilon_i, $$
Monte Carlo design inspired by GM2017:

• (x,y) locations drawn from uniform [0,1]

• W → 4 nearest-neighbours, row
normalized

- 4 covariates including intercept, some spatially correlated,

• Mixed $\beta$: some spatially varying $\beta_v(u_i, v_i)$ and some constant over the space $\beta_c$
Beta0

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Beta1

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Beta2

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Beta3

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# Monte Carlo: Submarket simulation

4 simulated submarkets cases:

* Different Beta for 4 submarkets for observable covariates

* One spatially correlated covariate is not observed for all submarkets

→ introduce additional Spatial Heterogeneity + dependence between submarkets

* It has null Beta value for one submarket

→ **3 dependent submarkets + one fully independent submarket**
# Monte Carlo: Submarket simulation

## 2 simulated submarkets case:

* One case with different Beta

* One case with same Beta

→ **false submarket segmentation**
# Monte Carlo preliminary results

Results based on this model:

$$Y_i = \lambda((u_i, v_i), T, S; h_d, h_t, k_s)WY + \beta_v((u_i, v_i), T, S; h_d, h_t, \rho_s)X_v + \epsilon_i,$$

* Bandwidths $\rho_s$ for the independent submarket is closed to zero and $\rho_s > 0$ for other submarkets (4 submarkets case)

* Bandwidth $\rho_s$ for "false" submarket segment is closed to one (2 submarkets case)

* $\beta_i$ and spatial parameter $\lambda_i$ unbiased
# RESULTS for Developable land Sales Data

* SAMPLE : 7 000 geolocalized sales of Developable land between 100 m² and 50000 m² in southern France (2007-2015)

* Simple didactic model:

$$ \log(prix) = surface + date + ndist_cbd + ndist_dense_urban_area + ndist_road | (u_i, v_i), T, S $$
RESULTS for Developable land Sales Data

* 4 potential submarkets based on different ways of classifying Developable Land Sales form different data sources:

* from fiscal administration database (1)

* from fiscal administration database (2)

* from digitalized Land Use Plan,

* from SAFER (french agricultural administration that monitors undevelopped land sales).
RESULTS on Developable land Sales

<table><thead><tr><td>MODEL</td><td>MAPE</td><td>CV (leave-one-out)</td><td>SSR</td></tr></thead><tbody><tr><td>SAR</td><td>32.36 %</td><td>1412.12</td><td>1286.27</td></tr><tr><td>GWR</td><td>30.71 %</td><td>1376.69</td><td>1213.11</td></tr><tr><td>GWR + TIME</td><td>30.75 %</td><td>1377.084</td><td>1217.09</td></tr><tr><td>GWR + Spatial Dependence (GWRSAR)</td><td>29.54 %</td><td>1263.224</td><td>1164.58</td></tr><tr><td>GWRSAR + endogenous categorical segmentation</td><td>24.11 %</td><td>862.38</td><td>947.72</td></tr></tbody></table>

W = Adaptive bisquare (with 10 first spatial neighbours belonging to the same submarket)

Optimal bandwidth (CV criteria) with adaptive gaussian kernel for distance and LR 2010 kernel for submarkets :

$$H^* = (h_d=520 \text{ neighbours}, \rho1=0.821, \rho2=0.662, \rho3=0.891, \rho4=0.085)$$
RESULTS on Developable land Sales

<table><thead><tr><td>MODEL</td><td>MAPE</td><td>CV (leave-one-out)</td><td>SSR</td></tr></thead><tbody><tr><td>SAR</td><td>32.36 %</td><td>1412.12</td><td>1286.27</td></tr><tr><td>GWR</td><td>30.71 %</td><td>1376.69</td><td>1213.11</td></tr><tr><td>GWR + TIME</td><td>30.75 %</td><td>1377.084</td><td>1217.09</td></tr><tr><td>GWR + Spatial Dependence (GWRSAR)</td><td>29.54 %</td><td>1263.224</td><td>1164.58</td></tr><tr><td>GWRSAR + categorical submarkets</td><td>24.11 %</td><td>862.38</td><td>947.32</td></tr><tr><td>GWRSAR with independent estimation of each segment</td><td>28.65 %</td><td>1093.35</td><td>828.35</td></tr></tbody></table>
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Thanks for your attention