Endogeneity in pharmaceutical knowledge generation: An instrument‐free copula approach for Poisson frontier models

Haschka, Rouven E.; Herwartz, Helmut Article — Published Version Endogeneity in pharmaceutical knowledge generation: An instrument‐free copula approach for Poisson frontier models Journal of Economics & Management Strategy Provided in Cooperation with: John Wiley & Sons Suggested Citation: Haschka, Rouven E.; Herwartz, Helmut (2022) : Endogeneity in pharmaceutical knowledge generation: An instrument‐free copula approach for Poisson frontier models, Journal of Economics & Management Strategy, ISSN 1530-9134, Wiley, Hoboken, NJ, Vol. 31, Iss. 4, pp. 942-960, https://doi.org/10.1111/jems.12491 This Version is available at: https://hdl.handle.net/10419/266766 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc-nd/4.0/ Received: 4 May 2021 | Revised: 21 December 2021 | Accepted: 13 May 2022 DOI: 10.1111/jems.12491 ORIGINAL ARTICLE Endogeneity in pharmaceutical knowledge generation: An instrument‐free copula approach for Poisson frontier models Rouven E. Haschka 1 |Helmut Herwartz 2 1 Institute of Econometrics and Statistics, University of Cologne, Cologne, Germany 2 Chair of Statistics and Econometrics, University of Göttingen, Göttingen, Germany Correspondence Rouven E. Haschka, Institute of Econometrics and Statistics, University of Cologne, Universitätsstr. 24, D‐50923 Cologne, Germany. Email: [email protected] Abstract This study provides an assessment of the R&D–patent relation of European pharmaceutical firms that are not flawed by endogeneity biases. Firms invest in R&D and generate latent knowledge which then manifests in observable patent outcomes through a Poisson model. The process of turning R&D into knowledge is described by a production process subject to inefficiency and endogeneity. To estimate a Poisson stochastic frontier model, the suggested novel copula‐based approach directly accounts for the dependence between the endogenous regressors and the inefficiency component. Hence, its implementation does not require any instrumental variables. Simulation results underline that the proposed estimator outperforms conventional instrumental variable estimators. Neglecting endogeneity leads to a substantial underestimation of the R&D elasticity of patents generated in the European pharmaceutical industry. 1|INTRODUCTION High innovation output likely influences the business success in the long run, while ongoing investments in R&D are necessary to continuously generate knowledge and remain innovative (Griliches, 1998; Jaffe, 1989; Siebert, 2017). Since innovation‐relevant knowledge generated is usually unobserved, it can only be approximated, for instance, by the number of patent grants (Griliches, 1998; Jaffe, 1989; Siebert, 2017). As patent counts are discrete, recent empirical models investigating the R&D–patent relationship rely on count data models, such as Poisson regressions (Fé & Hofler, 2013). Considering inefficiencies within innovation processes has also gained increasing attention in the recent literature (Fu & Yang, 2009; Pieri et al., 2018). Regarding each firm as a “producer”of knowledge, knowledge generation can be described by means of a production process (Daub, 2008; Dittmer & Strätz, 2012; Griliches, 1998; Siebert, 2017). At the same time, this embedding allows for determining firm‐specific efficiency within the R&D–patent relationship (for literature reviews, see Abbas et al., 2014; Czarnitzki & Toole, 2011; Holgersson, 2013). Quantitative assessments of innovative efficiency are often performed in the framework of stochastic frontier analysis (SFA, for methodological reviews, see Amsler et al., 2016; Coelli et al., 2005; Kumbhakar & Lovell, 2003). When modeling stylized innovation processes, a particular model specification issue arises if firms have some a priori information on eventually inefficient knowledge generation, and they adjust the level of R&D expenditures accordingly J Econ Manage Strat. 2022;31:942–960.942 | wileyonlinelibrary.com/journal/jems This is an open access article under the terms of the Creative Commons Attribution‐NonCommercial‐NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non‐commercial and no modifications or adaptations are made. © 2022 The Authors. Journal of Economics & Management Strategy published by Wiley Periodicals LLC. (Siebert, 2017). However, such dependence is unobserved to the analyst and has become one of the most common forms of endogeneity in the context of innovation efficiency modeling, which makes drawing valid causal conclusions virtually impossible. 1 Maximum‐likelihood‐based instrumental variable (ML‐IV) estimation methods have recently been proposed to handle endogeneity in SFA models (Amsler et al., 2016; Kutlu, 2010; Prokhorov et al., 2020; Tran & Tsionas, 2013). However, a general drawback of ML‐IV methods is, that they rely upon the availability of consensual outside information to construct the instruments (Windmeijer & Santos Silva, 1997). In search of suitable instrumental information, empirical studies typically condition on “internal”information, such as lagged explanatory variables, which are often weak and fail to fully cope with endogeneity (N. Wang & Hagedoorn, 2014). Seeing that suitable—that is, “external”—instrument information in frontier models is often scant, unavailable, or weak, this study proposes a frontier model for Poisson distributed outcomes that we conceptualize as an extension of the IV‐free joint regression model using copulas introduced by Park and Gupta (2012). Given that copula techniques have been successfully adopted for Gaussian SFA's (Tran & Tsionas, 2015), we extend it towards a Poisson SFA model. Although joint estimation models have been intensively studied (Becker et al., 2021; Haschka, 2022; Tran & Tsionas, 2021) and used in various empirical settings (Datta et al., 2017; Haschka et al., 2021; Manchanda et al., 2004; Zhang et al., 2017), their general applicability in Poisson SFA models has not yet been addressed. The method is based on a copula function to directly model dependencies of endogenous regressors and inefficiency and thus does not require any external information. As an additional model feature, we adopt scaling properties (H.‐J. Wang & Schmidt, 2002), which enable a structured perspective on potential determinants of inefficiency. We conduct Monte Carlo experiments and assess the performance of the proposed estimator in comparison with stylized ML as in Fé and Hofler (2013), and IV estimation with “weak”and “strong”instrumentation (N. Wang & Hagedoorn, 2014). Simulation‐based evidence suggests that the copula estimator (i) provides unbiased estimates if model regressors are correlated with unobserved inefficiency, (ii) copes with reverse causality, and (iii) delivers quantitative assessments of the correlation between inputs and inefficiencies. The empirical analysis aims at an unbiased understanding of knowledge generation in a cross section of 137 unaffiliated European pharmaceutical firms. The focus on the pharmaceutical industry follows similar rationales as the analysis of N. Wang and Hagedoorn (2014) and Fé and Hofler (2013). On the one hand, the empirical evidence on knowledge generation has highlighted important sector‐specific characteristics. Hence, approaching knowledge generation conditional on the pharmaceutical industry does not suffer from unobserved cross‐industry heterogeneities as a potential channel of endogeneity. Moreover, the cross‐sectional perspective immunizes the analysis against interindustry variation in other factors such as technological opportunities and the effectiveness of patents as a means of appropriating returns from R&D. On the other hand, the patenting rate in the pharmaceutical industry is among the highest in high‐technology sectors (Chen & Chang, 2010; Danguy et al., 2019; Griliches, 1998). Significant differences in gross margins have been primarily attributed to the superior records of pharmaceutical companies in protecting their innovations by patents. For instance, pharmaceutical gross margins are nearly twice as large as those in the semiconductor industry (Chen & Chang, 2010). Moreover, it is worth highlighting that a major fraction of costs is incurred in the R&D stage, with the “R proportion”holding outstanding relevance for this sector. In this regard, endogeneity biases that originate in the personal attributes of the drug developers hold specific relevance for the analysis. For instance, experienced researchers could be expected to reduce trial‐and‐error experiments in laboratories. In a similar vein, a research staff familiar with patenting processes and authorities has the potential to position a firm among the top performers within the pharmaceutical industry. Finally, given that drug development touches sensitive societal fields, such as public health and health policy, both the composition and experience of the research staff could hold particular importance for effective knowledge generation in the pharmaceutical industry in comparison with other sectors. Going beyond endogeneity concerns, it is worth noting that for this industry research taking place in universities—that is, pharmaceutical faculties—could act as a major source of knowledge transfer and the acquisition of key business skills (Haschka & Herwartz, 2020; Powell, 1998). Hence, the description of knowledge generation subject to intraindustry competition is likely to benefit from the potential to scale inefficiencies in SFA models, as in H.‐J. Wang and Schmidt (2002). Our empirical results enable two major conclusions. First, we diagnose with high significance regressor endogeneity challenging the estimation of the marginal causal effect of R&D expenditures for steering patenting outcomes in the European pharmaceutical industry. Specifically, this effect is underestimated by almost 50% under exogeneity assumptions. Given empirical evidence of R&D‐inefficiency dependence, it should be acknowledged that firms adjust their R&D inputs according to their own inefficiency. As this dependence is unobserved to analysts or competitors, active management should crucially take account of feedback between R&D and inefficiency for the HASCHKA AND HERWARTZ | 943 purposes of market screening and unraveling competitive efficiency levels. Second, pointing to important knowledge spillovers, pharmaceutical firms in proximity to a university are on average 56% more efficient than firms without such a neighbor. Accordingly, the acquisition of knowledge from neighboring universities largely improves innovativeness and complements in‐house innovative activities. In Section 2, we establish an intraindustry model of patent generation at the firm level, outline the Poisson SFA model with inefficiency scaling, and describe the copula‐based estimation approach. In Section 3, we examine the finite sample performance of alternative estimators by means of a Monte Carlo study. Section 4provides a detailed comparative analysis of patent outputs of the European pharmaceutical industry, before Section 5 concludes. Detailed simulation results and a set of complementary simulations are provided in the Supporting Information. 2|LATENT KNOWLEDGE GENERATION AND IV‐FREE ESTIMATION The efficient management of knowledge generation is crucial for long‐term business success and requires an unbiased assessment of the translation from R&D expenditures into knowledge manifestations, such as patents. In this section, we first sketch a production model to describe the transition of innovation inputs into knowledge subject to inefficiencies that emerge from an intraindustry perspective covering a cross section of competing firms. We establish an empirical model that further advances the approach taken by Fé and Hofler (2013) and apply a production function approach in which patents are the realization of a latent knowledge generation process. Second, we turn to an encountering of potential sources of endogeneity as pertinent threats to an unbiased analytic assessment of causal effect structures. In the third instance, we develop an explicit SFA count data model and suggest an IV‐free approach to model estimation. 2.1 |Approximating the firm‐level innovation process An extensive body of literature has shown that innovation processes are more complex than stylized translations of R&D into knowledge (for a systematic review, see Hall et al., 2005). On the one hand, knowledge generated is unobserved and thus remains latent by nature. When using the number of patents as an approximation, it is necessary to take into account the discrete nature of patent outcomes (Hall et al., 2005). On the other hand, it has become pertinent to allow for inefficiencies within the process turning R&D into knowledge (Griliches, 1998; Siebert, 2017). Against this background, we model innovation processes by means of Poisson frontier models to explicitly consider these particularities (Fé & Hofler, 2013). Stylized characteristics of industrial R&D activity suggest that it is reasonable to assume that patent counts follow a Poisson distribution (for pioneering work, see Hausman et al., 1984; Jaffe, 1986; Pakes & Griliches, 1984). Especially in the pharmaceutical industry, patents can be thought of as measuring the number of successful outcomes from a large but unobserved number of projects within a firm's R&D lab, each of which has a small probability of success. Furthermore, intraindustry companies are subject to similar restrictions within the market, for example, to the likelihood that the patent office responds to an inquiry from a patent applier. Nevertheless, the success in patenting likely depends on the experience of researchers or personal relations in dealing with the patent office. Accordingly, firms with less experienced researchers in this regard are less successful in patenting, giving rise to inefficiencies in patent generation. To make these considerations explicit, consider the following model for iN =1,2,…, firms: patents ~ Pois(latent knowledge )with ii(1) fX βTElatent knowledge = ( ; ) × . iii (2) In (2), latent knowledge i denotes the knowledge generated from a vector of innovation inputs X i subject to a production function f Xβ(;) i , for example, of Cobb–Douglas type (Haschka & Herwartz, 2020; Siebert, 2017), and technical efficiency ∈TE [0, 1] i . Both knowledge and technical efficiency are known to the firm, but unobserved by the analyst. Since patents can be seen as an observable manifestation of knowledge, we assume that latent knowledge i turns into patents i through a Poisson distribution, that is,  XTE[patents , ] = Pois(latent knowledge ) iii i. Further, the 944 | HASCHKA AND HERWARTZ Poisson model comes with the advantage that latent knowledge i does not need to be observed and the number of patent applications does not need to be known. 2.2 |Endogeneity in knowledge generation models A particular empirical challenge arises if firms have some a priori information on eventually inefficient knowledge generation and adjust the level of R&D expenditures accordingly (Siebert, 2017). In (2), ≠XTECov[ , ] 0 ii introduces endogeneity to the model. However, unbiased assessments are pivotal to unravel efficiency differences for the strategic positioning of firms to improve the processes of knowledge generation and patenting. Regarding the origin of endogeneity in knowledge generation models, one might consider three nonexclusive channels. First, patent generation is typically seen as a reflection of R&D expenditures. However, patents themselves might also lead to R&D activities (Correa, 2004; Scherer, 2001). For instance, Czarnitzki and Toole (2011) argue that patents reduce uncertainty and thereby influence R&D decisions. Accordingly, endogeneity might result from patterns of reverse causality (Arora et al., 2003;Cincera,1997). Technology shocks that affect investment decisions provide a second origin of endogeneity. For instance, Alexopoulos (2011) has shown that technological shocks are positively linked to inputs of knowledge production, such as R&D. Since such shocks are unobserved to analysts, they show up in model terms assessing productive efficiency. As a result, technology shocks induce correlation between model residuals and explanatory input variables (Alexopoulos, 2011; Schilling, 2015). Third, the experience of researchers or personal relations in dealing with the patent office are also likely to shape efficiency. Seeing a major fraction of costs being incurred in the R&D stage with outstanding relevance of the “R proportion,”this channel appears to be particularly important in the pharmaceutical industry. Obtaining unbiased estimates under endogeneity invokes further complications if productive output is quantified in terms of a count variable, since discrete distributions are often more complex to handle in comparison with stylized continuous output measures (Fahrmeir & Lang, 2001). For instance, Fé and Hofler (2013) adopt a Poisson frontier model to estimate a knowledge generation function for a number of patents awarded to pharmaceutical firms conditional on supposedly exogenous R&D expenditures. Cincera (1997) estimates the effect of R&D expenditures on discrete patent outcomes by means of lagged explanatory variables as instruments. However, after testing instrument validity, the author finds that any (weak) exogeneity hypothesis for R&D is rejected with conventional significance. Nonetheless, seeing strong arguments for the prevalence of endogeneity (Cincera, 1997; Correa, 2004; Scherer, 2001), endogeneity could have biased the understanding of the R&D–patent relation obtained within the framework of standard Poisson frontier regressions. 2.3 |Copula‐based Poisson frontier modeling Figure 1presents a flowchart of the proposed model. When conducting market or competitor analyses, (i) the R&D‐ inefficiency dependence (indicated by the double arrow) must be taken into account and (ii) approximating observed patent outcomes by means of a count data model (e.g., Poisson regressions) is crucial, since otherwise both estimated R&D elasticities and firm‐specific inefficiency measures are biased. Determining competitors' efficiency holds particular relevance for the strategic positioning of the firm, since inefficient firms are likely to perform weaker regarding long‐run business success. Further, the consideration of external determinants of inefficiency allows one to disentangle firm‐specific unobserved inefficiency from observed environmental influences, such as university spillovers (Siegel et al., 2003). Especially in the pharmaceutical industry, similar research taking place in universities with pharmaceutical faculty FIGURE 1 Stylized illustration of the proposed model HASCHKA AND HERWARTZ | 945 likely promotes local knowledge that can be absorbed and thus benefits firm performance (Szücs, 2018;Yu‐ ming, 2009). For instance, knowledge externalities originating in universities likely complement the in‐house research of firms located in geographic proximity (Blume & Fromm, 2000; Oshri et al., 2015). In this respect, unraveling sources of (in)efficiency is pivotal when conducting market or competitor analyses. To estimate the relationship in (1) and (2) summarized in Figure 1, we start from the copula approach of Tran and Tsionas (2015) that copes with endogenous interrelations, and generalize it such that it is applicable to Poisson distributed outcomes. The approach is particularly useful, as it avoids the search for suitable instrumental information. More specifically, we construct the joint distribution of the endogenous regressors and the inefficiency component conditional on the assumption that the outcome variable is Poisson distributed. As an additional model feature, we adopt scaling properties and thereby allow for observable determinants of inefficiency. 2.3.1 |The Poisson stochastic frontier model For the general model representation, let Y idenote the discrete output variable, for example, the number of patents granted to firm i . By assumption, Y iexhibits a Poisson distribution for a sample of N producers, that is, YλiN~Pois( ), =1,2,…, , ii (3) where λ i is the latent mean of the Poisson distribution, which is presumed behind the issuance of observable patent counts of firm i , that is, latent knowledge. The structured SFA model relates the unknown mean to observable covariates, that is, innovation inputs, and a stochastic component that accounts for deviations from fully efficient knowledge generation. Specifically, {} xαzβ λ κu= exp + + − , iiii ′′ (4) uhu=* , ii i (5)  () uμσ *~,. iu +* 2(6) Equation (4) describes the innovative production process with λ i capturing the unobserved knowledge generated. Exogenous inputs are collected in the vector x i . Extending the model in Fé and Hofler (2013), we allow for endogenous input variables that are correlated with u * i and collected in the p ‐dimensional vector z i . For both categories of input variables, we assume linear influences on λ i with parameters α and β . Finally, the intercept κ accounts for total factor productivity (Hausman et al., 1984). 2 Similar to common approaches in SFA modeling, the term u i describes production inefficiencies, that is, ∈∞u[0, ) i.In(5), we extend the basic Poisson SFA by introducing the function hito scale the distribution of u * i , that is, uhu= * ii i , where the underlying stochastic component u * i exhibits a truncated normal distribution (Kumbhakar & Lovell, 2003). By virtue of the scaling index sδh= exp( ′ ) ii, it is possible to relate inefficiencies to (observable) environmental influences, for example, spillover effects, collected in the vector s i (H.‐J. Wang & Schmidt, 2002). 3 2.3.2 |Maximum‐likelihood estimation Under the assumption that all inputs are exogenous, that is, β=0 in (4), the conditional distribution of Y igiven x i , s i ,and u * i can be derived by multiplying the respective marginals, that is, the probability mass function of Y iand the probability density function of u * i . Accordingly, the likelihood for a sample of N observations is 946 | HASCHKA AND HERWARTZ ⏟ ∞                                                  () () θxsLY λλ Ygu du(,,)= exp(−) !** , i NiYi i i fuY i ML =1 0 **, i ii (7) where () θ αδκμσ=,′,′,, ′ u* 2is the vector of model parameters to be estimated and () gu * i is the density of the half‐ normal distribution (see Equation 6). Owing to the latency of u * i , it has to be integrated out from the joint density for each observation i . Since the likelihood lacks a closed form, the integral has to be evaluated by means of numerical integration procedures. 4 The estimator resulting from the maximization of L l nML mimics the estimator in Fé and Hofler (2013) and is henceforth denoted as θ ˆML . 2.3.3 |Joint estimation by means of copulas Now consider the case when endogenous covariates enter the model, that is, ≠β 0 in (4). Let ( ) f zzuY,…, , *, ipi ii1 denote the joint density of zz ( ,…, ) ipi1 , u * i and Y ifor a single firm i . The joint distribution captures the entire dependence of the endogenous regressors and the inefficiency term (Park & Gupta, 2012). Hence, the endogeneity problem can be solved by maximizing ( ) f zzuY,…, , *, ipi ii1instead of ( ) f uY **, iiin (7). Noticing that the conditional probability mass function of Y icorresponds to () f zzupY,…, , *( ) ipi ii1, only the joint density ( ) f zzu,…, , * ipi i 1has to be determined in advance. Following Tran and Tsionas (2015), we suggest a copula function to construct and estimate ( ) f zzu,…, , * ipi i 1 conditional on the observed data. According to Sklar's theorem (Sklar, 1959), the joint density can be given as  () ( ) ()() f zzu cFz FzGugu fz,…, , *=(),…,(),**() , ipi iipipi ii j p jji111 =1 (8) such that ⋯⋯ ∞ ∞ ∞ ∞∞   () f z z u dz dz du,…, , ** = 1 ipi iipi i −−011 .In(8), capital and small letters denote marginal cumulative distributions and density functions, respectively, and c is the copula density, which links its components by means of the probability integral transform (Li & Racine, 2007; Nelsen, 2006). When replacing () gu * i in (7)by ( ) f zzu,…, , * ipi i 1 from (8), the likelihood accounts for endogenous interrelations between zz ( ,…, ) ipi1 and u * i , since the dependence structure is completely characterized by means of the joint density. As a result, the copula‐based likelihood is ∝ ∞ ∞                                                     () () ()() ()() θxzsLY λλ YcFz Fz Gu gu fz du λλ YcFz Fz Gu gu du (,,,)= exp(−) !(),…, (), ** () * exp(−) !(),…, (), *** . i NiYi i ippi ii j p jji i i NiYi i ippi iii Cop =1 011 =1 =1 011 i i(9) Before proceeding with the implementation, we briefly describe model identification (for a detailed outline of identification in copula‐based joint estimation models, see Haschka, 2022). Recall that the copula function disentangles joint variation among endogenous regressors from remaining unexplained variation due to u * . Accordingly, the model is identified as long as the marginal distributions of the endogenous regressors differ from the half‐normal distribution assumed for u * . Subsequently, the conditional expectation of u * given z becomes a nonlinear function and this nonlinearity allows us to identify the linear regression coefficients in (4) through the joint distribution. However, if z j is indeed half normal, the model is not identified because the copula can no longer distinguish the variations as a result of endogenous regressors from the variation caused by u * (Tran & Tsionas, 2015). Consequently, the identification problem has important implications for empirical applications. If one assumes a half‐normal distribution for the inefficiency term, it has to be examined whether the endogenous regressors exhibit half‐normal distributions before application. As the copula further HASCHKA AND HERWARTZ | 947 requires sufficient variation in regions where the conditional expectation is nonlinear, binary endogenous regressors also cannot be modeled. 5 2.3.4 |Implementation of the copula‐based estimator Regarding the copula c , we use the Gaussian copula, which is applicable in many settings and has several desirable theoretical properties (Danaher & Smith, 2011). For instance, the Gaussian copula is more flexible for multidimensional modeling than Archimedean copulas (for further arguments in favor of the Gaussian copula, see Park & Gupta, 2012;Tran & Tsionas, 2015). 6 The density of the Gaussian copula () () cFz Fz Gu(),…, (), *;Σ ippi i 11 in (9)isgivenby ∕ { } () () () () () () cFzFzGu IFz FzGu (·) = (det(Σ)) × exp −1 2Φ(()),…,Φ(()),Φ*′ 1 2×(Σ− )Φ(()),…,Φ(()),Φ*, ippi i pippi i −12 −111 −1−1 −1+1 −111 −1−1 (10) where Φ − 1 is the quantile function of the standard normal distribution and Σ is the correlation matrix formalizing the dependence among the respective components in c . As a particular merit of the Gaussian copula, the off‐diagonal entries in Σ coincide with the Pearson correlations. Hence, the copula delivers quantitative assessments of the correlation between endogenous regressors and inefficiency, which is a valuable diagnostic when modeling economic production patterns. The parametric forms of ( ) G u* iand () gu * i follow from the assumption in (6). The marginal densities f z() jji in (9) do not contain any parameter of interest and can be dropped from the likelihood, since they enter as normalizing constants. Building on Tran and Tsionas (2015), we replace Fz Fz(),…, () ippi11 by their respective empirical counterparts before maximizing the likelihood. Given observed samples of zji, jp =1,…, ; iN =1,…, , we use the empirical cumulative distribution function of z j ,thatis, ≤  Fzz ˜=() jNi Nji j 1 +1 =1 0. 7 Seeing that the empirical cumulative distribution function is a step function by definition, the model is at odds with the assumptions behind Sklar's theorem (Sklar, 1959), which states that the copula in (10) is only uniquely defined if its margins are continuous. However, we show in the Supporting Information that such a violation is not problematic, as the model can be identified regardless of the noncontinuity of F ˜j, even in small samples. Finally, L l n Cop (see Equation 9) is maximized with respect to () θ αβδκμσ=,′,′,′,, ,vechl[Σ]′′ u* 2, where vechl [ Σ] stacks the lower diagonal parameters of Σ in a column vector comprising p unknown parameters. 8 To test for the presence of regressor endogeneity, either model selection criteria such as Akaike's Information Criteria or Bayesian Information Criteria or likelihood‐based tests can be used. Regarding the latter, the correlation coefficients in vechl[Σ]capture the endogeneity of the regressors in z . Under regularity conditions, the ML estimators of model parameters are asymptotically normally distributed. Thus, the Wald statistic for testing the null hypothesis of the exogeneity of regressor zp,thatis, Hρ:= 0 z 0pis ∕ ρ ρ ˆSD(ˆ ) zz pp. The likelihood‐ratio and score tests for endogeneity can also be derived in a straightforward manner under the copula model. In the following, the resulting estimator is denoted θ ˆCop . In summary, the proposed approach is IV‐free and copes with interrelationships between input selection and productive inefficiency without any additional distributional assumptions (Tran & Tsionas, 2015). 3|SIMULATION STUDY To examine the finite sample properties of the proposed copula estimator ( θ ˆCop ), we conduct some Monte Carlo experiments. The IV‐free estimator promises unbiased estimation under correlation between observed production inputs and unobserved stochastic inefficiency. however, in the case of exogeneity, it might suffer from inefficient parameter assessment in comparison with stylized ML estimation ( θ ˆML ). Alternatively, ML‐IV estimators could hold interest to handle endogeneity in Poisson frontier models (Drivas et al., 2014). In sum, our comparative investigation 948 | HASCHKA AND HERWARTZ involves four estimators, namely, the copula estimator, the common ML estimator, and two variants of ML‐IV estimation, denoted θ ˆ IV1 and θ ˆ IV2 below, which differ in the choice of the external information employed. We next explain the generation of Poisson distributed stochastic outcomes and encounter the alternative estimators entering our comparative analysis. We evaluate distributional properties of rival parameter estimates by means of stylized box plots in the main text of this section. More detailed simulation results on the average performance of alternative estimators, as well as discussions on inferential results are provided in the Supporting Information and briefly discussed. 3.1 |Data generation Discrete outcomes YλiN~Pois( ), =1,2,…, , ii are drawn after structuring the Poisson mean λ i (see Equation 6)as () λ καxβzhu= exp + + −* , iiii i(11) where κ is an intercept parameter, xN~(0,1 ) iis an exogenous input variable and potential endogeneity is channeled through a correlation between the second input variable z i and productive inefficiency u * i . To be explicit about the origin of potential endogeneity, it holds that                 zγwηuηφ=+ andCorr *,= , ii iii(12) where w N~(0,1) iis strictly exogenous and might be considered as “external”information (if ≠γ0 )to enable ML‐IV estimation (for a similar approach, see Drivas et al., 2014). Apparently, the joint generation of u * i and η i is crucial for the emergence of endogeneity in the form of a simultaneous input adaptation to technology shocks. Specifically, we draw u * i and η i jointly from a normal copula with correlation                 uη φ Corr *,= iiand a half‐ normal marginal distribution for u * i ,thatis,  () uσ *~0, iu +* 2. The choice of the marginal distribution of η i allows us to distinguish two frameworks of data generation and model estimation. On the one hand, we consider a benchmark scenario for which the underlying true distributions are in line with density specifications required for the purposes of ML‐IV estimation. On the other hand, we consider a scenario of intrinsic model misspecification in the sense that actual shocks η i are drawn from a skewed distribution with excess kurtosis. Subsequently, η i is falsely considered Gaussian for ML‐IV estimation. For benchmarking purposes and the consideration of density misspecification the marginal distributions of η i are ∕ () ησ ηχ σ ( i) ~ 0, and (ii) ~ (( (2) −2) 2) , iηiη 22(13) respectively. To enhance model flexibility, the inefficiencies u * i enter the Poisson mean in scaled form uuh=* iii, hδssN= exp( ), where ~ (0, 1). ii i (14) Regarding parameter settings, we set the values of φ to analyze the performance of alternative estimators under exogeneity ( φ =0 ) and degrees of moderate ( φ =. 4 ) and strong endogeneity ( φ =. 8 ). The remaining model parameters are set as κ = 2 , α βδ===.5 , σ σ== 1 ηu 2* 2 and γ= 1 . We consider finite samples of alternative sizes N = {100, 250, 500} , and regard N = 500 as sufficient to unravel the asymptotic properties of the considered estimators. Each simulation experiment is replicated 1000 times. All computations were performed in R. We next encounter the alternative estimators and their adaptation to the simulated data‐generating models (DGMs). HASCHKA AND HERWARTZ | 949 that pharmaceutical research outcomes are related to the population's health (Baines et al., 2018). Consequently, stronger production restrictions or sharper regulations extended to drug approvals might be considered as potential manifestations of technological shocks (Baines et al., 2018). To complement our assessment of knowledge generation of European pharmaceutical firms in 2016, we compare our results to those found in Fé and Hofler (2013) and P. Wang (1998) for the case of US producers in 1976. Both studies use the same US data and a Poisson model that is similar to our Model I in terms of the covariate information employed. Since Fé and Hofler (2013) neglect (potential) endogeneity, a comparison with our ML results is natural. Our ML estimate of the elasticity of innovative output with respect to R&D spendings ( .252 1 ) is substantially smaller. In P. Wang (1998), the associated coefficient ranges from .671 6 up to 1.588, whereas Fé and Hofler (2013) estimate 1.207, which is more than four times larger than our estimate. Despite cross‐country heterogeneity, a sizeable reduction of R&D effectiveness in the over course of more than 40 years is consistent with literature findings diagnosing a steady decline in pharmaceutical payoffs to R&D (Hashimoto & Haneda, 2008, report a decline of 50% from 1983 to 1992). Such diminishing returns in knowledge generation might be ascribed to intensified competition (Pammolli et al., 2011), or the fact that pharmaceutical innovations have become incrementally spread over a large variety of products (Aghion & Howitt, 1998;Ha&Howitt,2007). 4.3.3 |Knowledge spillovers Seeing a steady decline in pharmaceutical R&D effectiveness (Hashimoto & Haneda, 2008; Pammolli et al., 2011), a promising complement to produce one's own innovations by investing in R&D is to absorb external knowledge—that is, “diffusion spillovers”(Nelson, 2009). A growing body of literature has highlighted that the capacity to absorb local external knowledge holds particular relevance for firms in the pharmaceutical industry to remain innovative (Furman et al., 2005; Harhoff, 2005; Runiewicz‐Wardyn, 2013). To approximate such effects, the dummy variable uni indicates if a university or public research institution with more than 1000 employees (excluding auxiliaries) is located within a 10‐km road distance from a firm (for a similar approach, see Hall et al., 2005).AscanbeseeninTable1, all three estimated Poisson frontier models lead to the same conclusion in qualitative terms. A negative and significant effect of uni indicates that geographic proximity to a university or public research institution is associated with higher levels of efficiency in knowledge generation. Thus, we can confirm the findings of Siegel et al. (2003), namely that university presence facilitates knowledge spillovers which stimulate innovative activity. To assess the model implications in an economically meaningful way, it is necessary to jointly evaluate the quantitative results with inefficiency variance. 19 The copula estimator reveals a variance estimate of 1.183 and an estimate of − .410 7 for the coefficient of uni. Since these estimates jointly determine innovative efficiency, we are able to attribute efficiency differences to local knowledge spillovers originating from universities. In detail, pharmaceutical firms located in geographic proximity to a university are on average 56% more efficient than firms without such a neighbor. 20 Since universities are powerful sources of external knowledge (Szücs, 2018), our results indicate that the capacity to absorb such local spillovers largely enhances innovativeness. From the firm perspective, similar research taking place in private research establishments and universities hosting a pharmaceutical faculty could be the major source of spillovers due to technological transfers and the acquisition of key business skills (Powell, 1998). 5|CONCLUSION Previous studies have highlighted five important aspects that should be taken into account when modeling firm‐level innovation processes, namely (i) innovation‐relevant knowledge generated by the firm is unobserved, (ii) the discrete nature of patents should be taken into account when using them as an approximation of innovative output, (iii) the process turning R&D into knowledge is likely affected by inefficiencies that (iv) are subject environmental factors, and (v) R&D inputs are typically endogenous. We establish an empirical model to simultaneously consider these aspects regarding the pharmaceutical industry. The suggested Poisson frontier model allows for inefficiency scaling, and accounting for regressor endogeneity does not require instrumental information. Rather, building upon the Gaussian SFA model of Tran and Tsionas (2015) feedback linking R&D expenditures and inefficiency is quantified by means of a (Gaussian) copula. We examine the finite sample behavior of the proposed estimator by means of Monte Carlo simulations. The results show that the coefficient estimates of the proposed method are not affected by endogeneity. From a set of complementary simulation experiments, we conclude that 956 | HASCHKA AND HERWARTZ the performance of the proposed estimator remains unaffected in case of departures from some underlying model assumptions, such as non‐Gaussian dependence structures. However, the proposed approach is not without limitations. First, although the Gaussian copula is highly flexible, especially when dealing with multiple endogenous inputs as in an empirical application, it still builds upon the assumption of linear dependency of its margins. Second, Monte Carlo results show that unbiased estimates come at the cost of high variances in small samples. Although instrumental variable estimation is the method of choice under endogeneity, weak instrumentation or density misspecifications are pertinent threats to such fully parametric estimation. Nevertheless, we expect that the proposed method provides useful alternatives to many empirical problems addressed in the form of Poisson frontier models. We employ the proposed estimator to quantify a knowledge production function for European pharmaceutical firms in 2016. In particular, innovation processes are subject to technological interactions between innovative efficiency and R&D. On the one hand, we can confirm the arguments of Alexopoulos (2011) that stylized technological shocks are positively linked to the inputs of knowledge production. On the other hand, endogeneity biases are pertinent, since restrictive assumptions of regressor exogeneity result in a marked underestimation of an R&D elasticity and efficiency. In addition to the assessment of endogenous input selection, we re‐examine the effect of university knowledge spillovers on innovative efficiency. Our results support the “absorptive capacity”hypothesis (Cohen & Levinthal, 1989,1990). Since pharmaceutical research is costly, innovation policy should explicitly develop the capacity to absorb local technology spillovers. In particular, universities hosting a pharmaceutical faculty stimulate beneficial externalities that foster innovativeness, which can be exploited for efficiency enhancements. Accordingly, a deeper understanding of the channels of external pharmaceutical knowledge acquisition would benefit the strategic positioning of firms to become more innovative. ACKNOWLEDGMENTS We gratefully acknowledge helpful comments and suggestions received from three anonymous reviewers and the editor Daniel Spulber. Financial support by the German Research Association (DFG) Research Training Group 1644 “Scaling problems in Statistics”(grant no. 152112243) is also gratefully acknowledged. Much of the research documented in this paper was conducted while the corresponding author was employed at the Chair of Statistics and Econometrics at the Georg‐August University of Göttingen, Germany. Open Access funding enabled and organized by Projekt DEAL. ORCID Rouven E. Haschka http://orcid.org/0000-0002-2916-9745 ENDNOTES 1 We discuss the important origins of endogeneity in knowledge generation models below in Section 2.2. The linkage of endogenous covariate information and stochastic inefficiency can lead to biased estimates for causal effects if applied methods build upon assumptions of regressor exogeneity. As a result, the outcome of managerial decisions can hardly be assessed under the presumption of independent actions combined with the invariance of all other factors. 2 To be precise, total factor productivity is related to κexp( ) . 3 Although scaling functions are widely employed in Gaussian SFA models (Kumbhakar & Lovell, 2003), they have not yet been used for the class of count data models. 4 In contrast to Fé and Hofler (2013,2020), who approximate the integral using Halton sequences, we follow Tran and Tsionas (2015) and use Gauss–Konrod quadrature with 50 points. 5 We assess adverse effects on the performance of the estimator when dealing with binary endogenous regressors in the Supporting Information. 6 We characterize the performance of the proposed estimator under misspecification of the copula function in the Supporting Information. 7 The rescaling factor ∕N 1 (+1 ) instead of ∕ N1 ensures that the empirical cumulative distribution is well bounded in ( 0, 1 ) . 8 We use the derivative‐free simplex method for numerical optimization of Nelder and Mead (1965) to obtain the ML estimates of model parameters θ . The proposed copula estimator can be classified as a semiparametric method, as it combines the nonparametric distribution of endogenous regressors with the parametric distribution of the inefficiency term. In semiparametric models, the derivation of Hessian‐based asymptotic standard errors can be extremely difficult such that bootstrap procedures have become a natural alternative to assess estimation uncertainty (Park & Gupta, 2012). Accordingly, we use simple bootstrap techniques to estimate standard errors and HASCHKA AND HERWARTZ | 957 confidence intervals. For fully nonparametric approaches in SFA models for continuous outcomes, we refer the reader to Simar et al. (2016) and Mastromarco and Simar (2021). 9 The correlation between the endogenous variable ziand the instrument w iis on average about .705. Accordingly, w ican be considered as a strong instrument for zi. The “weak”instrument w* i exhibits an average correlation, which is smaller by a factor of .6715. 10 The coefficient attached to the exogenous variable x ( α ) did not exhibit a bias (see Tables S1 and S2 in the Supporting Information). For this reason, boxplots for α ˆare not included, although they can be provided by the authors upon request. 11 Mansfield (1982,1986) showed that in the pharmaceutical industry the patenting rate of inventions is about 84%. More recently, Danguy et al. (2019) document patenting rates between 79% and 74%. Noticing that not all knowledge output is patented, patents are only an imperfect indicator to approximate knowledge production. Nonetheless, they hold particular relevance in the pharmaceutical industry, since the patenting rate in this sector is among the highest in any high‐technology sectors (Danguy et al., 2019; Griliches, 1998). 12 We restrict our analysis to European patents, since the PATSTAT database only provides information about patents granted by the European Patent Office. 13 External agency workers were disregarded due to data limitations. 14 Input variables lagged by 1 year are available for only 94 firms out of the full sample of 137 firms. Accordingly, ML‐IV estimates are based on the reduced sample. 15 The copula estimator is set up with a three‐dimensional Gaussian copula to model the interdependence between R&D,MP, and the (unobserved) inefficiency term, which is assumed to be half normally distributed. Regarding the ML‐IV estimator, we additionally assume that the endogenous inputs R&D and MP are marginally normal. 16 As an alternative to considering Poisson distributed patent outcomes, discrete counts could also be assumed to follow a more flexible negative binomial distribution. Explicit estimation results for the negative binomial model are available upon request from the authors. Except for intercept estimates, ± 2 standard errors rule‐of‐thumb confidence intervals constructed around copula‐based estimates retrieved under the negative binomial model include the estimates as documented for the Poisson models I and II in Table 1. 17 Specifically, it holds that                  uσ *=* iu ϕ(0) Φ(0) , where ϕ(·) and Φ(· ) denote the probability density function and the cumulative distribution function of the standard normal distribution, respectively. 18 By taking the ratio, that is, ∕ 1 −.8444 1.30 2 . 19 Since ( ) uN σ *~0, * iu +2, then uδsu=exp( ) * ii i is ( ) Nδsσ ~ 0, exp(2 ) * iu +2, see Haschka et al. (2020). 20 By taking the ratio, that is, ∕ 1 −exp(2 × (−.4107) × 1) exp(2 × (−.4107) × 0 ) . The remaining two estimators provide nearly identical results. In detail, they report efficiency gains of 54.6% (ML) and 56.1% (ML‐IV). REFERENCES Abbas, A., Zhang, L., & Khan, S. U. (2014). 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