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Journal of Geographical Systems

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01.12.2009 | Original Article

A spatio-temporal model of housing prices based on individual sales transactions over time

verfasst von: Tony E. Smith, Peggy Wu

Erschienen in: Journal of Geographical Systems | Ausgabe 4/2009

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Abstract

A spatio-temporal model of housing price trends is developed that focuses on individual housing sales over time. The model allows for both the spatio-temporal lag effects of previous sales in the vicinity of each housing sale, and for general autocorrelation effects over time. A key feature of this model is the recognition of the unequal spacing between individual housing sales over time. Hence the residuals are modeled as a first-order autoregressive process with unequally spaced events. The maximum-likelihood estimation of this model is developed in detail, and tested in terms of simulations based on selected data. In addition, the model is applied to a small data set in the Philadelphia area.

Vorheriger Artikel The neighborhood effects of foreclosure

Nächster Artikel Hedonic approaches based on spatial econometrics and spatial statistics: application to evaluation of project benefits

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The number of affordable housing units provided (built or renovated) by the CDC is also commonly used as a measure of “success.” However, it has been argued by many housing researchers that increased housing supply is only a measure of input, and thus is not a fair assessment of neighborhood revitalization as an outcome (see for example Smith (2003) and the many studies cited therein). Hence the assumption implicit in the present approach is that improved neighborhood quality should increase local demand for housing, and thus local housing prices.

Certain technical appendices have been omitted to save space, and can be found in the Electronic Supplementary Material online.

Alternatively, it may often be more appropriate to use the log of sales price as y _i.

Note that there may in fact be some minimal time lag required before a given sales price can influence subsequent prices (such as the time required for this sale to be published in the local paper). Hence the inclusion of all prior sales is a simplifying assumption.

It is worth noting that even if more elaborate spatial kernel functions were to be used, the bandwidth, d, of each kernel is well known to be the single most critical determinant of spatial dependence (see for example Silverman 1986).

Note also that if housing prices are in log form, then this term corresponds to a geometric average of housing prices rather than an arithmetic average.

If time intervals, Δ _t, are allowed to become “arbitrarily small,” and are denoted by δt, then in the formulation of Jones (1993), ϕ(δt) = exp(−α ₀ δt) = ρ ^2δt where ρ = exp(−α ₀).

This does not arise in the continuous model where it is natural to assume that simultaneous events occur with probability zero.

It is shown in (A1.2) of Appendix 1 (see footnote 2) that det (I _n − D) = 1, so that (I _n − D)⁻¹ always exits and B is well defined.

It is also shown in (A1.2) of Appendix 1 that det (A) = 1, so that A ⁻¹ always exits.

Indeed, such oscillation behavior loses all meaning in the continuous version of AR(1), where the autocorrelation parameter is required to be nonnegative [as is evident from the positivity restriction, α ₀ > 0, in Jones (1993) for the identity, ρ = exp(−α ₀), of footnote 7 above].

Following Anselin and Moreno (2003), one might also interpret such negative estimates of ρ to be evidence that the present spatio-temporal specification is simply not supported by the data.

Here it should be emphasized that the following simulations are intended only to illustrate the “typical” small sample properties of this model in a single situation. Systematic simulation studies of model performance under a range of space-time structures and parameter values are left for future work.

Here parameter choices (d, Δ) for W were chosen to yield a degree of space-time interaction among housing sales that roughly matched that of the Philadelphia application below, where about 66% of the houses were influenced by previous sales (based on the space-time bandwidths used).

In view of this, it is somewhat surprising that estimates of λ appear to behave quite well by comparison. Moreover, since this relation persists in all simulations studied thus far, it raises an interesting (open) question as to why the expression for \( \hat{\lambda }(\hat{\rho }) \) in (28) above should remain more stable than \( \hat{\rho } \).

During this time period there was a significant increase in housing prices throughout the entire Philadelphia area.

This address-level sales data was extracted from the Philadelphia Board of Revisions of Taxes (BRT) Properties File from 1990 to 2006.

It should be clear from Fig. 5 that the areas PEC and CA are sufficiently far apart to ensure that no space-time dependencies occur between houses in separate areas.

It should be noted here that since this model involves both lagged dependent variables and a time trend term, it is technically an instance of a “autoregressive process around a deterministic time trend.” But while the rates of convergence for OLS estimates are more delicate in this case, it can be shown that the standard model significance tests continue to be asymptotically valid. [See for example Hamilton (1994, Sect. 16.3)].

In particular, OLS estimates in the presence of lagged dependent variables are consistent and asymptotically normally distributed about their true values. However, since the lagged dependent variables and residuals are not fully independent, these estimates are typically biased for small samples. [See for example Davidson and MacKinnon (2004, Sect. 3.2)].

The standard R-square is known to be somewhat problematic in the case of lagged dependent variables. Hence for comparability with the spatio-temporal formulation below, we choose to define pseudo R-square here to be the squared correlation between y and it prediction, \( \hat{y}\, = \,\hat{E}(y|X)\, = \,(I_{n} \, - \,\hat{\lambda }W)^{ - 1} X\hat{\beta } \) with OLS estimates \( \hat{\lambda } \) and \( \hat{\beta } \). However, it is also of interest to note that in this particular application the unadjusted R-square (0.372) is almost the same as the pseudo R-square (0.362).

Here it should be noted that CDCs are local non-profit organizations whose funding is devoted almost entirely to housing projects, and not to data collection. Hence all housing data was drawn from the Philadelphia BRT (footnote 17 above). Moreover, while this BRT data did include provisions for a number of key housing attributes (such as “number of bedrooms” and “interior and exterior condition”), most of this data was either missing or unusable for other reasons.

In the present context of maximum-likelihood estimation, the AIC measure is considered by many to yield more reliable goodness-of-fit comparisons than pseudo R-square. However, the latter is somewhat easier to interpret.

We are indebted to an anonymous referee for pointing out the close similarities between our current model and that of Pace et al. (2000).

This is a special case of the STLM model in expression (7) of Pace et al. (2000), where the interaction terms (TX, SX, TSX, STX) are missing, along with matrix of non-lagged variables, Z.

It should be remarked that Pace et al. (2000) motivate the form of their model by constructing [in expressions (1), (2) and (5)] a spatio-temporal extension of the spatial Durbin model (Anselin 1988) which does indeed account for spatio-temporal autoregressive dependencies in the unobserved residuals. But this development is somewhat misleading in the sense that their final model, STLM [expression (7)], ignores the crucial “common factor” constraints on coefficients that preserve these autoregressive dependencies. Hence, while STLM could in principle be used to test this “common factor hypothesis” (as implied by their discussion on p. 234), the model itself is simply a more elaborate version of the spatio-temporal lag model in (34) above.

A full argument is given in Appendix 3 (see footnote 2 above).

In fact this “special case” provides the motivation for essentially all linear autoregressive models, both in time and space. For the spatial case, this is clear from the motivating examples in the original papers of Whittle (1954) and Ord (1975).

Proofs of this result can be found in any standard text, such as Hamilton (1994, Sect. 3.4).

For further discussion of this point see Green (2003, Sect. 12.2).

It is also worth noting that this result depends only on the first two moments of the independent innovations (ɛ _i: i = 1,…, n) so that the normality assumption in (7) is not required.

For example, if ρ > 0 then it is clear from the cumulative nature of (50) that sales residuals u _i with many sales in the recent past, i.e., with many positive dependencies (τ _ij: j < i), will tend to have much higher variances than those with very few sales in the recent past.

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Titel: A spatio-temporal model of housing prices based on individual sales transactions over time
verfasst von: Tony E. Smith
Peggy Wu
Publikationsdatum: 01.12.2009
Verlag: Springer-Verlag
Erschienen in: Journal of Geographical Systems / Ausgabe 4/2009
Print ISSN: 1435-5930
Elektronische ISSN: 1435-5949
DOI: https://doi.org/10.1007/s10109-009-0085-9

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