Dealing with location in the valuation of office rents in London Multilevel and semi-parametric modelling Aniel Anand, 1 st July 2015

Background Data collected by the Valuation Office Agency Developing analytical tools to assist in the valuation of property Valuation professionals indicate that location is a major driver of property valuation What analytical challenges does this present?

Why is location important? Properties within the same location are likely to have similar rents

Impact if location is not captured Excluding location from the regression can result in: -Omitted Variable Bias -Spatially correlated error terms -Under-estimated standard errors Y = β 0 + β 1 X 1 + β 2 X 2 + ε Y = β 0 + β 1 X 1 + ε C(ε i,ε j ) ≠ 0

OLS regression – England and Wales Rent premiums in major cities and coastal towns Clustered residuals show there is a location effect How can this location effect be measured?

Identifying Spatial Autocorrelation - Variograms Shows relationship between the average correlation between residuals and the geographic distance between residuals. Spatial autocorrelation occurs at the upward part of the graph, before the residuals tail off.

Identifying Spatial Autocorrelation - Moran’s I test -Values range from −1 (indicating perfect dispersion) to +1 (perfect correlation). - A zero value indicates a random spatial pattern. H 0 : There is no spatial autocorrelation H 1 : There is spatial autocorrelation Convert Moran’s I values to a Z score. Test at 5% level of significance and compare the p values.

OLS regression – including location Spatial autocorrelation still evident. More complex regression models are required. Including locational characteristics in regression

Multilevel modelling Controlling for different levels of spatial hierarchy Decomposes the variance such that the Level 1 residuals, ε, are uncorrelated

Building the multilevel model Model selection Choose model with the lowest AIC, using forward and backwards selection methods. 2 level random intercept / slope model (LA level) RI:Y ij = β 0 + β 1 x ij + β 2 x j + u 0j + ε ij RS:Y ij = β 0 + β 1 x ij + β 2 x j + u 0j + u 1j x j + ε ij Extending to 4 levels (down to LSOA level) Y ijkl = β 0 + β 1 x ijkl + β 2 x jkl + β 3 x kl + β 4 x l + u 0jkl + u 0kl + u 0l + u 1jkl x jkl + ε ijkl

Chosen multilevel model – Inner London Multilevel model has reduced spatial autocorrelation effect. Spatial autocorrelation occurs for properties up to about 130 metres apart from each other. Can we do any better?

Generalised Additive Model Relaxing the constraint of modelling location by a linear function. Use the x,y co-ordinates of each property to model the spatial component more flexibly. Y i = β 0 + β 1 x i + s(x 1i, x 2i ) + ε i where s(x 1i, x 2i ) is the smoothing effect for location. Reduces to the standard OLS model when there is no smoothing effect.

Non-linear effect of location The deviations of the spatial smoothing parameter away from zero indicate the extent of the non-linear effect that location has in the model. There are several distinct pockets of high rent areas in London, which is why a non- linear model is more appropriate.

Chosen GAM – Inner London Spatial autocorrelation reduced to properties up to about 20 metres apart from each other However, GAM shows evidence of spatial heterogeneity (negative Moran’s I). GAM possibly over-compensates for the spatial effect.

Conclusion The root mean square error (RMSE) is used to assess the prediction precision of the model. Multilevel model over- fits the spatial element of the model. Rents of neighbouring properties does little to add to the predictive power of the model. Which model should we choose?

Questions