Empirically, the definition of a sub‐center is often complex, case dependent and ambiguous because suburban employment tends to be highly dispersed. The literature has applied two criteria to identify sub‐ centers. One, a sub‐center is defined as an area with significantly higher employment densities than surrounding areas. Two, a sub‐center should be large enough to have a significant effect on the overall spatial structure of the urban area, leading to local rises in population density, land prices, and perhaps housing prices. Early urban empiricists applied these definitions and extended the usual empirical expression of a monocentric city to its multi‐nodal counterpart but not without drawbacks.<sup>51</sup> Alternative empirical methods for capturing the internal distribution of economic activity in a polycentric setting include clustering models and nonparametric models.<sup>52</sup> Clustering method bunches together adjacent tracts to build sub‐centers based on ad hoc definitions of employment density cut‐offs and total job thresholds. For example, a cut‐off that has been used in the literature defines two types of centers: “10– 10” and “20–20.” The former is an agglomeration of contiguous tracts having a minimum job density of 10 jobs per acre combined with a total number of jobs of at least 10,000 while the latter is a tract with at least 20 jobs per acre and 20,000 total jobs in each sub‐center.<sup>53</sup> These models are criticized for the arbitrary nature of the density and total employment cutoffs implying that the number of sub‐centers can be increased by decreasing the threshold and vice versa.<sup>54</sup> The technique proposed in the above research have been applied to identify sub‐centers in a number of large US cities, including Atlanta, Baltimore‐Washington, Boston, Dallas, Houston, Indianapolis, Los

> 51 E.g. the model requires a priori specification of the candidate subcenters implying that the gradient from each center contributes to employment density at every point in the metropolitan area. This complicates estimation especially if there are a larger number of subcenters. (See, Gordon et al.,1989 and Gordon and Richardson,1996).

> 52 The most critical challenge associated with studying urban morphology is the difficulty in obtaining data at fine spatial scales. Cities chosen to study this model have based on the ease of collecting this data and these are notably the ones with strong priors on their urban forms. For instance, not many cities have population and employment data available at the level of a tract. Land registry or any other source that is able to identify parcels is far less available. Thus, issues such as cross‐city heterogeneity within a country or the behavior of density gradients over time have been largely ignored (See Combes et at., 2012 for a comparison of land price gradients across cities in France and McMillen, 1996 for the dynamics of price gradient over 150 years in Chicago). Secondly, there have been measurement issues relating to not just identifying subcenters but also when making predictions for land parcels based on area density gradient. Thirdly, what we want to ask the literature is not whether cities are monocentric or not, a hypothesis that can be rejected in nearly all cities, but exactly how monocentric is a city. The key difficulty with this analysis is that the R2 of gradient regressions do not provide a good metric by which to measure this and reasonable methods in this field are yet to be developed (Duranton and Puga, 2015). Similarly, the predictions of the monocentric model on various aggregate need not be taken literally, but generally, empirical methods must be developed to measure how different is reality from such predictions. Monocentric model makes a number of predictions about some aggregate quantities and land rent at the centre (e.g. proportionality between total land rent and total commuting costs). One study that makes such an assessment is by Combes et al. (2012) who examine the proposition that land rent at the centre is proportional to city population.

> 53 Giuliano and Small (1991), Small and Song (1994), Bogart and Ferry (1999), Anderson and Bogart (2001), Giuliano et al. (2007). 54 Daniel McMillen along with his co‐authors has sought to address the misspecification of monocentric models and the lack of statistical tests for the clustering methods through nonparametric regression‐based models (McMillen, 2001a; McMillen and Smith, 2003; McMillen and McDonald, 1997, 1998). These models look at the deviations from general spatial trends in employment density and identify outliers relative to a spatial average that uses half of all the observations. This, however, substantially obscures the topography of local employment. Recently, methods have been devised such that the estimation of nonparametric employment density surface occurs locally— with a subsample that is kept sufficiently small to keep intact the structure of local employment density. This is necessary in order to identify subcenters, as the general spatial trends in employment density using a larger window size largely mask independent local concentrations of employment. Local maxima on the density surface become candidate centers (Redfearn, 2007a).

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