Forecasting the housing vacancy rate in Japan using dynamic spatiotemporal effects models

Abstract

This study attempts to predict and forecast the future heterogeneous increase in the vacant house ratio among prefectures in Japan using spatial panel models with unobserved dynamic spatiotemporal effects. The study formulated models with autoregressive and random-walk spatiotemporal effects, referring to the dynamic spatiotemporal effects (DSE) models. We estimate the model parameters and latent spatiotemporal effects via Markov Chain Monte Carlo algorithm. Simulation studies demonstrated the superior performance of the DSE model in terms of future prediction when spatial and temporal correlation exists. The model is then applied to the prefecturewise ratio of vacant houses to the non-rental housing stock in Japan, and the results imply existence of spatiotemporal correlations that cannot be captured by explanatory variables. Furthermore, it is revealed that the DSE models can provide better forecasting than the existing spatial panel models.

Appendices

Appendix 1

Data generating process

	Scenario I	Scenario II	Scenario III
	Spatial autoregressive error	Spatiotemporal effects (moderate serial autocorrelation)	Spatiotemporal effects (strong serial autocorrelation)
Data generating process	${y}_{t}=\alpha +\beta {x}_{t}+u+{\varepsilon }_{t}$, where $u\sim N(0, cH\left(\phi \right))$, ${\varepsilon }_{t}=\rho W{\varepsilon }_{t}+{v}_{t},$ ${v}_{t}\sim N(0,{\tau }^{2}{I}_{n})$	${y}_{t}=\alpha +\beta {x}_{t}+u+{\xi }_{t}+{\varepsilon }_{t}$ where $u\sim N(0, cH\left(\phi \right))$, ${\xi }_{t}|{\xi }_{t-1}\sim N\left(\rho {\xi }_{t-1},{\sigma }^{2}H\left(\phi \right)\right),$ ${\varepsilon }_{t}\sim N(0,{\tau }^{2}{I}_{n})$	${y}_{t}=\alpha +\beta {x}_{t}+u+{\xi }_{t}+{\varepsilon }_{t}$ where $u\sim N(0, cH\left(\phi \right))$, ${\xi }_{t}|{\xi }_{t-1}\sim N\left(\rho {\xi }_{t-1},{\sigma }^{2}H\left(\phi \right)\right),$ ${\varepsilon }_{t}\sim N(0,{\tau }^{2}{I}_{n})$
Regions	$i$($i=1,\dots ,n$ with $n=47$)
Time periods	$t$($t=1,\dots ,T$ with $T=5$)
Fixed parameters	$\alpha =2$, $\beta =5$, c = 9, $\rho =0.5$, and $\tau =0.3$
Geographic weight matrix	$H\left(\phi \right)$: Correlation matrix, where each element is ${\rho }_{G}\left({s}_{i}-{s}_{j};\phi \right)=\text{exp}(-||{s}_{i}-{s}_{j}|{\left.\right|}^{2}/{\phi }^{2})$, where $\phi =1$ and $||{s}_{i}-{s}_{j}||$ is a geographical distance between region i and j
	$W=H\left(h\right)$ with $h = 2$	-
Explanatory variable	${x}_{it}$ is generated by uniform distribution on $[t-1, t+1]$ The mean value of ${x}_{it}$ is $t$ $(t=1,\dots ,T)$

Appendix 2

Details of the SAC and the SDM models

Let ${y}_{t}$ be an $n$ dimensional response vector and ${X}_{t}={\left({x}_{1t},\dots ,{x}_{nt}\right)}^{T}$ be an $n\times p$ matrix of explanatory (exogenous) variables at time $t$, where $n$ is the number of regions or individuals. Further assume that both ${y}_{t}$ and ${X}_{t}$ are geographically located, and the interaction among locations can be described by a user-specified spatial weight matrix W, which is a time-invariant $n\times n$ matrix. The model considering the effect of spatial proximity and its correlation among geographical locations is commonly known as the spatial autoregressive model, such as in LeSage (2009). There are many variations for modeling spatial correlation for panel data by assuming the spatial effect on explanatory variables and error terms, and the subsequent studies, including LeSage and Pace (2009), Elhorst (2014), LeSage (2014) and Kawabata and Abe (2018) utilize the model called the spatial Durbin model (SDM) formulated as follows:

$$\left(\text{SDM}\right) {y}_{t}= \rho W{y}_{t}+{X}_{t}{\beta }_{\left(1\right)}+W{X}_{t}{\beta }_{\left(2\right)}+{\varepsilon }_{t},$$

where ${\varepsilon }_{t}$ is an independent error term, $\beta$ is an unknown regression parameter, and $\rho$ is a correlation parameter satisfying $\left|\rho \right|<1$. Another specification that is often utilized in spatial econometrics is the Spatial Autoregressive Model with Auto Regressive disturbances (SAC), which is formulated as follows:

$$\left(\text{SAC}\right) {y}_{t}= \rho W{y}_{t}+{X}_{t}\beta +{u}_{t}, {u}_{t}=\lambda W{u}_{t}+ {\varepsilon }_{t},$$

where $\lambda$ is an additional spatial correlation parameter in the error term, satisfying $\left|\lambda \right|<1$. When $\lambda =0$, the above SAC model reduces to the standard spatial autoregressive model. The SAC model allows correlation with specific spatial weights and this paper utilizes this model as one of the representative models in the spatial econometric models. The methods used to identify and estimate the SAC model are formulated in such as Anselin (1988) and Kelejian and Prucha (1998), which are essentially through the maximum likelihood (ML) and the generalized method of moments (GMM). In the simulation studies and data analysis, we used statistical software STATA to apply the SAC and SDM models for panel data analysis.

See Figs.

Fig. 5

Trace plots of the posterior samples for the DSE-AR. (Estimation time periods 1988–2018). (Parameter $\beta$ (coefficient for age 0–29 ratio), $\tau$, $\sigma$, $\phi$, $\delta$, in order)

5 and

Fig. 6

Trace plots of the posterior samples for the DSE-RW. (Estimation time periods: 1988–2018). (Parameter $\beta$ (coefficient for age 0–29 ratio), $\tau$, $\sigma$, $\phi$, in order)

6.