PGR Conference Presentation

Using spatial hierarchical models to estimate place attractiveness

Sam Comber

Objective: build a national indicator of town centre boundary attractiveness.

How do we define place attractiveness?

Cultural infrastructure (Oner, 2017)?

Public services and goods (Florida, 2000)?

Consumer amenities (Glaser et al. 2001)?

How do previous studies approach this?

Although quality of life has no natural unit per se, previous research has shown a large proportion of the variation in place attractiveness, as measured through the lens of the residiential housing market and that variation in housing prices can be explained by the provision of shops, goods and services.

In the urban economic literature, research has been motivated by the theoretical framework of spatial equilibrium; that the willingness to reside in a location can be explained by exploring housing prices and wage levels. More recently, however, cities and urban environments have demonstrated housing prices to increase faster than wage levels, implying a premium for particular locations (Oner, 2017; Glaeser et al. 2001).

In this way, a number of studies estimate the relevance of consumption opportunities to the attractiveness of places, with a focus on home-buyer decisions towards urban amenities.

Therefore, the spatial heterogeneity in the price paid for housing can be seen as a function of amenities such as climatic conditions, open park spaces, and the agglomerative forces of public and private services.

How do we approach this?

Business rates: a property-based tax levied on the estimated value of all non-residential properties such as shops, offices, warehouses and factories.

As business rates fluctuate with local economic market conditions, they reflect not only the size and scale of retail economies, but also the performance of the surrounding areas (Astbury and Thurstain-Goodwin, 2014).

How do we frame our methodological approach?

Spatial FE:

$$\text{log }y_{ij} = \sum_{j=1}^{j} \theta_{j}D_{ij} + \beta_{k}x_{ij} + \epsilon_{ij}$$

Hierarchical model:

$$\text{log }y_{ij} = \alpha_0 + X_i\beta_k + u_j + \epsilon_{ij} $$

$$r_{j} = \frac{1}{n} \sum_{i=1}^{n} y_{i' j} - \hat{y}_{i' j}. $$

Multiplies raw residual for group $j$ with shrinkage factors.

$$u_j = r_{j} \frac{\sigma_u^2}{\sigma_u^2 + \sigma_e^2 / n_j}.$$

A limitation of our spatial FE model is that it conflates unobservable influences that operate at the property-level and town centre boundary scale. If allowed, this leads to inefficient model estimation through heteroscedastic and/or spatially-correlated error in the covariance structure.

Multilevel models remedy this by treating the boundaries as part of the explanation for geographically varying outcomes. This allows two stores within the same town centre boundary to be more alike in their outcomes than would be expected given their individual characteristics alone.

Correlation within boundaries is expected because stores are assumed to be affected by the same aggregate effects, also known as group dependence.

MLM therefore makes use of two-level hierarchical structure (stores and TCBs) to explain variation in the business rate.

Spatial Hierarchical model:

$$\text{log }y_{ij} = \alpha_0 + X_i\beta_k + \theta_j + \epsilon_{ij} $$

$$\theta_j = \lambda M \theta_j + u_j$$

Reduced form of Equation (6) creates spatial covariance across system (Dong and Harris, 2015):

$$\theta_j = (I_j - \lambda M)^{-1}u_j$$

Spatial Moving Average (SMA) model:

$$\theta_j = (I + \lambda M) u_j$$

No inversion of the spatial filter, meaning spatial covariance is constrained to only first- and second-order neighbours (Anselin, 2003).

Findings

What next?

Learn to write log-likelihoods for each model to generate goodness-of-fit measures such as AIC and LL.
Begin write-up.
Present at conferences.

Thanks for listening!