Hitotsubashi University
Institute of Innovation Research

Testing for localization: A new approach

Yasusada Murata
Ryo Nakajima
Ryuichi Tamura

IIR Working Paper WP#16-11

Aug. 2016

Institute of Innovation Research
Hitotsubashi University
Tokyo, Japan
http://www.iir.hit-u.ac.jp

Testing for localization: A new approach∗Yasusada Murata† Ryo Nakajima‡ Ryuichi Tamura§
August 19, 2016

Abstract
Recent empirical studies document that knowledge spillovers attenuate and industry
localization decays with distance. It is thus imperative to detect localization accurately
especially at short distances. We propose a new approach to testing for localizationthat corrects the ﬁrst-order bias at and near the boundary in existing methods while
retaining all desirable properties at interior points. Employing the NBER U.S. Patent
Citations Data File, we illustrate the performance of our localization measure basedon local linear density estimators. Our results suggest that the existing kernel density
methods and regression approaches can be substantially biased at short distances.
Keywords: localization; knowledge spillovers; local linear density; boundary bias;
micro-geographic data
JEL codes: R12; O31∗

The authors thank Gilles Duranton, Hidehiko Ichimura, Bill Kerr, Henry Overman, as well as participantsin Conference on Regional Economics and Public Policy (University of Bern), 9th Meeting of the Urban
Economics Association (Washington, D.C.), 4th European Meeting of the Urban Economics Association(Saint Petersburg), Third International Conference “Industrial Organization and Spatial Economics” (SaintPetersburg), III Workshop on Urban Economics (Barcelona), 3rd European Meeting of the Urban Economics
Association (Palermo) for helpful comments and suggestions. Tamura acknowledges ﬁnancial support from
JST/RISTEX.
†
Nihon University Population Research Institute (NUPRI); and University Research Center, Nihon University. E-mail: murata.yasusada@nihon-u.ac.jp‡
Department of Economics, Keio University. E-mail: nakajima@econ.keio.ac.jp§
Institute of Innovation Research, Hitotsubashi University. E-mail: ryuichi.tamura@iir.hit-u.ac.jp1

1

Introduction

Localization – concentration over and above some reference distribution of economic activity– is perhaps the most salient feature of the spatial economy. Since the seminal works of Jaﬀeet al. (1993) and Ellison and Glaeser (1997), the localization of knowledge spillovers and thatof industries have been detected largely within aggregate geographic units such as states andmetropolitan statistical areas (MSAs).1 It is now widely recognized that localization results
diﬀer depending on the chosen spatial units, which is known as the modiﬁable areal unit
problem (Gehlke and Biehl, 1934).
The recent availability of micro-geographic data has spurred the development of distancebased methods that do not rely on geographic units. Duranton and Overman (2005) and
Murata et al. (2014) conduct kernel-density (henceforth K-density) estimations to smooth the
distribution of distances between establishments or between cited-citing patents. Belenzon
and Schankerman (2013) and Kerr and Kominers (2015) take regression approaches using
dummy variables for distance rings, each of which is generated by discretization of distances.
While both distance-based methods are useful to uncover detailed location patterns, Kdensity estimations have the ﬁrst-order bias at short distances, and distance-ring dummyvariable (DD) regressions are sensitive to the way of discretizing the micro-geographic data.2
Since a large number of recent empirical studies document that knowledge spillovers attenuate
and industry localization decays with distance,3 it is imperative to detect localization moreaccurately especially at short distances.
We thus propose a new approach to testing for localization using micro-geographic data,
which allows us to overcome the problems with DD regressions and K-density estimations.
The idea is to combine key ingredients of the existing discretizing and smoothing methods.
1

Put diﬀerently, the extent of localization to be limited within states or MSAs, and the analysis abstracts
from the relative position of spatial units by making, for example, the distance from New York to Boston
equivalent to that of New York to Los Angeles.
2
For example, to detect localization at 4 km, we need to focus on the second bin if the binwidth is 3 km.
However, if the binwidth is 5 km, we need to examine the ﬁrst bin. Hence, the localization result diﬀers
depending on the binwidth, which may be viewed as a variant of the modiﬁable areal unit problem.
3
Indeed, these results are well documented in the literature (e.g., Rosenthal and Strange, 2003; Durantonand Overman, 2005, 2008; Arzaghi and Henderson, 2008; Belenzon and Schankerman, 2013; Murata et al.,2014; Buzard et al., 2015; Carlino and Kerr, 2015; Kerr and Kominers, 2015).

2

Speciﬁcally, we ﬁrst “bin” the micro-geographic data by distance as in DD regressions andcount the number of observations in each bin to construct a histogram. Using the distance ateach bin center and the corresponding height of the histogram obtained from discretization,
we then conduct a local linear regression, which is a smoothing technique as in K-density
estimations. Applying the same procedure to a control sample generated by randomizationand taking the diﬀerence between the two estimates – one capturing the spatial distributionof an observed sample and the other capturing that of a control sample – we construct a new
localization measure that enables us to test if there is a signiﬁcant departure from randomness.
The new measure derived from such local linear density (L-density) estimations – local
linear regressions using the pre-binned data – has advantages over the existing discretizing
and smoothing methods. Indeed, one could obtain a localization measure from the ﬁrst stepof binning per se. Such a measure based on histograms is, however, shown to be essentiallythe same as the localization measure generated by DD regressions, which is sensitive to theway of discretizing the micro-geographic data.4 The second step of the local linear regression
smooths the histogram, thereby coping with the problem arising from discretization. One may
argue that K-density estimations could be used as they also rely on a smoothing technique.
While the K-density method displays only the second-order bias at interior points, it suﬀers
from the ﬁrst-order bias at short distances.5 In a nutshell, our localization measure addresses
the issue of discretization and has a better property at and near the boundary while retaining
all desirable properties of the existing methods at interior points.
To illustrate the performance of our L-density measure relative to the K-density measure,
we employ the NBER U.S. Patent Citations Data File (Hall et al., 2001). Using the dataon observed citation distances for all patent classes, we ﬁrst show that the estimated Kdensity with or without reﬂection is far below the estimated L-density at the boundary while
4

Furthermore, the histogram estimators have the ﬁrst-order bias even at interior points when the derivativeof the true density is not zero (Simonoﬀ, 1996, Chapter 2.1).
5
Even when adopting the reﬂection method as in Duranton and Overman (2005), the K-density estimator
has the ﬁrst-order bias at the boundary (Jones, 1993; Marron and Ruppert, 1994; Simonoﬀ, 1996, Chapter 3.2).
As shown in Section 5.1, the underestimation at the boundary induces overestimation near the boundary since
the densities must sum to one over the entire domain of distances. Furthermore, if the kernel is symmetricand diﬀerentiable like the Gaussian kernel that is often used in the literature, the K-density estimator with
reﬂection must have zero derivative at the boundary even when the true density does not (Silverman, 1986,
Section 2.10).

3the reverse is true near the boundary. One may argue that our localization tests take thediﬀerence between the actual and counterfactual distributions, so that the ﬁrst-order bias
generated by the K-density estimations may be oﬀset because both distributions are biased.
However, this need not be the case since as shown below the actual distributions tend to be
more right skewed (i.e., the mass of each distribution is concentrated in relatively shorter
distances) than the counterfactual ones, and thus are more likely to suﬀer from the ﬁrst-orderbias at short distances.
To see this more clearly, we turn to the disaggregated analysis at the patent class level.
The class-speciﬁc distance-based tests reveal that the L-density tests detect localization for
signiﬁcantly more patent classes than the K-density tests. We further rank patent classes by
the magnitude of the localization indices, and ﬁnd that the ranking diﬀers between the Kand L-density methods. This result suggests that the localization indices based on the Kand L-densities provide a diﬀerent piece of information. Our L-density approach will help
to revisit the determinants and shapes of agglomeration and coagglomeration, given that theK-density indices have been incorporated into empirical speciﬁcations in that literature (see,e.g., Ellison et al., 2010; Kerr and Kominers, 2015).
Since our main focus is on the diﬀerence between the K- and L-density methods at shortdistances, we explore at which distance the K-density tests underestimate the number of
localized patent classes as compared to the L-density tests. It turns out that 40 out of 61
underestimated classes fall within the domain between 0 km and 100 km, which account for
66 % of the number of underestimated classes. The K-density tests thus underestimate thenumber of localized classes especially at short distances. Hence, eliminating the ﬁrst-orderbias at short distances is of ﬁrst-order importance. This is especially so because various
empirical studies show that knowledge spillovers attenuate and industry localization decays
within short distances. Correcting the K-density tests leads to a better understanding of
localization of economic activity as they have been widely used for detecting localization eversince Duranton and Overman (2005).
The L-density estimator depends on the binwidth in the ﬁrst step of binning and on the
bandwidth in the second step of the local linear regression. We thus examine how sensitive4

our L-density tests are to their choices. We ﬁnd that neither the localization result for
all distances nor our main result for short distances is aﬀected by diﬀerent binwidth andbandwidth selections. Hence, the result that the K-density tests underestimate the numberof localized patent classes than the L-density tests especially at short distances is fairly robust.
The rest of the paper is organized as follows. Section 2 proposes a new approach using
L-density estimators and compares it to the existing density-based localization measure. In
Section 3 our L-density measure is related to alternative regression-based localization measures. Sections 4 and 5 describe the data and the estimation procedures, respectively. The
estimation results and their robustness are reported in Section 5. Section 6 concludes.

2

Density-based localization measures

We propose a new approach to testing for localization using micro-geographic data. We ﬁrst
present our localization measure based on the L-density estimators. We then compare it to theexisting density-based localization measure, namely the K-density measure of localization. We
ﬁnally introduce possible regression-based localization measures that we examine in Section 3.

2.1

A new approach

Following the seminal work by Jaﬀe et al. (1993), we consider the case of localized knowledge
spillovers as evidenced by patent citations for ease of exposition. Our approach, however, canbe readily applied to other contexts such as industry localization and co-localization. As inmany studies on patent citations and knowledge spillovers, our analysis relies on the premisesthat patents proxy new ideas, and that patent citations proxy ﬂows of these ideas.
Our localization measure involves the diﬀerence between observed and control samples
as in Jaﬀe et al. (1993) and Duranton and Overman (2005). To begin with, we obtain the
L-density estimator for an observed sample from the following two steps. In the ﬁrst step,
we “bin” the micro-geographic data by distance and count the number of observed citations
in each bin to construct a histogram. In the second step, using the distance at each bin
center and the corresponding height of the histogram, we run a local linear regression to5yield the L-density estimator. Applying the same procedure to a control sample generated byrandomization, and taking the diﬀerence between the two density estimators – one capturing
the spatial distribution of an observed sample and the other capturing that of a control sample,
we construct a localization measure to see if there is a signiﬁcant departure from randomness.
The new localization measure based on such L-density estimations – local linear regressions
using the pre-binned data – performs better than the existing ones at and near the boundary,while retaining all desirable properties at interior points.
To formally describe the procedure, let N c denote the set of cited-citing relationships.6
We deﬁne the distance dij between an originating patent i and a citing patent j for eachcitation link (i, j) ∈ N c . Let r = 1, . . . , R denote the index of distance rings. We deﬁne ther-th distance ring as (dr − ∆/2, dr + ∆/2], where dr is the midpoint of the r-th distance ringand ∆ is the binwidth. For example, d1 corresponds to 5 km, d2 to 15 km, d3 to 25 km andso on, in which case the binwidth ∆ is 10 km.
The L-density for observed citation distances can be estimated as in the following two
steps. In the ﬁrst step, we classify the distance data {dij }(i,j)∈N c into R diﬀerent rings. To
this end, we deﬁne for each cited-citing relationship (i, j) ∈ N c an indicator variable as
Dijr = I (dij ∈ (dr − ∆/2, dr + ∆/2]) ,(1)

which takes one if dij is in the r-th distance ring and zero otherwise. For each distance ring r,∑∑cwe then count the number Nrc = (i,j)∈N c Dijr of citations. Let N c = R
r=1 Nr be the total
number of citations. The height of the histogram for distance ring r is then given by theshare Nrc /N c of observed citations falling within distance ring r divided by the binwidth ∆as follows:
cr =

Nrc
.
N c∆

(2)

As a result of the ﬁrst step, the micro-geographic data on citation distances, whose sample
size is N c , boils down to the binned data {dr , cr }R
r=1 with sample size R.
6

In the context of testing for industry localization using micro-geographic data (e.g., Duranton and Overman, 2005), one can reinterpret N c as the set of pairs of all possible establishments in the industry in question.

6

In the second step, we conduct a local linear regression using the pre-binned data, whichamounts to solving the minimization problem as follows:min{β0 ,β1 }R∑[cr − β0 − β1 (dr − d)]2 f ((dr − d)/h),

(3)

r=1

where f is a kernel function and h is a bandwidth. Then, the L-density estimator, which isdeﬁned as β̂0 at a given distance d, can be expressed as follows (Cheng, 1994, eq.(2.18)):∑R [L̂c (d) =where Ŝq (d) =

]
Ŝ
(d)
−
Ŝ
(d)(d
−
d)
f ((dr − d)/h)cr
2
1
r
r=1
Ŝ2 (d)Ŝ0 (d) − Ŝ1 (d)2

,

(4)

∑R

q
r=1 (dr − d) f ((dr − d)/h).

Having obtained the L-density estimator for the observed sample, we turn to the Ldensity estimation using a control sample. For each observed cited-citing relationship (i, j) ∈N c , we identify a set of control patents, from which we randomly draw a control patentto generate a counterfactual citation. This allows us to deﬁne the set Ñ c of hypotheticalcited-citing relationships. By construction of counterfactual citations, the total number Ñ c ofcounterfactual citations equals that of actual citations, i.e., Ñ c = N c . Computing the number
˜ = Ñrc /(N c ∆),˜Ñrc of counterfactual citations in each distance ring, we obtain c̃r = Ñrc /(Ñ c ∆)which is similar to (2). Using {d˜r , c̃r }R̃
r=1 thus generated, we obtain the L-density estimatorL̃c (d) for counterfactual citations.7
We ﬁnally deﬁne a localization measure based on the L-density estimators by taking thediﬀerence between the observed and control samples as in Jaﬀe et al. (1993) and Durantonand Overman (2005). Our new measure of localization at distance d is thus given by thediﬀerence between the two L-density estimators as follows:γLD (d) ≡ L̂c (d) − L̃c (d).

7(5)

See Section 4.3 for more details on the construction of counterfactual citations. In the context of testingfor industry localization using micro-geographic data (e.g., Duranton and Overman, 2005), one can reinterpretÑ c as the set of pairs of establishments in a hypothetical industry generated by randomization.

7

We will describe how this measure can be used for testing for localization in Section 5.2.8

2.2

Comparison

We now compare our localization measure γLD to the existing density-based measure of localization, namely, the K-density measure of localization. We then introduce possible regressionbased localization measures that we examine in Section 3.
It is easy to see that our localization measure γLD is similar to the localization measure
based on K-density estimations (Duranton and Overman, 2005):

γKD (d) ≡ K̂ c (d) − K̃ c (d),whereK̂ c (d) =∑1f ((dij − d)/h)
N ch
c

(6)

(7)

(i,j)∈Nis the K-density estimator for actual citations, and the counterfactual estimator K̃ c (d) is
analogously deﬁned by using the set Ñ c of counterfactual citations. It is worth emphasizingthat both γLD and γKD are given by the diﬀerence of the density estimators.
However, it is recognized that the K-density estimator, when applied to bounded data(like our distance data, which is bounded from below at zero), can be severely biased at
the boundary. To be more precise, even with the reﬂection method that is often adopted
in the localization literature, the bias of the K-density estimator is O(h) at the boundary,while it is O(h2 ) at interior points (Jones, 1993; Marron and Ruppert, 1994; Simonoﬀ, 1996,
Chapter 3.2). Thus, in ﬁnite samples, the boundary bias can be substantial. As shown in
Section 5.1 the underestimation at the boundary induces overestimation near the boundary
since the densities must sum to one over the entire domain of distances.9
8

The L-density estimator has become increasingly more popular in recent empirical studies, especially inthe context of the regression discontinuity design (e.g., McCrary, 2008). Our novelty lies in constructing
the localization test based on the L-density estimators that do not suﬀer from the ﬁrst-order bias at shortdistances, and that are not sensitive to the way of discretizing the micro-geographic data.
9
Furthermore, in the special case where the kernel function is symmetric and diﬀerentiable like the Gaussian
kernel that is widely used in the literature, the K-density estimator must have zero derivative at the boundary
even when the true density does not (Silverman, 1986, Section 2.10). See Wand and Jones (1995) for further
discussion on boundary bias in nonparametric methods.

8Given that knowledge spillovers attenuate and industry localization decays within shortdistances, the L-density estimator has both theoretical and practical advantages over the
existing estimators (Cheng, 1994; Cheng et al., 1997).
Theoretically, it has the same asymptotic rate of convergence at the boundary as in theinterior, which is O(h2 ), and thus displays a better boundary property than the K-density
estimator. One may argue why we combine the existing discretizing method and smoothing
technique. Indeed, one could derive from the ﬁrst step of binning per se the localization
measure γHIST (d) based on the histograms at distance d ∈ (dr − ∆/2, dr + ∆/2] as follows:
γHIST (d) ≡ cr − c̃r .

(8)

We will show in Section 3 that γHIST is essentially the same as the localization measureobtained from the DD regressions (e.g., Belenzon and Schankerman, 2013; Kerr and Kominers,2015). Without a smoothing parameter h, however, γHIST is sensitive to the choice of ∆ andhas the ﬁrst-order bias even at interior points unless the derivative of the true density is zero
(Simonoﬀ, 1996, Chapter 2.1).
Practically, the implementation of the L-density estimation is simple and fast in computation. The reason is that the sample size of the second step of smoothing equals the number
R of bins, which is much smaller than the number N c of citations that is required for theK-density estimation. Note that the kernel function f in (7) relies on dij for (i, j) ∈ N c ,rather than on dr for r = 1, ..., R in the L-density estimator (4).
Alternatively, one could skip the ﬁrst step of binning and construct a localization measure
based only on the second step of the local linear regression. Although we will also discuss
this possibility in Section 3, the analysis requires the information on the set of all possible
pairs of patents (not just on the set N c of pairs of patents having cited-citing relationships).
Thus, the sample size for the local linear regression without pre-binning is much larger thanN c that is required for the K-density estimation.
Our L-density approach, which combines the existing discretizing method and smoothing
technique, allows us to address these theoretical and practical problems simultaneously.

9

3

Regression-based localization measures

Our localization measure γLD requires pre-binning and the local linear regression. Therefore,
it is natural to think that alternative measures can be derived from nonparametric regressions.
Indeed, one could construct a localization measure based on the DD regression. We will showthat such a localization measure is essentially the same as the localization measure γHIST
obtained from the ﬁrst step of binning per se. Alternatively, one could conduct the local
linear regression without pre-binning to obtain a localization measure. In what follows, we
explore these possibilities building on the nonparametric regression literature and argue thatthese approaches have diﬃculties from either a theoretical or a practical point of view.
Let N denote the set of all possible pairs of patents. It is worth emphasizing that unlike
N c , the set N includes pairs of patents that do not have a cited-citing relationship. Thus,
the number N of all possible pairs of patents is much larger than the number N c of citations.
Denote by cij the citation dummy variable which takes one if patent i is cited by patent j,i.e., if (i, j) ∈ N c , and zero otherwise, i.e., if (i, j) ∈ N \N c . We start with the regression
model as follows:
cij = m(dij ) + εij ,(9)where εij is an idiosyncratic error term with zero mean.10 The function m can be viewed as
the likelihood of citation between two patents with distance dij since m(dij ) = E(cij |dij ) =Prob(cij = 1|dij ).11

3.1

A special case: Distance-ring dummy-variable regression

We ﬁrst analyze a special case in which a functional form for m in the regression model (9)is known. A popular choice of m is the sum of dummy variables for distance rings (e.g.,
In the context of industry localization, one may reinterpret N as the set of all possible pairs of establishments in all manufacturing industries and dij as the distance between establishments i and j. Setting
cij = 1 if establishments i and j are in the same industry and cij = 0 otherwise, then yields essentially thesame localization test as in Duranton and Overman (2005) once the regression model (9) is estimated via the
Nadaraya-Watson regression (Nadaraya, 1964; Watson, 1964). We will come back to this point in Section 3.2.
11
There is another variant of the regression model based on diﬀerent observation units. However, as shownin Appendix A, this alternative approach leads to the same function m as in the regression model (9). Hence,
without loss of generality, we focus on the regression model (9).
1010Belenzon and Schankerman, 2013; Kerr and Kominers, 2015). Replacing m(dij ) in (9) withthe sum of the dummy variables, the DD regression model is given by

cij =

R
∑

µr Dijr + εij ,

(10)

r=1

where Dijr is the dummy variable deﬁned as (1) and µr is the coeﬃcient on Dijr .12 Let∑
Nr = (i,j)∈N Dijr be the number of all possible pairs of patents in distance ring r. The OLS
estimator for the DD regression model can then be expressed as follows (see Appendix B)
Nrc
,
µ̂r =
Nr
where Nrc =

(11)

∑

(i,j)∈N c Dijr is the number of citations in distance ring r as in Section 2.1.

Having analyzed the observed sample, we turn to the DD regression using a control sample.
For each observed cited-citing relationship (i, j) ∈ N c , we identify a set of control patents,from which we randomly draw a control patent to generate a counterfactual citation. As
before, let Ñ c be the set of counterfactual cited-citing relationships. By construction, thetotal number of counterfactual citations equals that of actual citations, i.e., Ñ c = N c . Denoteby c̃ij a counterfactual citation dummy, which takes one if (i, j) ∈ Ñ c and zero otherwise.
Then, as shown in Appendix B, the OLS estimator for the r-th distance ring is given byµ̃r = Ñrc /Ñr = Ñrc /Nr , where the counterfactual number Ñr of all possible pairs of patents
in the r-th distance ring is the same as the denominator Nr in (11) because the set N of all
possible patent pairs that we use to estimate µ̂r and µ̃r is common.
Finally, we deﬁne the measure of localization based on the DD regressions at distanced ∈ (dr − ∆/2, dr + ∆/2] as the diﬀerence between the actual and counterfactual coeﬃcients:][
γDD (d) ≡ µ̂r − µ̃r = wDD × Nrc − Ñrc ,

(12)

∑R
A few remarks are in order. First, r=1 Dijr = 1 holds for any patent pair (i, j). We thus exclude a
constant term from (10). Second, the dummy variables are orthogonal. Letting Dr be a column vector of ther-th distance ring with a typical element Dijr , we have Dr′ Ds = 0 for distance rings r and s ̸= r. Finally, theDD regression model is a linear probability model in the context of binary dependent variable model.
1211where wDD ≡ 1/Nr . Note that, for µ̂r and µ̃r to be comparable, the binwidths for the observed
˜ Under this assumption,and control samples are assumed to be the same here, i.e., ∆ = ∆.
the localization measure (8) based on the histograms at distance d ∈ (dr − ∆/2, dr + ∆/2]can be rewritten as γHIST (d) ≡ cr − c̃r = wHIST × [Nrc − Ñrc ] = (wHIST /wDD ) × γDD (d), where
wHIST ≡ 1/(N c ∆). Thus, the localization measure (12) obtained from the DD regressions
is proportional to the localization measure (8) derived from the ﬁrst-step of binning of the
L-density estimations. Hence, these two localization measures provide qualitatively the same
localization results: Knowledge spillovers are localized at distance d ∈ (dr − ∆/2, dr + ∆/2]
if the diﬀerence Nrc − Ñrc is positive and signiﬁcantly diﬀerent from zero.

3.2

A general case: Local linear regression

We consider a general case of the regression model (9) without imposing a speciﬁc functional
form on m. A typical estimator is then the local linear (LL) regression estimator.13 Formally,
the LL regression estimator m̂(d) is given by β̂0 that solves the minimization problem:∑min{β0 ,β1 }[cij − β0 − β1 (dij − d)]2 f ((dij − d)/h)(13)(i,j)∈N

as follows:m̂(d) =Ŝ(d)Ŝ2 (d)Ŝ0 (d) − Ŝ1 (d)2

,

(14)

where

Ŝ(d) =∑ []Ŝ2 (d) − Ŝ1 (d)(dij − d) f ((dij − d)/h)(15)(i,j)∈N c

Ŝq (d) =∑(dij − d)q f ((dij − d)/h)(16)(i,j)∈N(see, e.g., Wand and Jones, 1995). As in the previous cases, we consider the estimator m̃(d)for counterfactual citations. Taking the diﬀerence between m̂(d) and m̃(d), the localization
13

Another estimator that is often used in the literature is the Nadaraya-Watson (NW) regression estimator.
We show in Appendix C that the localization measure obtained from the NW regressions is essentially thesame as γKD that is derived from the K-density estimations.

12

measure based on the LL regressions can be deﬁned as follows:[]
γLL (d) ≡ m̂(d) − m̃(d) = wLL × Ŝ(d) − S̃(d) ,(17)

where wLL ≡ [Ŝ2 (d)Ŝ0 (d) − Ŝ1 (d)2 ]−1 . Note that the actual and counterfactual bandwidths are
assumed to be the same here. One may think of the diﬀerence Ŝ(d) − S̃(d) as a localization
measure. We will discuss such a possibility in the next subsection.

3.3

Diﬃculties with regression-based localization measures

In Section 2 we have compared the two diﬀerent localization measures, γLD and γKD , bothof which are based on density estimators. We now discuss the relationship between our
localization measure γLD and the localization measures based on nonparametric regressions.
Our localization measure γLD obtained from the L-density estimations shares a similarity
with γHIST in (8) and γDD in (12) since they rely on the binwidth ∆. However, the diﬀerenceis that γLD depends also on the smoothing parameter h. This allows us to cope with the
problem that γHIST and γDD are sensitive to the choice of ∆. Furthermore, our measure
is similar to γLL in (17) because both measures are obtained from local linear regressions.
Notwithstanding the similarities, these alternative approaches to γLD have diﬃculties fromeither a theoretical or a practical point of view.
Theoretically, the localization measure obtained from DD regressions is essentially thesame as that derived from the histograms. However, the histogram estimator has the ﬁrstorder bias even at interior points unless the derivative of the true density is zero (Simonoﬀ,
1996, Chapter 2.1). Thus, the localization tests using γHIST and γDD tend to underperform
those using γKD . In contrast, the LL regression estimator has the O(h2 ) bias at the boundary
as well as at interior points (Fan and Gijbels, 1996, Section 3.2). Therefore, the localization
measure γLL could perform better than γKD that is subject to the O(h) bias at the boundary.
Despite such a desirable boundary property, the LL regression estimator has a drawbackin terms of computational time. As shown in equations (14)–(16), the implementation of the
LL regression requires the summation Ŝq (d) with q = 0, 1, 2, each of which involves the kernel
13

function evaluation as many times as the sample size N . Thus, when testing for localization
using micro-geographic data, the required sample size is often quite large. For example, in thecase of the NBER U.S. Patent Citations Data File that we use in this paper, the number ofpatents eligible for the LL regression analysis is 937, 469, so that the number of observations,which is the number of all possible pairs of patents, can be up to 8.8 × 1011 . Such a large
number of observations make estimation infeasible in practice.14
It should be noted that a computational burden is much less severe for the K- and Ldensity estimations. Indeed, the localization measures γKD and γLD can be computed without
much diﬃculty. The reason is that the K-density estimator requires the kernel function
evaluation only for N c times rather than N times as shown in (7), where the summation istaken over the set N c of patent pairs having a cited-citing relationship, not the set N of all
possible patent pairs. Similarly, as discussed in the previous section, the L-density estimatorrequires only R (< N c ) times, where R is the number of bins. One may think of the diﬀerenceŜ(d) − S̃(d) in (17) as an alternative localization measure using the LL regressions. However,the computational burden remains heavy since, as shown in (15), the computation of Ŝ(d)still involves the summation of Ŝq (d) with q = 1, 2, each of which requires the kernel function
evaluation for N times rather than N c or R times.

4

Data

4.1

Patents and patent citations

Our data are based on the NBER U.S. Patent Citations Data File by Hall et al. (2001), which
covers all patent applications between 1963 and 1999 and those granted by 1999, as well ascited-citing relationships for patents granted between 1975 and 1999. For each patent, the listof inventors, the addresses of inventors, and the technological category are recorded, alongwith other information such as the year of application, assignees, and the type of assignees.

14

Another example can be taken from Duranton and Overman (2005). Since there are 176, 106 establishments in their data, the sample size for the LL regression, i.e., the number of all possible pairs of establishments,
will be more than 1.5 × 1010 , where we use (cij , dij ) = (cji , dji ) because their data is undirectional.

14

The information of patent application month and patent class (3-digit) and subclass (6-digit)
codes is supplemented with the United States Patent and Trademark Oﬃce (USPTO) Patent
BIB database.
We begin with 142, 245 U.S. nongovernmental patents granted between January 1975 andDecember 1979. The sample period is chosen to be comparable to the previous studies. We
focus on the “U.S. patents” whose assignees are in the contiguous United States. We observethat 115, 905 (81.5%) of them were cited at least once by other U.S. patents (see Column 1 ofTable 1). We call them the originating patents and identify the citing patents that cited theoriginating patents by examining all patents granted between January 1975 and December
1999.
Insert Table 1
To focus on knowledge ﬂows between diﬀerent inventors of diﬀerent assignees, we exclude“self-citations”. A citing patent is classiﬁed as self-citing (i) if it had the same assignee as its
originating patent; or (ii) if it was invented by the same inventor as its originating patent.
To distinguish unique inventors, we use the computerized matching procedure proposed by
Trajtenberg et al. (2006). We ﬁnd that 15.0% of citing patents are classiﬁed as self-citations.
After excluding self-citations, we obtain 647, 983 citing patents (see Column 1 of Table 1).

4.2

Geographic information

We identify the location of each invention at the census place level. The U.S. Census Bureaudeﬁnes a place as a concentration of population. There are 23, 789 places in the 1990 census,
which we use below. Restricting patent inventors who reside in the contiguous U.S. area, we
ﬁrst match the address of each inventor to its 1990 census place by name. If the name match
fails, we locate it via the populated place provided by the U.S. Geographic Names Information
System (GNIS). We match the inventor’s address with the GNIS populated place, which ismore ﬁnely delineated than the census place, and then ﬁnd the census place that is nearest
to the identiﬁed GNIS populated place by using their spatial coordination information. This
procedure identiﬁes 18, 139 census places for 97.0% of all inventors in the sample.
15

We measure the distance between an originating patent and a citing patent by the greatcircle distance between the census place of the originating inventor and that of the citing
inventor.15 When there is more than one inventor of a patent, we compute all possible
great-circle distances between the census places of the originating inventors and those of theciting inventors. We then consider two types of distances – minimum and median distances– between the cited and citing patents.

4.3

Control patents and counterfactual citations

To test whether knowledge spillovers are localized, we must control for the existing spatial
distribution of technological activities. Following Jaﬀe et al. (1993), we consider control
patents satisfying the following two conditions.16 First, control patents should belong to thesame technological area as the citing patent under consideration. In the baseline case we
select control patents at the 3-digit level and check the robustness of the result by choosing
ﬁner controls at the 6-digit level. We refer to the former as 3-digit controls, and call the latter6-digit controls as in Murata et al. (2014). Second, control patents should be in the samecohort as the citing patents. As in the previous studies, we use one-month and six-monthwindows for the 3-digit and 6-digit controls, respectively.
The second and third columns of Table 1 report the numbers of originating and citing
patents having at least one control. Note that citing patents do not always have controls,and, even if they do, the control is not necessarily unique for each citing patent. As shown,
60.2% of the citing patents have 3-digit controls. The rate of the citing patents having 6digit controls is lower, at 18.7%. The citing patents with no controls assigned (and their
originating patents) are dropped out of the sample. We also drop patent classes in which
originating patents are distributed across less than 10 census places to obtain well-behaved
15

Since census places are not spatial points, this deﬁnition poses a “zero distance” problem, i.e., even when
the actual distance between the originating and citing inventors is not zero, it is measured to be zero if they
happen to live in the same census place. As in Murata et al. (2014) we correct this by using the distance√
between the two randomly chosen points in census place ℓ with area Aℓ , which is given by [128/(45π)] Aℓ /π(Kendall and Moran, 1963). The median and average of within-area distances for census places are 1.3 km
and 1.7 km, respectively.
16
The way we control patents in the K- and L-density estimations is also similar to that in the DD regressionsin Belenzon and Schankerman (2013) and Kerr and Kominers (2015).

16

estimated density functions. As a result, 92.6% of the originating patents remain “in-sample”
for the 3-digit controls, and the corresponding number is 51.0% for the 6-digit controls. We
use these in-sample patents in the subsequent analysis.
Once the relevant control patents are identiﬁed, we can construct the counterfactual citations, with which we compare the actual citations, as follows. For each observed cited-citing
relationship, we deﬁne an admissible patent set that consists of the citing and control patents
at the 3- or 6-digit level, i.e., the patents that either actually cited or could have cited theoriginating patent. We then allocate a counterfactual citation between the originating patent
and a patent that is randomly drawn from the corresponding admissible patent set.

4.4

Actual versus counterfactual citations

Table 2 (a) reports the descriptive statistics for the actual citations, where we use 390, 104
citing patents reported in Table 1. As shown, both the minimum and median distance distributions deﬁned in Section 4.2 are right skewed, i.e., the mass of the distribution is concentrated
in relatively shorter distances. Furthermore, the distribution of minimum citation distances
is more right skewed than that of median citation distances.17
Insert Table 2
Table 2 (b) illustrates the descriptive statistics for the counterfactual citations, where
we report the averages for 1000 draws from 33, 472, 826 control patents. Comparing panels(a) and (b) of Table 2, we ﬁnd that the counterfactual citation distances tend to be greaterthan the actual citation distances for both the minimum and median cases. Since the actual
distribution is more right skewed than the counterfactual one, the former is more likely to
suﬀer from the ﬁrst-order bias at short distances generated by K-density estimations than thelatter. Thus, the boundary bias need not be oﬀset even when we take the diﬀerence between
the actual and counterfactual K-densities for citation distances as shown below.

17

We ﬁnd that the median citation distance is larger than the minimum citation distance at every percentile
point. Table 2 (a) reports the distances at several lower percentile, namely, 1, 5, 10, 25, 50 percentile, points.

17

5

Estimation

We now describe the estimation procedure in more detail and provide the estimation results.
We start with the K- and L-density estimations using the aggregate data and illustrate theﬁrst-order bias at short distances generated by the K-density estimation. We then turn to thedisaggregate analysis at the patent class level. We construct class-speciﬁc localization tests
based on K- and L-density estimators and summarize the localization results. We ﬁrst report
the localization results for all distances for the sake of completeness, and then illustrate
our main results for short distances. We ﬁnally examine the robustness of the results byconsidering diﬀerent binwidths and bandwidths.

5.1

K-density and L-density estimations

It is well known that the K-density estimators can be severely biased when the domain of thedata is bounded. As mentioned above, the ﬁrst-order boundary bias caused by the K-density
estimation still remains even when adopting the reﬂection method. In contrast, the L-density
estimators have the second-order bias only. We begin with the analysis by illustrating theperformance of these density estimators for all distances. We then focus on the behavior atand near the boundary.

5.1.1

Density estimations for all distances

Figure 1 (a) presents the density estimation results — the K-density estimates with and without reﬂection, and the L-density estimate — for the actual citation distances. For comparison
purposes, we superimpose the histogram generated by the ﬁrst-step binning of the L-density
estimation. To obtain these results, we stick to the standard practice in the existing studies.
More precisely, as to the K-density estimation, we follow Duranton and Overman (2005) who
use the Gaussian kernel function and the rule-of-thumb (ROT) bandwidth in Silverman (1986,
Section 3.4.2). With regard to the L-density estimation, we use the binwidth ∆ = 2σ(N c )−1/2
for the ﬁrst binning step as in McCrary (2008). For the second step of the LL regression,
we choose the triangular kernel and consider two diﬀerent ways of selecting the bandwidth
18

discussed in Fan and Gijbels (1996).18 We ﬁrst employ a cross-validation method as a baseline.19 In Section 5.4 we then consider the ROT bandwidth in Fan and Gijbels (1996) as arobustness check.20
Insert Figure 1
Comparison of the density estimates reveals that the K-density estimates, with and without reﬂection, and the L-density estimate are quite similar at distances greater than about
300 km. This result is in accordance with the theoretical prediction that the order of biases
diﬀers only at and near the boundary between the two estimators. In what follows, we thus
focus on the behaviors of the estimated densities within shorter distances between 0 km and300 km in Figure 1 (b), which is an enlargement of Figure 1 (a).

5.1.2

Density estimations at and near the boundary

Figure 1 (b) illustrates the estimated densities at and near the boundary. Several remarks
are in order. First, the K-density estimate without reﬂection is substantially lower than thatwith reﬂection and the L-density estimate at the boundary. This can be explained by thefact that the K-density estimator, if not corrected by a reﬂection method, is inconsistent at
short distances (Simonoﬀ, 1996, Chapter 3.2).
Second, the K-density estimate with reﬂection is located under the L-density estimate
at the boundary. The discrepancy between the two density estimates can be ascribed tothe diﬀerence in the order of boundary biases, which is the ﬁrst order for the K-densitywith reﬂection and the second order for the L-density. Interestingly, the underestimation18

It is widely recognized that the choice of the kernel function is not so important for the performance of theresulting LL regression (see Fan and Gijbels, 1996, p.76). Yet, Cheng et al. (1997) prove that the triangularkernel, deﬁned as ft (x) = 1 − |x| for x ∈ (−1, 1] and ft (x) = 0 otherwise, is optimal at the boundary. We
thus use it in the subsequent analysis.
19
More concretely, let L̂ch (d) denote the L-density estimator in equation (4), involving a speciﬁc value ofthe bandwidth parameter h. For each r = 1, · · · , R, we use data {ds , cs }s̸=r to build a “leave-one-out” L∑R
density estimator L̂ch,−r (d) and then determine the bandwidth parameter ĥ so as to minimize (1/R) r=1 [cr −L̂ch,−r (dr )]2 w(dr ), where w(dr ) is a weight function.
20
The ROT bandwidth in Fan and Gijbels (1996) is more involved as it requires estimating derivatives
nonparametrically. Let g be the estimated fourth-order polynomial for the pre-binned data {dr , cr }R
r=1 . Denoteby g ′′ and σ 2 the second derivative and the mean squared error ofthepolynomial
regression.
Then,according∫∑R
to Fan and Gijbels (1996), the bandwidth is given by h = κ[σ 2 w0 (t)dt/ r=1 {[g ′′ (dr )]2 w0 (dr )}]1/5 , wherew0 (t) is an indicator function on the interval [0, dR ] with dR being the midpoint of the R-th distance ring,and the constant κ depends on the kernel function.

19

at the boundary induces overestimation near the boundary since the densities must sum toone over the entire domain of distances. Indeed, the K-density estimate even with reﬂection
underestimates severely at the boundary up to around 45 km. This underestimation at theboundary is compensated by the overestimation near the boundary, which is at distances
between 45 km and 300 km in Figure 1 (b).
Third, the slope of the density at the boundary varies across estimators. It is positive,
zero and negative for the K-density without reﬂection, the K-density with reﬂection and the
L-density, respectively. Indeed, without reﬂection, the derivative of the K-density estimate
at the boundary must be theoretically positive since there is no data with negative distances,while with reﬂection, the estimate using the symmetric and diﬀerentiable Gaussian kernel
is subject to the constraint that the derivative at the boundary is zero. By contrast, theL-density estimation does not impose such restrictions on the derivative.
Last, although the L-density estimation requires pre-binning, the estimated L-density does
not perfectly mimic the histogram, especially at and near the boundary. The diﬀerence arises
because the L-density estimator at distance d, which by construction smooths the histogram
obtained from the ﬁrst step of binning, puts some non-negative weights on distances awayfrom d, as seen from equation (4).

5.2

Localization tests and indices

We now address whether knowledge spillovers as evidenced by patent citations are localized.
To this end, we start with the localization measures γKD and γLD , which are based on the Kand L-density estimations, respectively. Those measures allow us to compare the estimateddensity of actual citation distances to that of counterfactual citation distances. However,testing for localization formally requires a confidence band of the estimated counterfactual
densities. We thus sample counterfactual citations repeatedly from the admissible patent set,
and regard the deviation of the actual density from the upper bound of the conﬁdence bandas evidence of localization.
To be more speciﬁc, let us denote the upper local 5% conﬁdence intervals by K c (d) and20

Lc (d) for the K- and L-density estimators, respectively. We run 1000 Monte Carlo simulations,and estimate the counterfactual K- and L-densities for each simulation. Then, K c (d) andLc (d) are given by the 950th of the estimated counterfactual K- and L-densities that are
ranked in ascending order. In turn, the upper global 5% conﬁdence intervals, denoted by
K c (d) and Lc (d), are deﬁned as the identical local conﬁdence intervals such that, when we
consider them across all distances between 0 km and the threshold distance d¯ km, only 5%
hit them. Analogous to equation (6) and (5), we can say that knowledge spillovers exhibit
global localization at 5% signiﬁcance level for the K-density (resp., L-density) estimators if¯ Thus, using theK̂ c (d) − K c (d) > 0 (resp., if L̂c (d) − Lc (d) > 0) for at least one d ∈ [0, d).
indices of global localization at distance d:ΓKD (d) = max{K̂ c (d) − K c (d), 0}
ΓLD (d) = max{L̂c (d) − Lc (d), 0},∑
we deﬁne the cross-distance global localization indices as ΓKD ≡d ΓKD (d) and ΓLD ≡∑¯d ΓLD (d), where the summations are taken up to the median distance d of all possible
actual and counterfactual citation distances. The K-density (resp., L-density) test detects
localization if ΓKD > 0 (resp., if ΓLD > 0).
As shown in Murata et al. (2014), the extent of knowledge spillovers diﬀers by technology. To take this heterogeneity into account, we detect localization of knowledge spillovers
originating from each patent class.21 To this end, we classify all originating patents into different patent classes by their primary class, and examine whether each patent class – to which
originating patents belong – displays localization.
Let A be the set of all patent classes. For each patent class A ∈ A, we estimate the K- andL- densities for the actual and counterfactual citation distances. The Monte Carlo simulation
procedure described above provides the class-speciﬁc local and global conﬁdence intervals byA
which localization is tested. Finally, for each patent class A, we denote by ΓA
KD (d) and ΓLD (d)21Note that citing patents, which cite an originating patent in one patent class, may or may not belong tothat patent class. Put diﬀerently, we allow for intra- and inter-class knowledge spillovers.

21

the class-speciﬁc global localization indices at distance d for the K- and L-density estimators,
respectively. The cross-distance global localization indices for patent class A are then given
∑ A
∑ A
A
by ΓA
KD ≡
d ΓKD (d) and ΓLD ≡d ΓLD (d), where the summations are taken up to the classA

speciﬁc median distance d . Hence, the K-density (resp., L-density) test detects localizationA
22
of knowledge spillovers originating from patent class A if ΓA
KD > 0 (resp., if ΓLD > 0).

5.3

Localization results

We now summarize the localization results obtained from the K- and L-density methods. We
ﬁrst report the localization results for all distances for the sake of completeness, and thenfocus on the main results for short distances.

5.3.1

Localization results for all distances

Table 3 summarizes the numbers and the percentages of localized patent classes. Columns 1
and 2 (resp., 3 and 4) report the case with the 3-digit (resp., 6-digit) controls. In each case we
consider the K-density with reﬂection and the L-density. Panels (a) and (b) show the results
for the minimum and median citation distances, respectively. In both panels, the K-density
tests with the 3-digit (resp., 6-digit) controls detect localization for about 70% (resp., 30%)of feasible patent classes.23 In the L-density tests, the numbers of localized patent classes
increase systematically for both the 3- and 6-digit controls. We may thus conclude that theK-density tests underestimate localized patent classes than the L-density tests.
Insert Tables 3 and 4
In what follows, we report the localization results using the minimum citation distances
and the 3-digit controls as the benchmark.24 Table 4 further presents the top 20 patent classes
22

To perform the localization tests for each patent class, we adopt a class-specific bandwidth and denote itA
by hA
KD (resp., hLD ) for the K-density (resp., L-density) case.
23
These results are the same as in Murata et al. (2014). As in the previous studies, we drop patent classes
in which originating patents are distributed across less than 10 census places to obtain well-behaved estimateddensity functions. Thus, the numbers of feasible patent classes diﬀer between the 3- and 6-digit cases.
24
As shown in Appendix D, the results for the median citation distances are quite similar to our main
results for the minimum citation distances given in Section 5.3.2. We also obtain similar results even with the
6-digit controls. However, Henderson et al. (2005) argue that the localization tests using the 6-digit controls
are subject to sample selection biases. The sensitivity analysis in Murata et al. (2014), which encompasses

22

A
with the highest degrees of localization, measured by ΓA
KD and ΓLD . Since about a half ofthem are not overlapped, the localization indices based on the K- and L-densities provide a
diﬀerent piece of information. Our L-density approach will help to revisit the determinants
and shapes of agglomeration and coagglomeration, given that the K-density indices have been
incorporated into empirical speciﬁcations in that literature (see, e.g., Ellison et al., 2010; Kerrand Kominers, 2015).
To see the diﬀerence in the number of localized patent classes between the two tests, let
AKD and ALD be the sets of localized patent classes by the K- and L-density tests, respectively.
Then, we take the following three disjoint sets: AKD∩LD , which consists of localized classes
detected by both the K- and L-density tests; and AKD\LD (resp., ALD\KD ), which consists oflocalized classes detected only by the K-density (resp., L-density) test. Given these disjoint
sets, we can decompose the sets of localized patent classes as AKD = AKD∩LD ∪ AKD\LD and
ALD = AKD∩LD ∪ ALD\KD . Thus, the increase in localized classes arises due to the diﬀerencein size between ALD\KD and AKD\LD . For the localization results in Table 3, we ﬁnd that 61
patent classes belong to ALD\KD while 5 patent classes belong to AKD\LD . Hence, the numberof localized classes increases by 56 (= 61 − 5).
5.3.2

Localization results at and near the boundary

Turning to our main results, Figure 2 illustrates the diﬀerence between the K- and L-density
tests using an example of a patent class belonging to ALD\KD (patent class 283: Printed
Matter).25 Panel (a) (resp., panel (b)) depicts the K-density (resp., L-density) test. As canbe seen from panel (a) (resp., panel (b)), the actual density does not (resp., does) exceed theupper bound of the global 5% conﬁdence interval. Thus, the K-density test does not detect
localization, whereas the L-density test does for this patent class. It is worth emphasizing thatthe diﬀerence in the localization result comes from the diﬀerence in the boundary behavior
between the two panels.
the 3- and 6-digit controls, shows that the 6-digit controls correspond to the case where the biases induced byimperfect matching between citing and control patents due to unobserved heterogeneity are inﬁnitely large.
See Carlino and Kerr (2015) for more discussion.
25
As mentioned above, we consider the minimum citation distances here. The result for the median citation
distances is similar and can be found in Figure D1 in Appendix D.

23

Insert Figure 2
How often do we observe this pattern? To obtain our main results at short distances, we
now illustrate the spatial pattern of the diﬀerence in the number of localized patent classes
between the K- and L-density tests. Figure 3 gives the frequency distribution of patent classes
for the distances at which the members in ALD\KD initially localize. This allows us to explore
at which distance the K-density tests underestimate the number of localized patent classes
as compared to the L-density tests. The top panel of Figure 3 shows that the patent classes
in ALD\KD are most frequently observed at shorter distances. Indeed, 40 out of 61 patentclasses in ALD\KD are localized within 100 km, which account for 66 % of the number ofunderestimated patent classes. We also illustrate in the bottom panel of Figure 3 the spatial
distribution of the members in AKD\LD , although it does not display a clear pattern.26
Insert Figure 3
We may thus conclude that the K-density tests underestimate the number of localized
patent classes than the L-density tests especially at short distances. Hence, eliminating theﬁrst-order bias at short distances is of ﬁrst-order importance. This is especially so becausea large number of recent studies document that knowledge spillovers attenuate and industry
localization decays within short distances. Thus, correcting the K-density tests leads to abetter understanding of localization of economic activity as they have been widely used for
detecting localization ever since Duranton and Overman (2005).

5.4

Robustness check

We ﬁnally check the robustness of our main result that the K-density tests underestimate
the number of localized patent classes than the L-density tests, especially at short distances.
First, we stick to the baseline cross-validation method and consider diﬀerent values of hA
LD /∆.
In particular, we impose hA
LD /∆ > τ with τ = 10, 20, 30, which is in line with McCrary (2008)
who provides the condition in terms of the ratio between the bandwidth and binwidth to ensure
26

Again, we consider the minimum citation distances here. See Figure D2 in Appendix D for the result
using the median citation distances. Both results are very similar.

24the robustness to diﬀerent binwidths. Second, instead of the cross-validation method, we use
the ROT bandwidth given by Fan and Gijbels (1996) while imposing diﬀerent thresholds
τ = 10, 20, 30.
In both cases, hA
LD /∆ > τ with τ = 10, 20, 30 need not be satisﬁed. If the conditions
are not met, we take the following heuristic iterative procedure: Starting from the baseline
binwidth ∆0 , we compute the corresponding bandwidth hA
LD,0 for each patent class A. Whilethe condition is not satisﬁed for at least one patent class A ∈ A, we decrease the binwidth by
10 percent iteratively, i.e., ∆s+1 = 0.9∆s , to compute the updated class-speciﬁc bandwidthhALD,s+1 .
Insert Table 5
Table 5 presents the robustness results and reports the percentages of patent classes forwhich the iterative procedure is implemented at least once. With the ROT bandwidth given
by Fan and Gijbels (1996), all patent classes satisfy hA
LD /∆ > 10, so that no iteration isneeded. When the threshold is set at τ = 20, 30, some patent classes violate hA
LD /∆ > τin the case of the 6-digit controls. Furthermore, when the cross-validation method is used,some patent classes do not meet the condition even with the smallest threshold τ = 10. In allcases, however, once the iterative procedure is adopted, we can show that the K-density tests
underestimate the percentage of localized patent classes than the L-density tests. Hence, ourresults for all distances do not seem to be aﬀected by the choice of binwidth and bandwidthparameters.
Insert Figure 4
Turning to the robustness of our main result at short distances, Figures 4 (a) and (b)replicate Figure 3 using the cross-validation method with hA
LD > 10 and the ROT bandwidth27
with hA
Again, we can see clear evidence on the downward biases of
LD > 10, respectively.

the K-density tests within short distances: We ﬁnd that in the former (resp., latter) case
66 % (71 %) of the patent classes underestimated by the K-density tests fall within 100 km.
Since these numbers are similar to the one shown in the benchmark, our main result that the27

We obtain quite similar results for diﬀerent values of τ . The results are available upon request.

25

K-density tests underestimate the number of localized patent classes than the L-density tests
especially at short distances is not sensitive to the binwidth and bandwidth selections.

6

Concluding remarks

Combining the existing discretizing method and smoothing technique, we propose a new
approach to testing for localization using micro-geographic data. Our localization measure
is based on local linear density estimators that have a better boundary property, and thus
suitable for analyzing localization that generally occurs within short distances.
Our L-density approach can be applied to ﬂows of people, goods, and ideas. In this paper
we illustrate the case of knowledge spillovers using the data on citation ﬂows. The analysis
can be readily used for commuting and commodity ﬂows, as well as for industry localizationand co-localization.
When applied to industry localization, our localization test satisﬁes all ﬁve criteria summarized in Duranton and Overman (2005), namely it (i) is comparable across industries; (ii)controls for the overall agglomeration of manufacturing; (iii) controls for industrial concentration; (iv) is unbiased with respect to scale and aggregation; and (v) gives an indication of
the signiﬁcance of the results.
While having these desirable properties, the L-density approach allows us to address boththe ﬁrst-order biases at short distances and the issue that localization results diﬀer dependingon the way of discretizing the micro-geographic data. Our results suggest that coping withthese problems is crucial for a better understanding of to what extent people, goods, andideas are localized in the spatial economy.

26

References[1] Arzaghi, M. and J.V. Henderson (2008), “Networking oﬀ Madison Avenue”, Review of
Economic Studies 75 1011-1038.
[2] Belenzon, S. and M. Schankerman (2013) “Spreading the word: Geography, policy, andknowledge spillovers”, Review of Economics and Statistics 95, 884-903.
[3] Buzard, K., G.A. Carlino, R.M. Hunt, J.K. Carr and T.E. Smith (2015), “Localized
knowledge spillovers: Evidence from the agglomeration of American R&D Labs andpatent data”, Working Paper #15-03, Federal Reserve Bank of Philadelphia.
[4] Carlino, G.A. and W.R. Kerr (2015), “Agglomeration and innovation”, Handbook ofRegional and Urban Economics 5, 349-404.
[5] Cheng, M.-Y. (1994), “On boundary eﬀects of smooth curve estimators (dissertation)”,Mimeo Series #2319, Institute of Statistics, University of North Carolina.
[6] Cheng, M.-Y., J. Fan, and J.S. Marron (1997), “On automatic boundary corrections”,
The Annals of Statistics 25, 1691-1708.
[7] Davidson, R. and J.G. MacKinnon (2004), Econometric Theory and Methods (Oxford:
Oxford University Press).
[8] Duranton, G. and H. Overman (2005), “Testing for localization using micro-geographic
data”, Review of Economic Studies, 72: 1077-1106.
[9] Duranton, G. and H. Overman (2008), “Exploring the detailed location patterns of U.K.
manufacturing industries using microgeographic data”, Journal of Regional Science, 48:
213-243.
[10] Ellison, G.D. and E.L. Glaeser (1997), “Geographic concentration of in US manufacturingindustries: A dartboard approach”, Journal of Political Economy, 105: 889-927.

27

[11] Ellison, G.D., E.L. Glaeser, and W.R. Kerr (2010), “What causes industry agglomeration? Evidence from coagglomeration patterns”, American Economic Review, 100:
1195-1213.
[12] Fan, J. and I. Gijbels (1996), Local Polynomial Modelling and Its Applications (London:
Chapman & Hall).
[13] Gehlke, C.E. and K. Biehl (1934), “Certain eﬀects of grouping upon the size of thecorrelation coeﬃcient in census tract material”, Journal of the American Statistical Association 29: 169-170.
[14] Hall, B., A. Jaﬀe, and M. Trajtenberg (2001), “The NBER patent citation data ﬁle:Lessons, insights and methodological tools”, NBER Working Paper #8498.
[15] Henderson, R., A. Jaﬀe, and M. Trajtenberg (2005) “Patent citations and the geographyof knowledge spillovers: A reassessment: Comment.” American Economic Review 95,
461-464.
[16] Jaﬀe, A., M. Trajtenberg, and R. Henderson (1993), “Geographic localization of knowledge spillovers as evidenced by patent citations”, Quarterly Journal of Economics 108:577-598.
[17] Jones, M.C. (1993), “Simple boundary correction for density estimation”, Statistics andComputing 3: 135-146.
[18] Kendall, M. and P. Moran (1963), Geometrical Probability (London: Charles Griﬃn &Company Limited).
[19] Kerr, W.R. and S.D. Kominers (2015), “Agglomerative forces and cluster shapes”, Review
of Economics and Statistics 97, 877-899.
[20] Marron, J.S. and D. Ruppert (1994), “Transformations to reduce boundary bias in kernel
density estimation”, Journal of the Royal Statistical Society. Series B 56: 653-671.

28

[21] McCrary, J. (2008), “Manipulation of the running variable in the regression discontinuity
design: A density test”, Journal of Econometrics 142, 698-714.
[22] Murata, Y., R. Nakajima, R. Okamoto, and R. Tamura (2014), “Localized knowledge
spillovers and patent citations: A distance-based approach”, Review of Economics andStatistics 96: 967-985.
[23] Nadaraya, E.A. (1964), “On estimating regression”, Theory of Probability and its Applications 9: 141-142.
[24] Rosenthal, S. and W. Strange (2003), “Geography, industrial organization, and agglomeration”, Review of Economics and Statistics 85, 377-393.
[25] Silverman, B.W. (1986), Density Estimation for Statistics and Data Analysis (New York:
Chapman and Hall).
[26] Simonoﬀ, J.S. (1996), Smoothing Methods in Statistics (New York: Springer-Verlag).
[27] Trajtenberg, M., G. Shiﬀ, and R. Melamed (2006), “The NAMES GAME: Harnessing
inventors’ patent data for economic research”, NBER Working Paper #12479.
[28] Wand, M.P. and M.C. Jones (1995), Kernel Smoothing (London: Chapman & Hall).
[29] Watson, G.S. (1964), “Smooth regression analysis”, Sankhyā: The Indian Journal of
Statistics, Series A 26: 359-372.

29

Table 1: Sample patent sizes
Total
3-digit
6-digitOriginatings
115,905
107,561
59,168(Percent)(100.00)(92.64)(51.04)
Citings
647,983
390,104
120,876(Percent)(100.00)(60.20)(18.65)Controls—
33,472,826 941,532
Note: The citing patents with no controls assigned(and their originating patents) are dropped out of
the sample.

30

Table 2: Descriptive statistics of minimum and median citation distances for actual andcounterfactual citations
Citation
Moments
distances
Min Max
Mean Std Dev Skewness(a) Actual citations
Minimum 0.47 4513.48 1373.67 1240.81
0.89Median0.49 4517.25 1483.43 1264.69
0.79(b) Counterfactual citations
Minimum 0.42 4516.11 1472.08 1241.77
0.79Median0.52 4539.75 1577.99 1261.29
0.69
Note: All distances are in kilometers.

31

1st

5th

Percentiles
10th 25th

50th

4.00 19.10 75.72 375.81 973.77
11.22 37.02 123.05 441.82 1098.28
6.53 42.78 149.88 453.12 1089.60
15.96 70.03 197.76 528.93 1187.97

Table 3: Localization test results
3-digit control
K-density L-density
(1)
(2)
384
384

6-digit control
K-density L-density
(3)
(4)
360
360

All classes∗(a) Minimum citation distance
Localized classes
273
329
109
149
Percentage of localization
71.09%
85.68%
30.28%
41.39%(b) Median citation distance
Localized classes
275
334
109
165
Percentage of localization
71.61%
86.98%
30.28%
45.83%
Note: (*) The numbers of feasible patent classes diﬀer between the 3- and 6-digit
control cases because the patent classes in which originating patents are distributed
across less than 10 census places are dropped out of the sample.

32

A
Table 4: Top 20 localized patent classes by ΓA
KD and ΓLD .

K-density case (ΓAKD )Patent class
Hydraulic and earth engineering
Boots, shoes, and leggings
Butchering
Surgery
Apparel apparatus
Communications, electrical: acoustic wave systems and devices
Land vehicles: bodies and top
Sugar, starch, and carbohydrates
Acoustics
Pipe joints or couplings
Roll or roller
Cutlery
Compositions: ceramic
Implements or apparatus for applying pushing or pulling force
FencesFluid sprinkling, spraying, and diﬀusing
Expanded, threaded, driven, headed, tool-deformed, or locked-threaded fastenerSpecial receptacle or package
Article dispensing
Fluid handling

3-digit
code
405
36
452
606
223
367
296
127
181
285
492
30
501
254
256
239
411
206
221
137

Rank by
ΓA
KD
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Rank by
ΓA
LD
1
3
14
4
6
37
2
23
26
7
8
91
29
62
5
38
22
28
65
71

3-digit
code
405
296
36
606
256
223
285
492
112
429
104
152
109
452
494
72
68
242
623
512

Rank by
ΓA
LD
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Rank by
ΓA
KD
1
7
2
4
15
5
10
11
35
85
190
111
89
3
198
26
24
22
71
49

L-density case (ΓA
LD )Patent class
Hydraulic and earth engineering
Land vehicles: bodies and tops
Boots, shoes, and leggings
Surgery
FencesApparel apparatus
Pipe joints or couplings
Roll or roller
SewingChemistry: electrical current producing apparatus, product, and process
Railways
Resilient tires and wheels
Safes, bank protection, or a related device
Butchering
Imperforate bowl: centrifugal separators
Metal deforming
Textiles: ﬂuid treating apparatus
Winding, tensioning, or guiding
Prosthesis, parts thereof, or aids and accessories therefor
Perfume compositions

33

Table 5: Robustness check: Localization results for all distances
3-digit control% Localized % Iterated†
71.09%−

6-digit control% Localized % Iterated†
30.28%−

K-density
L-density
Cross-Validation:hA
85.68%
8.33%
40.83%
24.17%LD /∆ > 10A
85.42%
44.27%
40.56%
44.72%hLD /∆ > 20
AhLD /∆ > 30
85.42%
59.38%
40.28%
51.39%FG ROT:hA
82.03%
0.00%
38.61%
0.00%
LD /∆ > 10
hA
/∆
>
20
82.03%
0.00%
38.61%1.11%
LD
A
hLD /∆ > 30
82.03%
0.00%
38.33%
25.83%
Note: While the inequality condition is not satisﬁed for all patent classes, thebinwidth, ∆, is iteratively decreased by 10 percent for that condition to be satisﬁed.
The percentage of patent classes for which the iterative procedure is implemented
at least once is reported.

34

0.00200.00000.0010Density

0.0030L−density
K−density with reflection
K−density without reflection

0

200

400

600

800

1000

1200

Kilometers(a) Density estimation results for 0 km to 1200 km

0.00200.00000.0010Density

0.0030L−density
K−density with reflection
K−density without reflection

0

50

100

150

200

250

300

Kilometers(b) Density estimation results for 0 km to 300 km

Figure 1: Density estimation results at the aggregate level. Panels (a) and (b) present the samedensity estimation results for diﬀerent domains of distances. The binwidth of the histogram
equals ∆ = 2σ(N c )−1/2 , which is obtained from the ﬁrst step of the L-density estimation.

35

0.00300.0010Density

0.00200.00300.0000Density

0.00200.00100.0000
0

200

400

600

800

1000

0

1200

200

400

600

Kilometers
K−density

Kilometers
L−density(a) K-density test(b) L-density test

800

1000

1200

Figure 2: K-density and L-density tests using patent class 283 (Printed Matter). The solid
curves are the estimated densities for the actual citations. The dashed (resp., dotted) curves
correspond to the global (resp., local) conﬁdence intervals obtained from the counterfactual
citations.

36

# of classes

50
30
20
10
0
50
# of classes

Frequency distribution of 61 patent classes
detected only by L-density tests

40

0

500

1000

1500

2000

2500

Frequency distribution of 5 patent classes
detected only by K-density tests

40
30
20
10
0

0

500

1000
1500
Kilometers

2000

2500

Figure 3: Frequency distributions of patent classes for the distances at which the members in
AKD\LD and ALD\KD initially display localization. The bars in dark grey (resp., light grey) are
the frequencies of localized patent classes detected only by the L-density (resp., K-density)
tests.

37

# of classes

50
30
20
10
0
50
# of classes

Frequency distribution of 61 patent classes
detected only by L-density tests

40

0

500

1000

1500

2000

2500

Frequency distribution of 5 patent classes
detected only by K-density tests

40
30
20
10
0

0

500

1000
1500
Kilometers

2000

2500

(a) Cross-validation method# of classes

50
30
20
10
0
50
# of classes

Frequency distribution of 49 patent classes
detected only by L-density tests

40

0

500

1000

1500

2000

2500

Frequency distribution of 7 patent classes
detected only by K-density tests

40
30
20
10
0

0

500

1000
1500
Kilometers

2000

2500

(b) Rule-of-thumb bandwidth

Figure 4: Frequency distributions of patent classes for the distances at which the members
in AKD\LD and ALD\KD initially display localization. We use the cross-validation method inpanel (a) and the rule-of-thumb bandwidth in panel (b). In both panels, hA
LD /∆ > 10 holds
for all patent classes A ∈ A, and the bars in dark grey (resp., light grey) are the frequencies
of localized patent classes detected only by the L-density (resp., K-density) tests.
38

Appendix A: An alternative regression model
In this appendix, we consider an alternative regression model with diﬀerent observation units.
Let z and z ′ be geographic sites (e.g., zip code area or county), and consider
czz′= ms (dzz′ ) + εzz′ ,
nzz′

(18)

where czz′ and nzz′ are the number of citations and the number of patent pairs between z andz ′ , respectively; and dzz′ is the distance between sites z and z ′ . Thus, for each pair of sites(z, z ′ ), the citation rate czz′ /nzz′ is regressed on the distance dzz′ between the two sites.
It can be veriﬁed that the function ms in (18) is identical to the function m of the regression
model (9). To see this, let zi and zj be the geographic sites to which patents i and j belong,∑∑respectively. Noting that czz′ = {i|zi =z} {j|zj =z′ } cij , we readily obtain the result as follows:(
ms (dzz′ ) = E

czz′
dzz′
nzz′

)
=

1 ∑
nzz′

∑

{i|zi =z} {j|zj =z ′ }E(cij |dzi zj ) =

1 ∑
nzz′

∑

m(dzi zj ).

{i|zi =z} {j|zj =z ′ }Since the value of m(dzi zj ) is constant for any i and j such that zi = z and zj = z ′ , we obtain
ms (dzz′ ) = m(dzi zj ). Hence, without loss of generality, we focus on the regression model (9).

39Appendix B: The OLS estimators for the DD regressions
Since the regressors are orthogonal in (10), the OLS estimator for distance ring r is given by28
∑

∑
D
c
ijr
ij
(i,j)∈N c Dijr(i,j)∈Nµ̂r = ∑
= ∑,2(i,j)∈N Dijr(i,j)∈N Dijr
where we use∑(i,j)∈N Dijr cij =∑2(i,j)∈N c Dijr and Dijr = Dijr .

Using the deﬁnitions of Nrc

and Nr , we obtain (11). Similarly, using the counterfactual citation dummy c̃ij and the setÑ c of counterfactual citations, the OLS estimator for the r-th distance ring is given by∑µ̃r =where we use∑(i,j)∈N Dijr c̃ij =(i,j)∈N Dijr c̃ij∑=
2(i,j)∈N Dijr∑∑∑(i,j)∈Ñc

Dijr(i,j)∈N Dijr

2(i,j)∈Ñ c Dijr and Dijr = Dijr .

,Using the properties of Ñrc

and Ñr , we obtain µ̃r = Ñrc /Ñr .

28

This result comes from the Frisch-Waugh-Lovell Theorem (see, e.g., Davidson and MacKinnon, 2004,
pp.65-66). When all regressors are orthogonal, the coeﬃcient of any single variable in the multiple regression
can be expressed as the coeﬃcient of the variable in the simple regression after dropping out all the otherindependent variables.

40

Appendix C: On the relationship between K-density estimation andNadaraya-Watson regression
In general the regression model (9) can be estimated via local polynomial regressions. Let pdenote the degree of the polynomial being ﬁt. At distance d, the local polynomial regression
estimator m̂(d) is obtained by ﬁtting the p-th order polynomial to the data using weighted
least squares. More speciﬁcally, m̂(d) is given by β̂0 that solves the minimization problem:min{β0 ,··· ,βp }∑[cij − β0 − β1 (dij − d) − · · · − βp (dij − d)p ]2 f ((dij − d)/h).

(19)(i,j)∈N

The commonly used values for the order of the polynomial are p = 0 and 1. In such cases,
we have simple explicit formulae (see, e.g., Wand and Jones, 1995). In Section 3.2, we deal
with the case with p = 1. When p = 0, the value β̂0 is called the Nadaraya-Watson (NW)
regression estimator and is given by∑(i,j)∈Nm̂NW (d) = ∑f ((dij − d)/h)cij(i,j)∈N f ((dij − d)/h)Noting that∑(i,j)∈N f ((dij − d)/h)cij =.

∑(i,j)∈N c f ((dij − d)/h), the NW regression estimatorcan be expressed in term of density estimators as follows:m̂NW (d) =N c K̂ c (d),N K̂(d)whereK̂ c (d) =∑1f ((dij − d)/h)N chc(i,j)∈N

K̂(d) =1Nh∑f ((dij − d)/h)(i,j)∈N

are K-density estimators. Note that the former expression, K̂ c (d), is the same as (7).
As in the main text, we denote by m̃NW (d) the NW regression estimator when the depen-

41dent variable is a counterfactual citation dummy c̃ij . Then, the localization measure is givenby the diﬀerence between the two estimators as follows:[]γNW (d) ≡ m̂NW (d) − m̃NW (d) = wNW × K̂ c (d) − K̃ c (d)where wNW ≡ (N c /N )[K̂(d)]−1 > 0. Note that the actual and counterfactual bandwidths
are assumed to be the same here. Under this assumption, the localization measure basedon the NW regression estimators can be rewritten in terms of the localization measure (6)derived from the K-density estimators as γNW (d) = wNW × γKD (d). Since wNW > 0 for any
distance, γNW (d) and γKD (d) have the same sign. Hence, they provide qualitatively the same
localization results: Knowledge spillovers are localized at distance d if γNW (d) and γKD (d) are
positive and signiﬁcantly diﬀerent from zero.

42

0.00100.0005

Density0.00100.00000.00050.0000Density0.00150.0015

Appendix D: The results for the median citation distances

0

200

400

600

800

1000

0

1200

200

400

600

800

1000

1200

Kilometers
L−density

Kilometers
K−density(a) K-density test(b) L-density test

Figure D1: K-density and L-density tests using patent class 283 (median citation distance).
The solid curves are the estimated densities for the actual citations. The dashed (resp.,
dotted) curves correspond to the global (resp., local) conﬁdence intervals obtained from thecounterfactual citations.

43

# of classes

50
30
20
10
0
50
# of classes

Frequency distribution of 63 patent classes
detected only by L-density tests

40

0

500

1000

1500

2000

2500

Frequency distribution of 4 patent classes
detected only by K-density tests

40
30
20
10
0

0

500

1000
1500
Kilometers

2000

2500

Figure D2: Frequency distributions of patent classes for the distances at which the members
in AKD\LD and ALD\KD initially display localization (median citation distance). The bars indark grey (resp., light grey) are the frequencies of localized patent classes detected only by
the L-density (resp., K-density) tests.

44