Regional Science and Urban Economics 48 (2014) 110–119

Contents lists available at ScienceDirect

Regional Science and Urban Economics

j ourna l homepage: www.e lsev ie r .com/ locate / regec

The floor area ratio gradient: New York City, 1890–2009

Jason Barr a, Jeffrey P. Cohen b

a Department of Economics, Rutgers University, Newark, United Statesb Economics, Finance, and Insurance Department, Barney School of Business, University of Hartford, United States

E-mail addresses: jmbarr@rutgers.edu (J. Barr), professo1 Anas et al. (1998) summarize the findings on the pop

on land prices include McMillen (1996) for Chicago an19th century NewYork.McMillen and Singell (1992) prov

http://dx.doi.org/10.1016/j.regsciurbeco.2014.03.0040166-0462/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 9 October 2012Received in revised form 6 February 2014Accepted 26 March 2014Available online 13 April 2014

JEL classification:C14R14R33

Keywords:New York CityFloor area ratioLocally weighted regression

An important measure of the capital–land ratio in urban areas is the Floor Area Ratio (FAR), which gives abuilding's totalfloor area dividedby the plot size. Variations in the FAR across cities remain an understudiedmea-sure of urban spatial structure.We examine how the FAR varies across thefive boroughs of NewYork City. In par-ticular, we focus on the FAR gradient over the 20th century. First we find that the gradient became steeper in theearly part of the 20th century, but then flattened in the 1930s, and has remained relatively constant since themid-1940s. Next we identify the slope of the gradient across space, using the Empire State Building as our corelocation. We find significant variation of the slope coefficients, using both ordinary least squares and geograph-ically weighted regressions. We then identify subcenters, and show that while accounting for them can bettercapture New York's spatial structure, by and large, the city remains monocentric with respect to its FAR. Lastly,we find a nonlinear relationship between plot sizes and the FAR across the city.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

There is a large literature in economics on measuring the nature ofurban spatial structure. The focus of these works tends to concentrateon density or price measures relative to the urban core.1 Collectively,this work has provided several key findings about the urban landscape.First, the price and density gradients drop off at an exponential ratemoving away from the center. Second, over the 20th century therehas been a steady and persistentflattening of the gradient, as central cit-ies have, to some degree, “hollowed out,”while suburban districts haveseen large population and employment increases. Third, especially sincethe mid-20th century, urban spatial structure has become poly-centric.With the rise of “edge cities” and smaller suburban agglomerations,population gradients—while still dropping rapidly from the center—tend to be “bumpy” across space.

The aim of this work is to explore an important, but understudied,form of urban density—the intensity of land use that comes from thebuildings themselves; that is, the structural density of the city(Brueckner, 1987). In particular we focus on the density of commercialstructures. The few studies on structural density have tended to focus on

rjeffrey@gmail.com (J.P. Cohen).ulation density gradient. Workd Atack and Margo (1998) foride evidence forwage gradients.

housing (McMillen, 2006), butmuch lesswork has explored the “shape”of commercial buildings across space and over time.

One key measure of structural density is the floor area ratio (FAR),which is the ratio of total usable floor space to the size of the plot. Forexample, a 10-story building constructed on the entire lot would havea FAR of 10, as would a 20-story building on half the lot. The FAR isalso a useful indicator of the capital–land ratio (McMillen, 2006;Clapp, 1980). As land values rise closer to the city center where trans-port costs are lower, the monocentric city model implies firms wouldsubstitute more capital (structure) for land (O'Sullivan, 2010).

As McMillen (2006) stresses, the monocentric model is static innature. It details how density should fall with distance from the center,at a given time, on the assumption that the parameter values are fixed.To address this issue we investigate the density gradient over both timeand space. However, it might be the case that density and building ageare related. If it is the case, for example, that older structures tend toexhibit a steeper FAR gradient with respect to the center, then notaccounting for the age of the structure might bias the effects of distancefrom the center. Thus if age and distance are positively correlated, omis-sion of the age variable will tend to bias the distance coefficients down-ward (or closer to zero). In this paper we control for possible vintageeffects by including in the regression the years of completion for eachbuilding.

In particular we focus on New York City from 1890 to 2009, andlook at the FAR gradient across the city. Using a data set over such along period allows us to investigate the shape of the gradient over

111J. Barr, J.P. Cohen / Regional Science and Urban Economics 48 (2014) 110–119

time. Second,we investigate the degree towhich the city ismonocentricor not. As we discuss in more detail below, we use the location of theEmpire State Building as the center of the city. New York City is com-prised of 469 miles2 of land, and contains a population of 8.2 millionpeople. Because it is so large, it can be considered ametropolitan regionworthy of investigation unto itself. Despite New York's importance, itsland use patterns remain understudied.

In this paper we ask: (1) What does the structural density gradientlook like relative to the center? (2) Has it changed appreciably overtime? (3) Is there evidence of poly-centricity within the city itself?(4) Does controlling for the building age affect the estimates of theFAR gradient?

Since we have data on the plot size, we also examine how it affectsstructural density across the city; investigating if lot size has a uniformeffect, and if it is positive or negative. On one hand a large lot size canpromote structural density, since it allows for a more efficient buildinglayout and helps avoid the “elevator problem” with skyscrapers,whichmust allocate a large fraction of internal space for elevator shafts.But this may apply only where land values are high. In areas were landvalues are lower, a large plot may encourage less density because of aconcave relationship between price per square foot and size (Colwelland Munneke, 1997). In other words, if the marginal cost of land isdecreasing, so will the marginal provision of structure. It remains anempirical question how plot size and structure density are related.

To answer these questions, we employ the methods of LocallyWeighted Regression (LWR). By and large, the density gradient litera-ture has assumed a constant gradient coefficient across space, anduses ordinary least squares (OLS) to estimate the size of this coefficient.This assumption, however, can lead to biased estimates or standarderrors since gradients can change over both time and space (McMillenand Redfearn, 2010). While OLS, for example, can include timedummy variables, they are not able to capture the subtle movementsof the gradient.

The LWR procedure estimates a separate coefficient for each obser-vation. In particular, we estimate geographically weighted regressions,which are based on the geographic distances of each building fromthe others.2 We also incorporate a time dimension into the weightmatrix that links all the buildings together. Since our data set includesbuildings completed over the 20th century, buildings completed ina certain year can only be influenced by completions in the same orprior years.

LWR models are a more general approach that nests OLS. If the OLSassumption about constant gradient coefficients is correct, then LWRand OLS will yield the same results. However, for New York City, wefind there is and has been a significant amount of variation in the FARgradient coefficients over both time and space, which demonstratesthat the OLS assumptions do not apply in this situation.

Based on our results, we find that the gradient for the city as a wholeflattened over the first half of the 20th century, then remained relativelysteady between the late-1940s and mid-1980s, and then flattened to anew “plateau” over the last quarter century. The evidence also suggestsa further flattening of the gradient since 2005.

On first approximation the monocentric model appears to be a goodrepresentation of structural density in the city, but LWRs reveal severaldifferent “centers of gravity,”which vary across boroughs. For example,the Empire State Building is a reasonable center for structures complet-ed within about eightmiles away. After that, however, the further awayonemoves from the center, the less likely its influence is on the densitygradient. In particular we find positive gradient coefficients at severalpoints outside the center of the city. This provides evidence of several

2 We note that geographically weighted regressions are one type of Locally WeightedRegressions, where theweights are a function of the Euclidean distance between observa-tions. For the remainder of the paperwe the use the term LWRwhen referring themethodused in this paper.

smaller “nodes” within the city or adjacent to it. In addition we findthat the exclusion of the year of completion variable does not apprecia-bly affect the distance coefficients, suggesting we need not be overlyconcerned with vintage effects.

As another test for spatial structure, we first run anOLS regression ofthe log of the FAR on several control variables, including the distancefrom the Empire State Building. Based on the OLS residuals we thenidentity five areas in the city with clusters of large positive residuals.We then include these “subcenters” in the regressions and find thatwhile distance to subcenter and airports are statistically significant,they do not increase the R2 by very much; this suggests that New Yorkremains largely monocentric. Finally we investigate the LWR coeffi-cients of plot size on the FAR relative to the center. We find that closerto the core, the effect is positive, and that after about four miles awayfrom the center the effect becomes negative, on average, and after tenmiles away virtually all the coefficients are negative. Further, since thefunctional form is non-linear, it suggests that using OLS to estimate itsaffects may not be appropriate.

The rest of the paper is as follows. The next section provides a briefliterature review. Then Section 3 discusses the data and the resultsfrom ordinary least squares. Section 4 discusses the results from thelocally weighted regressions. Section 5 discusses how we identifysubcenters and the results of OLS regressions with the subcenters.Section 6 presents the estimated plot size coefficients for the FAR acrossthe city. Finally, Section 7 provides some concluding remarks.

2. Literature review

There is a large literature on land rent and population gradients(Anas et al., 1998), but little known work that directly addresses theFAR gradient. Some works have contributed to the FAR gradient litera-ture indirectly, by examining the intra-urban location of tall buildings.One of these is Frankel (1997), who studies the determinants of loca-tions of tall buildings in the Tel-Aviv municipal region. He finds, forexample, a rising probability that 25+ floor buildings will constructedin the core district; and the likelihood of these very tall buildings fallsas one moves away from the center. Barr (2012) finds that skyscraperheight falls about 25 to 35 ft with each mile away from the core inManhattan. Clapp (1980) looks at the determinants of office space inthe Los Angeles metropolitan region, by estimating hedonic office rentequations. He finds that while subcenter access and distance toemployees are important factors, they are weak relative to the pull ofthe central business district, where the benefits of face-to-face contactsremain strong.

For New York City, Atack and Margo (1998) investigate land valuegradients during the 19th century, and find a general flattening of theland price gradient after the Civil War. Haughwout et al. (2008) investi-gate the land value gradient from 1999 to 2006 in the New York Citymetropolitan area; their aim is to create a land value index over theperiod. However, their data show a substantial amount of land valuevariation in Manhattan, which suggests a standard gradient model istoo simple.

McMillen (2006) is the only knownwork that directly addresses theFAR gradient. This research includes estimation of the FAR gradient forindividual homes in the city of Chicago, based on both OLS and Splinefunction approaches. McMillen (2006) notes that closer to the core,lower commuting costs should raise the value of land, which thenshould cause the FAR to be higher. This is a crucial reason for studyingthe FAR gradient — that is, to confirm this common underpinning ofurban economics. If one confirms that the FAR gradient is downwardsloping and monotonic in a city with only one core, this is consistentwith the monocentric city model (O'Sullivan, 2010).

One of our aims in this paper is to demonstrate that the FAR gradientfor Manhattan implies a variation of the monocentric city model withperhaps one core and at least one subcenter – the Empire State Building,and lower Manhattan – is applicable to the city. Furthermore, we

112 J. Barr, J.P. Cohen / Regional Science and Urban Economics 48 (2014) 110–119

demonstrate that several of the boroughs of New York City have sub-centers of their own, implying a somewhat more intricate model thanthe simple monocentric city model might imply for a smaller city.

AsMcMillen and Redfearn (2010) discuss, the use of OLS tomeasureeffects that vary across geographic space can lead to biased results. Onecan include both spatial and time related fixed effects in the FAR gradi-ent equation but these coefficients cannot capture the complex natureof a spatial structure. Thus it is necessary to employ more spatially-oriented estimation procedures. In particular, locally weighted regres-sions (LWR), by generating separate coefficient estimates for each ob-servation, allow for a much more nuanced measurement of an urbanarea's spatial patterns. Specifically, McMillen and Redfearn (2010) dem-onstrate that LWR leads to a much smoother distribution of residualsthan fixed effects estimation, as well as helps to mitigate against omit-ted variable bias.

McMillen (1996) uses LWR to analyze land values in Chicago. Hisresults show how the spatial distribution of land values evolved fromthe 1830s to the 1990s. In the earliest period of Chicago's history, landvalues were generally rising uniformly approaching the city center.However, LWR estimation shows that by the early twentieth century,land values took on a much more diverse picture, with multiple andsmaller peaks throughout the city. By 1990 the land value gradientswere quite varied throughout the city. A standard OLS approachwould have difficulty in capturing this complex pattern.

Meese and Wallace (1991) estimate hedonic housing price modelswith LWR to compare to standard OLS estimates in several municipali-ties in California. By using LWR they can see how index values are deter-mined, once freed from the functional form restrictions of OLS. They findthat the LWR-generated indexes offer both considerable flexibility andprecision. McMillen and Redfearn (2010) estimate hedonic price func-tions in Chicago using LWR. Their results show that the effect of distanceto an elevated subway line in Chicago can have either a positive or neg-ative effect on housing price in different neighborhoods.

While this paper focuses on comparing and contrasting results fromOLS regressions and LWRs, other works illustrate that there are severaldifferent approaches to measuring both time- and spatial-varyingeffects on prices or density measures. We mention a few exampleshere, and also note that we leave the use of these techniques, as itrelates to New York's FAR gradient, for future work.

McMillen and Thorsnes (2003) use semiparametric methods toconstruct a time-varying index of the effect of a Superfund designationof a copper smelter site in Tacoma,Washington. They find that proxim-ity to the smelter lowered housing prices before the Superfund programbegan, but, over time, as the program advanced and the sitewas clearedof its waste, that proximity became positive (i.e., being close to the sitebecame an amenity).

Cameron (2006) incorporates a more general specification in herOLS regressions for the effect of a Superfund site on housing prices inthe vicinity of Woburn, Massachusetts. In particular, the regressionsincluded measures of the geographical direction of distance effects.In this case, Cameron shows that the effect of the Superfund site onhousing prices depends on the coordinates relative the site, becausewind blew the site's odors predominately in one direction.

3 See Barr and Tassier (2013) for the reasonwhy midtown emerged as a separate busi-ness district.

3. Data

The data set comes from the 2012 “PLUTO” file provided by theNew York Department of City Planning (http://www.nyc.gov/html/dcp/html/bytes/dwn_pluto_mappluto.shtml, 2013). This file lists exten-sive information for every property in the City of New York, includinga unique identification number (the “BBL”—borough, block and lotnumber), building address, latitude and longitude coordinates, yearcompleted, number of floors, the floor area ratio, the lot size, the build-ing type (and sub-type), and other information related to zoning, andland use.

For our analysis, we included observations from 1890 to 2009whichonly had a single structure on the property. In addition, we removedsome properties that had some extreme outlier characteristics, such aslots less than 100 square feet, and floor area ratios greater than 100.This created a base data set with 662,161 total properties of all types.Of this we only investigate commercial properties, i.e., those thathoused a business of some kind.We also note that the data set only con-tains extant buildings, and not ones that were torn down. This mightlead to selection bias if older, torn-down structures, for example, consis-tently had greater FARs. We do not think this is a major issue, however,since commercial structures, by and large, tend to be larger buildingsand less likely to be torn down; secondly, we do control for the age ofthe structure in our regressions, which should mitigate any selectionbias, to some degree. Finally we note that some 65% of the buildings inour sample were completed before 1950.

The work on spatial structure aims to look at density changes rela-tive to the central core. For this paper, we chose the location of theEmpire State Building (ESB), at West 34th Street and 5th Avenue asthe “center.” We choose this location because the ESB is generally con-sidered the economic “center of gravity” of the city (Haughwout et al.,2008). It is in between lowerManhattan and the high-rise office districtbetween 42nd and 59th Streets. Manhattan, in particular, in terms of itsstructural density is poly-centric.3 Because we are interested in the cityas a whole (which extends some 20 miles outward from the ESB), weuse it as an approximate center.

We also note that the Empire State Building was completed in 1931,and our data set includes buildings from1890 to2009, so a large fractionof structures were completed before the ESB. However, the ESB repre-sents a kind of central base location, from which the gradient can becalculated. Moving the center a mile or two in one direction does notappreciably affect the conclusions. We also explore the issues of multi-ple centers in the paper as well.

3.1. Descriptive statistics

Table 1 lists the different types of commercial properties in our sam-ple. There are approximately 41,400 structures in our data set. Storescomprise the greatest proportion of commercial structures, with atotal of approximately 16,000 or 40%. Garages/gas stations, and offices,are the next most common groups of commercial structures, withapproximately 6000 of each type. The remaining types of propertiesinclude warehouses (approximately 5200), factories (approximately3600), and a relatively small number of lofts, hospitals, hotels, andtheaters.

Table 2 lists the distribution of structures across the five boroughs.Brooklyn (approximately 33%) and Queens (approximately 30%)have the greatest amount, with Staten Island (slightly under 7%) last.Manhattan (17%) and Bronx (13%) are in the middle.

Table 3 provides descriptive statistics for the FAR. As expected dueto its relatively high land values compared with the other boroughs,Manhattan has the greatest average FAR (and the greatest standarddeviation). Also, as predicted by its relatively low land values, StatenIsland has the lowest structural density in the city. Also, there are39,353 properties for which we have data on their FAR.

3.2. The FAR gradient

Fig. 1 shows the scatter plot of the log of FAR versus distance fromthe Empire State Building. On average the gradient is negative, aswould be expected; thought it does not appear to be linear, as thereappears to be a general flattening of the gradient further from the

Table 4OLS regression results. Dependent variable is ln(FAR) for commercial structures. Absolutevalue of robust t-statistics below coefficient estimates.

(1) (2) (3) (4)

Dist. to empire state −0.129 −0.507 −0.053 −0.215(109.10)⁎⁎ (64.98)⁎⁎ (37.60)⁎⁎ (22.65)⁎⁎

(Dist. to ESB)2 0.044 0.019(42.67)⁎⁎ (16.87)⁎⁎

(Dist. to ESB)3 −0.001 −0.001(34.13)⁎⁎ (15.41)⁎⁎

Ln(Lot Area) −0.168 −0.177(3.14)⁎⁎ (3.32)⁎⁎

Ln(LotArea)2 0.005 0.005(1.48)⁎ (1.63)

Year −0.46 −0.477(20.25)⁎⁎ (21.05)⁎⁎

Year2 0.0001 0.0001(20.22)⁎⁎ (21.02)⁎⁎

Brooklyn dummy 0.353 0.424(20.06)⁎⁎ (22.48)⁎⁎

Bronx dummy 0.305 0.366(16.16)⁎⁎ (18.00)⁎⁎

Manhattan dummy 1.27 1.163(52.03)⁎⁎ (46.23)⁎⁎

Queens dummy 0.266 0.309(15.14)⁎⁎ (16.84)⁎⁎

Constant 1.05 1.79 452.1 468.9(109.35)⁎⁎ (106.49)⁎⁎ (20.35)⁎⁎ (21.16)⁎⁎

# Observations 39,953 39,953 39,944 39,944R2 0.26 0.31 0.42 0.42Building type dummies Yes Yesp-Val for dummies 0.00 0.00

⁎⁎ Stat. sig. at 99%.⁎ Stat. sig. at 95%.

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16 18 20 22

ln(FA

R)

Distance from EmpireState Building (miles)

Fig. 1. Scatter plot of ln(FAR) versus distance from the Empire State Building for commercialstructures, with linear trend line.

Table 1Distribution of building types across NYC (excludes residential buildings, asylums andnursing homes, houses ofworship, condominiums, parks buildings, andutility properties).

Building type Count Percent

Stores 16,726 40.4Garages/gas stations 6738 16.3Offices 5959 14.4Warehouses 5240 12.7Factories/industrial 3644 8.8Lofts 1287 3.1Hospitals/health 987 2.4Hotels 662 1.6Theaters 154 0.4Total 41,397 100

Table 2Distribution of commercial structures in the five boroughs of NYC.

Borough Frequency Percent Cumulative

Brooklyn 13,942 33.68 33.68Bronx 5407 13.06 46.74Manhattan 6857 16.56 63.3Queens 12,447 30.07 93.37Staten Island 2744 6.63 100Total 41,397 100

113J. Barr, J.P. Cohen / Regional Science and Urban Economics 48 (2014) 110–119

core. Except for the “bump” for downtown Manhattan, there doesnot appear to be any obvious other sub-centers or deviations fromthe negative slope, but we will return to this issue in more detail inSection 5.

3.3. OLS results

In this sectionwe present evidence on the nature of the FAR gradientfrom OLS equations. Table 4 presents the results. We present four equa-tions. Eq. (1) is the distance gradient from a simple OLS regression. Itshows that on, average, the FAR decreases about 13% per mile. Eq. (2)is the gradient with quadratic and cubic terms. As the scatter plotshows, adding these terms provides a better fit, in terms of capturingthe “shape” of the gradient.

Eq. (3) provides an additional set of controls. We add in the log ofthe plot size and log of plot size squared, the year and the yearsquared, borough dummies, and building type dummies. For thiswe included dummies for the sub-categories of building. For exam-ple, “warehouses” are further divided by the city into “fireproof,”“semi-fireproof,” “frame, metal,” etc. (see Pluto Data Dictionary forfull list of types.)

Eq. (1) shows that the simple OLS regression is able to account forabout 25% of the variation in the FAR gradient across the city. Addingthe squared and cubed terms—Eq. (2)—brings it up to 31%.

For the plot area, we see a negative relationship overall. For the yearbuilt variable there is a negative relationship, but the quadratic terms

Table 3Descriptive statistics for FAR across the five boroughs of NYC.

Borough Mean Std. Dev Min. Max. Nobs.

BK 1.38 1.20 0.01 23.96 13,567⁎

BX 1.13 0.94 0.01 12.14 5132MN 5.84 5.87 0.02 87.67 6831QN 1.20 1.37 0.01 37.6 11,781SI 0.65 0.50 0.01 6.03 2642NYC 2.00 3.18 0.01 87.76 39,953

⁎ Difference is nobs. are due to missing FAR values.

show a leveling off over time. Finally, we can see differences in averageFAR levels across the city. As expected Manhattan has the greatest far(Staten Island is the omitted borough); Brooklyn, the Bronx andQueens,have roughly similar FAR values, on average, while Staten Island has thelowest structural density in the city.

Eq. (4) is the same as Eq. (3) but with the addition of polynomialterms for the distance variable. They add a marginally better fit. In gen-eral, the addition of controls reduces the size of the gradient coefficient.The polynomial terms show monotonicity is preserved.4

4 Note that we also ran OLS regressions that included zip code fixed effects (in additionto building type dummies). For brevity, we do not present the results here since the gra-dient coefficients similar to those in Table 4. Results are available upon request.

Table 5Descriptive statistics for coefficient estimates from the locally weighted regressions.

Variable Mean St. dev. Avg. St. Err. Min. Max. # Obs.

New York CityDist. ESB −0.11 0.12 0.02 −1.27 1.77 39,944Ln(Area) −0.06 0.22 0.03 −1.68 1.30 39,944Year −0.01 0.03 0.01 −0.83 2.38 39,944

BrooklynDist. ESB −0.10 0.09 0.01 −0.55 0.35 13,566Ln(Area) −0.10 0.11 0.03 −1.03 1.30 13,566Year −0.01 0.01 0.00 −0.68 1.00 13,566

BronxDist. ESB −0.09 0.05 0.02 −0.80 0.76 5132Ln(Area) −0.08 0.10 0.04 −1.19 0.62 5132Year −0.01 0.02 0.00 −0.23 0.41 5132

ManhattanDist. ESB −0.23 0.13 0.02 −0.67 0.25 6829Ln(Area) 0.32 0.15 0.03 −1.41 0.49 6829Year 0.00 0.05 0.01 −0.10 2.38 6829

QueensDist. ESB −0.09 0.13 0.02 −1.27 1.77 11,780Ln(Area) −0.15 0.13 0.03 −1.68 0.73 11,780Year −0.01 0.02 0.01 −0.83 0.66 11,780

Staten IslandDist. ESB −0.08 0.13 0.04 −1.26 0.56 2637Ln(Area) −0.32 0.11 0.06 −1.00 0.96 2637Year 0.00 0.02 0.01 −0.59 0.40 2637

Table 6OLS regressions with distance to closest subcenter and airport. Robust t-statistics in parenth(with p-values = 0.00).

(1)

Dist. to Empire State Bldg. −0.215(22.86)⁎⁎

(Dist. to ESB)2 0.019(17.54)⁎⁎

(Dist. to ESB)3 −0.001(13.78)⁎⁎

Ln(Area) −0.19(3.60)⁎⁎

Ln(Area)2 0.006(1.91)

Year −0.454(20.24)⁎⁎

Year2 0.0001(20.21)⁎⁎

Brooklyn dummy 0.239(11.50)⁎⁎

Bronx dummy 0.105(4.40)⁎⁎

Manhattan dummy 0.872(30.56)⁎⁎

Queens dummy 0.018−0.78

Dist. to closest subcenter −0.064(21.04)⁎⁎

(Dist. to subcenter)2

(Dist. to subcenter)3

Dist. closest airport × less than 5 miles

Dist. closest airport × (5–10 miles)

Constant 446.4(20.37)⁎⁎

R-squared 0.43

⁎⁎ Significant at 99%.⁎ Significant at 95%.

114 J. Barr, J.P. Cohen / Regional Science and Urban Economics 48 (2014) 110–119

4. Locally weighted regressions

Here, we use a version of weighted least squares, as suggestedby McDonald and McMillen (2007). Implementation of themodel gives an estimated parameter for each target observation(i.e., building):

βi ¼XX

wijX jX’ j� �−1 X

wijX jY j

� �;

where Xj is a vector of control variables including the constant andthe distance to the core for each observation except i; Yj the depen-dent variable (log of FAR) for all observations except i; wij is theweight that building j is given for building i; and the summationsgiven by ∑ are taken over all buildings, j, and wii = 0.

We use a Gaussian (standard normal) weighting function (kernel)given by

wij ¼ Kdijb

� �¼ e−

12

dijb

� �2

if yeari≥year j0 otherwise

( );

where dij is the Euclidian distance between building i and j (asmeasuredin degrees latitude and longitude). McMillen (2010) notes that thechoice of the kernel has little effect on the results since most kernelchoices have rapid decay with distance. b N 0 is the bandwidth param-eter, discussedmore below. If the year of completion of the target obser-vation, building i, was after the other buildings in the data set then the

eses. Note all regressions have 39,944 observations; all include building types dummies

(2) (3) (4)

−0.238 −0.203 −0.213(18.71)⁎⁎ (21.34)⁎⁎ (16.76)⁎⁎

0.026 0.017 0.022(13.97)⁎⁎ (15.67)⁎⁎ (12.01)⁎⁎

−0.001 −0.0004 −0.001(12.18)⁎⁎ (11.75)⁎⁎ (10.48)⁎⁎

−0.14 −0.198 −0.145(2.65)⁎⁎ (3.73)⁎⁎ (2.74)⁎⁎

0.003 0.006 0.003(1.00) (2.10)⁎ (1.15)

−0.453 −0.455 −0.452(20.38)⁎⁎ (20.28)⁎⁎ (20.37)⁎⁎

0.0001 0.0001 0.0001(20.34)⁎⁎ (20.25)⁎⁎ (20.33)⁎⁎

0.579 0.274 0.627(21.75)⁎⁎ (12.42)⁎⁎ (22.83)⁎⁎

0.458 0.212 0.594(15.64)⁎⁎ (8.41)⁎⁎ (19.44)⁎⁎

1.193 0.93 1.27(37.03)⁎⁎ (31.97)⁎⁎ (38.81)⁎⁎

0.411 0.138 0.571(14.03)⁎⁎ (5.76)⁎⁎ (18.79)⁎⁎

−0.211 −0.072 −0.237(20.67)⁎⁎ (23.30)⁎⁎ (23.07)⁎⁎

0.026 0.03(12.76)⁎⁎ (14.51)⁎⁎

−0.001 −0.001(6.24)⁎⁎ (7.71)⁎⁎

−0.057 −0.064(16.35)⁎⁎ (18.31)⁎⁎

−0.015 −0.013(9.34)⁎⁎ (8.61)⁎⁎

444.8 447.0 443.9(20.50)⁎⁎ (20.42)⁎⁎ (20.50)⁎⁎

0.44 0.43 0.44

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Avg Dist Coeff +1.96*SE -1.96*SE 5 per. Mov. Avg. (Avg Dist Coeff)

-0.5

-0.4

-0.3

-0.2

-0.1

01910 1920 1930 1940 1950 1960 1970 1980 1990 2000

MN BX BK QN

Fig. 2. Top (2a): average LWR coefficients over time for NYC, 1910–2009. Bottom (2b):average coefficients over time for each borough (Staten Island not shown), 1910–2009.

115J. Barr, J.P. Cohen / Regional Science and Urban Economics 48 (2014) 110–119

weight is a function of the geographic distance. If not, then the weightwas set to zero.5

The bandwidth parameter determines the “variance” of theweights.A larger b means that, ceteris paribus, observations further away willhave large weight values. For the LWRs, the bandwidth value wasselected using the standard cross-validation (C-V) method. The C-V al-gorithm runs a LWR for each observation for a specific bandwidth value.Then a statistic is generated that is the mean squared residual of theLWR, where the residual is the difference between the target value(i.e., lnFARi) and the predicted target value, after omitting the ith obser-vation from the model. The bandwidth that minimizes this statistic isused. See McDonald and McMillen (2007) for more information.

For each estimated coefficient we also calculate a standard error,given by Eq. 2.21 in Fotheringham et al. (2002). This approach is similarto the F-statistics calculations procedure described in the appendix ofMcMillen and Redfearn (2010).

4.1. LWR results

For the LWRs, we regressed the log of the floor area ratio (lnFAR) onthe log of plot size (lnArea), the year the building was constructed(Year) and the distance to the Empire State Building (DESB). Note thatthe results are from using a bandwidth that minimize the C-V, whichin this case gives b = 0.02. We also compare the results from a regres-sion of LnFAR on DESB and LnFAR on DESB and lnArea, omitting theyear of completion.

Table 5 gives the descriptive statistics for the results. We separatethe coefficient results into each borough for comparison. First, focusingon the distance coefficients we see that, on average, Manhattan has thelargest (in absolute value) coefficients, and that the average andmedianare negative. (See Table 6.)

4.2. FAR gradient over time

Fig. 2 shows the averages of the coefficients for each year. Fig. 2ashows the (unweighted) averages of all the coefficients with standarderrors bands equal +/−1.96 ∗ (average of the standard errors). Theyshow that starting in the 1920s, there was a general flattening of thegradient across the city. Then between around 1940 and 1985 the gradi-ent remained relatively constant, hovering in the−0.11 range, when itstarted to flatten again. However, since the mid-2000, the average hasstarted to decline a bit.

Anas et al. (1998) report estimates for the population density gradi-ent at three points in time (in miles), 1900, 1940, and 1950. They find itto be 0.32, 0.21, and 0.18, respectively. McDonald and McMillen (2007)report population a density gradient of 0.12 for New York in 2000.While our results show the same direction, the FAR gradient for thecity of New York appears to have small coefficients (in absolutevalue). For the decade of 1900 to 1909 our coefficient averages about0.11, for the 1940s, the average is 0.05, 1950s is 0.06, and for 2000s is0.04. Note that some of the difference across studiesmight also be relat-ed to the size of regions. Herewe focus on only NewYork City, butwhenother studies include metropolitan areas it is likely to increase the sizeof the gradient, if density falls off rapidly outside of the central city prop-er. Furthermore population density and FAR density are two differentvariables and there's no a priori reason to believe they would be in thesame range. But we leave for future work an examination of how theFAR and population densities might be linked andwhat this would sug-gest for relative gradient magnitudes.

Fig. 2b shows the averages of the coefficients over time for eachborough separately. It shows that there are different patterns between

5 Setting some weights equal to zero is analogous to specifying a “window” (seeMcMillen and Redfearn, 2010).

Manhattan and the outer boroughs over time (Staten Island is omittedto increase clarity). By and large, the outer boroughs showed a generalgradient flattening between 1900 and 1940, and have remained rela-tively constant since then.

Manhattan, on the other hand, has shown a different pattern. First,the average coefficients have consistently remained large (in absolutevalue) as would be expected given its importance in the national econ-omy and its higher land values. Second we see that an opposite trend ascompared to the other boroughs: the gradient in Manhattan appears tohave steadily fallen until the early 1960s. From there, over time, the gra-dient has flattened out. The gradient coefficients also showmuch morevariation from year to year in Manhattan. In addition, Manhattan's gra-dient seems much more sensitive to the business cycle—falling in yearsinwhich the economy is growing, andmoving closer to zero in the yearsthe economy is in recession or depression. For example, the 1930s toearly 1940s, shows the coefficientsmoving closer to zero, then droppinguntil the mid-1960s. Another “spike” toward zero occurs in the mid-1970s and the 1990s. The FAR's movements over the course of the busi-ness cycle suggests that land values are quite sensitive to economicactivity at the national level.

Fig. 3 shows the coefficient estimates versus distance from theEmpire State Building. The results of each borough are presented inseparate graphs. The overall pattern shows that the coefficients veryclose the Empire State Building are quite large (in absolute value) andmoving away they get closer to zero. In general they flatten out, be-tween mile 5 and 15. Then after that there are two areas, one at mile15 and one atmile 20where the coefficients go above zero This suggeststhat in these two areas there are smaller subcenters, on the fringes ofthe city. We have more to say about the subcenters in Section 5below. Fig. 3 suggests that OLS estimates are misleading with regardto the shape of the coefficients across the city.

Fig. 4 presents a type of intensity map for the distance coefficient es-timates (Fotheringham et al., 2002). Each point on themap is a t-statistic

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 5 10

BX MN

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0 2 4 6 8 10 12

BK

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1QN

-0.4

-0.3

-0.2

-0.1

0

0.1

0 5 10 150 5 10 15

20

SI

Fig. 3. LWRcoefficient estimates vs. distance from the Empire State Building (miles). Top left (3a):Manhattan andBronx. Top right (3b): Brooklyn. Bottom left (3c):Queens. BottomRight (3d):Staten Island. Note vertices may have different lengths across graphs.

116 J. Barr, J.P. Cohen / Regional Science and Urban Economics 48 (2014) 110–119

equal to the coefficient estimate divided its standard error. It shows therelative magnitude of the coefficients. Here we see concentric bands ofcoefficients; with the regions of positive coefficients. The dark red dotsshow the largest t-stats (in absolute value) and they are a semi-circulararea around the Empire State Building.

The next semi-circle around that has smaller t-statistics, but allabove 10 in absolute value. Two clusters of black dots (positivet-statistics) occur in three locations (the one in south-western cornerof Staten Island is not shown on the map). One is located in Jamaica,Queens; the other is Far Rockaway, Queens. These suggest local“centers of gravity.” Jamaica is a local transportation hub for the LongIsland Railroad; Far Rockaway, perhaps, attracts business nearJohn F. Kennedy Airport. The third cluster is in south west StatenIsland, across the river from Perth Amboy, New Jersey (not shownon map).

Since the LWR estimates give the slope of the density gradient, wecompare them to the slope coefficients from OLS. In particular, we firsttook a moving average (25) of the LWR coefficients versus the distanceto the ESB and plotted the scatter plot in Fig. 5. Also presented is theslope of the estimated coefficients from the OLS regression Eq. (4)Table 4. In particular, we graph

d∂lnFAR∂ DESBð Þ ¼ −:215þ 2 :019ð ÞDESB−3 0:001ð ÞDESB2

:

Fig. 5 shows the results. Over the graph suggests a kind of glass-half-full/glass-half-empty outcome. On average, the OLS slope coefficientsshare similarmovement comparedwith the LWR coefficients; however,the LWR coefficients are able to capture more subtle variations in thedensity landscape.

Finally, we reran the LWRs without the year to see if its omissiondramatically changed the distance to the ESB coefficients. We foundthat it did not. In fact, the correlation coefficient for the two sets of coef-ficients was .99, and a regression of coefficients without the year onthe coefficients with year gave an estimated slope coefficient of 1.01

and p-value = 0.00. This suggests that LWRs might offer an “antidote”to the problem of omitted variable bias.

5. Subcenter identification

As a method to identify possible subcenters in New York City, weused the residuals from Eq. (4), Table 4. From this list of residuals wethen mapped the clusters of residuals that (a) were greater than 1.47(two standard deviations away from zero), and had more than 15 ofthese residuals within a quarter mile of each other. From this procedurewe identified 6 centers or subcenters in order of size (based on size ofcluster). Fig. 6 shows the results.

In Manhattan we identified the area north of Grand CentralStation (Park and 53rd Street) and Wall Street. In Brooklyn, there isdowntown Brooklyn, and in Queens there is Flushing and Jamaica,and in the Bronx there is The Hub neighborhood (at East 149th andThird Avenue). The process did not yield any centers for StatenIsland. We note that technically midtown and downtown are not sub-centers, but actual centers in their own right, but for exposition werefer to them as subcenters because of the way we identify them inthis section.

A word is in order about the criteria we used. We recognize thereis some arbitrariness in determining the subcenters across the city.However, our aim is to find the largest and most important ones. If wereduced either one of the thresholds, we risked the possibility of includ-ing false positive identifications, in the sense that a cluster of positive re-sidualsmight simply reflect, for example, a public housing developmentrather than a true concentration of structural density.

In the end the rule we chose seemed to be a reasonable balancebetween identifying subcenters while not being overly liberal. The OLSregressions bear this out in the sense that the coefficient estimates forthe distance to subcenters are statistically significant, but even includ-ing the largest ones don't increase the R2 by a large amount. Also notethatwe obtained the exact same centerswhen positive residuals greaterthan one standard deviation, and 45 or more were clustered within aquarter mile of each other.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0 5 10 15 20

OLS Slope Avg. Coeff.

Dist.from ESB

Fig. 5.Moving Avg. (25) of LWR distance coefficient estimates vs. slope from OLS regres-sion (Table 4, Eq. (4)).

Fig. 4.Map of t-stats of LWR distance to ESB coefficients. Note Staten Island not shown.

117J. Barr, J.P. Cohen / Regional Science and Urban Economics 48 (2014) 110–119

We also note that work byMcMillen (2001) has used LWRs to iden-tify subcenters. With our data set, we found that we were able to iden-tify the same subcenters when using the LWR residuals. In particular,when we took positive LWR residuals that were one standard deviationaway and had 15 or more clustered within a quarter mile of each other,the same six subcenters appeared.

Lastly, we also included measures of the distance to the closestairport. New York City has two airports, LaGuardia and John F. Kennedy.Distance to thesemight also affect density, since they are transportationhubs. In the end, we found that the best specification was to divide thedistance to airport variable into two. The first variable is the distance tothe closest airport times a dummy variable that takes on the value ofone if the distance is less thanfivemiles, and zero otherwise. The secondvariable is the distance to the airport times a dummy variable that takeson the value of one if the distance is between five and ten miles away,and zero otherwise. The results show that moving further away the

Fig. 6. Clusters of positive residuals used to identify sub-centers in New York City.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 5 10 15 20

MN BK BX QN SI

Fig. 7. Scatter plot of lot area coefficients fromLWRs versus distance from the Empire StateBuilding.

118 J. Barr, J.P. Cohen / Regional Science and Urban Economics 48 (2014) 110–119

effect of the airports diminishes. For locations close the airport the FARdensity gradient is close to 0.06 on average. For the next group it falls toabout 0.014, on average.

6. The effect of plot size across the city

In both theOLS regressions and LWRswe included the plot size as anexplanatory variable. The OLS results show that, on average, there is anegative relationship between the lot size and the FAR, though the func-tional form appears nonlinear. Fig. 7 shows a scatter plot of the plot areaestimated coefficients versus the distance from the Empire State Build-ing. Here we can see that the LWRs paint a more subtle picture ascompared to the OLS results. Each borough is given a different color tosee how the coefficients vary across the city.

The results show that for most of Manhattan and the westernportions of Brooklyn and Queens, the coefficient estimates are positive,on average. After about four miles away, the average becomes negative,

119J. Barr, J.P. Cohen / Regional Science and Urban Economics 48 (2014) 110–119

and slowly decreases, so that by 10 miles away virtually all of the coef-ficients are less than zero. As well the magnitude of the coefficients(in absolute values) steadily becomes larger as one moves away fromthe center.

The coefficient estimates are implicitly measuring the elasticityof capital (structure) with respect to land. To see this, note thatln(FAR) = ln(Capital) − ln(Area); therefore the regression implicitlyestimates the equation lnCapital= (1+β1)lnArea,where the estimatesof β1 are presented in Fig. 7.

By and large, the estimates range between 0.5 and −0.6. For esti-mates greater than zero, capital is elastic with respect to lot size, whilefor values less than zero, lot size has an inelastic effect. In other words,a coefficient greater than one means a 1% increase in lot size is metwith a greater than 1% increase in capital, thereby increasing the FAR;an inelastic effect means that less than 1% of capital is added with a 1%increase in lot size and thus the FAR diminishes.

While we leave a more detailed examination of these results for fu-ture work, the relationship is most likely due to three factors: landvalues, the types of businesses, and zoning. Because land values aremuch higher closer to the center, each plot of land is likely to be usedmore intensively. Having a larger plot of land allows the developer toovercome theproblems associatedwith elevators. Since a skyscraper re-quires that a large fraction of its internal space is devoted to elevatorshafts, a larger lot can allow a developer to build a taller building andreap a greater return (Barr, 2012).

Moving away from the center, as land values drop, there is less eco-nomic pressure to use the plot more intensively, and our results showthat, in fact, at somepoint, larger plots are associatedwith less structuraldensity. This negative effect might also be related to the types of busi-nesses that are likely to locate further out from the center. In particular,in the outer boroughs there are more likely to be retail services thatcater to the neighborhood populations, including supermarkets, shop-ping centers, and automobile repair shops. These types of stores mostlylikely prefer a flatter, rather than taller, configuration because it is easierfor shoppers or workers to navigate through the space.6

Lastly, it is also likely that zoning rules play a role. First if the zoninglaws reduce the FAR limits in the suburbanparts of the city, itwill legallypromote the relationship we see in Fig. 7. Second, if the zoning rules re-quire developers to provide on-site parking, then this will force a less-intensive use of space, all else equal. Where land values are relativelycheap, developers prefer to provide grade parking, rather than decksor underground garages, because of the additional expense.

7. Conclusion

This paper explores a less commonly investigated component ofurban spatial structure—the Floor Area Ratio (FAR), which is a measureof the capital–land ratio. In particular, our goal is to investigate the FARgradient across both time and space. We focus on New York Cityfrom 1890 to 2009, incorporating practically every extant commercialbuilding in the city—some 40,000 structures completed over the last120 years. By controlling for the age of the structures, we are able todetermine the FAR gradient over time, with little concern for possiblevintage effects.

We estimate the FAR gradient by both ordinary least squares (OLS)and locally weighted regressions (LWRs). We find that the LWRs arebetter able to better capture the changes in the gradient over time, ascompared to OLS. In particular we find that the gradient steepened inthe early part of the 20th century, and then began to flatten in themid part of century, with a plateau around 1945.

6 Consider that even inManhattanmost supermarkets are on one floor, and occasional-ly, only in the most busy districts do you sometimes find a supermarket with a secondlevel.

Wealsofind a very different gradient pattern over time inManhattan,relative to the other boroughs. In particular, Manhattan's FAR gradientnot only is much steeper, as would be expected, but it also appears tobe very sensitive to the business cycle—flattening during downturnsand becoming steeper during boom times. This most likely reflects thesensitivity of Manhattan's land values to the general level of U.S. output.

Next we compare the LWR gradient coefficients to the slope of theOLS coefficients. We find that while a cubic functional form for OLS is areasonable measure of the FAR gradient, it does not capture the moresubtle patterns across space. In particular we find that in some locationsin the outer boroughs the gradient with respect to the center is positive.

This suggests then that there are other “centers of gravity” through-out the city. To explore this in more detail we identify possible subcen-ters by looking at positive clusters of OLS residuals. This procedureidentifies six subcenters (two in Manhattan, one in Brooklyn, one inthe Bronx and two in Queens). We then rerun the OLS equations withdistance to nearest subcenter and distance measures to the city'sairports. We find that while these additional controls are statisticallysignificant, they do not appreciably increase the R2, suggesting thatbeyond Manhattan, subcenters are not as important.

Finally, we investigate the relationship between the plot area coeffi-cients from the LWRs and distance from the center.We find that close tothe center, the coefficients are positive, suggesting that large plots allowland to be usedmore intensively there. But moving away from the cen-ter the relationship becomes negative, on average, suggesting largeplots are used less intensively further away from the center. We leavefurther exploration of this relationship for future work, but we hypoth-esize that it is related to land values, the mix of businesses in differentparts of the city, and zoning regulations.

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