روشی برای پیش بینی فضای زمانی با تقاضای قیمت دارایی واقعی

Using 5243 housing price observations during 1984–92 from Baton Rouge, this manuscript demonstrates the substantial benefits obtained by modeling the spatial as well as the temporal dependence of the errors. Specifically, the spatial–temporal autoregression with 14 variables produced 46.9% less SSE than a 12-variable regression using simple indicator variables for time. More impressively, the spatial–temporal regression with 14 variables displayed 8% lower SSE than a regression using 211 variables attempting to control for the housing characteristics, time, and space via continuous and indicator variables. One-step ahead forecasts document the utility of the proposed spatial–temporal model. In addition, the manuscript illustrates techniques for rapidly computing the estimates based upon an interesting decomposition for modeling spatial and temporal effects. The decomposition maximizes the use of sparsity in some of the matrices and consequently accelerates computations. In fact, the model uses the frequent transactions in the housing market to help simplify computations. The techniques employed also have applications to other dimensions and metrics.

Data sets have often been organized by units of time such as quarters or years as well as by geographical constructs such as regions, states, or counties. In reality, the data often represent an aggregation of individual observations which have more precise temporal and spatial characteristics. Much of the governmentally collected economic, medical, and social data falls into this category. However, the increasing capabilities of information systems and especially geographic information systems (GIS) have greatly aided work with disaggregated data having precise spatial and temporal references. For example, automated point-of-sale data from individual stores have a precise identification in time and space. Even governmentally collected data, where privacy concerns dictate minimum levels of aggregation, have become more precise in their identification of space or time. For example, the Home Mortgage Disclosure Act (HMDA) data for 1993 contained approximately 15 million individual transactions identified by over 60 000 locations (census tracts). The trend towards large data sets with substantial spatial and temporal detail raises the issue of how to forecast such data. Moreover, such data raise computational issues as well as conceptual issues of how to plausibly model the spatial and temporal dependence. Housing prices provide another example of this type of data. First, over 4.5 million houses sold during 1997 alone. Most data sources (multiple listing services or assessor databases) record the day, month, and year of the transaction. Given an address, geographic information systems can provide the corresponding latitude and longitude (or other locational coordinates) for 80% or more of the records (Johnson, 1998). At least five common applications employ housing transaction data. First, most houses in the US (and many in other countries) have their assessed value for tax purposes determined by the predictions from statistical models calibrated using individual housing transactions (Eckert, 1990, p. 27). Second, the movement of many primary and secondary lenders to some form of automated appraisal places an added premium on prediction accuracy (Gelfand, Ghosh, Knight & Sirmans, 1998). Third, the spatial and product differentiated nature of housing makes it difficult to compare prices across time and space. The desire to make such comparisons has spurred substantial activity in creating constant quality price indices by location (Hill, Knight & Sirmans, 1997). Fourth, hedonic pricing models use housing data to estimate the costs and benefits associated with such items as pollution, growth controls, and tax policies (Case et al., 1993 and Brueckner, 1997). Fifth, as a house comprises a large fraction of an individual’s wealth, a number of parties follow local price forecasts. Collectively, these applications involve the goals of accurate prediction, efficient coefficient estimation, valid inference, and the desire to understand both the temporal and spatial dependencies in prices. For housing data, the benefits from modeling the error dependence over time are well-known and the benefits from modeling the error dependence over space are becoming better known.1 Such benefits include more efficient asymptotic parameter estimation, less biased inference (positive correlations among errors artificially inflate t-statistics), and more precise predictions. Joint modeling of errors in both time and space offers the potential for further gains.2 However, these techniques have not been applied extensively to economic data and the best ways of specifying spatial, temporal, and spatial–temporal interactions do not appear obvious. As Gelfand et al. (1998) state, ‘Spatial–temporal interaction presents a difficult modeling challenge since it is unclear how to reconcile the two scales.’ Consequently, Gelfand et al. used time indicators which made one-step ahead forecasting difficult.3 Ideally, the joint specification of the errors in both space and time would perform well and provide an interpretable framework for viewing the error dependence. We approach this specification question from a compound filtering perspective. Variables filtered first for time and subsequently for space could display less error dependence in either time or space. The same, however, could apply to variables filtered first for space and subsequently for time. We allow the model to nest both of these alternatives. We find the optimum is a convex combination of the two compound filters. This provides an interpretable error dependency structure which also allows for possible spatial–temporal interactions. To demonstrate the potential of this means of joint modeling the spatial–temporal errors, we pit such a model against the opposite extreme of a model attempting to handle the effects of space and time through an extensive set of indicator variables. Both models share a common set of housing variables and data of 5243 observations on housing prices during 1985–92 from Baton Rouge, Louisiana. These data include each house’s location in latitude and longitude as well as the day, month, and year the transaction closed. In this contest the spatial–temporal model with 14 variables outperforms by 8% in terms of SSE the traditional indicator based model with 211 variables. Moreover, it maintains its accuracy in one period ex-sample forecasts. The spatial–temporal model relies upon a rich set of spatially and temporally lagged variables for its power. It represents a hybrid between the autoregressive distributed lag model common in time series and the mixed regressive spatially autoregressive model in spatial econometrics (Ord, 1975 and Anselin, 1988, p. 227). The techniques tend to support some of the procedures employed by appraisers who examine a limited set of ‘comparables’ in estimating the value of a house. In addition, we introduce techniques for rapidly computing the estimates based upon a novel decomposition of the spatial and temporal effects. The decomposition maximizes the use of sparsity (proportion of zeros) in some of the matrices and consequently greatly accelerates computations. In addition, we structure the data in a way which makes it feasible to use optimized linear filter routines to simplify some of the problems of dealing with irregularly spaced data over time. Finally, the model takes advantage of the frequent transactions in the housing market to eliminate the awkward normalizing constant in the likelihood which has traditionally impeded spatial error modeling. The culmination of these improvements allows us to compute individual spatial–temporal regressions in a few seconds, thus making these practical for applied work. Section 2 discusses the spatial–temporal model employed and provides details on an improved algorithm for computing the spatial–temporal estimates; Section 3 applies the spatial–temporal model to the Baton Rouge data, while Section 4 concludes with the key results.

This paper demonstrated some simple, easily computed techniques for taking into account both temporal and spatial information. Relative to using an extensive set of indicator variables, modeling correlations among the spatial and temporal errors produced a better goodness-of-fit coupled with much higher levels of parsimony in the regression equation. Specifically, the spatial–temporal regression with 14 variables exhibited 8% less sum-of-squared errors than the trend surface model regression with 211 variables. In this example, the extensive set of indicators over time and space did result in low long-range correlations among errors but still exhibited substantial short-range correlations among errors. In contrast, the spatial–temporal regression displayed lower levels of correlation among errors at all ranges. The spatial–temporal regression also displayed good ex-sample forecasting, an important desideratum in many contexts. For example, most local tax assessments in the US rely upon statistical predictions of housing values. In addition, primary and secondary mortgage lenders have begun exploring the use of statistical forecasting of housing prices as opposed to employing appraisers. Indicator set approaches do not lead as naturally to such predictions. Other applications such as creating constant quality indices could also benefit from jointly modeling the errors jointly over time and space. The continuous nature of the spatial variables in the spatial–temporal models allows the creation of a constant quality index surface as opposed to a set of separate indices. Naturally, specification of the correlations among errors across time and space could improve the quality of estimates in hedonic pricing applications. Structuring the problem to employ lower triangular weighting matrices comprised of individual sparse neighbor matrices and linear filter routines resolves the seemingly difficult problem of differencing each error from numerous other errors and avoids lengthy determinant computations. As many data in economics have specific spatial and temporal characteristics, these techniques could prove useful in other contexts as well. The becomes especially true when considering alternative spatial metrics which involve variables other than physical location. For example, Case, Rosen and Hines (1993) used separate metrics involving physical location, race, and income to improve on a standard regression’s ability to explain public expenditures. They found significant correlations among errors based upon both ordering by physical location and by race. Note, their data spanned the period 1970–85 and hence had a temporal dimension as well. As a potential extension to our work, decomposition of the weight matrix W into more than the physical location and temporal components examined herein could potentially produce additional econometric benefits without incurring significantly greater computational costs.