Knowledge Flow Among U . S . Metro Areas : Innovative Activity , Proximity , and the Border Effect *

Knowledge spillovers are critical for innovation and new value creation in an increasingly knowledge-intensive economy. The substantial scholarly attention on knowledge spillovers has shown that there is a rapid distance decay associated with knowledge spillovers and that there is a positive state border effect. We show that the effects of distance, technology proximity, and the state border effect on knowledge flows are dependent on the size of the regions (MSAs) involved in the knowledge flow. Not accounting for innovation size (innovative communities and social relationships) in the flow of knowledge across origin-destination regions results in aggregation bias in the parameter estimates. Knowledge spillovers are more localized for small innovation MSAs than for large ones. Distance is not as much of a resistance factor in knowledge flow for larger innovation metro areas compared with smaller regions. Spatial origin and destination effects due to technology compatibility of neighboring regions do not affect the knowledge flow among large innovation MSAs, but do have an effect when small MSAs interact with large MSAs.


Objective
The knowledge flow between citing and cited patents has been used extensively to test for knowledge spillovers since the availability of the NBER Patent Citations Data File (Hall et al., 2001).Papers that study knowledge flows between regions using patent citation data employing a spatial interaction regression model include Maurseth and Verspagen (2002), Singh et al. (2010), Li (2014), Griffith et al. (2007), LeSage et al. (2007), Peri (2005), Hussler (2004), Fischer et al. (2009), and Mukherji and Silberman (2013).We extend this literature by estimating knowledge flows separately for groups of metropolitan areas based on their innovative activity (alternatively called size) as measured by the number of patents issued to the area.This study builds on past works which ignored the potential impact of regions with high (low) concentrations of innovative activity on the absorption and diffusion of knowledge.The flow of knowledge across origindestination communities is hypothesized to depend on the size of innovative communities.Not accounting for innovation size (innovative communities and social relationships) in the flow of knowledge across origin-destination regions will result in aggregation bias in the parameter estimates.We anticipate that the resistance or separation factors in a spatial interaction model of

Literature Review
Knowledge spillovers are critical for innovation and new value creation in an increasingly knowledge-intensive economy and have generated substantial scholarly attention (Paul and Kogut, 1999;Jaffe, 1989;Rosenthal and Strange, 2001;Jaffe et al., 1993;Audretsch and Feldman, 1996).Two approaches are employed to investigate knowledge spillovers using patent citations.The spatial interaction regression (see Maurseth and Verspagen, (2002)), and the matching or choicesampling method (see Jaffe et al., (1993)).
The spatial interaction model describes origin-destination flows, in this instance the flow of knowledge measured by patent citations.The origin-destination areas are typically political boundaries such as U.S. states, U.S. metropolitan areas, or countries.Kerr and Kominers (2015) study citation flows among postal codes to consider how distances between postal codes influence patent citation rates.The knowledge flow between the two locations depends on the mass at each location measured by the stock of patents and the separation between them.Separation is measured by physical distance and technology compatibility.The origin-destination knowledge flow declines with increasing distance between them, increases with technology compatibility, and is positively associated with the amount of activity (patents) at each location.The use of technology compatibility between the two locations is analogous to controlling for the distribution of technology activities in the choice-sampling method.The focus in the spatial interaction model is on the interaction patterns at the aggregate level rather than the individual level as in the choicesampling approach.Compared to the matching method, the spatial interaction model avoids selecting a control group and directly estimates distance and border effects.© Southern Regional Science Association 2018.
The key variable in the spatial interaction model to measure localization of knowledge spillovers is the physical distance between the origin-destination pairs.Two measures of distance or mileage between the pairs are typically used: a continuous variable that gives the average effect across all distances and a series of indicator variables for distance bands between origin-destination pairs.The indicator variables show how the effects of knowledge spillovers dissipate with increasing distance.Previous literature has found that proximity, measured by the mileage between origin-destination pairs, is an important determinant of knowledge flows measured by patent citations (Maurseth and Verspagen, 2002;Li, 2014;Peri, 2005;Mukherji and Silberman, 2013).
Our spatial interaction regression model examines knowledge flows among U.S. metropolitan areas using patent citation data.The factors in our model are the separation between the knowledge origin and destination regions measured by physical distance and technological compatibility, the existing stock of knowledge measured by the maximum number of patent citations, and a number of indicator variables that account for factors that have not been measured by the other variables.We include fixed effects for all cited and all citing regions and a variable for the intra-state or border effect.Following LeSage and Fischer (2010), LeSage and Pace ( 2008), LeSage and Llano (2007), and Scherngell and Barber ( 2011), we incorporate into the spatial interaction model of knowledge flows spatial effects that captures origin and destination based dependence.
Regions with higher innovative activities are likely to have increased contact rates with other inventors and their ideas.The contact rate for knowledge spillovers or diffusion will be influenced by the number of research communities (i.e.R&D labs, universities, and the number of inventors) that find out about a new discovery (Warnick, 2006).Metropolitan areas with large concentrations of research communities will have higher contact rates within and across regions thereby speeding up the spread of an idea.In the spirit of Carlino et al. (2007), who demonstrated that urban density has a positive impact on innovation productivity, the innovation contact rate represents another view of the economic benefits of agglomeration.A similar explanation is provided by Agrawal et al. (2006) who found that knowledge flows to an inventor's prior location are approximately 50 percent greater than if they had never lived there, suggesting that social relationships, not just physical proximity, are important for determining knowledge flow patterns.It is more likely that inventors relocate to metropolitan areas with similar innovation research infrastructure or other large innovation regions.Consequently, distance and other barriers to knowledge transfer are likely to be less pronounced for such groups.
In contrast to the spatial interaction models, the matching or choice-sampling method compares the citations to a set of patents and control patents controlling for the preexisting geographic distribution of technological activities.Jaffe et al. (1993) found evidence supporting localized knowledge spillovers at the state and metropolitan statistical area (MSA) levels.Thompson and Fox-Kean (2005) challenged the Jaffe et al. (1993) findings in the selection of the control patents.Jaffe et al. (1993) used three-digit technology classes while Thompson and Fox-Kean (2005) used a finer six-digit technology class.When restricting the control patents to those that have any subclass code in common with the originating patents, Thompson and Fox-Kean (2005) found no evidence supporting localization at the state and MSA levels.Murata et al. (2014) bridged the debate between Jaffe et al. (1993) and Thompson and Fox-Kean (2005) by developing a distance-based test of localized knowledge spillovers that embeds the concept of control patents.They found solid evidence supporting localization even when using finely grained technology controls.
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Recent papers using the choice-sampling method confirm the localization of knowledge spillovers and identify a positive state border effect in the U.S. Singh and Marx (2013) simultaneously measured the impact of different geographical borders and proximity on knowledge spillovers using patent citations.They employed a regression framework that uses a choice-based sampling method to estimate the likelihood of citation between random patents.A surprising finding is that U.S. state borders have a robust same-state effect on knowledge spillovers.State borders serve as constraints on knowledge diffusion even after accounting for geographic proximity.Singh and Marx (2013) did not provide an explanation for the significant state border effect and suggested that further insight into the puzzling state border effect will require more research.Belenzon and Schankerman (2013) documented state-level localization effects for knowledge spillovers for patents assigned to only universities.They employed the choice-based sampling method comparing the characteristics of corporate patents that cite university patents to a control group that does not.Controlling for distance, inventors in the same state as the cited university patent are substantially more likely to cite the university's patents than an inventor outside of the state.Possible explanations for the state border effect is the importance of local information in dealing with tacit knowledge and state policy that promotes local commercial development of university innovations.
The rest of the paper is organized as follows.Section 2 describes the data and framework for the analysis.Section 3 presents the knowledge flow regression model and results.Section 4 concludes.

Description of Data
Measuring knowledge flows is fraught with difficulties since they often leave no paper trail (Krugman, 1991).Absent a perfect measure, researchers frequently use patent citation statistics (Jaffe et al., 1993).Our study of knowledge flows uses the NBER patent database that includes citations to all patents issued between 1976 and 1999. 1 We use all patents granted to an inventor in the U.S. between 1990 and 1999.The NBER database provides geographical information of the inventors of the issued and cited patents.Information on cited patents is available since 1963.Thus, citing patents are between 1990 and 1999 but the cited patents (backward citations) are between 1963 and 1999.This database does not distinguish between inventor and examiner added citations. 2The addresses of first inventors are used to aggregate the patent data by metropolitan statistical areas (MSAs or PMSAs). 3Where ZIP code information is available, the ZipList5 software is used to locate the patent to an MSA or PMSA.In the absence of ZIP code information, © Southern Regional Science Association 2018.the name of the MSA and state is used to make the MSA or PMSA assignments.Patents for which addresses are not available are not included in the analysis.The use of metropolitan areas is consistent with the geographic boundary within which knowledge flows, as suggested by Breschi and Lissoni (2001).
We chose for analysis the MSA/PMSAs (referred to as MSA in the remainder of the paper) that were granted more than 1,000 patents in the 1990s, or an average of 100 new patents per year.One hundred and six MSAs/PMSAs satisfied this criterion.Patents granted to these 106 metropolitan areas account for 74 percent of all patents granted during the 1990s in the U.S. by the U.S. Patent and Trademark Office (USPTO), and 87.5 percent of patents granted to all metropolitan areas in the U.S. San Jose, CA had the largest number of patents granted between 1990 and 1999 at 24,986 patents.Nashville, TN ranked 106 th with 1,032 patents granted from 1990 to 1999.For each of the 106 MSAs, we calculate the total number of citations made by patents issued in the MSA from 1990 to 1999 to patents granted to each of the other 105 MSAs.This results in 11,130 observations, or citation flows between pairs of metropolitan areas.Only 24 out of the 11,130 observations reported zero citations, which means that 99.8 percent of the pairs of MSAs absorbed knowledge from each other.Since 99.8 percent of the pairs of regions absorbed knowledge from each other, we eliminated the 24 zero citation observations.4 To test our hypothesis regarding the impact of a region's innovative activity size on knowledge flows, the 106 metropolitan areas are split into large innovative regions and all other regions.Large innovative metropolitan areas are defined as those whose total number of patents issued from 1990 to 1999 was more than one standard deviation from the average.The mean number of patents granted in the sample is 4,115 and one standard deviation from the mean is 4,446.Thus, the cut off for a large innovative region is 8,561 patents.Thirteen metropolitan areas satisfy this criterion. 5All other regions are referred to as "small."Four sub-samples are analyzed: large citing and large cited; large citing and small cited; small citing and large cited; and small citing and small cited.Table 1 show the 106 regions, identifies them as either an MSA or a PMSA, displays their number of patents granted in the 1990s, and their patent number rank. 6 literature exists that examines inter-metropolitan patenting behavior (Uallachain, 1999;Carlino et al., 2007).Our purpose is to examine the impact of innovative region size on knowledge spillovers measured by patent citation flows across regional pairs.We take as given the intermetropolitan patenting behavior and control for it in our model by including the maximum number of citations feasible and fixed effects dummy variables for citing and cited regions.We expect that variables that are statistically significant in the inter-metropolitan patenting behavior literature will be statistically different when comparing our large-small innovative regions.Carlino et al. (2007) found that employment density (jobs per square mile) is a positive and statistically significant determinant of inter-metropolitan patenting behavior (patents per capita).In our sample, density is The Review of Regional Studies 48(2) © Southern Regional Science Association 2018.The Review of Regional Studies 48(2) © Southern Regional Science Association 2018.measured as the number of assignees to a patent per square mile, a more precise measure than jobs per square mile.A test of differences in means confirms that the density for large innovative regions ( 16) is greater than and statistically different from the density for small innovative regions (4) at the 1 percent level or better.

Framework
Given an available stock of knowledge, Kj, in region j, region i with pi innovations can make a maximum of piKj references to that knowledge when each innovation uses all available knowledge.The actual use or flow of knowledge, measured as citations made by MSA i (destination) of patents granted to MSA j (origin), that takes place from the stock in region j (knowledge generating) to the innovations in region i (knowledge absorbing) during a period of time depends on piKj and additional factors as follows: (  1) is a classical spatial interaction model that describes origin-destination flows.Distij is the geographical distance between the two regions, Indexij is a measure of the technological compatibility between the two regions, DState,ij is a regional border indicator variable, and DCiting,i and DCited,j reflect other regional characteristics that may influence an MSA's ability to absorb knowledge from other regions (DCiting,i) and diffuse knowledge to other regions (DCited,j).Examples of these characteristics are the industrial structure, the number of inventors, the diversity of the economy, and the entrepreneurial culture.Since the indicator variables take the values of 0 or 1, they are modeled as exponential functions.Furthermore, the literature on spatial interactions with origin-destination effects shows that the interaction between a dyad not only depends on the properties of the individual units comprising the dyad, but the regions neighboring them as well.These spatial effects can have a significant impact on the interactions between the two units.These additional spatial origin and destination © Southern Regional Science Association 2018.
effects emanating from regions surrounding the knowledge generating and absorbing regions are captured by the variables Bj and Ai, respectively.α, β, Ω are elasticities of knowledge flow to spatial destination effects, spatial origin effects, and maximum possible references to knowledge generating region's knowledge, respectively.
Measurement of the technology compatibility index, Indexij, follows the one developed by Maurseth and Verspagen (2002).In calculating the compatibility index, we use the two-digit patent classification system used in the NBER patent citation data. 7If two MSAs are technologically close, the compatibility index between them will be high.In regression analysis, the coefficient of this index reflects the number of citations that are due to the similarities between the MSAs in technological space.To investigate distance in the context of knowledge flow between MSAs, we use a continuous variable and distance dummy variables.Distance, measured as the mileage between pairs of MSAs, gives the average effect across all distances.Distance dummy variables find the impact of different distances on knowledge flow, for example, whether the impact of being within 50 miles of each other is different than being between 2,500 miles.
Types of spatial cross-section models that account for spillovers from neighboring regions include the spatial autoregressive model ( =  +  + ), the spatial lag of X model ( =  +  + ), the spatial error model ( =  +  + ), and the general spatial model ( =  1  +  +  2  + ), where y represents the vector of dependent variables, X represents the matrix of explanatory variables, W is a spatial weight matrix,  and  are error terms.In this paper, we use the spatial lag of X model to capture the notion that citations of patents made by a region depend not only on the economic conditions, such as technological compatibility of the dyad, but also the characteristics of the neighbors of the citing and cited MSAs.That is, if an MSA is surrounded by other MSAs that are similar, the neighbors are likely to have similar types of innovators and economic relationships, and this can lead to a higher amount of knowledge flows between the dyad.This is a local spillover where endogenous interaction and feedback effects are not present.Endogenous interaction and feedback refers to when an impact on a neighbor region also impacts the neighbors of the neighbor, LeSage (2014).
To introduce the spatial factors in the estimation of the flow of knowledge between the MSAs we follow the origin-destination filtering approach used by LeSage and Pace (2008).To select the MSAs that neighbor the origin and destination MSAs, a spatial weight matrix, W, is used.This matrix uses the traditional inverse distance approach to model that interactions and spatial effects are stronger at closer distances.Using k to represent an MSAj that neighbors the origin MSAj, dkj to represent the inverse distance between MSAs k and j, the origin effects for origin j is given by   = ∑

𝑧𝑧
. The origin and destination effects are obtained from the multiplication of the Kronecker products W ⊗ I and I ⊗ W, respectively, and the vector of the compatibility index.For the full-sample, W is a 106x106 inverse distance spatial weight matrix and I is a 106x106 identity matrix.For smaller subsets such as large MSAs only, the matrices are © Southern Regional Science Association 2018.
13x13.Weight matrices typically assign a value of 0 to regions beyond a certain geographical space surrounding the destination and origin regions.To determine a reasonable estimate of the region within which such spatial effects may be operational, the full model is first estimated without the spatial effects.The results from the regression with the distance dummy variables will be used to inform the distance beyond which the weight matrix assigns a 0 value. 8o reduce heteroskedasticity in the patent product term, pi * Pj, the dependent variable is transformed to the logarithm of the actual citations as a share of the entire pool of potential citations, that is ln . The ratio of actual to all potential citations is also pi * Pj used by Duguet and MacGarvie (2005), among others.Although our citation data is count data, the transformation of the data converts the citation counts into a continuous variable.Consequently, the use of OLS is justifiable in our case.
To obtain an estimate of the direct effect of the distance dummy variables directly when multiple dummy variables are included, we use the variable transformation suggested by Hirschberg and Lye (2001).The transformed equation is described below.
( In this transformed equation, Dd denotes the distance dummy variables D50, D100, D200, D 300 , D 400 , D 500 , D 600 , D 700 , D 800 , D 900 , D 1000 , D 1500 , D 2000 , D 2500 , and D g2500.D50 takes the value 1 when the cited MSA is within 50 miles of the citing MSA, D100 takes the value 1 when it is within 51 to 100 miles from the citing MSA, and so on for the other distance dummies.The variable D g2500 takes the value 1 for distances greater than 2500 miles from the citing MSA.This variable is used for the variable transformations.The term    2500 gives the proportion of cited MSAs that are d miles away from the citing MSA to the cited MSAs that are greater than 2500 miles away.Similarly, M, denotes the MSA (Akron or Boston) used to transform the MSA dummy variables.In the above equation,  ����������� is the mean of the logarithm of the compatibility index and  ����� and  ����� denote the means of lnA and lnB, respectively.    gives the ratio of MSA pairs that belong to the same state to those that do not belong to the same state (DiffState).The constant of this regression equals the mean of the dependent variable. � in Equation ( 2) denotes the mean of the dependent variable.Furthermore, the coefficient of the transformed dummy variable exactly equals the coefficient of the variable without the transformation thus allowing us to generate direct estimates of each of the dummy variables.This transformation also allows for an easier interpretation of the coefficients of the dummy variables since such interpretations are difficult when multiple categories of dummy variables are included in a regression.Given that the constant of the regression equals the mean of the dependent variable, the values of the coefficients are interpreted as the amounts they add to or subtract from the mean of the dependent variable.Definitions and descriptive statistics for the regression variables are reported in Tables 2 and 3.
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KNOWLEDGE FLOW REGRESSION RESULTS AND IMPACT OF MSA SIZE
The estimated regression coefficients from Equation (2) are reported in Tables 4 through  8. Results for the full sample are in Table 4. Table 5 shows patent citation flow between pairs of large metropolitan areas.Table 6 displays patent citation flow for pairs of large MSAs citing patents from small MSAs.The patent citation flow for pairs of small MSAs citing patents from large MSAs is shown in Table 7. Table 8 displays the patent citation flow among pairs of small MSAs.Columns 1 and 2 provide regression results without local spatial origin and destination spillover impacts and Columns 3 and 4 show the results with local spatial origin and destination effects.Columns 1 and 4 include the distance between the areas measured as a continuous variable; Columns 2 and 3 have distance between the areas measured as a series of dummy variables.The main results from the regression estimation relate to the border effect, the impact of proximity and technology compatibility, and the effect of local spatial origin and destination spillovers.The results are discussed below.© Southern Regional Science Association 2018.The Review of Regional Studies 48(2) © Southern Regional Science Association 2018.© Southern Regional Science Association 2018.

Border Effect
The results for the full sample show a positive and statistically significant border effect, a finding consistent with prior research.After the full sample is split by MSA innovation size, we  find in the basic model with no spatial spillover and a continuous distance variable, Column 1, that the border effect dummy variable is positive and statistically significant in all regressions except for large MSAs citing large MSAs' patents.Once the distance dummy variables and spatial origin © Southern Regional Science Association 2018.
and destination spillover are included in the regression model, Column 3, the border effect is not statistically different from zero in two of the sub-samples: large citing, large cited and small citing, The Review of Regional Studies 48(2) © Southern Regional Science Association 2018.small cited.These results show that being in the same state has no impact on knowledge flow or that crossing a state border does not reduce knowledge flow when the city pairs are large-large or small-small.This result is in contrast with the positive and significant border effect found by Singh © Southern Regional Science Association 2018.
and Marx (2013), Belenzon andSchankerman (2013), andLi (2014).Our results control for MSA innovation size, distance measured as dummy variables, and local spatial spillovers which reduces the border impact when comparing the full sample to the split samples.
The size of the border regression coefficient is smaller in the split samples and not statistically different from zero in two instances.The lesser impact of the border effect is consistent with the hypothesis that crossing a U.S. state should not matter for knowledge flow because, unlike European regions, no significant differences in language, culture, and political institutions exist across U.S. states.The small impact of the border dummy suggests that physical distance has a stronger impact on knowledge flow than the presence of a regional border for a country like the U.S. State borders do constrain knowledge diffusion for the large citing, small cited and small citing, large cited city pairs.This finding is consistent with previous literature and shows that when city pairs are different in their innovation size, being in the same state increases knowledge flow.The positive, statistically significant border effect means that unobserved forces are at work that represent spillovers within a state for these MSA pairs.

Distance
The literature supports the localization of knowledge spillovers and suggests that there is an extremely rapid distance decay associated with local knowledge spillovers (Audretsch and Feldman, 1996;Jaffe et al., 1993).Singh and Marx (2013) found that knowledge flows are greatest when the source and recipient are co-located within the same MSA and that the distance effect gradually falls with distance, but is positive at all distance categories even for distances between 4,000 and 6,000 miles.Belenzon and Schankerman (2013) reported that the estimated distance effects are robust and the impact of distance dies out after about 100 to 150 miles, and Li (2014) reported a statistically significant negative distance when it is measured as a continuous variable.
A summary of the distance effects across the four sub-samples and the full sample is displayed in Table 9.The full sample results show a negative and significant distance continuous variable and positive indicator distance variables up to 500 miles with spatial spillovers.The results for the sub-samples using distance dummy variables, without local spatial spillovers (Column 2), indicate an above average number of citations made to patents granted within 500 to 800 miles.These results show that MSAs within a broad geographic area surrounding an MSA positively impact the number of patents granted to it, a finding consistent with prior research.Once spatial origin and destination spillovers are included, the impact of distance on knowledge flows is diminished.Importantly, the impact of distance on knowledge flow is found to be dependent on the relative sizes of the MSAs involved.When two large MSAs are involved, the positive effect of proximity exists up to 800 miles, not accounting for spatial spillovers.Once spatial spillovers are accounted for, the effect of distance remains positive up to 800 miles, but the impact of distance is smaller.When small MSAs are involved, either as citing or as cited MSAs, distance plays a more significant role.Without spatial dependency effects taken into consideration, citations are positive when small MSAs are within 700 miles.When all MSAs involved in knowledge transfer are small and spatial spillovers are included, the positive effect of distance extends to 700 miles but the impact is smaller.These results demonstrate that when considering the effect of physical distance on knowledge transfers, MSA innovation size must be taken into consideration.Knowledge spillovers are more localized for small MSAs than for large ones.Distance is not as much of a resistance factor in knowledge flow for larger metropolitan areas compared to smaller regions as shown by the log of distance continuous variable being smaller and less significant for larger regions compared with smaller regions (-0.057 vs -0.131 from Column 4).© Southern Regional Science Association 2018.Due to the semi-log nature of the regression, a coefficient of β for a dummy variable implies a (e β − 1) * 100 percent change for the dependent variable.Being within 50 miles has a substantial positive impact on knowledge flow for small MSAs citing patents from small MSAs, increasing the flow by 209 percent.The localization of knowledge flows for these small MSAs decays rapidly, increasing knowledge flow by 37 percent for distances between 51 and 100 miles, and by 24 percent for distances between 101 and 200 miles.For large MSAs, citing patents from other large MSAs within 50 miles increases knowledge flow by 73 percent and within 51 to 100 miles increases knowledge flow by 22 percent.

Technology Compatibility
The coefficient for technological compatibility, as measured by the variable Index, is lowest (0.435) for a pair of large metropolitan areas and greatest (1.372) for a pair of smaller ones.The full sample estimate is 1.239.A possible explanation is that larger MSAs are more technologically diverse and hence capable of absorbing more diverse knowledge than smaller ones.To test this hypothesis we compare the mean technology compatibility index in both samples.Our expectation is that the technology compatibility index will be less in the large to large sample compared with the small to small sample.If two MSAs are technologically close, the compatibility index between them is high.A means difference test confirms that the large to large mean technology compatibility index, 0.467, is statistically smaller that the small to small mean index, 0.512.As expected, the standard deviation is substantially greater in the large to large sample compared with the small to small sample (0.146 vs 0.123).
The appropriate level of classification of patents is an issue in the selection of control patents using the choice-sampling method of measuring knowledge spillovers.The use of threedigit or finer six-digit technology class impacts the findings on localization of spillovers (Thompson and Fox-Kean, 2005;Murata et al., 2014).We examine the impact of the level of © Southern Regional Science Association 2018.classification in the spatial interaction regression method by comparing the results of one-digit and two-digit technology classification in calculating the technology compatibility index and the resulting impact.The results using the two-digit classification (reported in the paper) compared with the one-digit classification are: the regression coefficient on the index variable is substantially larger; the regression coefficients on the distance indicator variables are slightly smaller; the regression coefficients on the spatial origin and destination effects are slightly smaller; and the border effect variables are slightly larger.9The overall results of the analysis remain unchanged.While technology classification is an important issue in the choice-sampling method, it does not materially impact the main findings using the spatial interaction regression method on the localization of knowledge spillovers.

Local Spatial Spillover Impacts
To address the potential for spatial spillover effects emanating from economic and technological conditions of neighboring regions, we use the spatial lag of X model.In this framework, spatial weights are applied to the model's explanatory variables.In the knowledge flow model, the key explanatory variables include the technological compatibility index, distance, a state border indicator, and citing MSA and cited MSA indicator variables.The results reported in Tables 4-8 include only results for the spatially weighted technological compatibility index, shown in columns 3 and 4. Including spatially weighted MSA indicator variables along with the unweighted indicator variables led to extreme multicollinearity problems.The coefficients of the other explanatory variables do not change with the inclusion of this spatial effect.Additionally, since including the own MSA indicator variable is more important than those of the neighbors, the results reported do not include any spatially weighted citing or cited MSA variables.
The origin and destination local spillover variables, based on the technology compatibility index, are positive but not statistically significant for the large-to-large MSA pairs when distance dummy variables are included.There are no local spillovers in knowledge flow between large MSA pairs.This result is consistent with the finding that technology compatibility exerts a small impact on knowledge flow between pairs of MSAs that are large.For the large and small dyads, the local spillover is stronger for regions surrounding the smaller MSAs.The spatial origin spillover is highest when large MSAs cite smaller ones, indicating that small MSAs surrounded by MSAs with high technological compatibility are more successful in disseminating their knowledge to larger MSAs.Researchers in the large MSA are likely more aware of knowledge in small MSAs if there are other surrounding small MSAs with similar technology compatibility.Reinforcing this impact is the spatial destination spillover being largest for small MSAs citing larger MSA knowledge indicating that small MSAs surrounded by others with high technology compatibility are more successful in absorbing knowledge from large MSAs.

CONCLUSION
The results for the sub-samples split by number of patents (innovation size) confirm our hypothesis that the geographic and technology resistance factors are less for knowledge flows between pairs of metropolitan areas that are both large and greater for pairs of MSAs that are both small.Innovative MSA size reduces the importance of spatial and technology proximity in mediating knowledge flows among regions.Moreover, technology compatibility local spillovers do not impact the knowledge flow among large MSAs, while they do impact knowledge flow when © Southern Regional Science Association 2018.
small MSAs are involved in knowledge flow with a large MSA.Knowledge spillovers are highly localized among smaller MSAs with a very rapid decay.In the small to small sample, distances less than 50 miles increase knowledge flows by 209 percent, while distances between 51 and 100 miles increase knowledge flows by only 37 percent.Not considering innovation size in knowledge flow among city origin-destination pairs results in aggregation bias that masks significant differences in the determinants of knowledge flow.
For the small-to-small and large-to-large MSA pairs, the state border effect of being within the state is not statistically different from zero when fine-grained distance indicator variables are included.While we do find a positive and significant border effect for the other two sub-samples, they represent only 22 percent of the MSA pairs and there may be a California effect since five of the large innovation MSAs are in that state.Existing studies of the state border effect in the U.S. have not had an explanation for the significant effect.Singh et al. (2010) concluded that "future research should investigate why state borders seem to play a surprisingly robust role in shaping knowledge diffusion patterns."Our results show that state borders do not play a substantial role in shaping knowledge diffusion patterns once fine-grained distance effects, spatial local spillover, and innovation size of metropolitan areas are accounted for.
Regions with an existing infrastructure that supports patent production at a high rate have an advantage in absorbing knowledge flow compared with smaller patent production regions.Our result is consistent with Packalen and Bhattacharya (2015) who found that larger city size provides considerable advantage in inventive activities.That advantage manifests itself in lowering the impact of proximity in mediating knowledge flows.The substantial difference in the localization of knowledge spillovers depending on innovative region size has implications for state innovation policy.State investment in innovation and research and development is based, in part, on the idea that knowledge spillovers are localized within a state.This idea is accurate only for investments in small innovative regions, but not for large innovative regions.Large innovative regions may, however, generate substantial knowledge spillovers within the region, an area for future research.The insignificant state border effect also suggests that state investments will not have as much knowledge spillovers as anticipated by policy makers.

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The destination effects can be addressed similarly.Using the same spatial matrix W, the neighbors of the destination MSA are identified.The destination effect gives the weighted values of the compatibility indexes of the neighbors of the destination MSA and is given by   = ∑

Table 1 : Metropolitan Regions Patents Granted in the 1990s
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