Measuring "Sprawl:" Alternative Measures of Urban Form in U. S. Metropolitan Areas
The Determinants of Urban Form and "Sprawl"
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- First Stage Results: Alternative Measures of Urban Form
- Second Stage Results: Determinants of Urban Form
| 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 towards or away from a CBD or other employment node equals the change in rent from such a movement. For such a representative consumer: w  here 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. Data 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 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. 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 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 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 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 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. 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. Directory: documents documents -> Concept stage documents -> Concept stage documents -> The great global switch-off documents -> Nonprofit grant application documents -> The Truth about the Rwandan Genocide documents -> Annual InStructional unit plan documents -> Chaos equipment included Download 0.75 Mb. Share with your friends:
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