# Declining influence over spatial distance

This blog post discusses one solution for models with declining influence over geospatial distance and its limitations in my exact use case. Furthermore, I want to stimulate discussion on how to improve the model and find alternatives.

## Background

At the end of last year (Dec. '17), I finished my Bachelor's thesis about the influence of the economic strength of geographic regions on their rental prices. For this hedonic regression, data on roughly 90000 rental flats within Germany was gathered, including indicators about the quality and configuration of these flats, as well as the exact geographic location. While the former were used as control variables, the latter found use in matching the economic strength of a region to said flats.

Measuring “economic strength” is not as trivial as it may sound. There is no clearly defined indicator for this, when it comes to sub-regions within a country. On a supra- or national level, economists usually use the GDP, but there often is no such measure for regions within a country. It is however possible to gather tax data on smaller regions within Germany, namely on countries, counties and even cities, due to its federalistic structure.

It is important to find a good balance between size of these regions (generally, you want as small regions as possible), supply of data as well as clarity. In the case of Germany, it makes sense to choose counties, as they usually provide a lot of useful data and do not vary as much in geographic size as other categorizations. Additionally, 401 segments seems like a manageable amount. Most importantly, counties vary greatly when it comes to economic strength, which means, that the econometric model is able to discriminate between these regions easily.

Choosing this partition does have its downsides however. For one, economic strength of a county may be influenced by the country it is in. Germany's “vertical revenue equalization” could strengthen a county, even though it does not contribute to that itself by providing good jobs and attractive living conditions. However, that would be the basis of the hypothesis that rental flats in stronger regions are more expensive than those in weaker regions.

Secondly, these predefined regions could influence one another, based on their spatial distance. A very strong county may raise rental prices not only within its own borders, but also in neighbouring counties and - depending on their size and “mobility” of the workforce - in even more than that. In order to counter this issue, "Zones of Mobility in the Workforce" have been defined multiple times before. These regions are crafted so they also reflect catchment areas. This results in regions with very high commuting within and almost none between them. The problem with this approach is its artificial nature. The zones are not identical to administrative regions, thus there is no data on their economic or financial state.

## Weighted Economic Strength per Inhabitant

n order to measure economic strength of a region, the “communal fiscal capacity” (ger.: “gemeindliche Steuerkraft”) was used. It includes not only the “real fiscal capacity” of said region, but also the communal share of the income and turnover tax, excluding the apportionment of the occupational tax. Authorities provide this data for each 401 county in every year. Since we are dealing with regions with varying population numbers, it makes sense to use the “communal fiscal capacity per inhabitant” instead of the total number for each county in order to reflect differences real strength. Hereinafter, this is labeled as “economic strength per Inhabitant (ESI)”.

As mentioned before, it is rather imprecise to simply match each rental flat to the economic strength of the county it is located in. It is safe to assume at least some level of influence between neighbouring counties.

As the real form of the effect is unknown, an assumption has to be made. It makes sense, that the level of influence of a region's economic strength on a flat's rental price declines with rising spatial distance between each other. Due to the lack of proper empirical evidence and research by others, a simple inverse form was used:

\begin{align*} \text{wesi}_{f} = \sum\limits_{c = 1}^{401} \frac{\text{esi}_{c}}{d_{fc}} \end{align*}

With:

• $wesi$: Weighted Economic Strength per Inhabitant
• $esi$: Economic Strength per Inhabitant
• $d$: spatial distance
• $f$: representing each rental flat in the sample
• $c$: representing each of the 401 counties

The “spatial distance” was defined as the geodesic distance between the spatial location of the flat (defined by its latitude and longitude) and the centroid of the county's geographic area.

This results in an indicator which is able to assign each flat of a county differing values regarding economic strength, depending on their exact location. If a flat within a weaker county is close to the border of a stronger one, it will have a much higher wesi than an identical flat within that weaker county which has a higher spatial distance to the stronger one.

It should be clear, that this indicator is problematic in multiple ways. The influence is assumed to be of the same form for all Germany, however it is not certain that this reflects the reality. Also, an inverse course was assumed, although this decision was not based on empirical research or theoretical indication, but rather convenience. Additional research has to be done in order to clarify, which form the gradual reduction of influence has. Thus WESI should be seen as an approximation and proof of concept at best.

In my case WESI proofed to be a reliable factor in the hedonic regression. I may be able to share the results in the near future, however this blog post simply discussed the initial issue and idea behind WESI, which has been crafted to tackle the initial issue.

If you have further comments, idea or inspiration, please mail me or find me on Social Platforms.