An Empirical Example

To illustrate the way theG_ijstatistic and the spatial econometric origin-destination flow models might be applied, patent citation data are used. Such data recorded in patent documents are widely recognised as a rich and fruitful source for the study of the spatial dimension of innovations and technological change⁸ (see, for example, Jaffe and Trajtenberg, 2002; Fischer et al., 2006b).

5.5.1 The Data

We use interregional patent citation flows as the dependent variable in the mod-els. The data specifically relate to citations between European high-tech patents.

By European patents we mean patent applications at the European Patent Office assigned to high-tech firms located in Europe. High-technology is defined to

8Each patent contains highly detailed information on the invention itself, the technological area to which it belongs, the inventors, the assignee and the technological antecedents of the invention.

Because patents record the residence of the inventors they are an invaluable resource for studying how knowledge flows are affected by the geography.

involve the ISIC-sectors aerospace (ISIC 3845), electronics-telecommunication (ISIC 3832), computers and office equipment (ISIC 3825) and pharmaceuticals (ISIC 3522). Self-citations, i.e., citations from patents assigned to the same firm, have been excluded, given our interest in pure externalities as evidenced by interfirm knowledge spillovers.

It is well known that the observation of citations is subject to a truncation bias, because we observe citations for only a portion of the “life” of an invention. To avoid this bias in the analysis we have established a five-years-window to count cita-tions to a patent.⁹The observation period is 1985–1997 with respect to cited patents and 1990–2002 with respect to citing patents. The sample used in this contribution is restricted to inventors located inn = 112regions, generally NUTS-2 regions, covering the core of “Old Europe” including Germany (38 regions), France (21 regions), Italy (20 regions), the Benelux countries (24 regions), Austria (8 regions) and Switzerland (1 region), resulting intoN= 12,432 interregional flows.¹⁰

Subject to caveats relative to the relationship between citations and spillovers, these data allow us to identify and measure spatial separation effects to interre-gional knowledge spillovers in this interaction system of 112 regions. Our interest is focused onK= 3measures:d⁽¹⁾is aN-by-1 vector that represents geographic distance measured in terms of the great circle distance (in kilometers) between the regions represented by their economic centres,d⁽²⁾ is aN-by-1 country dummy variable vector that represents border effects measured in terms of the existence of country borders between the regions.

As we consider the distance effect on interregional patent citations it is important to control for technological proximity between regions, as geographical distance could be just proxying for technological proximity. To do this we use a techno-logical proximity indexs_ij that defines the proximity between regionsiandj in technology space. We divide the high-technology patents into 55 technological sub-classes following the International Patent Code classification system. Each region is assigned a (55, 1)-technology vector that measures the share of patenting in each of the technological subclasses for the region. The technological proximity indexs_ij between regionsiandjis given by the uncentred correlation of their technological vectors. Two regions that patent exactly in the same proportion in each subclass have an index equal to one, while two regions patenting only in different subclasses have an index equal to zero. This index is appealing because it allows for a continuous measure of technological distance by the transformationd_ij = 1−s_ij. Appropriate ordering leads to theN-by-1 vectord⁽³⁾.

The productA_iB_j in (5.2) may be interpreted simply as the number of distinct (i, j)-interactions which are possible. Thus, it is reasonable to measure the origin

9For details on data construction seeFischer et al. (2006b). The trouble is that to obtain citations by any one patent application in year_t, one needs to search the references made by all patent applications after year_t. This is called the inversion problem that arises due to the fact that the original data on citations come in the form of citations made, whereas we need dyads of cited and citing patents to construct interregional patent citations flows.

10Note that intraregional flows are left out of consideration. In the case of cross-regional inventor teams the procedure of multiple full counting has been applied.

factor in terms of the number of patents in the knowledge producing regioniin the time period 1985–1997, and the destination factor in terms of the number of patents in the knowledge absorbing regionj in the time period 1990–2002 to produce the N-by-1 vectorsaandb.

5.5.2 Application of the G

Statistic

We briefly illustrate the application of theG_ijstatistic as a tool for identifying non-stationarities, using^oW as spatial weights matrix and error flows generated by the conventional log-additive spatial interaction model (5.6). The parameter estimates and their associated probability levels are summarised in Table 5.1, along with some performance measures. The estimated coefficients indicate that the origin, destina-tion and separadestina-tion variables are highly significant with appropriate signs of the coefficients. The results provide clear evidence that geographical distance is impor-tant, but less so than national borders. Most important is technological proximity.

This suggests that interregional knowledge flows seem to follow particular techno-logical trajectories, and occur most often between regions that are located close to each other in both technological and national spaces.

Table 5.1 The log-additive spatial interaction model: parameter estimates and performance measures (N= 12,432)

Ordinary least squares estimation Parameter estimates (p-values in brackets)

Constant [_α0] ₋4.851 (0.000)

Origin variable [α1] 0.594 (0.000)

Destination variable [α₂] 0.562 (0.000) Geographical distance [β1] ₋0.181 (0.000)

Country border [β₂] ₋0.592 (0.000)

Technological distance [_β3] ₋2.364 (0.000) Performance measures

AdjustedR² 0.563

Log likelihood ₋21,024.128

Sigma square 1.723

Notes: The spatial interaction model is defined by (5.6) where the stan-dard assumptions for least squares estimation hold.ais measured in terms of the log number of patents (1985–1997) in the knowledge pro-ducing regioni,bin terms of the log number of patents (1990–2002) in the knowledge absorbing regionj,d⁽¹⁾represents geographic distance measured in terms of the great circle distance (in kilometers) between the economic centres of the regionsiandj,d⁽²⁾border effects mea-sured in terms of the existence of country borders between regionsiand j, andd⁽³⁾technological distance measured in terms of the technolog-ical proximity indexsij. Model performance is given in terms of the adjustedR², the log likelihood and sigma square (the error variance).

High residual flows from neighbours of origin i to the destination region Ile-de-France

^

(a) Low residual flows from neighbours

of the region Leipzig to a destination region j

(b)

Fig. 5.2 Flows with selected significant_Gij(^o_W)statistic: (a) indicating high residual flows from neighbours of origin_ito the destination region ˆIle-de-France; and (b) indicating low residual flows from neighbours of the region Leipzig to a destination region_j

An origin-destination pair (i, j)with a significant G_ij(^oW)statistic indicates that there is local non-stationarity in flows from a neighbourhood of origin region ito destination regionj. A closer look at theG_ij statistic scores reveals that some destination regions and some origin regions have many significant statistics of the same sign. Figure 5.2 provides a visualisation for two specific cases. Figure 5.2 represents the case where ˆIle-de-France is the destination region, and Fig. 5.2 the case where Leipzig is the origin region. These two figures clearly indicate that there is spatial non-stationarity in flows when these regions are the destination and the origin, respectively.

Given regions with many significantG_ij(^oW)origin or destination factors in the spatial interaction model may be misspecified. We apply the G^∗_i•(^oW) and G^∗_•j(^oW) statistics¹¹ which may be – as Berglund and Karlstr¨om (1999) have shown – considered as test statistics for local non-stationarities in the origin and des-tination factors, respectively. Examining theG^∗_i•(^oW)andG^∗_•j(^oW)statistics we find that the regions ˆIle-de-France and Leipzig indeed have significant highG^∗_•jand significant lowG^∗_i• statistics, respectively. This might indicate that the significant G_ij(^oW)statistics for these regions could be attributed to non-stationarity in the destination and origin factors. The heterogeneity in the residual flows is confirmed by a Breusch–Pagan test. Its value of 568.1 is fairly significant (p= 0.000).

11For the definition of theG^∗_ij statistic see (5.18) together with Footnote 6. The subscript dot signifies that a sum is taken with respect to the subscript replaced by the dot.

5.5.3 ML Estimates of the Origin-Destination Spatial Econometric Flow Models

The pattern of residuals examined by means of theG_ijstatistic clearly indicates spa-tial autocorrelation, so that it makes sense to fit the models given by (5.25)–(5.26).

This section reports the ML estimates of the three spatial econometric model spec-ifications that reflect origin, destination and origin-destination spatial dependence of flows, respectively. We used the spdep package¹² running on a Sun Fire V250 with 1.28 GHz and 8 GB RAM to create the spatial weight matrices from polygon contiguities, and the errorsarlm procedure based on Ng and Peyton’s (1993) sparse matrix Cholesky algorithm to generate the ML estimates for the models. Using this algorithm, computation of the maximum likelihood estimates of the spatial econo-metric models that reflect origin, destination and origin-destination dependence of flows required only between 56 and 836 s, a remarkably short time considering that each iteration required calculating the determinant of a 12,432-by-12,432 matrix.

Table 5.2 contains the parameter estimates of the three model specifications and their associated log likelihoods. Moving from the log-additive spatial interaction model to the flow model reflecting spatial dependence at the origins (destinations) raises the log likelihood from−21,024.13 to−20,676.15 and−20,516.01, respec-tively. This is to be expected given the indication of the Lagrange multiplier test for spatial error dependence.¹³ It is clear that least squares which ignores spatial dependence and assumes residual flows to be independent produces a much lower likelihood function value. Capturing the dependences greatly reduces the residual variance and strengthens the inferential basis affiliated with the models. Moving further to the model specification that reflects spatial dependence at both origins and destinations raises the log likelihood further to−20,212.01.

The ML estimates display the expected signs, as the ordinary least squares esti-mates do. The estiesti-mates reported in Table 5.2 are not significantly different from each other, and, moreover, lie within the 95% confidence limits of the least squares estimates. So, in accordance with spatial econometric theory, mere spatial depen-dence in disturbances does not impact the point estimates, just the precision of the parameters. Turning to the spatial autoregressive parameter, we see that the estimates are highly significant. They clearly point to origin-based, destination-based and origin-to-destination-destination-based spatial dependence. The strength of depen-dence for destinations seems to be slightly more important than that for origins. But the estimate for the spatial autoregressive parameter reflecting dependence based on interaction between origin and destination neighbours is clearly most important.

Turning to the spatial autoregressive parameter ρ, we see that the estimates are highly significant. They clearly point to origin-based, destination-based and

12Source package: spdep 0.3-17 which may be retrieved fromhttp://cran.r-project.org/src/contrib/

Descriptions/spdep.html.

13Its value is at 833.6 and 1,254.8 respectively, when using^o_Wand^d_W, and fairly significant in both cases (p= 0.000).

Table 5.2 Spatial econometric flow models based on different spatial weights matrix specifica-tions: ML estimates using Ng and Peyton’s Cholesky algorithm (N= 12,432 observations)

Model specifications based on

oW ^dW ^oW+^dW

Parameter estimates (_p-values in brackets)

Constant [α0] ₋7.041 (0.000) ₋6.817 (0.000) - 4.658 (0.000) Origin variable [_α1] 0.598 (0.000) 0.576 (0.000) 0.593 (0.000) Destination variable [α2] 0.548 (0.000) 0.560 (0.000) 0.553 (0.000) Geographical distance [β₁] ₋0.194 (0.000) ₋0.223 (0.000) ₋0.224 (0.000) Country border [β2] ₋0.641 (0.000) ₋0.600 (0.000) ₋0.651 (0.000) Technological distance [β₃] ₋2.395 (0.000) ₋2.040 (0.000) ₋2.183 (0.000) Spatial autoregressive 0.311 (0.000) 0.365 (0.000) 0.613 (0.000) parameter [ρ]

Performance measures

Log likelihood ₋20,676.142 ₋20,516.006 ₋20,212.013

Sigma square 1.595 1.541 1.442

Computational time (s) 62 56 836

Diagnostics

(_p-values in brackets)

Likelihood-ratio test 3,695.970 (0.000) 1,016.243 (0.000) 1,624.23 (0.000)

Moran’s I ₋0.008 (0.929) ₋0.014 (0.990) ₋0.006 (0.939)

Notes: The origin-destination models are defined by (5.25)–(5.26).ais measured in terms of the log number of patents (1985–1997) in the knowledge producing regioni,bin terms of the log number of patents (1990–2002) in the knowledge absorbing regionj,d⁽¹⁾represents geographic distance measured in terms of the great circle distance (in kilometers) between the economic cen-tres of the regions_iand_j,_d⁽²⁾border effects in terms of the existence of country borders between iand_j, and_d⁽³⁾technological distance in terms of the technological proximity index_sij. The origin-based spatial weights matrix^o_Wis defined by (5.14), the destination-based spatial weights matrix^d_Wby (5.15), while the origin-destination-based weights matrix^o_W+^d_Wby (5.14)–

(5.15). Model performance is measured in terms of the log likelihood, sigma square (the error variance) and computational time in seconds (running on a SunFire V 250 with 1.28 GHz and 8 GB RAM)

origin-to-destination based spatial dependence. The strength of dependence for des-tinations seems to be slightly more important than that for origins. But the estimate for the spatial autoregressive parameter reflecting dependence based on interaction between origin and destination neighbours is clearly most important.