Housing Submarkets and the Lattice of Substitution

Abstract

This paper aims to stimulate interest in the early micro-economic conceptualisation of housing submarkets proposed by Rapkin and Grigsby, which defined market areas in terms of dwelling substitutability. Three key questions need to be addressed if a return to the Rapkin–Grigsby approach is to be achievable and worthwhile. First, what are the practical benefits? (The paper highlights a range of potential research applications that would benefit from the substitutability approach.) Secondly, in what way are existing approaches likely to be inadequate for demarcating substitutability-based submarkets? (Four criteria are proposed for assessing submarket estimation methods which existing approaches fail to satisfy.) Thirdly, what are the prospects for developing a substitutability metric that will make the Rapkin–Grigsby definition empirically feasible? (A new measure is proposed, based on the cross-price elasticity of price (CPEP), which is illustrated using data for Glasgow, Scotland.)

1. Introduction

In the seminal work of Rapkin et al. (1953) and Grigsby (1963), submarkets were defined in terms of dwelling substitutability: “A housing market area is the physical area within which all dwelling units are linked together in a chain of substitution” (Rapkin et al., 1953, pp. 9–10; quoted in Grisby, 1963, pp. 33–34). While this definition is intuitively appealing, it has been difficult to implement empirically, largely due to problems in finding a workable measure of dwelling substitutability. Nevertheless, were such a measure to be developed, it could potentially enrich a range of research topics in housing economics.

It might, for example, help us to explore the links between market segmentation and social segmentation. Using regression analysis, with substitutability as the dependent variable, we could estimate the extent to which substitutability is driven by the physical characteristics of dwellings and the extent to which it is driven by the social/ethnic/religious composition of the neighbourhood. Clearly, we cannot do this sort of analysis by way of a priori clustering of dwellings (according to administrative area, house type or neighbourhood mix, for example)¹ as we would be none the wiser in terms of understanding which drivers are most important in determining dwelling substitutability. We need a measure of substitutability that reflects the actual decisions that households make, decisions that reveal their preferences for dwelling characteristics, racial/religious/ethnic composition, access to amenities, etc.

Having a substitutability metric would also allow us to define submarkets in the spirit of Rapkin and Grigsby. Grouping dwellings into submarkets on the basis of substitutability would provide a way of summarising the preferences and choices of consumers. Submarkets, so defined, could have a range of potential applications. For example, substitutability-based submarkets may help to improve the performance of automatic valuation models (AVMs), revealing the hidden fault-lines in the local housing market, and capturing the connections between price movements of dwellings in different locations. They could also open up underdeveloped research avenues, such as the extent to which the shapes of substitutability-based submarkets tend to have compact well-defined geographies, revealing the aversion of households to living at the boundaries between residential enclaves (see Criterion 4, in section 2).

Being able to group dwellings on the basis of substitutability might also lead to important applications in house price index calculation. Dean et al. (2012), for example, find that grouping properties that do not have similar price trajectories can lead to considerable distortion in national price indices. Because the prices of close substitutes tend to move together, a reliable measure of substitutability could offer a means of grouping dwellings in a way that preserves the stability of within-group price relatives, leading to a set of index commodity weights that minimises aggregation bias. (Such a method could also find applications in the context of other complex durable goods such as cars and antiques.)

Finally, having a robust measure of the substitutability of dwellings could help us to develop more meaningful W (weights) matrices in spatial econometric house price models. A problem with traditional distance or contiguity definitions of W (Anselin, 1988) is that they do not have an explicit economic rationale. They are essentially proxies for some unobserved behavioural variable—such as substitutability—that connects price movements across different dwellings. Prices of close substitutes tend to move together because they respond in similar ways to demand and supply shocks. Yet proximity and contiguity are not necessarily reliable proxies for substitutability. Dwellings that are distant in Cartesian space can be close in substitutability space if they constitute similar bundles of dwelling and location characteristics. Were an appropriate substitutability metric available, it would (in principle) provide a more meaningful basis for W than distance or contiguity.

Unfortunately, finding such a metric has proved elusive. In its absence, the micro-econometric analysis of submarkets² has turned either to a priori clustering (grouping dwellings according to dwelling or neighbourhood attributes); or to hedonics; or to some combination of the two. Hedonic methods have, of course, been extensively used in the wider housing economics literature, typically in one of two forms, labelled forthwith as types 1 and 2.³ Type-1 hedonic analysis uses regression to control for variation in the mix of dwelling attributes when developing a house price index or valuation model. The method can be interpreted as a practical tool and does not require a particularly restrictive interpretation of regression coefficients (covariates are treated as controls/proxies for a mixture of observed and unobserved factors that affect house prices). Type-2 hedonic analysis, on the other hand, relies rather heavily on interpreting regression coefficients as measures of economic value and this interpretation has been subject to formidable critiques from leading economists (Epple, 1987; and Bartik, 1987, for example).

Unfortunately, it is very much the type-2 hedonic approach that has been utilised in the submarkets micro-econometrics literature,⁴ which typically clusters areas according to similarity of attribute prices. As demonstrated later, this approach is not only flawed in a number of fundamental ways, it is also difficult (if not impossible) to reconcile with the Rapkin–Grigsby notion of substitutability-based submarkets.

How, then, might we find an appropriate metric that will place submarkets research on a more solid footing and also open up these fresh avenues of research? In section 2, the type-2 hedonics method of submarket demarcation, and also the a priori attribute-cluster method, are scrutinised through the lens of substitutability to establish whether either can be reconciled with a Rapkin–Grigsby approach. A new method of measuring substitutability is then proposed in section 3 based on CPEP (cross-price elasticity of price). Section 4 considers how CPEP could be used to explore the existence and spatiality of submarkets and section 5 provides an empirical application. Section 6 concludes with a summary of findings and suggestions for future research.

2. Problems with Type-2 Hedonics and a priori Clustering as Methods of Defining Substitutability-based Submarkets

Consider the following inventory of housing market entities and definitions

Household: b = 1, 2, …B

Dwellings (or blocks of dwellings): $d_{i}$ , where i = 1, 2, …V

Attribute vector for dwelling i: $z_{i} = z_{(1) i}, z_{(2) i}, \dots, z_{(A) i}$

Attribute price vector for dwelling i: $P (z_{i}) = P_{(1) i}, P_{(2) i}, \dots P_{(A) i}$

Homogeneous attribute price vectors for i and j: $P (z_{i}) = P (z_{j})$

Euclidean distance between i and j: $D_{ij}$

Cross-price elasticity of price (CPEP) between i and j: $η_{ij}$

Homogeneous attribute price vector: HAPV

Let M denote the family of submarkets that make up the urban housing market as a whole. Each dwelling is an element of a submarket ( $i \in S_{k}$ ) and of the wider housing market ( $i \in M$ ). Where K > 1, there is more than one submarket, and $S_{k}$ is defined by some criterion that allocates dwellings to each subset of M. One criterion that has become popular in the micro-economics submarket literature is that of homogeneous attribute price vectors (HAPVs). Applying type-2 hedonics, dwellings are grouped into submarkets if they have similar hedonic attribute prices on the basis of the ‘law of one price’. If HAPV is applied as a sufficient condition, it is implicitly assumed that any two dwellings i and j with the same attribute prices will belong to the same submarket. However, the argument put forward in this paper is that similarity of attribute prices does not, in fact, imply that two dwellings are in the same submarket—the equivalence may be coincidental or simply irrelevant to the existence of submarkets.

If HAPV is applied as a necessary condition, it is implicitly assumed that dwellings in the same submarket will have the same attribute prices

ij \in S_{k} \Rightarrow P (z_{i}) = P (z_{j})

This allows us to deduce (using modus tollens) that, if i and j do not have the same attribute prices then they will not be elements of the same submarket

P (z_{i}) \neq P (z_{j}) \Rightarrow ij \notin S_{k}

This, too, is problematic. As explained later, HAPV is unlikely to hold as a necessary condition because it is perfectly possible to observe different attribute prices on two dwellings even though they belong to the same submarket. In summary, homogeneous attribute prices can be a false indicator of common membership of a submarket and heterogeneous attribute prices can be a false indicator of market segmentation. Similar problems arise with regard to a priori clustering of attributes: to demarcate substitutability-based submarkets we really need some measure of how the market groups dwellings in terms of substitutability, rather than assuming a particular substitutability function a priori.

These arguments are now developed more fully, woven around a set of four robustness criteria for developing SEMs (substitutability-based submarket estimation methods).

2.1 Criterion 1: SEMs should be Robust to Complex Interaction Effects

Rationale: The existence of many to many mapping of means and ends (MMMMEs) and transformative interaction effects (TIEs) implies that properties can be in the same submarket but have different attributes and different attribute prices.

MMMMEs exist because the same human need can be met in different ways, and the same housing bundle will meet different needs for different people. Consequently, homes having few common attributes and very divergent attribute prices could nevertheless be thought of as close substitutes. An important driver of MMMMEs, and one that makes predicting the global set of MMMMEs considerably more elusive, is the existence of transformative interaction effects (TIEs). TIEs occur when the effect of an attribute (whether geographical or structural) is fundamentally transformed when placed in a particular context, such as being combined with another attribute (either geographical or structural).

To illustrate, consider the options for crossing the English Channel.⁵ Suppose ferries, planes and trains are all genuinely close substitutes—a change in the price or availability of one will have a large effect on the demand for the others (they are all in the same market for transferring passengers across the English Channel).⁶ While each mode of transport has contrasting physical features, these features combine to offer a similar service because of TIEs. This leads us to question the meaningfulness of a priori clustering of attributes or relying on attribute price differences as a gauge of substitutability. Whereas wings on trains are of no value, wings on aeroplanes are essential. The attribute price of wings could differ dramatically between trains and planes because the utility of wings is transformed when connected to a jet engine and aerodynamic fuselage (an example of TIEs). Similarly, while vital to the functioning of trains, wheels will be of little value on ferries.

Note that, if we were to persist with the application of HAPV as our means of transport submarket delineation, the divergent attribute prices on wings and wheels across transport modes would lead us to the erroneous conclusion that the different modes belong to different submarkets. The error arises because, although the physical attributes of an aeroplane are quite different from those of a ferry, the final service bundles (comfortable, speedy travel) may be very similar.

Decomposing dwelling prices into the prices of constituent parts is unhelpful when measuring substitutability—what matters is the final service bundle. Different dwellings in different locations can provide remarkably similar levels of utility to a particular consumer despite the structural differences. Empirical estimation is frustrated because the service bundle provided by a house and its location is so multifaceted. MMMMEs and TIEs may be subtle and unforeseen, causing different combinations of physical and locational attributes to interact in different ways for different people. Whether attribute prices are observed as homogeneous or not becomes irrelevant to the question of substitutability, in the same way that the price of wings on automobiles vs the price of wings on aeroplanes is an absurd comparison and irrelevant to the cross-price elasticity of demand between transport modes.

Note also that dwellings can be in the same attribute-cluster group but not be in the same substitutability-based submarket; and they can be in the same substitutability-based submarket but not be in the same attribute-cluster group. Statistical clustering of dwellings by physical attributes (and grouping of attribute variables into product bundles) has become popular in the submarkets literature (for example, Maclennan and Tu, 1996; Leishman, 2009), but may not reflect how consumers group them, which may be nuanced and difficult to anticipate, not least because of the interplay between MMMMEs and TIEs.

We would like to group properties according to a variable that captures market behaviour (i.e. substitutability), rather than an a priori categorisation of the dwelling stock (for example, by property type or social mix). In transactions data, the only behavioural variable typically recorded is the selling price, but this is the dependent variable in hedonic analysis and therefore precluded from the clustering process.

Note that grouping properties by type of resident is also problematic: while two consumers may have different preference maps, this does not mean they will disagree on whether two properties are close substitutes—they may consider the dwellings to be substitutable but for different reasons. And even if we were able to anticipate individual rankings of substitutability, the outcome at the level of the market is made fundamentally unpredictable by the Condorcet paradox—transitivities that hold at the level of the individual do not always hold in the aggregate.⁷

Ideally, one would like to cluster by market-level substitutability, but this requires a way of measuring the degree of substitution, which has hitherto proved elusive.

2.2 Criterion 2: SEMs should be Robust to the Continuity of Substitutability Space

Rationale: Substitutability-based submarkets can happily exist along a frictionless continuum and so continuous differentiability of the substitution function linking i and j in Cartesian space will neither require nor preclude i and j from belonging to separate substitutability-based submarkets.

Goodman and Thibodeau (1998) argue that attribute price differences persist because of unobserved amenity effects. However, this suggests that heterogeneous attribute prices (and hence submarkets) in hedonic regression only exist because of omitted amenity variables. In principle, therefore, if one were to construct a model of an efficient market that captures all amenity affects (both social and economic), there would be no difference across dwellings in the marginal prices of the structural and locational attributes and their interactions; and (by their definition) no such thing as submarkets.

But why disentangle locational and structural attributes for the purposes of submarket definition? Using Rosen’s (1974) terminology, we are considering a class of commodities—i.e. homes—that are described by A attributes, $z = (z_{(1)}, z_{(2)}, \dots ., z_{(A)})$ . The conventional definition of submarkets is weak because the entire notion of submarkets can be subsumed by simply allowing z to include a mixture of structural and locational attributes, along with interactions between the two. Shifts in attribute prices across space would then only occur in empirical models due to omitted variables—unobserved attributes or uncaptured interaction effects. Under the HAPV definition, therefore, submarkets exist only as a misspecification error.

An alternative justification of persistent attribute price differences is to assume that they are caused by market inefficiencies and frictions (such as imperfect competition among households and the inelastic demand and supply of housing service; see Schnare and Struyck, 1976). While this is feasible, it leads again to a weak definition because it means that submarkets only exist if markets are inefficient or inflexible. This is problematic if one is seeking to derive substitutability-based submarkets because variation in substitutability of dwellings could persist even in a world of perfectly efficient markets. Heterogeneous dwellings, heterogeneous locations, heterogeneous preferences, MMMMEs and TIEs can all exist in a frictionless world and would cause substitutability to vary across dwellings, even if the supply of each dwelling type was perfectly elastic and all market participants were perfectly informed.

In summary, variations in substitutability within the urban system can co-exist with the presence or absence of market inefficiencies (as in Schnare and Struyck, 1976) and/or discrete breaks in the land rent surface (as in Goodman and Thibodeau, 1998). HAPV type-2 hedonic methods for identifying submarkets, however, rely heavily on such inefficiencies and discontinuities and so are not well suited to the task of delineating substitutability-based submarkets.

2.3 Criterion 3: SEMs should be Robust to Unobserved Attribute Variation

Rationale: If dwelling characteristics are spatially clustered, but not fully described in our hedonic data, then observed differences in attribute prices can occur even when actual attribute prices are homogeneous and observed spatially correlated errors in hedonic price regressions may simply reflect unobserved attribute heterogeneity.

Dwellings are heterogeneous, but so are their attributes. For attribute price differentials to evince HAPV submarkets, we must know all variations in attribute quality and quantity. That half a tank of petrol costs less than a full tank is not evidence of the existence of submarkets in the supply of petrol. Likewise, apparent differences in price per room between tenements and modern flats may reflect unmeasured differences in room size (for example, tenement rooms may have higher ceilings), rather than HAPV submarket boundaries. Unfortunately, full information on the quality and quantity of every attribute of every dwelling is rarely, if ever, available. The measurement errors that result will not be random but correlated with building types, which in turn are likely to be clustered across space. Coefficient shifts in hedonic regressions may therefore be more likely to be co-terminous with the spatial pattern of measurement errors than market segmentation. This means that the boundaries extrapolated from spatially autocorrelated errors (as in Tu et al., 2007) may have little or no correspondence with substitutability-based submarkets.

Note that attribute measurement errors are unlikely to affect a substitutability metric based on the price of the overall housing bundle; nor would they affect definitions of submarkets based on such a metric. If two dwellings genuinely belong to the same substitutability-based submarket, one would expect the price of the overall housing bundle to respond in a similar way to demand and supply shocks, irrespective of attribute prices. Focusing on the dynamics of the sale price of the entire housing bundle (which is generally measured with precision) rather than individual attribute prices (which are not) as the basis for submarket analysis, is likely to be a more fruitful way for submarket research to develop (see section 3).

2.4 Criterion 4: SEMs Should Neither Impose nor Assume Convexity, Compactness, or Contiguity

Rationale: If substitutability is granular (non-contiguous), non-convex or non-compact in Cartesian space, Euclidean distance will not adequately describe the spatiality of submarkets.

Clapp and Wang (2006) explicitly impose a convexity restriction on the shape of submarkets in Cartesian space. That is, they draw submarket boundaries in such a way as to ensure the shape does not have any holes, intrusions or protrusions, ensuring that any straight line drawn between two points in a submarket will be entirely contained within it. Most other authors ignore the shape of submarkets altogether, or inadvertently impose a degree of convexity by the type of submarket delineation method employed.

This means that, while the submarkets literature has explored the degree to which submarkets are structural or spatial (see review by Watkins, 2001), they have almost entirely overlooked the role and meaning of submarket shape, which could profoundly affect the reliability of Euclidean distance as an approximation of substitutability. For example, if substitutability-based submarkets are elongated, fragmented or contain holes (as in the concentric circles of the access-space model), dwellings can be far apart but still close substitutes.

This has profound implications for the reliability of using Euclidean distance to approximate submarket effects, which in turn affects how we might compute W matrices in spatial econometric models. If we use distance to define the spatial weights matrix when submarkets are highly non-convex, then the model residuals will reflect a mixture of attribute and amenity mis-measurement (which is likely to be clustered across space—see earlier) and errors that arise from the failure to account for non-convex patterns of substitutability (which are also likely to be distributed non-randomly and non-linearly in Cartesian space).⁸ We therefore seek a methodology that will do justice to the complexity of submarket spatiality—revealing the shape of substitutability-based submarkets, rather than imposing, restricting or ignoring it.

Being able to reveal the true geography of submarkets would also allow us to explore what determines their shape, such as the impact (intended and unintended) of planning decisions, or the sorting effect of the housing market (Schelling, 1971; Meen and Meen, 2003). Preference for racial and social homogeneity may minimise submarket perimeter length because of aversion to living at the boundary (Rose-Ackerman, 1975), leading to compact, convex shapes. However, there may be other factors (the cumulative history of residential planning decisions, local amenities, radial and orbital transport links, heterogeneous preference for mix, etc.) that mitigate the minimum border length hypothesis. Indeed, the concentric rings of access-space theory would produce highly non-convex sets in Cartesian space. Maclennan, however, argues that

In the early phase of urban development, the most affluent and influential social and economic group were not sufficiently numerous to occupy a complete residential ring of the city. Instead, they tended to gather within a well-defined area or sector on one side of the city centre (Maclennan, 1982, p. 23).

As the city develops, one might therefore expect the city to comprise a patchwork of residential enclaves, each with its own core and periphery. Submarkets of the type described by Maclennan may be equally non-convex in Cartesian space, but made up of many sets of concentric circles centred on multiple cores, rather than a single sequence centred on the CBD.

If we could measure the true configuration of the urban housing market, we would presumably be able to verify which of these theories dominates in particular urban contexts, leading to a taxonomy of submarket structures for world cities.

3. Deriving a Substitutability Approach to Submarkets

The traditional approach to measuring the substitutability of two goods is to estimate their cross-price elasticity of demand (CPED). Unfortunately, estimating demand functions for dwellings is problematic. “Observed marginal hedonic prices … reveal little about underlying supply and demand functions” (Rosen, 1974, p. 50; see also Epple, 1987; and Bartik, 1987). Rothenberg et al. (1991) made a concerted effort to measure substitutability without estimating CPED, but unfortunately their approach also relied heavily on the meaning and stability of hedonic coefficient estimates.

The alternative proposed here attempts to approximate CPED using the cross-price elasticity of price (CPEP). CPEP does not rely on type-2 hedonics but exploits, instead, the dynamics of the market to reveal substitutability.

Proposition: If demand and supply curves slope downwards and upwards respectively, the cross-price elasticity of price will have a strictly positive one-to-one relationship with the cross-price elasticity of demand.

Intuitively, the CPEP approach to substitutability can be understood as follows. Dwellings i and j are substitutes if a rise in the price of j leads to an increase in the demand for good i; hence, CPED > 0. Conversely, if i and j are complements, then CPED < 0. Now consider the following corollary. If j is a close substitute, a rise in the price of i causes a large increase in the demand for dwelling j and, if the supply of homes is less than perfectly elastic, the short-run effect of the increase in demand for j will be an increase in the price of i. That is, $↑ p_{j} \Rightarrow ↑ Q_{Di} \Rightarrow ↑ p_{i}$ (ceteris paribus).

The argument can be expressed more formally by considering the following equilibrium condition in the market for dwelling type i

Q_{Si} (p_{i}, Y) - Q_{Di} (p_{i}, p_{j}, Z) = 0

(1)

where, Y and Z are vectors of exogenous factors affecting demand $Q_{D}$ and supply $Q_{S}$ respectively, and $p_{i}$ is price of the inseparable housing bundle i.

By implicit differentiation of equation (1), the cross-price elasticity of price is derived as

η_{ij} = (\frac{d p_{i}}{d p_{j}}) (\frac{p_{j}}{p_{i}}) = (\frac{\partial Q_{Di} / \partial p_{j}}{(\partial Q_{Si} / \partial p_{i}) - (\partial Q_{Di} / \partial p_{i})}) (\frac{p_{j}}{p_{i}})

(2)

Provided all prices are positive ( $p_{i}, p_{j} > 0$ ), the demand curve for i is downward sloping ( $\partial Q_{Di} / \partial p_{i} < 0$ ), the supply is upward sloping ( $\partial Q_{Si} / \partial p_{i} > 0$ ) and i and j are substitutes rather than complements ( $\partial Q_{Di} / \partial p_{j} > 0$ ), it is clear that CPEP will be positive.

Now compare equation (2) with the formula for CPED (derived again by implicit differentiation of equation (1))

ε_{{Q_{D}}_{i}, p_{j}} = (\frac{d Q_{Di}}{d p_{j}}) (\frac{p_{j}}{Q_{Di}}) = (\frac{\partial Q_{Di}}{\partial p_{j}}) (\frac{p_{j}}{Q_{Di}})

(3)

Again, provided prices and quantity are positive ( $p_{i}, Q_{Di} > 0$ ), demand slopes downwards and i and j are substitutes rather than complements ( $\partial Q_{Di} / \partial p_{j} > 0$ ), it is clear that CPED will also be positive. Rearranging equation (3) in terms of the numerator partial derivative, we get $\partial Q_{Di} / \partial p_{j} = (Q_{Di} / p_{j}) η_{QDi, pj} .$ Substituting this expression into equation (2), we obtain CPEP as a function of CPED

η_{ij} = θ \cdot ε_{Q_{D_{i}}, p_{j}}

where,

θ = \frac{Q_{Di} / p_{i}}{(\partial Q_{Si} / \partial p_{i}) - (\partial Q_{Di} / \partial p_{i})}

The numerator will always be positive, as will the denominator, so long as the demand and supply curves for dwelling i slope downward and upward respectively. It follows that the CPEP will be monotonically increasing in the CPED

\frac{d η_{ij}}{d ε_{Q_{D_{i}}, p_{j}}} > 0

Since the CPED is a measure of substitutability, it also follows that CPEP can be interpreted as a proxy. Crucially, however, CPEP does not require us explicitly to decompose the demand function or rely on type-2 hedonics. Instead, we can approximate $η_{ij}$ using the slope coefficient from a regression of $π_{ti}$ on $π_{tj}$ , where $π_{ti}$ is the proportionate change over time in the price of dwelling i; and $π_{tj}$ is the proportionate change over time in the price of dwelling j

CPE P_{ij} = η_{ij} = \frac{d p_{i} / p_{i}}{d p_{j} / p_{j}} \approx \frac{\partial π_{ti}}{\partial π_{tj}}

If $η_{ij} > 0$ , then i and j are substitutes. $CPE P_{ij}$ , represented by $η_{ij}$ , increases with the level of substitutability to the point where $η_{ij} = 1$ , which indicates that i and j are perfect substitutes and proportionate changes in the price of i are always matched by proportionate changes in the price of j. If $CPE P_{ij} < 0$ then i and j are complements. There is no obvious reason why $CPE P_{ij} > 1$ should occur other than as a result of market friction. For example, there may be contemporaneous overshoots of $p_{i}$ in response to a change in $p_{j}$ , reflecting idiosyncrasies in the transactions process (arising from extreme bids or local ‘bidding conventions’—see Levin and Pryce, 2007; Pryce, 2011), which can be treated as white noise for current purposes. In the long run, and in the absence of market frictions, however, it is implausible that CPEP would be greater than unity, so $\max [E (η_{i j})] = 1$ .

4. Using CPEP to Investigate Sub-market Existence and Spatiality

4.1 Existence

CPEP leads to a natural test for the existence of submarkets. If CPEP = 1 for all pairs of dwellings, then all dwellings are perfect substitutes and there is no market segmentation,

if $S_{1} = M$ then

η^{*} = max [E (CPE P_{ij})] = 1

$\forall i, j,$ where $i, j \in M$

We can represent the non-existence of submarkets by plotting substitutability against $D_{ij}$ , the distance between i and j. In $η_{ij}, D_{ij}$ space by a horizontal scattering of points all exactly equal to (or randomly scattered around) $η^{*}$ , the value representing perfect substitutability (see the line labelled $η^{*}$ in panel (a) of Figure 1).

Figure 1.

Non-spatial and spatial submarkets.

4.2 Spatiality: The Global Effect of Distance

It would be useful for intercity comparison to have a summary measure of the spatiality of the entire urban submarket system. Using our price-dynamic approach to measuring substitutability, a global indicator of Euclidean substitutability (GIES) for an urban area is given by gradient $ϕ$ of the relationship between $η_{ij}$ and Euclidean distance $D_{ij}$ between pairs of dwellings (i, j),

ϕ = \partial η_{ij} / \partial f (D_{ij})

(4)

For a simple generic measure of the effect of distance on substitutability, one could assume CPEP to be approximately linear in logged distance: $η_{ij} = α + ϕ \ln D_{ij}$ . If proximity is not an important aspect of substitutability, then one would expect $η_{ij}$ to be unrelated to distance, resulting in a spherical scatter of $η_{ij}$ (measured by $ϕ = 0$ , as in Figure 1, panel (a)). On the other hand, if proximity is an important determinant of substitutability (due to access to the same amenities and disamenities, for example), then one would expect $η_{ij}$ to decline with distance, most probably at a decreasing rate (illustrated in Figure 1, panel (b)).

Why might we expect there to be a scatter, rather than a line, of points in $η_{ij}$ , $D_{ij}$ space? If strong substitutability occurs between distant dwellings because of elongated and non-convex shapes of spatial submarkets, then dwellings at the extreme ends of that submarket may be highly substitutable, but far apart. Also, there may exist scattered clusters of substitutable bundles due to similar dwelling, neighbourhood and amenity combinations occurring at different points in the city (Rothenberg et al., 1991, p. 64). Thirdly, there may exist non-causal (i.e. coincidental) contemporaneous movements in distant pairs of inflation time-series. This leads to type-II errors: dwellings being allocated to the same submarket when in fact they belong to a different submarket. We can reduce type-II errors by adding conditions that screen-out spurious CPEP estimates (for example, by ruling out dwellings that are very distant), but this is likely to increase type-I errors. That might not matter in some applications (we may want to establish a shortlist of properties that are definitely close substitutes without worrying too much about erroneously excluding others). So, the optimal point along the trade-off between type-I and type-II errors will be determined by the particular application.

4.3 Lattice of Substitution and the Shape of Submarkets

Having decided on a measure that allows us to gauge the substitutability between a given dwelling $i = a$ and all other dwellings $j \neq a$ in the urban area, we can conceive of this set of bilateral links for the dwelling $i = a$ as a lattice of substitution for a, denoted by $L_{a}$ , where $L_{a} = {η_{i j} : i = a}$ . We can think of the lattice as a surface in Cartesian space, plotted for a particular dwelling $i = a$ , where the hills of this surface represent dwellings that are considered by the market to be close substitutes to a and the valleys represent dwellings that are not considered close substitutes to a. Note that a separate surface could be plotted for each and every dwelling in the housing stock. The family of lattices $L = {L_{i} : i = 1, 2, \dots, V}$ therefore fully describes the substitutability set in a given urban economy.

Submarkets can then be thought of as schema for summarising and simplifying the family of substitution lattices for an urban area. The simplest derivation of a set submarkets is one that categorises a single lattice of substitution into groups with similar levels of substitution with respect to a particular dwelling i = 1. We label this a first-order categorisation (FOC) and it is essentially a matter of identifying contour lines of substitutability with respect to a particular dwelling. Second-order categorisation (SOC) entails clustering according to two substitution lattices, $L_{1}$ and $L_{2}$

S_{1}, S_{2}, \dots S_{K} \subseteq M = {i : cluster (η_{1 j,} η_{2 j}) where, η_{1 j} \in L_{1} and η_{2 j} \in L_{2}}

We might choose randomly the dwelling that constitutes the basis for $L_{2}$ , or we might be more judicious and deliberately select a dwelling that is not a close substitute to dwelling i = 1. Third-order categorisation would involve clustering according to three substitution lattices $L_{1}$ , $L_{2}$ and $L_{3}$ , and so on. Using more than one substitutability lattice offers a means of triangulating the results. Ideally, one would like to perform a Vth order categorisation but that may be computationally challenging.

Note that these clustering processes do not impose spatiality because dwellings are clustered in substitutability space not Cartesian space. This is important because, while the outcome of this process may well lead to systematic patterns in Cartesian space, any apparent spatiality of clustering outcome will not have been caused by the method and therefore should reflect the true spatiality of housing substitution.

5. Empirical Illustration

To illustrate, consider now the application of the CPEP method to 33 680 GSPC residential property transactions in Glasgow, Scotland, all spatially coded⁹ and with attribute information, for the period 1999–2007. Quarterly house price inflation time-series were computed for 10 000 individual dwellings using the following procedure.

First, estimate a third-order Taylor series approximation of the house price surface in Cartesian space for each year. Extending the parlance of Fik et al. (2003),¹⁰ time–location value signatures (TLVS) were estimated using flexible functional forms that include interactions between attributes, x and y co-ordinates for the dwelling, area dummies¹¹ (based on a priori information on where likely shifts in the price surface may lie) and quarter dummies for each year’s surface estimation. Insignificant variables and dummies were then eliminated using a stepwise procedure. Note that each TLVS was estimated independently for each year, allowing coefficients complete freedom to vary over years. Coefficients were also allowed to vary over space through interactions with x and y co-ordinates and area dummies. Dwelling attributes are included to control for the mix of properties selling in a particular time-period. Quarterly time dummies are also included (and allowed to interact with dwelling type and location). The adjusted R² results for all nine TLVS regressions were as follows: 0.73 (1999), 0.73 (2000), 0.76 (2001), 0.71 (2002), 0.63 (2003), 0.58 (2004), 0.61 (2005), 0.64 (2006) and 0.71 (2007). Having estimated a TLVS for each year with quarterly slope and intercept dummies, an estimated price, P_i, can then be computed for each i in every time-period (quarter): $P_{i} = TVL {S_{i}}_{(t)}$ .

Secondly, estimate inflation surfaces for each intervening time-period and extract time-series for each of the constant-quality dwellings. By calculating the vertical distance for each i between each successive TLVS, we obtain the absolute price change ( $Δ P_{i} = TVL {S_{i,}}_{(t = 2)} - TVL {S_{i}}_{(t = 1)}$ ). From this we can obtain the proportionate price change ( $π_{i (t = 2)} = Δ P_{i} / TVL {S_{i}}_{(t = 1)}$ ) at every point in Cartesian space which can be plotted as a house price inflation surface for each period. Note that, once the inflation surfaces have been estimated, it is possible to read off inflation values for any point in the geographical space covered by the model. In the event, 10 057 hypothetical dwellings were selected at points where house sales had actually occurred during our data period (this ensured that the geographical points chosen reflected real-life residential locations).

5.1 Application (a): Testing for the Spatiality of Housing Submarkets

Calculating $ϕ = \partial η_{ij} / \partial ϕ (D_{ij})$ , the global indicator of Euclidean substitutability (GIES), is not a trivial exercise. If there are 10 057 dwellings for which we have computed inflation time-series, then there are 10 057 x 10 057 potential correlations between inflation time-series to be estimated and 10 057 x 10 057 distances to be calculated. Including correlations/distances from i to itself, and those correlations/distances from i to j when $D_{ij}$ has already been calculated, results in over 100 million pairs of dwelling units, for which we need to compute $\ln η_{ij}$ and $D_{ij}$ . To make computation more manageable, we estimated $η_{ij}$ = the cross-price elasticity of price between dwellings i and j, for a random selection of 100 000 pairs, derived from the slope coefficient from regression of $π_{ti}$ on $π_{tj}$ , where $π_{ti}$ is the annual constant quality price inflation time-series for dwelling i. The scatter plots of $η_{ij}$ (CPEP) against $D_{ij}$ (distance) are plotted against each other in Figure 2. The graph indicates that not all dwelling units are perfect substitutes—the values of CPEP do not lie along the horizontal line of unity. However, as distance increases, it clearly has a diminishing effect in determining substitutability. Beyond 8 km, the slope of the line of best fit does indeed become horizontal.

Figure 2.

Scatter plot of cross-price elasticities and distance between dwellings with fractional polynomial line of best fit.

Given the downward-sloping relationship between substitutability over short distances, there is prima facie evidence in Figure 2 that submarkets have a spatial component. The value of $ϕ$ for the system as a whole is negative: $ϕ = \partial η_{ij} / \partial \ln D_{ij}$ = −0.017 (Robust confidence interval (CI) = −0.0179825, −0.0164733; R² = 0.02; n = 100 000). The use of logged distance in computing $ϕ$ was imposed as a simplification. When we run a linear spline regression of $η_{ij}$ on $D_{ij}$ , we find that the slope declines in absolute terms. Up to 1 km, the slope = −0.45 (robust CI = −0.4879, −0.418; adjusted R² = 0.0785). That is, for every 1 km increase in distance between dwellings, the cross-price elasticity, $η_{ij}$ , falls by 0.45 units. From 1 km to <2 km, $\partial η_{ij} / \partial \ln D_{ij}$ = −0.35 (robust CI = −0.3691, −0.3323; adjusted R² = 0.0785). From 2 km to <4km, $\partial η_{ij} / \partial \ln D_{ij}$ = −0.01 (robust CI = −0.0128, −0.0037; adjusted R² = 0.0785). Beyond 4 km, the distance effect on substitutability becomes negligible, $\partial η_{ij} / \partial \ln D_{ij} \approx 0$ (robust CI = −0.0050, −0.0030; adjusted R² = 0.0785).

Nevertheless, it is clear from the very low R² values that the substitutability between dwellings must have a large non-spatial component—at least in terms of the capacity of simple Euclidean distances to capture spatiality (even in the spline regression, 92 per cent of the variation in CPEP is due to factors other than Euclidean distance). This provides support for the non-spatial conception of submarkets (an important theme in Rothenberg et al., 1991) and an imperative to explore further the shape of submarkets—the existence of convexity, granularity and non-compactness renders distance an incomplete measure of submarket spatiality.

5.2 Application (b): Improving Automatic Valuation Models (AVMs)

One of the potentially useful benefits of developing substitutability-based submarkets is that they could potentially improve AVMs. To test whether CPEP submarkets have the potential to add anything to the goodness of fit of AVMs, we compare the adjusted R² results from three simple hedonic AVMs¹² estimated on 2007 GSPC data. In the first AVM, no submarket dummies were included and the adjusted R² was around 0.51. In the second AVM, realtor jurisdiction dummies were included to demarcate submarkets (as in Palm, 1978; Michaels and Smith, 1990, and others) and the adjusted R² rose to 0.58. In the third AVM, submarket dummies based on CPEP third-order clustering were included instead of the realtor jurisdiction dummies and the adjusted R² rose to 0.63, with all CPEP dummies having significance and <0.01 and all but one having significance and <0.001. These simple diagnostics are far from conclusive proof, but they do tentatively suggest that substitutability-based submarkets could be a useful development in the evolution of AVMs.

5.3 Application (c): Estimating the Shape of Substitutability-based Submarkets

Criterion 4 suggests that the shape of submarkets could itself be informative, revealing the degree of aversion to living at the boundary between residential enclaves. Shape also has potentially important implications for the design of spatial econometric house price models/AVMs because non-convex spatially complex submarkets are likely to undermine the ability of traditional weights matrices to capture spatial effects in house price regressions.

When we plot in Cartesian space the contours of the substitution lattice for randomly selected dwelling $d_{120}$ (Figure 3), we see clear evidence of non-convex spatiality, demonstrated by the irregular shapes of the contours (they contain intrusions, petrusions and holes). In order to allocate dwellings to discrete submarkets based on more than one substitution lattice, we need to apply cluster methods that permit multiple cluster criteria. Two further dwellings were selected ( $d_{9206}$ and $d_{3247}$ ) as the basis for two further substitution lattices, chosen on the basis of being a distant substitute of $d_{120}$ ; and being physically distant in terms of location from $d_{120}$ . If we examine the scatter plot of the Cartesian co-ordinates of dwellings,¹³ colour coded by CPEP submarket¹⁴ we again find that submarkets are not a simple function of Euclidean distance.

Figure 3.

Substitutability lattice for dwelling $d_{120}$ plotted as a contour map.

In summary, there is evidence of non-convexity, non-compactness and non-contiguity both in the one-lattice contour plot and in the three-lattice cluster analysis. Moreover, there is no evidence of the concentric rings predicted by the standard urban economic model—if anything, the clustering of dwellings is more radial than orbital.

6. Conclusion

This paper has sought to present a case for developing an empirical measure of dwelling substitutability. The case rests partly on the potential benefits of obtaining a substitutability metric (for example, it could offer new ways of exploring old questions and potentially highlight new ones); and partly on the problems associated with using either homogeneous attribute prices, or a priori attribute-clustering, to derive substitutability-based submarkets. To derive such submarkets, one ideally needs a method that is robust to: (1) complex interaction effects between structural and locational attributes; (2) the continuity/discontinuity of substitutability space; (3) unobserved attribute variation; and (4) complexities in submarket shape (particularly non-contiguity, non-convexity and non-compactness).

The cross-price elasticity of price (CPEP) was proposed as a measure of substitutability. CPEP, as defined here, does not rely on type-2 hedonics. It does not decompose dwellings into constituent attributes, nor does it cluster by attributes. Dwellings are instead treated as inseparable bundles—so it is likely to be more robust both to transformative interaction effects and to attribute measurement errors. Moreover, CPEP does not rely on discontinuity in house price surfaces (because substitutability—and hence submarkets—can exist along a continuum); and it does not appear to impose or assume convexity, compactness or contiguity in submarket shape (dwellings are clustered in substitutability space rather than Cartesian space, hence revealing, rather than imposing, the geographical pattern of market areas).

CPEP was defined in terms of market equilibrium (see equation (1)). Yet would the measure have any meaning in a world characterised by disequilibria, where prices are in a perpetual state of flux, never quite reaching equilibrium before having to adjust to some new perturbation to demand or supply; or responding to local/submarket perceptions and bidding conventions (Pryce, 2011; Levin and Pryce, 2007, 2011)? In such a world, $CPE P_{ij}$ could still be interpreted as a numerical indicator of the extent to which prices for any pair of dwellings are likely to move together, and this would remain a useful measure of substitutability. Even in a world characterised by disequilibria, one would nevertheless expect changes in the prices of i and j to be highly correlated if those dwellings are perceived by the market to be close substitutes. Substitutability is conceptually robust, therefore, not only to the existence or absence of discontinuities in the house price surface, but also to the existence or absence of equilibrium (see the earlier discussion under criterion 2).

This is all well and good in theory, but is CPEP worth the effort in practice? It is too early to provide a conclusive answer, but there are some encouraging signs from the illustrative analysis presented here that the method could be of genuine practical use. Preliminary evidence suggests, for example, that CPEP submarkets could improve the performance of AVMs, but more thorough testing using a comprehensive set of model types is needed to establish whether CPEP submarkets consistently outperform other ways of defining submarkets in terms of boosting valuation accuracy. It also raises the possibility of an iterative approach to CPEP estimation. For example, one could start with an a priori set of submarkets (for example, based on realtor jurisdictions) in order to estimate an initial set of CPEP submarkets. One would then use these CPEP submarkets to improve the type-1 hedonic price change estimation. This in turn would lead to a more robust set of CPEP estimates, and so on. Note that none of this relies on type-2 hedonics.

Tentative work by Brown et al. (2012) suggests that CPEP could also be used to analyse the determination of substitutability—exploring, for example, the extent to which substitutability is determined by distance between dwellings, differences in dwelling attributes and differences in location attributes. (For example, a regression of $CPE P_{ij}$ on the distance between i and j, and a variety of structural and locational differences between i and j, suggests that substitutability in Glasgow is significantly reduced by differences in social mix—such as differences in the proportion of households in the surrounding neighbourhood that are Catholic.) Future research could compare results for different cities, gauging how the relative roles of different drivers vary across urban economies.

In the current paper, graphical inspection of results suggested a degree of non-convexity, non-compactness and granularity in submarket shape. However, more work is needed to develop a set of quantitative measures of convexity, compactness and granularity, which would pave the way for establishing a taxonomy of cities, with the long-term goal of comprehensively classifying structures of residential segmentation in cities across the world. Further work also needs to be done, however, to establish how stable are the boundaries of substitutability-based submarkets and how drivers of substitutability might change in relative importance (both over time and across cities).

Finally, while this paper has attempted to utilise type-1 hedonics to overcome the problems of type-2 submarket delineation, it should be noted that type-1 hedonics are not the only way of estimating the dwelling-level or block-level house price time-series needed to compute the CPEP metric. Yet another avenue for future research, therefore, would be to develop alternative ways of computing CPEP and related measures (using repeat sales methods or trajectory clustering, for example). Moreover, continued advances in regression estimation methods mean that there will be many opportunities to improve the type-1 hedonics used here to estimate the CPEP lattices. All hedonic estimation methods are subject to error, particularly with respect to how one captures unobserved heterogeneities between dwellings. Nevertheless, these errors could, in principle, be reduced by employing highly flexible methods, such as geographically weighted regression, to give more locally nuanced estimates of home values in each period.

Footnotes

Acknowledgements

The author is grateful to Duncan Maclennan and the seminal writings of Bill Grigsby for inspiring his initial interest in housing submarkets. The author would also like to express thanks to George Galster, Nema Dean, Chris Leishman, Eric Levin and numerous anonymous referees who have helped to refine and clarify the ideas presented here. The author is grateful to GSPC and Experian for supplying the house price and neighbourhood data. An earlier version of this paper (entitled: ‘Housing market segmentation: the theory and measurement of submarkets’) was awarded the 2009 Hypoteční Banka Prize for the Best European Network of Housing Research Conference Paper.

Funding

Thanks are due to the Department of Communities and Local Government for funding the research on submarkets that led to this work. The research was also part-funded by the EPSRC Socio-economic Model and Community Impact Simulators project (EPSRC grant reference: EP/F037716/1).

Notes

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