Measuring "Sprawl:"Alternative Measures of Urban Form in U.S. Metropolitan Areas

By

Stephen Malpezzi and Wen-Kai GuoRevised, January 15, 2001

The Center for Urban Land Economics ResearchThe University of Wisconsin

975 University AvenueMadison, WI 53706-1323smalpezzi@bus.wisc.edu

wgou@bus.wisc.eduhttp://wiscinfo.doit.wisc.edu/realestate

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Stephen Malpezzi is Associate Professor and Wangard Faculty Scholar in the Department of Real Estate and Urban Land Economics, and an associate member of the Department of Urban and Regional Planning, of the University of Wisconsin-Madison. Wen-Kai (Kevin) Guo is a Ph.D. candidate in Food Science at the University of Wisconsin-Madison.

Comments on this and closely related work have been provided by Alain Bertaud, Michael Carliner, Mark Eppli, Richard Green, James Shilling, Kerry Vandell, Anthony Yezer and participants at the Homer Hoyt Institute/Weimer School's January 1999 and January 2000 sessions as well as the June 1999 Midyear meeting of the American Real Estate and Urban Economics Association. Comments and criticisms of this paper are welcome.

The research we describe was supported by the University of Wisconsin's Graduate School, the Wangard Faculty Scholarship, and the UW Center for Urban Land Economics Research. Opinions in this paper are those of the authors, and do not reflect the views of any of the above individuals, or of any institution.

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Introduction

Economists and other social scientists have tried to study urban form more or less rigorously for the better part of two centuries. The earliest commonly cited work is that by German scholars such as von Thunen (1826) and somewhat later work by Lösch (1944). In this century pioneering English-speaking scholars include Clark (1951), Hoyt (1939, 1966), and Burgess (1925). The 60s and 70s saw a further flowering with work such as Alonso (1964), Mills (1972), Muth (1969), and Wheaton (1977) among many others. Song (1996) provides a particularly nice discussion of some alternative measures. Excellent reviews and extensions of this large literature can be found in Anas, Arnott and Small (1998), Fujita (1989), and Turnbull (1995).

The economics of location has been a fertile academic field for some forty years, and is enjoying resurgence due partly to the high profile of recent work on the economics of location by generalist economists such as Krugman (1991). But this academic resurgence is nothing compared to the eruption of interest in urban form by a wider range of political and social commenters. One watershed was certainly journalist Joel Garreau’s excellent popularization Edge City (1991), which broadened the audience for discussion of urban form. But the real explosion of interest in urban form has been due to the growing concern and exploding reference to urban sprawl.

What environmental activists and now many others call sprawl is certainly not new to urban economists. The phenomenon of rapid growth on the periphery of the city is something that has been a core feature of most of the literature mentioned above, and the much larger literature that lies behind it. To give just one example, Edwin Mills’ classic 1972 book studies the decentralization of urban population in a large sample of U.S. cities going back to the latter part of the 19th century. Sam Bass Warner’s Streetcar Suburbs, coming out of a different scholarly tradition, is another well-known early examination of decentralization.

Perhaps the first dividing line between urban economists and many other urban observers is the use of the word sprawl, with its pejorative connotations. While a number of authors have used the term sprawl in the academic literature (see references below), most urban economists have preferred less value-laden terms, such as urban decentralization (Mills 1999) or accessibility (Song 1996). But economists have lost the lexicographic battle. To give just one example, we did a simple Internet search on the term "urban decentralization" using Infoseek.com. The search engine returned 27 hits. We repeated an identical search using the term "urban sprawl." The engine returned 5,946 hits.

With all the extraordinary attention paid to sprawl, it is quite

interesting that only recently have some of those involved in the policy discussion attempted to define it. Consider the following quotation:

...sprawl in all its forms is seldom satisfactorily defined. Urban sprawl is often discussed without an associated definition at all. ... Some writers make no attempt at all at definition, while others engage in little more than emotional rhetoric, as in "the great urban explosion (which) has scattered pieces of debris over the countryside for miles around the crumbling centre ... a destruction of the qualities of the city"

Despite its contemporary relevance, the quotation is from a paper by Robert Harvey and W.A.V. Clark, written 35 years ago.1

Our view is that discussions about sprawl, whether academic or policy oriented, are greatly hampered by loose definition and inadequate measurement. Our intention in this paper is to contribute toward improving the measurement of sprawl. Using a consistent data source, we compute some familiar measures such as average population density of the metropolitan area, and the familiar negative exponential density gradient. We also compute some less often used measures, such as those based on gravity models, and some which are fairly new, including measures based on order statistics, measures of fit of various models, and measures that incorporate the notion of autocorrelation.

We also investigate the use of data reduction techniques to collapse some of these disparate measures into a univariate index. We also evaluate how well each one of our measures incorporates the information contained in the others, i.e., to get some sense of "which measure is best." We also estimate simple models of the determinants of each of the measure, using right-hand side variables suggested by the urban economics literature as well as some of the sociological explanations for decentralization.2

While we investigate a large number of measures, we certainly do not exhaust all possible measures. For example, our data tell us little directly about how much space within a tract is devoted to one land use or another,

1 Harvey and Clark (1965). Harvey and Clark took the internal quotation from Pearson (1957).2 The simple models are estimated with an eye towards facilitating comparisons and validating measures of urban form. They are best viewed as exploratory. See our companion paper, Malpezzi (2000) for a more detailed model, albeit with the estimation focused on a single measure.

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or how much space is privately owned vs. publicly owned. These are important components of some people’s views of sprawl.

Our explanatory models are exploratory and simple. We mention here and discuss again below the fact that more complete models would deal with the potential endogeneity of some of our right hand side variables, as well as utilize a more complete vector of determinants. For example, for tractability this paper abstracts from the effect of housing prices as an equilibrating mechanism, in contradistinction to Malpezzi and Kung (1998), which argues that housing price gradients and location may be jointly determined. We also limit ourselves to data from the 1990 Census, that is, our measures are developed from a single cross section. Measures that focus on tract level changes are only one class of many possible extensions to our measurement effort here.

Previous Research

The literature on urban form is huge. Our intention in this paper is to focus on measurement, so our review here is extremely selective. Readers interested in a broader review of the literature on sprawl are referred to Malpezzi (2000), Ewing (1997), and Gordon and Richardson (1997) among others. Those who wish a more detailed review of the academic approach to urban form should consult Anas, Arnott and Small (1998), as well as McDonald (1989), Wheaton (1979), and Ingram (1979).

Surely the simplest measure of sprawl, and one used any number of times by urban economists and others, is the average density of the metropolitan area. Brueckner and Fansler (1983) and Peiser (1989) are among well-known papers by urban economists that use this measure.

Far too many papers to cite focus on the negative exponential density gradient and its many derivatives and extensions. According to Greene and Barnbrock (1978), the first to use the negative exponential function was the German scholar Bleicher (1892).3 In many respects, the function was popularized by the work of Colin Clark in geography, and later by Edwin Mills, Richard Muth, and others in economics. Many authors have noted that the monocentric negative exponential is not always a terribly good fit for many metropolitan communities; see Richardson (1988), Kau and Lee (1976), and Kain and Apgar (1979) for examples. Here we note the following key points. (1) Without doubt the univariate negative exponential fits some cities reasonably well, and others quite badly. (2) Despite this, it is still often used partly because of the advantages of having a univariate 3 Although McDonald, in his excellent (1989) review, suggests that Stewart (1947) apparently first fit the negative exponential form described here. Most observers would agree that it was Clark (1951) who popularized the form among modern urban scholars.

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index of decentralization or sprawl (see, for example, Jordan, Ross and Usowski 1998). (3) It is apparently neglected in the literature that the measure of fit of such a simple univariate model is in its own way a measure of sprawl, as we will discuss below.

Most individual papers that measure decentralization, or 'sprawl,' focus on one measure or at most a few measures. There are exceptions, of course. Several papers have examined differences among functions theoretically and using simulation methods. Ingram (1971) examined average distances, negative exponential functions, a linear reciprocal function, among others, and Guy (1983) examined a number of accessibility measures in a broadly similar fashion. Broadly speaking, these papers clarify differences among candidate functions, and tell us which might best capture stylized facts of observed patterns, but do not offer empirical tests per se. Some papers have tested a limited number of specific hypotheses, e.g. whether a single parameter exponential function performs as well as some flexible form (e.g. Kau and Lee).

In many respects Song (1996) is a paper that parallels this one, in that it uses actual data to test a wide range of alternative forms. Song estimates a variety of gravity, distance and exponential models using tract level data from Reno, Nevada. Best-fit criteria suggest that gravity measures and, especially, a negative exponential measure, perform much better than linear distance measures. As Song is careful to note, results from a single metro area are suggestive, but it remains to examine other forms, and especially to test forms across other metropolitan areas. Most analysts would admit the possibility of differing performance for a given estimator in, say, Los Angeles compared to Boston or Portland, for example. In our paper, we follow Song in examining a range of possible measures of urban form, but rather than focus on a single location, we examine a wide range of U.S. metropolitan areas.

More recently, Galster et al. (2000) have independently undertaken an exercise in some respects similar to ours. Galster and colleagues estimate a series of measures of urban form for a dozen large MSAs (in contrast to our measures, for some 300). Later, we will briefly compare our results to Galster et al.'s, and to Song's.

The Measurement of Urban Form

In this section we discuss some measures of sprawl and related measures of urban form.

First we introduce some notation. We use a capital P to indicate the population of a metropolitan area, and small p to indicate the population of a

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tract. Capital A denotes the area of the metropolitan area, and small a denotes the area of the tract. Capital D refers to the density of the metropolitan area, i.e. D=P/A, and small d, d=p/a, is the density of a tract. Distance from the city center is denoted by the letter u. Letters i and j index tracts within a metropolitan area, and k indexes metropolitan areas themselves. Generally we construct a measure for each metropolitan area, and for notational simplicity we usually drop the subscript k.

Average Density

The most common measure is average density in the metropolitan area.:

(1) Average MSA density; for each MSA, D=P/A. In our database of MSA results, described below, this variable is denoted MSADENS.

While widely used, the limitations of the measure are obvious. Consider two different single-county MSAs of equal area and equal population. Suppose the first contains all of its population in a city covering, say, a fourth of the area of the county, the rest of which is rural and lightly settled. Suppose the other MSA has a uniform population distribution. Our measure, average density, is the same. But most observers would consider the second MSA as exhibiting more "sprawl" than the first.

Of course there is no reason to limit ourselves to average densities. Other moments, and nonparametric measures can also be considered, as below.

Alternative Density Moments

In this paper we construct several new indicators of population density gradients, based on the densities of the Census tracts in each MSA. The starting point for each MSA is to compute these tract densities, and then to sort tracts by descending density. We then construct several indicators of "sprawl", one for each MSA:

(2) Maximum tract density, DENMAX = max(di)

(3) Minimum tract density, DENMIN = min(di)

(4) DENMED: the density of the "median tract weighted by population," that is, median(di) when tracts are sorted by density, the tract containing the median person in the MSA. For example, suppose the population of the MSA is 100 people, in 7 tracts:

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Tract TractN Density Population 1 10 302 9 303 8 104 7 105 6 106 5 57 4 5

The median person is "contained" in tract 2, so DENMED=9.

(5), (6): DENQ1 and DENQ3, the corresponding 1st and 3rd quartiles of tract density, constructed as above.

(7), (8): DENP10 and DENP90, the corresponding 10th and 90th percentiles of tract density, constructed as above.

Measures of Dispersion in Tract Densities

(9): DENCV, the coefficient of variation of tract densities.

(10). DENGINI, a Gini coefficient measuring variation in tract density, constructed as follows. Sort tracts within MSA in descending density. Compute cumulative population in percent, CPP and cumulative area in percent, CAP, as you move down tracts. Compute difference between CPP and CAP for each tract. Then sum over all tracts within each MSA.

(11). Theil's information measure, an alternative to the Gini coefficient:

where ai is the area of the ith tract, A is the MSA area, pi is the population of the ith tract, and P is the MSA population.

Population Density Gradients

The measure of city form that has been most often studied by urban economists is the population density gradient from a negative exponential

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function, often associated with the pioneering work of Alonso, Muth and Mills, but as noted earlier first popularized among urban scholars by the geographer Colin Clark. More specifically, the population density of a city is hypothesized to follow:

where d is tract population density at distance u from the center of a city; d0 is the density at the center; e is the base of natural logarithms; gamma is "the gradient," or the rate at which density falls from the center. The final error term, ε, is included when the formulation is stochastic.

Among the other attractive properties of this measure, density is characterized by two parameters, with a particular emphasis on γ, which simplifies second stage analysis. The function is easily estimable with OLS regression by taking logs:

ln d(u) = ln d0 - γu + ε

which can then be readily estimated with, say, density data from Census tracts, once distance of each tract from the central business district (CBD) is measured. Thus, we construct measures

(12) our gradient, gamma (denoted KMB1_1 in the database), and

(13) density at the center, d0 (denoted INTB_1).

The exponential density function is sufficiently important to warrant brief discussion. This particular form has the virtue of being derivable from a simple model of a city, albeit one with several restrictive assumptions, e.g. a monocentric city, constant returns Cobb-Douglas production functions for housing, consumers with identical tastes and incomes, and unit price elasticity of demand for housing.

As is well known, the standard urban model of Alonso, Muth and Mills predicts that population density gradients will fall in absolute value as incomes rise, the city grows, and transport costs fall. Extensions to the model permit gradients to change with location-specific amenities as well (Follain and Malpezzi 1981).

The negative exponential function often fits the data rather well, for such a simple function in a world of complex cities. Sometimes it does not fit well, as we will confirm. Many authors have experimented with more flexible forms, such as power terms in distance on the right hand side (of which more below).

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The world is divided up into two kinds of people: those who find the simple form informative and useful, despite its shortcomings (e.g. Muth 1985), and those who believe these shortcomings too serious to set aside (e.g. Richardson 1988).4 In fact, given the predicted flattening of population density gradients as cities grow and economies develop, it can be argued that the monocentric model on which it rests contains the seeds of its own destruction; and that a gradual deterioration of the fit of the model is itself consistent with the underlying model.

Extensions of the Simple Exponential Gradient

As already noted, the simplest, and most widely used model for estimation is:

ln d = a + b ln u

where d is the tract's density, and u is distance from the center. We are relying on this simple model for our second stage work, but we have also computed three additional models, with right hand side variables:

(14) A quadratic model, i.e. with terms u and u2.(15) A cubic model, u, u2, and u3.(16) A fourth power model, u, u2, u3 and u4.

In our database of results, these coefficients are represented by variables KMBa_b, where a represents the order of the model, and b represents which term. For example, KMB3_1 is the coefficient of linear distance in the cubic model. The intercepts from these models are denoted INTB_a, where a is again the order of the model.

Measuring Discontiguity

A simple and natural measure of discontiguity is:

(17) The R2 statistic from the univariate density gradient regressions, denoted RSQ_1.

Consider the two panels of Figure 1. Panel A shows a very highly stylized city with a given density gradient, as does Panel B. In Panel A, we

4 The world is also divided up into people who divide the world into two kinds of people, and people who don't, but that's another paper.

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have drawn a pattern consistent with very contiguous density patterns as one moves from the center of the city outwards. The second panel shows a city with the same density gradient, but a much more discontiguous pattern. The R-squared from the density gradient regressions is a natural measure of this discontinuity. However it should be noted that a low R-squared is a sufficient but not necessary condition for such discontinuity.

To see this, consider a city where the density gradient is very contiguous from tract to tract, but assume that the gradient varies by direction as well as distance from the CBD. As example would be a metro area in which the gradient declines very rapidly with distance in one direction, but very slowly in another. Suppose this difference is very systematic, and density changes slowly as one rotates from left to right around the central point of the city. Such a city would not be truly discontiguous by most people’s thinking, but would have a low R-squared for a simple two-parameter density regression of the usual kind, where it is maintained that density varies with distance but not direction. Of course it would be possible to estimate distance density gradients that vary by direction as well as distance (see Follain and Gross 1983), but undertaking such an exercise would require resources beyond our present ones.

(18) The difference in R2 statistic from the univariate density gradient regressions, and the R2 statistic from the fourth power density gradient regressions, DRSQ1_4. Another variation on the preceding theme; if the univariate model is a good one, then adding successive power terms adds little to the explanatory power of the regression. If, on the other hand, the improvement in fit is large, the simple model is less satisfactory.

Measures of Spatial Autocorrelation

The r-squared measure from the negative exponential regression will capture some of this, but we can construct examples where, for example, spatial auto correlation is high but the R-squared is low. Consider a case where spatial auto correlation is high and positive (i.e., very little sprawl on this element) but where density varies tremendously by radial location (direction north or south, for example). Since our simple negative exponential model imposes that density is a function of distance but not direction, we would find a low R-squared even though the spatial autocorrelation in such a city could be high. A more generalized measure of such autocorrelation is therefore desirable.

(19) Moran’s I (denoted MORAN_I). One commonly used measure is Moran’s I, which is effectively a correlation coefficient, constructed using a weighting matrix where weights depend upon location. More specifically, the formula as usually written is:

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where n is the number of tracts, and C is an n by n matrix that incorporates the information of which tracts are contiguous. Specifically, each row represents a tract, and contiguous tracts have ones entered in their corresponding columns. Other elements of C are zero.

One practical difficulty in computing Moran’s I for a large number of places (tracts and MSAs) is to develop an algorithm to compute the matrix C. In this version of our paper we use an algorithm from Isaaks and Srivastava (1990) which uses a quadratic approximation. That is, when distances are small, the elements of C are approximately 1, but as they get larger given the elements are quadratic in distance, they rapidly approach 0.

Other measures of spatial autocorrelation are possible, of course. Moran's I is isotropic, i.e., it depends on distance from the tract in question, but not direction. To the extent that direction as well as distance does matter, an anisotropic measure that accounts for direction would be a natural extension. Dubin, Pace and Thibodeau (1998) and Gillen, Thibodeau and Wachter (1999) discuss the use of such anisotropic indexes for single metropolitan areas. Computational difficulties have so far kept us from producing such an index for our full set of metropolitan areas, but given resources, this would be an appropriate extension for future work.

Compactness

In Bertaud and Malpezzi (1999), Alain Bertaud developed a compactness index, rho, which is the ratio between the average distance per person to the CBD, and the average distance to the center of gravity of a cylindrical city whose circular base would be equal to the built-up area, and whose height will be the average population density:

where rho is the index, d is the distance of the ith tract from the CBD, weighted by the tract's share of the city's population, w; and C is the similar, hypothetical calculation for a cylindrical city of equivalent

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population and built up area. A city of area X for which the average distance per person to the CBD is equal to the average distance to the central axis of a cylinder which base is equal to X would have a compactness index of 1.

In this paper we use the simpler weighted average of distances from one set of points in a metropolitan area to another to compute two distance measures. Distances are corrected for the earth's curvature.

(20) DCENTAVG is the weighted average distance to the center, where tract populations are the weights. (21) DCENTMED is the weighted median distance to the center, where tract populations are the weights.

Of course, in the modern city, many if not most employment and shopping destinations are no longer in the city. Gravity based measures are conceptually similar measures that are less CBD-focused.

Gravity Based Measures

Gravity measures were popularized by Lowry (1964) among others. Song (1996) presents several such measures. The general form of a gravity model takes its form from Newton’s Law of Gravitation. However, several variants can be obtained by various choices of the power terms and so on involved. Song discusses and estimates a wide range of these for a single metropolitan area. Given the difficulty of estimating and comparing gravity measures across metropolitan areas if one permitted the exponents to be chosen by the data, we prefer to pick two common and simple assumptions, one where terms are linear and the other exponential.

(22) The linear gravity function, GRAVLIN, is is the weighted average distance from the center of each tract to every other tract, in turn in fact the same as above:

(23) The exponential gravity function, GRAVEXP, can be written:

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Combining Measures

Given the multidimensionality of sprawl, sprawl is a natural candidate for data reduction techniques. We used the well-known method of principal components.

Principal components analysis derives a vector p = Za', where Z is a matrix whose columns consist of n observations on K variables and a' is a K-element vector of eignenvalues. We choose the a's so that the variance of p is maximized subject to the normalization condition that a1 + a2 + ... + aK = 1. Given inevitable collinearity, we choose 12 of our 22 spatial measures as elements of Z:

INTB_1, the intercept from the simple univariate exponential model;

KMB1_1, the coefficient of distance in the simple exponential model;

RSQ_1, the R-squared from the simple exponential model;

DRSQ1_4, the improvement in R-squared from the univariate exponential model to the fourth power model;

DENCV, the coefficient of variation of tract densities;

DENMED, the density of the median tract, when tracts are ordered by density;

DENP90, the density of the 90th percentile tract, ordered by density;

DENGINI, the Gini coefficient of tract densities;

MSADENS, the average density of the entire MSA;

DCENTMED, the weighted median distance to the center of the MSA;

GRAVLIN, the linear gravity measure;

GRAVEXP, the exponential gravity measure.We extract three principal component measures from this data, and label them PC1, PC2 and PC3.

Other Possible Measures

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We have seed that in addition to the traditional gradient measure, many measures of urban form have been put forward and studied. The simplest, of course, is the average density of the city or metropolitan area. We have proposed a fairly large set of other measures to include here, but we have not exhausted the possibilities. Others include measures such as functions based on densities other than the negative exponential, such as the normal density (Ingram 1971; Pirie 1979; Allen et al. 1993).

Many additional measures could be developed using techniques developed by urban geographers and others for the analysis of data exhibiting spatial autocorrelation. Moran's I, which we have computed, is one such measure but there are others. Anselin and Florax (1995), Pace and Gillen (forthcoming) and Pace, Barry and Sirmans (1998) describe these techniques in greater detail.

A few papers have examined land use conditions on the fringe as opposed to the metropolitan area as a whole. (See Brown, Phillips and Roberts (1981)).

Also, we note that the American Housing Survey has data on land area for single-family houses. To our knowledge, no one has used this data in the analysis of sprawl. For example, median lot size of single-family homes built in the last five years would be one possible indicator that could be constructed. We did undertake some preliminary work with this data. Problems arose from the fact that such data are only reported for single-family units. But the biggest problem with this potential measure is that preliminary analysis of AHS data tells us that their are many missing observations, and further (and more worryingly) missing plot area is correlated with other housing characteristics and income, suggesting potential biases in measures created from this dataset. Further work along these lines remains for future research.

The Determinants of Urban Form and "Sprawl"

In order to evaluate different measures, we intend to estimate simple least squares models of the determinants of sprawl. What does theory, and previous research, suggest those determinants might be?

The well-known "standard urban model" of Alonso (1964), Muth (1969) and Mills (1972) postulates a representative consumer who maximizes utility, a function of housing (H) and a unit priced numeraire nonhousing good, subject to a budget constraint that explicitly includes commuting costs as well as the prices of housing (P) and nonhousing (1). It is easy to show that equilibrium requires that change in commuting costs from a movement

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towards or away from a CBD or other employment node equals the change in rent from such a movement. For such a representative consumer:

where u is distance from the CBD and t is the cost of transport. This equilibrium condition can be rearranged to show the shape of the housing price function:

Now consider two consumers, one rich and one poor. Assume H is a normal good. If (for the moment), t is the same for both consumers but H is bigger for the rich (at every u), the rich bid rent function will be flatter. The rich will live in the suburbs and the poor in the center. Even if t also increases with income (as is more realistic), as long as increases in H are "large" relative to increases in t, this result holds.5 Also, as incomes rise generally, the envelope of all such bid rents will flatten. Also, clearly, as transport costs fall, bid rents will flatten.

The standard urban model has a competitor, which is sometimes called the "Blight Flight" model (Follain and Malpezzi 1981). As presented in the U.S. literature, the Blight Flight Model has a negative tone. People have left the cities not because they preferred suburban living a la the standard model, but because the cities themselves have become less desirable places to live. As U.S. cities became more and more the habitat of low-income households and black households, the argument goes, housing and neighborhood quality declined and white middle-to-upper income households flew to the suburbs.

While "Blight Flight" explanations focus on negative amenities such as crime and fiscal stress, the models are easily generalizable to positive amenities such as high quality schools. Blight Flight can be generalized and formalized by adding a vector of localized amenities (and disamenties) to the standard urban model above. See, for example, Li and Brown (1980), Diamond and Tolley (1982), and many subsequent applications.

Of course other causes can be considered, for example the degree of monocentricity, opportunity cost of land in non-urban uses, and the industrial structure of a city (manufacturing implies different land use patterns than office work, for example). Mills (1999) has a nice discussion of these.

Data5 But see Wheaton (1977) and Glaeser et al. (2000) for dissenting views.

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Our basic data source for our measures of urban form is 1990 Census tract -level data on population and tract area. We construct tract density as a simple ratio. Our measures are "gross" at the tract level, although at the metropolitan area level the effect of some large features are controlled for.

Unless otherwise noted, all second-stage variables are metropolitan are level variables taken from the 1990 Census. Geographic variables (adjacent to another metro area, or adjacent to a large body of water in the suburbs) are described in Malpezzi (1996).

First Stage Results: Alternative Measures of Urban Form

Basic Measures

Table 1 presents the measures we have constructed to date, for 35 large metropolitan areas (MSAs/PMSAs) using 1990 Census data. Metro areas are sorted by descending population. A database with the results for all 330 metro areas can be downloaded at:

http://wiscinfo.doit.wisc.edu/realestate/urban_indicators.htm.

The measures in Table 1 are, in order presented:

Average population density;

Measures based on order statistics of the individual tract densities, when sorted by population; namely, the median, the first and third quartiles, and the 10th percentile;

Population density gradients, and their fit (R-squared);

Compactness as measured by the average and median distances between tracts and the CBD, said averages and medians weighted by tract population;

Gini coefficients and Theil indexes of the dissimilarity of tract densities

Gravity measures

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Spatial Autocorrelation, as measured by Moran's I

Principal components summaries of key indicators (based on a method described below)

Figures 2 through 5 present scatterplots of representative indicators by metropolitan area population.6 Some patterns are quite obvious. For example, larger metropolitan areas tend to be denser. Larger metropolitan areas tend to have flatter density gradients, or more precisely large metropolitan areas tend to have flat gradients whereas small metropolitan areas have a wide range of gradients. While this would yield a positive linear relationship that suggests that a nonlinear, possibly heteroskedastic-corrected, model would be appropriate.

We also examined the pairwise correlation among these measures.7 Not surprisingly, correlations are high between conceptually related measures, e.g., between the average density in a metropolitan area and the intercept from the negative exponential model, as well as the average density of the median Census track. Correlations are similarly high between gravity measures using different exponents.

We found the following broad patterns. First of all of our measures based on percentiles – whether the median, the first quartile, or the 10th percentile or tracks- are highly correlated. The pairwise correlations among these variables are generally about 0.9. Correlations are less high with the simple person’s per square mile density variable used in so many other studies. The Gini and Theil measures are correlated with each other but not terribly correlated with most other measures.

Generally denser metro areas, by whatever measure using order statistics, are positively correlated with positive population density gradients; but the correlation is modest. Denser metro area by most measures have lower R-squared (worse fits) for the density gradient regressions. Interestingly, it makes more of a difference if one take the logarithm of a given density measure than it does to choose among percentile measure. The correlations among linear percentile based measures are typically about 0.9 as mentioned above; the same high correlation is obtained among the several logarithms of these measures. But across linear and logarithmic measures the correlation is much less, typically from .5 to .6. That is not terribly surprising since there is such a very wide range of densities among metro areas that the log transformation gives a significantly different shape to the data. 6 A larger set of scatterplots, and the key to the three-letter codes for MSAs in Figures 2 through 6, are available at http://wiscinfo.doit.wisc.edu/realestate/urban_indicators.htm7 The matrix of correlations is also available at our website.

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Summarizing Key Indexes Using Principal Components

In this section, we discuss the construction of reduced form measure using the method of principal components. We ran principal components against the following individual measures: average population density, density gradient, density of median track, linear gravity measure, exponential gravity measure. Our initial exploration of the eigenvectors associated with these suggested that many signs were inconsistent, in the following sense. If our factor score measures sprawl, we would expect it to be increasing in the population density gradient (a negative number) and decreasing in average population density. The measure would be a consistent one as long as the signs were reversed. Our expectation of the signs of each of these variables is listed as follows:

Upon reflection, we realize that to a large extent what these initial eigenvectors were measuring was the size of the metropolitan area. Large metropolitan areas tend to be denser than smaller ones, and the population density gradients flatter for both for reasons that are well understood. One way to deal with this on an ad hoc manner is to first estimate a regression of each element against population and then construct a scale using principal components on the residuals from this regression i.e., with the effect of city size purged. We present both sets of principal components in Table 2, and the scores for each metropolitan area at the end of Table 1, above. When the effect of city size is purged, signs are more in line with expectations, although not perfectly so.

Regression Analysis of Each Indicator vs. Other Indicators

Table 3 presents a set of least squares regressions that we can use to examine how correlated each index is with the others. We place no particular economic meaning upon these regressions; they are simply summary ways of describing multiple correlations among the data.

The heading of each column represents the sprawl indicator considered in that column. The rows are the other indicators used as "independent variables." Notice that we do not use the principal components measures from Table 1, because they are by construction linear combinations of other indicators.

As argued above, everything else equal, an indicator that has a high r-squared when regressed against the other indicators is a better summary of the information contained in the rest of the set. However, such a high r-squared could come from a very, very close relationship with only one or

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two functionally related measures. We suggest as an alternative criterion the number of significant coefficients in the "model." Everything else equal, an indicator which is statistically related to a larger number of other indicators (each capturing some element of sprawl as discussed above) is preferred to an indicator whose correlation depends on one or a few indicators.

The best way to analyze this quickly is to refer to the r-squared statistics at the bottom of the table, as well as the count of the number of significant coefficients in each model. We do not count significant intercepts. We use a probability of 0.10 as our cutoff.

It's clear that from the point of view of fit the best performers are the density of the tract containing the median household and its related indicator, the tract containing the 90th percentile, as well as the exponential gravity measure. Close behind in terms of fit are the linear gravity measure and the simplest measure, average density (person per square kilometer). In terms of the number of significant coefficients, the best performer is the intercept from the single parameter negative exponential model: ten of eleven coefficients are significant. Close behind are the density of the track containing the median household, and the linear gravity measure. The standard urban model's distance coefficient, the density of the track containing the 90th percentile, and the exponential gravity measure also performed quite well by this criterion.

In terms of combination of good fit and large number of significant coefficients, we like the density of the tract containing the median household when tracks are sorted by density. The gravity measures also performed well. Perhaps the real result from this table is by this criterion, most indicators do very well indeed. Those related to the intercept and coefficient of the standard urban model, various moments of tract and metropolitan area density, and the gravity measures all performed creditably. In fact, from these results, one must feel very sanguine about the simplest and oft-criticized measure of average metropolitan area density. We did not have this prior going in, but it appears that average MSA density creditably serves as a proxy for many other elements. The simple negative exponential distance coefficient, another oft-criticized but heavily used statistic, also performs very creditably according to these criteria.

Comparisons to Previous Studies

Our first stage results can be briefly compared to results from Song (1996) and from Galster et al.(2000), each in turn. First we reiterate that Song's study examines a single metro area, from Reno, Nevada; his criteria

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for choice of a "good" measure are based on fit within a single metro area, while ours focus mainly on cross-MSA criteria. Naturally Song does not consider the simplest measures, like average density and our order-statistics measures, because within a single MSA there is nothing to "fit" with such a measure.

As we mentioned above, Song found that in Reno gravity measures and, especially, a negative exponential measure, perform much better than linear distance measures. Song is careful to note that such results may or may not generalize to other metropolitan areas. Our Table 5, above, suggests that the fit of similar models will in fact be quite different in one metro area or another. R-squared of a univariate exponential density model varied from near nil to about 90 percent; and we argued above that this fit was in and of itself valuable information about urban form. While there is a noticeable tendency for larger MSAs to have lower r-squared, there are small metro areas with low r-squared, and among large MSAs the fit ranges from circa 0.2 for New York and Los Angeles to circa 0.6 for Atlanta and Boston, with Chicago in between.

Another way to examine how the "best model" varies by MSA is to consider what, if anything, is gained from the simplest negative exponential model, to one where a flexible fourth power polynomial is fit to the log of density (the basis for DRSQ1_4 described above). Figure 6 plots the unadjusted R-squared for the fourth power model against that for the simplest model, for all MSAs in our study. Metro areas along the 45° line, such as Jersey City and West Palm Beach at the bottom, and Laredo near the top, see little improvement from fitting a more flexible form. On the other hand, Jamestown-Dunkirk NY, Grand Rapids MI, and Florence SC, among others, see substantial improvement from adding power terms. These results confirm that Song's conjecture that "best fit" within MSAs varies by MSA is quite correct.

Galster et al.'s more recent study has proceeded independently from this one but along somewhat similar lines, that is estimating a series of measures, in their initial study for a set of 13 large metropolitan areas. In their careful effort Galster et al. divide each metro area up into half-mile grids, rather than relying on Census tracts. They define a number of elements of "sprawl" or decentralization: density, continuity (how much "leapfrog" development), concentration (whether land use is used uniformly across the MSA or in a few locations), compactness (the size of the built-up area's footprint), centrality (whether most development is close to the CBD or far away), nuclearity (whether an MSA tends toward monocentricity or polycentricity), diversity (whether land uses are mixed within subareas of the MSA) and proximity (whether different land uses are close to each other within an MSA). Galster et al. calculate specific measures of each of these concepts using their grid system; computational details are provided in

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their paper. They then rank each MSA by each element, and compute an overall index of sprawl by adding the six component indexes.

For most of these concepts, we have one or more measures that are related, although given the differences in our data structure the details of construction are different. In brief, Galster et al.'s density is conceptually similar to our MSADENS;, their continuity is similar to our RSQ_1;, concentration is related to our DENMED, compactness is proxied by our DCENTAVG; and centrality is measured by our KMB1_1. We have no direct measure of nuclearity, but RSQ1_4 will presumably be high if nuclearity is low. We have no direct analogue of diversity or proximity because we have no information on other land uses.

Figure 7 shows that our measures are related. The vertical axis is our preferred single measure, the density of the tract containing the median person, when tracts are sorted by density. The horizontal axis is Galster et al.'s sprawl index, based on the sum of the ranks of each of their six components. New York is the "least sprawled" MSA according to both measures; Atlanta the "most sprawled;" in general the two measures show substantial correlation.

Second Stage Results: Determinants of Urban Form

This section describes a set of ordinary least squares regressions of each potential sprawl measure against a consistent set of right-hand side variables. This time we include the factor scores as left hand side variables. These results are contained in Table 4. (We did not include the factor scores in Table 3 because they are linear combinations of the other dependent variables by construction).

The first three rows of Table 4 represent independent variables that reflect the physical constraints that a metropolitan area faces. The first is a dummy variable for being adjacent to a metropolitan area. The next two are dummy variables for whether the suburbs or the central city are adjacent to a large body of water, such as an ocean or one of the Great Lakes. These geographic variables were constructed by reviewing maps of each metropolitan area.

Many of our measures stem from the monocentric model of the city, which is oft criticized. In fact, many metropolitan areas have more than one central city. This has led some authors such as Jordan, Ross and Usowski (1998) to limit their analysis to metropolitan areas that have a single central city. But for our purposes, this is undesirable because having multiple

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central cities may well be a key aspect of sprawl, so we wouldn't want to restrict the sample this way. As an ad hoc adjustment, we have included the number of central cities in each metropolitan area as an independent variable.

Follain and Malpezzi (1981) and Mills and Price (1985) among others, argue that central city externalities such as high poverty, crime, and bad schools will tend to engender blight flight. We picked a single representative blight flight measure, namely the central city murder rate, to represent these negative externalities.

In a number of preliminary regressions, Jersey City, New Jersey, and the New York metropolitan area consistently came up as outliers in many respects. We've already seen that these are extremely dense places, especially Jersey City. We therefore included dummy variables in these regressions, although an alternate set without them yielded qualitative similar results.

The first thing to notice from Table 3 is that there is a wide range in overall predictive performance. Adjusted r-squares range from less than 10 percent for the indicator representing "Improvement In Unadjusted R-Squares Between Linear And Four-Power Standard Urban Models," to almost 0.9 in some of the density moments and gravity measures.

The most consistent performer of all the independent variables is clearly the size of the metropolitan area. This is really not terribly surprising. Generally, the larger the metropolitan area, the denser, and the flatter the gradient (although, as noted above, the relationship is not necessarily a simple one; there are a number of smaller metropolitan areas with very flat gradients). Generally, fast growing metropolitan areas tend to be less dense and more dispersed, as do metropolitan areas with multiple central cities.

Perhaps the most surprising result is that for the log of median MSA income. The tendency is for higher income metropolitan areas to be denser and to have steeper gradients. Now when we examine unadjusted two-way plots of gradients and income, such a result would not be surprising, since larger metropolitan areas tend to have higher incomes, and we've already noted that higher densities at least would be expected for metropolitan areas of greater size. But we would expect the result to be opposite in sign once we've controlled for population in the regression, based on the precepts of the standard urban model. In particular, we would expect flatter gradients for higher income metropolitan areas. It remains to be seen whether this contrary result holds up in more carefully specified models in future work.

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Metropolitan areas with higher central city murder rates tend to be less dense, everything else equal, with flatter gradients as the blight flight view of the world suggests. The performance of the geographic variables is somewhat mixed, but in general, greater geographic constraints are related to higher density moments and somewhat greater dispersion.

Again returning to comparisons across dependent variables, examining Table 4 we find that in many respects the order statistics measures performed very well. The measures of spatial autocorrelation including r-squared and the gravity coefficients performed less well. Average population density, the simplest measure and one used by papers such as Brueckner and Fansler and Peiser, among others, performs surprisingly well. Based on Table 4's results we would say that the simple average density measure so commonly used performs fairly well, although we would prefer our order statistics measures given a choice.

Conclusions

In this paper we have attempted to move the discussion of "sprawl" ahead by considering a range of measures that can be constructed from a single year’s Census tract-level data. We have not exhausted the possible set of measures that could be constructed; in particular, we have not constructed any dynamic measures based on changes. But we have constructed and compared a fuller set than has been analyzed before in a cross-metropolitan context.

We generally find, perhaps not surprisingly, that many alternative measures exist and that widely speaking, most tell the same story. The indicators most commonly used by urban economists, average population density and the density gradient, perform reasonably well. We do propose an alternate measure based on order statistics that we think has some advantages.

We also discussed some extensions to our current work, including the use of spatial autocorrelation based measures and measures based on changes. Armed with these and other indexes, academic research into "urban sprawl" can place today’s policy debates on a firmer footing.

Armed with the indicators we've constructed and detailed in this paper, cross-MSA empirical research can proceed on a range of issues. These include a more detailed treatment of the costs and benefits of "sprawl," including a particular focus on the interplay between transportation and urban form. However, there are still significant gains to be had by further research on these measurement issues. In addition to applying more of the recent technology from urban geography, such as

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alternative models of spatial autocorrelation, urban decentralization is a dynamic phenomenon. As 2000 Census data becomes available there could be large gains to carrying out similar exercises for more recent data (and for other Census years), permitting the modeling of decentralization in changes.

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