Pedro Cunha Neves, Tiago Sequeira

Transcription

1 CEFAGE Working Paper 2017/03 The Production of Knowledge: A Meta-Regression Analysis Pedro Cunha Neves, Tiago Sequeira Universidade da Beira Interior and CEFAGE-UBI Universidade de Évora, Palácio do Vimioso, Largo Marquês de Marialva, 8, Évora, Portugal Telf: cefage@uevora.pt - Web:

2 The Production of Knowledge: A Meta-Regression Analysis Pedro Cunha Neves Univ. Beira Interior and CEFAGE-UBI Tiago Neves Sequeira Univ. Beira Interior and CEFAGE-UBI Abstract The production of knowledge has been subjected to quantitative analysis following the development of the R&D based endogenous growth theory. There is a spirited discussion about the empirical validity of the semi-endogenous or endogenous (Shumpeterian) growth theories, with clear policy relevance. While the first theory points out ineffective policies in the long run, the latter allows for important effects of fiscal policies, namely subsidies or taxes to R&D. We survey the empirical literature on this topic and implement a meta-analytic regression for the spillover effect, which crucially determines the validity of those theories. We discover that the average spillover effect is lower than one but with the upper bound of the confidence interval above one. We also find that the spillover effect tends to be higher once patents are used to measure knowledge and the estimation of knowledge production accounts for foreign inputs and time-effects; it tends to be lower when the sample includes more countries, only rich economies and industrial data, and when estimations use instrumental variables. Although most recent contributions argue in favor of the Shumpeterian growth theory, we show that when looking at a more complete picture, semi-endogenous theory cannot be neglected. Keywords: semi-endogenous growth, Shumpeterian growth, knowledge production function, R&D, national innovative capacity, research policy. JEL Classification: O10, O30 1 Introduction The knowledge or ideas production function is the cornerstone of the modern growth theory and is the crucial element of the policy implications of this theory. Depending on its specification, policies such as subsidies to R&D and population growth may or not have effects in the long run. Moreover, it also governs the effects the knowledge accumulation may have on growth and across borders. Despite its importance, there is not yet a consensus on the functional form of the knowledge production function. Recent empirical papers have focused on distinguishing between the predictions of the first generation endogenous growth models of Romer (1990), Grossman and Helpman (1991), and Aghion and Howitt The authors acknowledge financial support from FCT, Portugal, and FEDER/COMPETE 2020, through grant UID/ECO/04007/2013 (POCI FEDER Departamento de Gestão e Economia and CEFAGE-UBI. Universidade da Beira Interior. Avenida Marques d Avila e Bolama, Covilhã, Portugal. Corresponding author. pcn@ubi.pt. 1

3 (1992), the second-generation growth models, i.e., the semi-endogenous growth model of Jones (1995) and Segerstrom (1998), and the Schumpeterian growth models of Aghion and Howitt (1998), Peretto (1998), Howitt (1999), and Peretto and Smulders (2002). Those theories differ in two crucial aspects: (i) the scale-effects (a long-run positive effect of the population size on growth), existing in the first generation endogenous growth models but dismissed by the most recent empirical evidence; (ii) the effect of population growth on economic growth present in the semi-endogenous theory. Shumpeterian models tend to reconcile theory and evidence eliminating effects of both the population size and the population growth on economic growth, recovering also the effects of endogenous variables and policies on growth. In particular, according to these theories, economic growth should depend on an interplay of firms, families and government choices. However, the empirical literature has not reached a consensus on the acceptance of semi-endogenous or Shumpeterian (endogenous) growth. In this paper we survey the literature on this issue and provide quantitative explanations for the heterogeneity found in the empirical works, using meta-regression analysis. Meta-regression analysis is a particular type of meta-analysis that allows summarizing and investigating the sources of variation in the results reported in a specific literature. It has been extensively used in several areas of economics research in recent years (Stanley et al., 2008). We discover that the average spillover (or standing-on-the-shoulders) effect tends to be smaller than one (tending to validate the semi-endogenous theory) but with the upper bound of the confidence interval higher than one (thus, not rejecting the Shumpeterian growth theory). It tends to be higher once patents are used to measure knowledge, as the knowledge production estimation accounts for foreign inputs (foreign knowledge and imports) and time-effects. It tends to be lower once the sample includes more countries, only rich countries, industrial data and estimation using instrumental variables (IV) methods, and accounts for the country distance to the technological frontier. We also obtain that semi-endogenous growth is more probable in regression studies (than in cointegration studies), in panel data studies, and in published articles. In Section 2 we survey the literature on the estimation of the knowledge production function. In Section 3 we describe the data for our Meta-regression analysis. In Section 4 discusses the evidence on the publication bias and on the average estimate of the spillover effect. In Section 5 we present and analyze the results of the multivariate Meta-regression analysis. Section 6 extends the analysis beyond the consideration of the spillover effect. Section 7 concludes. 2 Estimation of the knowledge production function Most contributions to the estimation of a knowledge production function depart from a functional form inspired by Romer (1990) as the following: 2

4 A = F (A, L A ) (1) where A is the stock of knowledge and L A represents a set of physical resources allocated to the knowledge production which may include physical or human capital, labor or final goods. This function can be enlarged to include the international effects (international knowledge, imports, foreign direct investment or convergence effects, usually named as distance to the frontier) and also difficulty or dilution effects usually considered in the Shumpeterian growth theory. Whatever its functional form, in order to allow for endogenous growth, asymptotically and in the absence of population growth, the derivative of f = g A = Ȧ A in order to A, must be lim t + f A > 0. This has been noted by e.g. Daalgard and Kreiner (2003). Porter and Stern (2000) were the first to estimate a knowledge production function similar to the logarithmic version of equation (1). 1 This paper claims to have found evidence for the validity of the Romer (1990) model, as the coefficient of A in the knowledge production function tends to be one (1) or close. The consequence of this is that the level of population would positively influence economic growth, which was rejected by the empirical evidence presented e.g. in Jones (1995). In Porter and Stern (2000), however, the semi-endogenous growth model is not clearly rejected in many specifications, as the Romer (1990) model demands that the coefficient of A in the knowledge production function is at least one and this does not happen in all the specifications in that paper. When considering variables linked with the national innovation capacity (e.g. industrial clusters, university capacities), Furman et al. s (2002) estimates favor the semi-endogenous growth, a conclusion also obtained by Furman and Hayes (2004) and Ramzi and Salah (2015) using the same framework. On the contrary, Hu and Mathews (2005) estimated the knowledge production function with similar controls for the Asian countries and found support for strong spillovers, such that the endogenous framework is validated. Using a more recent time span than Porter and Stern (2000), and a larger set of countries, Pessoa (2005) obtained estimates consistent with the Jones (1995) semi-endogenous model. Luintel and Khan (2009) use a new dataset of patents, highly intensive in research, and obtain high values for the domestic spillover effects, thus supporting the endogenous growth model. Luintel and Khan (2012) estimate a knowledge production function for a set of emerging countries and reach conclusions along the same line. Until now, the empirical literature has concentrated on the comparison between first-generation endogenous growth model (implying the existence of scale-effects) and the semi-endogenous growth model. Other papers test the validity of the second-generation scale-free endogenous growth model, which is the same as saying that they test the validity of semi-endogenous growth against the Shumpeterian endogenous growth. Dinopoulos and Thompson (2000) support scale-free versions of the Romer (1990) endogenous growth model. Laincz and Peretto (2006) analyze several time-series in developed countries. 1 This is crucially different from the literature that estimates the determinants of the TFP growth rate, initiated by Coe and Helpman (1995). Typically that literature departs from the functional form of production of the final good. 3

5 They do not find time trends for either employees or R&D workers per establishment or that average establishment size is positively and significantly related to growth, thereby supporting the Shumpeterian endogenous theory. Ulku (2007) evaluates a non-scale endogenous growth model in which the scale-effects are diluted by a difficulty effect measured by the population size. Their results tend to validate the model as they obtain increasing returns to scale toward A in equation (1) and also validate the relative research intensity term. However, this last condition holds only in the larger OECD countries, meaning that for other samples the first generation endogenous growth model is as described by the data. Ang and Madsen (2015) extend the knowledge production function to consider several international effects as well as a larger time span, from 1870 to They found a domestic spillover very close or higher than one and a difficulty effect statistically significant and with the sign that is predicted by the non-scale endogenous theory. Thus, this paper strongly supports the Shumpeterian theory. The same authors (Ang and Madsen, 2011) reach the same conclusion for a set of six Asian countries. The contributions surveyed so far estimate different functional forms for equation (1) and present point estimates for the degree of the returns to scale in order to past knowledge the standing-on-the-shoulders or spillovers effect. However, some other empirical contributions do not directly estimate the degree of the returns to scale to past knowledge and rely on the predicted relationships within a given theory to test it through unit root or cointegration techniques. Ha and Howitt (2007), Botazzi and Peri (2007), Madsen (2008) and Sedgley and Elmslie (2010) are in this group and their results tend to support Shumpeterian theory. Generally, the authors observe that while TFP growth has been stationary, R&D expenditure has shown a downward trend, and thus this relationship contradicts semi-endogenous growth. Botazzi and Peri (2007) also take into account the international spillovers, and Madsen (2008) applies the same type of tests to a longer time-series. Sedgley and Elmslie (2010) extend previous work to account for transitional dynamics. A third group of papers (Zachariadis, 2003; Venturini, 2012, and Minniti and Venturini, 2014) estimate a framework based on Aghion and Howitt (1992) such as: ι = Z L A x (2) where ι is the rate of innovation (or knowledge production), Z is an exogenous productivity parameter, L A is the amount of resources allocated to knowledge production, and x is the difficulty associated with the knowledge production. These studies use firm data instead of country data. If a significant relationship between the rate of knowledge production ι and the relative resources to the difficulty effect L A x is obtained, then the non-scale fully endogenous growth model is supported. In general, this is the result obtained by these papers. Table 1 depicts the reviewed articles or papers according to how they claim to support first generation models (e.g. Romer, 1990), semi-endogenous growth (e.g. Jones, 1995) or Shumpeterian growth (e.g. Aghion and Howitt, 1998; Peretto, 1998). Overall, a great majority of papers claim to support the 4

6 Shumpeterian endogenous growth theory. However, as noted above, among them are those that also do not reject the validity of semi-endogenous growth theory. An example can occur when a variety of equation (1) is estimated as Ȧ = Aφ (L A /x β ) λ. If the spillover or standing-on-shoulders effect, φ, is estimated as statistically significant and less than 1 and λ 1/β and statistically significant, then we may have some incomplete dilution of scale effects and still end up with elements of semi-endogenous growth. In that case, g A = λ(g L βg x ) (1 φ) (where g stands for growth rates), thus preserving elements of semi-endogenous growth. Table 1: Classification of Articles or Papers by the Theories Supported First Generation Theories Semi-Endogenous Theories Shumpeterian Theories Regressions Porter and Stern (2000) Furman, Porter, and Stern (2002) Ulku (2007a,b) with coefficient Furman and Hayes (2004) Luintel and Khan (2009) for the stock Hu and Mathews (2005) Venturini (2012) of knowledge Pessoa (2005) Luintel and Khan (2012) à la Ramzi and Salah (2015) Ang and Madsen (2011, 2015) Romer (1990) Madsen (2016) Regressions Zachariadis (2003) à la Aghion and Venturini (2012) Howitt (1992) Minniti and Venturini (2014) Unit-Root and Dinopoulos and Thompson (2000) Cointegration Ha and Howitt (2007) Tests and Other Botazzi and Peri (2007) Madsen (2008) Sedgley and Elmslie (2010) 3 The data In order to select the studies to include in the meta-analysis of the estimates of the spillover effect, we searched in the Econlit, Web of Science and Google Scholar databases for papers estimating a knowledge production function. We also searched the references and the citations of the most recent and cited studies on the topic. From the papers obtained, we selected those that present estimates for the coefficient and the standard deviation (or the t-statistic) associated with the stock of knowledge, i.e., an estimate of φ, as mentioned in the previous section. Both published articles and working papers were considered. Our search came up with a total of 12 studies. This set of studies comprises the papers mentioned in the previous section (listed in the first line of Table 1), with the exception of Ang and Madsen (2011) and Ramzi and Salah (2015), which were excluded because they do not report the values of the standard-error of the coefficient. Table 2 provides information about the main results and characteristics of each of the 12 selected papers. Our database is composed of 144 estimates of our effect size - the spillover effect, φ. They are graphically represented in the forest plot in the Appendix (Figure A.1). 5

7 Table 2: Summary of studies main characteristics Study Porter and Stern (2000) Furman, Porter, and Stern (2002) Furman and Hayes (2004) Hu and Mathews (2005) Code Nr. of estimates Mean of estimates Mean of SD of estimates Sample of countries Data level Data structure Mixed Country Panel Type of publication coefficients Working- Paper Rich Country Panel Journal Mixed Country Panel Journal East Asia Country Panel Journal Pessoa (2005) Mixed Country Panel Cross- Section Journal Ulku (2007)-a Rich Country Panel Journal Ulku (2007)-b Rich Industry Panel Journal Luintel and Khan (2009) Rich Country Panel Journal Luintel and Working Emerging Country Panel Khan (2012) Paper Venturini (2012) Rich Industry Panel Journal Ang and Madsen (2015) Mixed Country Panel Journal Madsen (2016) Rich Country Panel Journal Note: Code is the identifier in the database and makes a correspondence with the Forest Plot presented in Figure A.1. The estimates of φ range between a minimum of (by Ulku, 2007-a) and a maximum of (by Furman and Hayes, 2004). In 56 observations (39%), φ is equal to or higher than one, while it is lower than one in 88 observations (61%). This indicates that although most of the papers reviewed in the previous section stand for the Shumpeterian growth theory, when looking at point estimates the picture changes, as in most cases elements of the semi-endogenous growth theory cannot be excluded from empirical evidence. However, we should note that the spillover coefficient is very close to one and in some cases the upper limit of the confidence interval is above one. In fact, of the 144 observations, 73 present confidence intervals in which the upper limit is equal to or above one, and for 71 the upper limit of the confidence interval is below one. This means that for only 71 of the 144 (49%) observations, the Shumpeterian theory is clearly rejected. The lower limit of the confidence interval is above one only for 41 observations (29%), meaning that semi-endogenous theory is rejected for those observations. For the remaining 32 observations (22%), elements of both theories can be accepted. This evidence clearly indicates that the discussion about the empirical validity of both economic growth theories is still open. To evaluate the degree of heterogeneity in the reported estimates of the spillover effect, we calculate the classical Cochran s Q-statistic and the I 2 index. The Q-statistic measures the weighted sum squares of the differences between study estimates and the fixed effects average estimate. The I 2 index is equal to (Q (n 1))/Q and quantifies the proportion of total variation in the estimates that is due to heterogeneity between studies, as opposed to sampling variability (Higgins and Thompson, 2002; Higgins et al., 2003). 6

8 In our dataset, Q = , 92 and I 2 = 99%, which denotes a very high degree of heterogeneity. This means that there is substantial variation between studies estimates that should be accounted for. Section 5 we employ Meta-regression analysis to provide explanations for this variation. 4 Publication bias and average estimate of the spillover effect Before explaining the variation in studies estimates, we compute the average estimate of the spillover effect and test for the presence of publication bias in this literature. In meta-analyses the combined estimate of the effect size is often obtained using either fixed effects or random effects estimators. They are both weighted averages of the effect sizes reported in the primary studies. The fixed effects estimator assumes that there is only one true effect size, common to all studies, and that the observed variability in the reported estimates comes only from sampling variation. On the contrary, the random effects estimator accounts for the presence of heterogeneity, as it considers that studies have different true effect sizes; consequently, the observed variability in the reported estimates comes not only from sampling error - within studies variation - but also from differences in studies true effect sizes - between studies variation. 2 Due to the heterogeneity detected in the previous section, we will employ the random effects specification in the estimations performed throughout the paper. Publication bias has long been recognized as an important problem in empirical research. In its most frequent form, publication bias arises when statistically significant results are more likely to be produced and published by authors and journals than non-significant results. This leads to a distortion in empirical results, as the effect under analysis tends to be overestimated. Publication bias has been abundantly addressed in meta-analyses in many research areas, including economics (Card and Krueger, 1995; Doucouliagos, 2005; Doucouliagos et al., 2005; Stanley, 2005; Stanley et al., 2008). The funnel plot is a widely used tool to detect the presence of publication bias and simultaneously to obtain an idea of the average effect. Popularized by Egger et al. (1997), the funnel plot is a scatter diagram that displays the estimates of the effect size in the horizontal axis and their precision (usually measured by the inverse of the standard errors reported in the primary studies) in the vertical axis. As thoroughly explained by Stanley (2005), in the absence of publication bias, estimates of the effect size will vary randomly and symmetrically around the mean, the dispersion being higher in studies with lower precision. In this case, the diagram will assume the shape of a symmetric inverted funnel. But if there is publication bias favoring a certain direction, studies with higher standard errors (lower precision) tend to present estimates with a higher magnitude and biased toward that direction. In this case, the diagram will be asymmetric, especially in its lower part. Thus, the (a)symmetry of the funnel plot is the key to assessing publication bias. Figure 1 shows the funnel plots for our dataset, one with precision = 1/SE i 2 For further details on the fixed and random effects estimators in the context of meta-analyses, please see Hedges and Olkin (1985) and Borenstein et al. (2009). 7

9 in the y-axis (Figure 1a) and another in which precision appear in log scale for better visualization due to its high amplitude (Figure 1b). Figure 1: Funnel Plots (a) Energy Shares by Income Level (b) Energy Shares by Democracy Level There seems to be no evidence of publication bias, as the point estimates of the spillover effect are symmetrically distributed around the average, which is slightly below one. The conclusions revealed by visual inspection of the funnel plot can be formally tested by running a simple regression of the effect sizes on the respective standard errors: φ i = α 0 + α 1 SE i + u i (3) In the presence of publication bias, authors of studies with small samples and higher standard errors will tend to search more intensively (from datasets, estimation techniques, and model specifications) for higher estimates of the effect size in order to report statistically significant results. Thus, φ will be correlated with SE. In the absence of publication bias, φ will be uncorrelated with SE, as the reported estimates will vary randomly around the average effect, α 0, regardless of the standard errors (Stanley, 2005). Equation (3) can thus be used to test for the presence of publication bias and simultaneously to estimate the average of the effect size after controlling for publication bias. However, its estimation by OLS has two main problems. First, given that each reported effect has its own standard error, the disturbances u i are heteroskedastic. This problem can be easily corrected by implementing the usual procedure of dividing both sides of equation (3) by SE (Stanley, 2005), which leads to: t i = α 0 precision i + α 1 + e i (4) where t i = φ i /SE i is the conventional t-statistic associated with φ i reported in the primary studies and precision = 1/SE i. Given that the coefficients are now reversed, testing for the intercept in equation (4), 8

10 α 1, being equal to zero is a test for the presence of publication bias (Funnel Asymmetry Test - FAT), while testing for the slope, α 0, being equal to zero is a test for the presence of a significant average effect beyond publication bias (Precision Effect Test - FAT) (Egger et al., 1997; Stanley, 2005; Ugur et al., 2016). The second problem in estimating equation (3) by OLS is the presence of statistical dependence. When several observations are drawn from the same study, they share the same datasets, specifications or estimation procedures, and therefore are likely to be correlated (Hunter and Schmidt, 1990; Nelson and Kennedy, 2009). In this case, OLS produces biased estimates. The easiest way to address this issue is to choose only one estimate from each study (Stanley, 2001; Lipsey and Wilson, 2001). However, this would generally lead to a considerable reduction in the size of the meta sample, which is not desirable when the number of studies is limited. If several observations from each study are to be used in the meta-analysis, then hierarchical models, panel data estimators, clustered data analysis, or bootstrapped standard errors can be employed to address the problem of within-study correlation (Nelson and Kennedy, 2009; Doucouliagos and Laroche, 2009). We choose to estimate equation (4) using hierarchical linear models, since they not only correct the standard errors for within-study correlation, but also estimate the regression coefficients allowing for the presence of heterogeneity between studies (Ugur et al., 2016). Examples of meta-analyses in economics that have used hierarchical linear models are Bateman and Jones (2003), Johnston et al. (2005), Ugur et al. (2016). In the hierarchical models, observations are nested into groups with different characteristics. Thus, differences in individual observations can be attributed to both within-group variation and between-group variation. The model s coefficients are allowed to vary randomly between groups. In its most generic form, a hierarchical linear univariate model of the dependent variable Y i,j on explanatory variables X i,j can be written as: Y i,j = (β 0 + γ 0,j ) + (β 1 + γ 1,j )X i,j + ε ij (5) where: subscript i refers to observations and subscript j refers to groups; β 0 and β 1 are the fixed-effects intercept and slope, respectively; γ 0,j and γ 1,j are the group-specific intercept and slope, respectively, which are assumed to follow a normal distribution. This generic version is called the random intercept and slope model, as it allows both the intercept and the slope to vary randomly across groups. If only the intercept is allowed to vary across groups (in which case the slope is assumed to be fixed and the variance of γ 1,j is zero), we have a random intercept model; if only the slope is allowed to vary across groups (in which case the intercept is assumed to be fixed and the variance of γ 0,j is zero), we have a random slope model. The hierarchical structure can be applied in meta-analysis, as the observations (estimates of the effect size) are nested in groups (studies), that have different characteristics (random variation). We thus estimate equation (4) using hierarchical linear models. The results are reported in Table 3. 9

11 Table 3: Estimation of equation (4): Dependent Variable: t Coefficients for: P recision 0.902*** (0.088) constant (1.114) RE variances V ar(p recision) [0.02; 0.21] V ar(residuals) [47.95; 78.29] N. obs. (N.Studies) 144 (12) Log likelihood Wald test (α 0 =0) *** LR test (random intercept and slope vs. random slope) LR test (random intercept and slope vs. random intercept) *** LR test (HM vs. OLS) 37.18*** Notes: Estimation by maximum likelihood. Standard errors for coefficient estimators are in parentheses (). 95% confidence intervals for random effects variances are in brackets []. Level of significance: *** for p-value<0.01; **for p-value<0.05;* for p-value<0.01. N. Obs is Number of Observations; N. Studies is Number of Studies. RE variances mean Random Effects variances. The lower part of the table shows the results of a battery of tests that lend support to our estimation choices. The likelihood ratio (LR) test presented in the last line clearly shows that the hierarchical models are preferred to a simple OLS model. The two other LR tests reported in the table provide evidence that favors the random slope model over other hierarchical specifications. This is an expected result, because, since the coefficients are reversed in equation (4), a random slope model applied to (4) means that we allow for random variation between studies in the average spillover effect, α 0. This is the essence of the heterogeneity detected in Section 3. The middle part of the table presents the estimate of the variance of the random slope and of the residuals and the respective confidence intervals at 95%. The confidence interval for the variance of the random slope suggests that it is significant, which further confirms the adequacy of the random slope model and the existence of heterogeneity in the reported effect sizes. 3 The upper part of the table shows the results of the estimation of coefficients α 1 and α 0 using the random slope model. Estimations are obtained by maximum likelihood. We do not reject that α 1 = 0, which confirms that there is no evidence of publication bias in the empirical literature estimating the spillover effect, as suggested by the visual inspection of the funnel plot. This can be explained by the fact that the spillover effect does not directly measure the impact of a specific variable on another variable, and thus the tendency for authors and journals to search for and publish statistically significant results is not so pronounced. The value of associated with α 0 is the random effects average of the spillover effect (accounting for the hierarchical structure of the data). The average is lower than one, which lends 3 We do not report the variance of the intercept since we are estimating a random slope model. However, the estimation of the random intercept and slope model reveals that the variance of the intercept is practically zero, which confirms that there is no random variation in the intercept of equation (4). 10

12 some support to the semi-endogenous theory. However, the limits of the confidence interval for 95% are and 1.074, so the hypothesis that α 0 = 1 not can be rejected. Thus, in line with the preliminary analysis of the data carried out in the previous section, we can conclude that, although the average of the spillover effect may indicate the presence of elements of the semi-endogenous theory, the validity of the Schumpeterian theory is not rejected by the data. 5 Multivariate Meta-regression analysis In this Section we estimate a multivariate Meta-regression in order to investigate how a set of moderating factors (analytical and empirical dimensions of the research field) can explain the variation in the spillover φ estimates reported in the primary studies. The moderating factors are classified as analytical or empirical dimensions and are mostly captured through dummy variables that reflect a specific feature of the research field vis-à-vis a reference category, as specified in Table A.1 in Appendix A (summary statistics for the covariates are also given in the same table). The moderating variables linked with the analytical dimension of the research field are all related to the specification of the knowledge production function (equation 1) used in the primary studies. They include (1) the estimate value for the duplication effect, λ, 4 and dummies for: (2) the use of patent data as a proxy of A; the inclusion of (3) foreign knowledge, (4) imports 5 and (5) distance to frontier as explanatory variables in the specification of equation (1). The moderating variables linked with the empirical dimension of the primary studies are dummies for: (1) the inclusion of time-effects 6 ; (2) the sample being composed of only rich countries; (3) the estimation employing instrumental variables (IV) methods; (4) the sample using industrial data; (5) the publication type being a published article. Finally, the number of observations and the number of countries included in each regression are also considered as moderating variables. For the estimation of the multivariate Meta-regression we use the random slope version of the hierarchical model, justified by LR tests. 7 Table 4 shows the estimation results. Column (1) presents the baseline estimation, obtained by maximum-likelihood. In column (2) estimation is performed by restricted maximum-likelihood. This approach tends to reduce the bias of the maximum-likelihood, especially for small samples. In column (3) estimation is performed as in (1), but excluding outliers. 8 With this we want to confirm that the effects of moderating variables are not due to the presence of outliers. For comparison, we add two more regressions in columns (4) and (5), estimated by OLS. In column (4) standard-errors are clustered within studies, and in column (5) 12 study dummies are added to the moderating variables. Our baseline specification shows a highly significant, positive and less than one precision effect (condi- 4 If λ < 1, there is some degree of duplication in the production of ideas in that some of the knowledge that is created may not be entirely new. 5 Or, in a more general view, openness to international trade or external financial dependence. 6 Using time dummies, time-trend variables or year-fixed effects. 7 LR tests were performed on individual covariates in order to decide which ones should enter with random slopes. The tests suggest that the specification in which covariates IV method and duplication enter with random slopes is the most 11

13 Coefficients for: Table 4: Estimation of the multivariate meta-regression: Dependent variable: t i (1) (2) (3) (4) (5) Precision 0.847*** 0.846*** 0.703*** 0.903** (0.252) (0.266) (0.246) (0.315) (0.365) Duplication * * * ** *** (0.181) (0.195) (0.206) (0.203) (0.058) Patent 0.367*** 0.367*** 0.429*** *** (0.127) (0.133) (0.119) (0.240) (0.165) Foreign know *** 0.024*** *** 0.022*** (0.006) (0.006) (0.037) (0.003) (0.006) Imports 0.012*** 0.012*** *** (0.004) (0.004) (0.037) (0.002) (0.004) Distance * * *** ** (0.003) (0.004) (0.014) (0.002) (0.004) Time effects 0.088** 0.089** 0.105*** *** (0.038) (0.039) (0.038) (0.087) (0.103) Rich countries *** *** *** (0.010) (0.011) (0.013) (0.024) (0.014) IV methods * * ** *** (0.213) (0.238) (0.218) (0.162) (0.099) Panel (0.175) (0.185) (0.167) (0.141) (0.196) Observations *** (0.000) (0.000) (0.000) (0.000) (0.000) Countries *** *** *** (0.001) (0.001) (0.002) (0.004) (0.002) Industrial data *** *** *** *** (0.022) (0.023) (0.029) (0.067) (0.201) Publication ** ** ** *** (0.079) (0.084) (0.087) (0.086) (0.232) constant ** RE variances (0.917) (0.968) (1.067) (1.740) (1.056) Var(duplication) [0.090; 0.692] [0.010; 0.841] [0.127; 0.932] Var (IV methods) [0.050; 0.764] [0.055; 1.119] [0.054; 0.796] Var(residuals) [20.26; 32.97] [22.18; 37.02] [16.54; 28.33] N. obs. (N.Studies) 144 (12) 144 (12) 121 (12) 144 (12) 144 (12) Log likelihood Wald test *** *** *** LR test 81.61*** 80.57*** 86.14*** R Notes: Moderator variables are divided by the standard errors reported in the primary studies. Estimation by maximum likelihood in columns (1) and (3), restricted maximum likelihood in column (2), OLS with clustered robust standard errors in column (4). Study dummies included in column (5) are significant. Standard errors for coefficient estimators are in parentheses (). 95% confidence intervals for random effects variances are in brackets []. Level of significance: *** for p-value<0.01; **for p-value<0.05;* for p-value<0.01. Wald Test is for global significance. N. Obs is Number of Observations; N. Studies is Number of Studies. LR Test is HM vs. OLS. RE variances mean Random Effects variances. tional on the effect of the moderating variables, detailed below), pointing, as previously, to the validation of the semi-endogenous growth theories. However, given that the upper limit of the confidence interval is always greater than one, evidence cannot exclude the verification of the Shumpeterian (endogenous) appropriate. Results are available upon request. 8 Observations for which t-values are below Q (Q 3 Q 1 ) or higher than Q (Q 3 Q 1 ) are considered outliers. 12

14 growth theory. We can also observe a negative and significant effect of the value of the duplication effect (on primary studies) on the estimation of the spillover effect. This is an expected result, as, for a given evolution of the knowledge dynamics and for given inputs, the higher the duplication effect (stepping-on-toes), the lower should be the spillover effect (standing-on-the-shoulders). It is remarkable that this effect is robust to different estimation methods and accounts for cross-study heterogeneity. Additionally, the use of patent data to measure the dynamics of knowledge production contributes to higher spillover effects. As a consequence of implying higher spillovers, it is worth noting that the use of patent data contributes to the validation of the Shumpeterian theory (vis-à-vis the semi-endogenous theory). In these studies, technology stock is alternatively measured recurring to patents or other measures of technological sophistication such as GDP or GDP per capita. This result may indicate that the standingon-the-shoulders effect is greater when the knowledge stock is directly associated with a measure of innovation (and less when it is measured through more indirect variables of technological sophistication). Moreover, accounting for foreign knowledge and imports in the domestic knowledge production increases the estimation of the standing-on-the-shoulders effect. On the contrary, accounting for the distance to the frontier tends to diminish that effect. These significant effects highlight the importance of considering interactions between countries in estimating the knowledge production function. Concerning the sample features in the primary studies, we find that the spillover effect tends to be lower when only rich countries are included, industrial data are used, and the number of countries in the sample is high. While the inclusion of time-effects in the estimation of the knowledge production function tends to increase the spillover effect, the consideration of instrumental variables in estimation tends to decrease it. This means that endogeneity tends to upwardly bias the coefficient of the spillover effect. Finally, published articles tend to report lower estimates of the spillover effect than working papers. It is worth mentioning that the results of the regression that excludes outliers show no-significance of previously significant coefficients for the open economies variables (foreign knowledge, distance to frontier and imports). Moreover, the regression without outliers also presents a somewhat lower value for the spillover (0.703) which puts it closer to the strict validation of the semi-endogenous theory. However, the upper limit of the confidence interval is still higher than one. In this regression, we also obtain for the first time some evidence of publication bias, meaning that it seems that researchers tend to publish significant results when potential outliers are excluded from the analysis. Given these results, we can obtain strongly-consistent (statistically significant at least in the three first columns) sources of spillover variations to be the duplication effect, the use of patent data to measure knowledge, the inclusion of time effects in the knowledge production function, the use of IV methods, the number of countries, the use of Rich Countries and Industrial Data in the sample, and the type of publication. Except for patents and time effects that have a positive and significant sign, all the other strongly- 13

15 consistent moderating factors tend to reduce the spillover effect. We can classify the variables linked with the openness of the economy (foreign knowledge, distance and imports) as moderately-consistent sources of the spillover variations, as they tend to be highly significant in most of the specifications but the specification that excludes potential outliers greatly decreases their statistical significance. Finally, the consideration of a panel database and the number of observations are not relevant in explaining the variation of the spillover effect, as the respective moderating variables are either nonsignificant in all regressions (in the case of the panel variable) or are significant in just one regression (as is the case of the variable observations). 6 What can we learn from a larger set of articles? With the baseline sample we disregard information coming from other empirical contributions that deal directly with the comparison between growth theories, but that do not estimate the coefficient for the spillover effect. This information is disregarded due to the use of very different econometric methods to explore the causal linkages in the knowledge production function (see Table 1). For these additional articles there is no information on the spillover coefficient and the associated standard errors, which invalidates a complete meta-analysis, as performed in the previous section. However, these contributions include some interesting information regarding the validation of the (endogenous) Schumpeterian growth theory or of the semi-endogenous growth theory. In order to include such information in the analysis, we constructed a dummy as a dependent variable that takes the value 1 if: (i) the lower limit of the confidence interval of the spillover estimate is above 1 in the baseline database; (ii) the Shumpeterian theory is accepted when it is tested for; (iii) the semiendogenous model is rejected when it is tested for. While criterion (i) is applied only to the papers included in the baseline database, criteria (ii) and (iii) can be applied to the articles using unit-root or cointegration tests and testing knowledge production functions à la Aghion and Howitt (1992). We call this variable the Dummy for the Shumpeterian Theories. In this larger database we first consider the 22 published articles and working papers presented in Table 1. They provide 460 observations of primary estimations. From these observations, 285 (62%) point to the validation of the Shumpeterian theory. Comparing this percentage with the figures presented in Section 3, it can be immediately noted that these latter methods tend to confirm the Shumpeterian theory more often than the estimations of the knowledge production functions à la Romer (1990). We estimate a multivariate Meta-regression with Dummy for the Shumpeterian Theories as the dependent variable and almost all the variables of the previous section as moderators. Variable duplication was excluded because the newly introduced studies do not estimate the parameter of the duplication effect, and variable small database was added to differentiate between studies belonging or not to the baseline database (it takes the value 1 if the study estimates a knowledge production function a la Romer (1990) 14

16 and thus presents an estimate of the spillover effect, and 0 otherwise). In one article (Ramzi and Salah, 2015), no information on the number of observations is reported; the same happens in two estimations in Botazzi and Peri (2007) and in three estimations from Dinopoulos and Thompson (2000). Because of that, our regressions include 454 observations for 21 articles. 9 We estimate the Meta-regression by logit and probit hierarquical methods (columns 1 and 2 of Table 5, respectively) and clustered standard-errors logit and probit linear methods (columns 3 and 4 of Table 5, respectively). When this larger database is taken into account we find that the presence of imports contributes to the rejection of the semi-endogenous theories (and validation of the Shumpeterian theories) with a highly significant positive sign across all the specifications. This reinforces the positive sign of the variable imports obtained in Table 4. However, foreign knowledge tends to decrease the probability of validating the Shumpeterian theory when it is considered in the estimation of the knowledge production function. Moreover, consideration of panel databases, and publication (as an article) tend to decrease the probability of validating the Shumpeterian theory. The negative and significant sign obtained for publication also reinforces the already negative and significant sign obtained in Table 4. As for the variable small database, it is statistically significant and its coefficient is negative, which means that the probability of validating the Schumpeterian theory is lower when studies estimate a knowledge production function à la Romer (1990) and thus obtain an estimate value for the spillover effect. These results are robust, as they are obtained in all four regressions performed. 7 Conclusions There is a spirited discussion about the empirical validity of the semi-endogenous or endogenous (Shumpeterian) endogenous growth theory, with clear policy implications. While the first theory points out to ineffective policies in the long run, the latter allows for important effects of policies, namely subsidies or taxes to R&D. Additionally, while the first theory points to a positive effect of population growth in long-run economic growth, the latter does not imply such a positive effect. The empirical literature on this topic is well defined, aiming primarily to estimate the knowledge production function. It began to appear in the first year of this millennium with the article by Porter and Stern (2000), following the emergence of the endogenous growth theory of the 1990s (e.g. Romer, 1990, Grossmann and Helpman, 1991 and Aghion and Howitt, 1992). As far as we know, our contribution is the first attempt to review this well-defined literature on the estimation of the knowledge production function. We can conclude that although the average of the spillover effect may indicate the presence of elements of the semi-endogenous theory, the validity of the Schumpeterian theory is not rejected by the data. 9 Tests excluding the variable Number of Observations and thus including the six missing estimates do not change our results. Results are available upon request. 15

17 Table 5: Estimation of the meta-regression for the larger dataset of primary studies Dependent variable: Dummy Shumpeterian Theories (1) (2) (3) (4) Coefficients for: Patent (0.861) (0.499) (0.583) (0.338) Foreign know ** ** ** ** (0.525) (0.308) (0.524) (0.307) Imports 1.315*** 0.775*** 1.124** 0.682** (0.514) (0.300) (0.493) (0.274) Distance (0.484) (0.286) (0.684) (0.386) Time effects (0.503) (0.291) (0.358) (0.207) Rich countries (0.683) (0.393) (0.484) (0.277) IV methods (0.887) (0.531) (0.839) (0.464) Panel ** ** *** *** (1.237) (0.706) (0.518) (0.303) Observations (0.000) (0.000) (0.000) (0.000) Countries (0.034) (0.020) (0.017) (0.009) Industrial data (1.357) (0.796) (0.605) (0.340) Publication ** ** ** *** (1.560) (0.916) (1.010) (0.599) Small Database ** ** *** *** (1.017) (0.591) (0.510) (0.294) constant 5.937*** 3.420*** 4.446*** 2.624*** (2.233) (1.289) (1.317) (0.767) Random effects variances V ar(constant) [0.78; 10.62] [0.27; 3.58] N. obs. (N. Studies) 454 (21) 454 (21) 454 (21) 454 (21) Log likelihood Wald test (global significance) 23.38** 24.51** *** *** LR test (HM vs. OLS) 11.39*** 10.73*** Pseudo R Notes: Estimation by maximum likelihood in all columns, with clustered robust standard errors in columns (3) and (4). Standard errors for coefficient estimators are in parentheses (). 95% confidence intervals for random effects variances are in brackets [].Level of significance: *** for p-value<0.01; **for p-value<0.05;* for p-value<0.1. Moreover, standard tests indicate that publication bias is not a serious issue in this literature. When moderating effects are taken into account, the statistically significant average spillovers estimates range from to Nevertheless, upper bounds of the confidence intervals higher than unity cannot lead to the rejection of the Shumpeterian growth theory. Through this analysis, we discover several variables that influence the standing-on-the-shoulders effect. Especially important are the strongly-consistent sources of spillover variations, which are the duplication (or stepping-on-toes) effect, the use of patent data to measure knowledge, the inclusion of time effects in the knowledge production function, the use of IV 16

18 methods, the number of countries, the use of rich countries and industrial data in the sample and the type of publication. Except for patents and time effects, which have a positive and significant sign, all the other strongly consistent moderating factors tend to decrease the spillover effect. We can classify the variables linked with the openness of the economy (foreign knowledge, distance, and imports) as moderately consistent sources of the spillover variations. Using a wider database that combines information from different primary studies methods, we clearly obtain that for given values of the other moderating factors, the fact that the estimation of the knowledge production function includes the estimation of a coefficient for the standing-on-the-shoulders (spillover) effect contributes to the validation of the semi-endogenous growth theory. Additionally, foreign knowledge included in the knowledge production function, consideration of panel databases, and publication (as an article) tend to decrease the probability of validating the Shumpeterian theory. In the opposite direction, the presence of imports contributes to the rejection of the semi-endogenous theories (and validation of the Shumpeterian theories). Despite the fact that the most recent contributions on the estimation of the knowledge production function argue in favor of the Shumpeterian (endogenous) growth theory, our meta-analytic review highlights that the global picture tends to favor the semi-endogenous growth theory. From the specifications of the knowledge production function, the use of patents as a measure of knowledge, the consideration of time effects, and, to a lesser extent, international linkages tend, however, to favor the Shumpeterian theory. Given these results, future research may combine elements of both theories together in endogenous growth models and test them empirically. Validation of such models may highlight the role of fiscal policies and population growth in long-run economic growth. 17