Evidence on an endogenous growth model with public R&D

Ziesemer, Thomas Working Paper Evidence on an endogenous growth model with public R&D UNU-MERIT Working Papers, No. 2024-002 Provided in Cooperation with: Maastricht Economic and Social Research Institute on Innovation and Technology (UNU-MERIT), United Nations University (UNU) Suggested Citation: Ziesemer, Thomas (2024) : Evidence on an endogenous growth model with public R&D, UNU-MERIT Working Papers, No. 2024-002, United Nations University (UNU), Maastricht Economic and Social Research Institute on Innovation and Technology (UNU-MERIT), Maastricht This Version is available at: https://hdl.handle.net/10419/326898 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Ziesemer Published 19 February 2024 Maastricht Economic and social Research institute on Innovation and Technology (UNU-MERIT) email: [email protected] | website: http://www.merit.unu.edu Boschstraat 24, 6211 AX Maastricht, The Netherlands Tel: (31) (43) 388 44 00 UNU-MERIT Working Papers ISSN 1871-9872 Maastricht Economic and social Research Institute on Innovation and Technology UNU-MERIT UNU-MERIT Working Papers intend to disseminate preliminary results of research carried out at UNU-MERIT to stimulate discussion on the issues raised. 1 Evidence on an endogenous growth model with public R&D Thomas H.W. Ziesemer, Maastricht University, UNU-MERIT. P.O. Box 616, 6200MD Maastricht, The Netherlands. ORCID 0000-0002-5571-2238. [email protected] Abstract The empirical inves�ga�on of proper�es of an endogenous growth model by Huang, Lai, and Pereto (2023) in this paper confirms important assump�ons and results of the model for OECD countries. Labour-augmen�ng technical change is enhanced through private and public R&D stocks in FMOLS and DOLS mean-group es�ma�ons, and pooled mean-group (PMG) es�ma�on, also when adding the number of enterprises. The CES spillover func�ons in the growth models func�ons for R&D stock dynamics are supported through nonlinear es�ma�on under the assump�ons of iden�cal or different spillover parameters for private and public R&D. We suggest strong public-to-private spillovers and weak private-to-public spillovers as well as high elas�ci�es of subs�tu�on for private-public R&D stocks for private R&D processes and low CES for public R&D processes. We confirm the existence of private-public researcher interac�on effects in the private R&D knowledge growth func�on and provide tenta�ve evidence for the linear rela�on between public researchers and firm-level R&D and the hump-shaped rela�on between public and private researchers (both as % labour force). A vector-autoregressive (VAR) panel model in growth rates produces results, which are in accordance with the impact of public R&D cuts on the steady state and the transi�onal dynamics of the HLP model. Keywords: Endogenous growth; public R&D; evidence. JEL code: O41; O38; O47. 1. Introduc�on Huang, Lai, and Pereto (2023) (henceforth HLP) are the first to have extended the most recent version of endogenous growth models to include public R&D leaving a model with only private R&D as special case. In this paper, the purpose is to es�mate and test some of the proper�es of their model for samples of OECD countries. The major elements of the model are (i) a technical change func�on depending on stocks of private and public R&D and the number of firms; (ii) dynamic equa�ons for private and public R&D containing business and government researchers respec�vely, CES spillover func�ons with stocks of knowledge, and private-public labour interac�on terms; (iii) a hump-shaped rela�on between government and business researchers, and (iv) an analysis of the model dynamics a�er a cut of the number of government researchers. The contribu�on of this paper is as follows. (i) We es�mate the technical change func�on, and (ii) also the private and public R&D accumula�on func�ons nonlinearly under the assump�ons of iden�cal and non-iden�cal parameters of the CES spillover func�ons. (iii) We build a VAR from GMM (orthogonal devia�ons version) in growth rates of all just men�oned variables (because of country-specific �me trends leading to fixed effects a�er differencing) to compare the empirical effects of public R&D cuts on private and public R&D capital, government and business researchers, the number of firms, and technical change to those 1 I am grateful for useful comments from Chien-Yu Huang. 2 of the HLP model; the reason is that the steady-state proper�es and transi�ons to a new steady state a�er a cut on public R&D explain the working of the theore�cal model. The evidence is preliminary in the sense that not all modern econometric methods can be used because the available data series are very short and nonlinear es�ma�on has its own difficul�es known from the es�ma�on of CES produc�on func�ons. Earlier literature on R&D spillover func�ons has used linear or log-linear models, mostly Cobb-Douglas or translog func�ons; one excep�on is the use of a generalized CES (briefly ‘VES’) func�on, staying in the comfortable realm of linear econometrics by way of using log-log es�mates based on first-order condi�ons (Ziesemer 2021a).2 The R&D accumula�on func�ons in this paper are only par�ally linear (see Greene 2012) and therefore cannot avoid the nonlinear es�ma�on of CES spillover func�ons. We indicate the problems in due course. However, the nonlinear es�ma�on of R&D spillover CES func�ons is an innova�on in this paper as much as its theore�cal modelling of HLP is an innova�on. 2. Theore�cal and empirical Modelling 2.1 The theore�cal model Labour produc�vity of the firms, 𝑇𝑇𝑖𝑖, in eq (7) of HLP (2023), 𝑍𝑍𝑖𝑖𝜃𝜃𝐷𝐷𝑖𝑖𝛾𝛾, is log linear with Z as private R&D capital stock and D as public R&D capital stocks for each firm i = 1, …, N. 𝑇𝑇𝑖𝑖=𝑍𝑍𝑖𝑖𝜃𝜃𝐷𝐷𝑖𝑖𝛾𝛾 (1) In order to go to the macroeconomic level, we write (1) in natural logarithms, replace Di by D/N, and Zi by Z/N and add logN on both sides. This yields logT = log 𝑇𝑇𝑖𝑖+𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 𝜃𝜃𝑙𝑙𝑙𝑙𝑙𝑙𝑍𝑍+𝛾𝛾𝑙𝑙𝑙𝑙𝑙𝑙𝐷𝐷+(1−𝜃𝜃−𝛾𝛾)𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 (1’) In case of decreasing returns in (1), the func�on for the firm level cannot be just rewriten for the macroeconomic level as it has been done in the empirical literature, but rather the theore�cal model of HLP suggests having the number of firms on the right-hand side of (1’). The firms’ R&D dynamics with CES spillovers from (8) and (9) in HLP leads to (2) and (3) 𝑍𝑍󰇗=𝛼𝛼𝛼𝛼(𝑠𝑠𝐺𝐺)�𝜒𝜒𝐾𝐾𝑖𝑖𝜂𝜂+ (1 −𝜒𝜒)𝐵𝐵𝑖𝑖𝜂𝜂�1𝜂𝜂 �𝐿𝐿𝑍𝑍 (2) 𝐷𝐷󰇗=�(1 −𝜒𝜒)𝐾𝐾𝑖𝑖𝛿𝛿+𝜒𝜒𝐵𝐵𝑖𝑖𝛿𝛿�1𝛿𝛿 �𝐿𝐿𝐺𝐺 (3) This is obtained when summing equa�ons (8) and (9) in HLP over all N firms with index i, under the simplifying assump�on that all terms before the labour variables 𝐿𝐿𝑍𝑍 and 𝐿𝐿𝐺𝐺 for business and government researchers, as well as the le�-hand side are iden�cal for all firms. 𝛼𝛼(𝑠𝑠𝐺𝐺) is a government-labor interac�on term explained below. χ is the own private to private and public-to-public spillover parameter. (1-χ) is the cross spillover parameter 2 Belderbos and Mohnen (2020) discuss spillovers in detail. 3 from public to private and private to public R&D. Symmetry in the spillover parameters is assumed for simplicity and will be generalized in the estimation later. Next, we divide the first equa�on by Z and the second by D. The result is as follows. 𝑍𝑍󰇗 𝑍𝑍=𝛼𝛼𝛼𝛼(𝑠𝑠𝐺𝐺)�𝜒𝜒𝐾𝐾𝑖𝑖𝜂𝜂+ (1 −𝜒𝜒)𝐵𝐵𝑖𝑖𝜂𝜂�1𝜂𝜂 �𝐿𝐿𝑍𝑍 𝑍𝑍 (2’) 𝐷𝐷󰇗 𝐷𝐷=�(1 −𝜒𝜒)𝐾𝐾𝑖𝑖𝛿𝛿+𝜒𝜒𝐵𝐵𝑖𝑖𝛿𝛿�1𝛿𝛿 �𝐿𝐿𝐺𝐺 𝐷𝐷 (3’) Produc�vity growth according to the theore�cal model (1) or (1’), (2’), and (3’) then is 𝑇𝑇 �=𝑇𝑇 �𝑖𝑖+𝑙𝑙 �=θ𝑍𝑍 󰆹𝑖𝑖+𝛾𝛾𝐷𝐷 �𝑖𝑖+𝑙𝑙 �= 𝜃𝜃�𝑍𝑍 󰆹−𝑙𝑙 ��+𝛾𝛾�𝐷𝐷 �−𝑙𝑙 ��+𝑙𝑙 � (1’’) =𝜃𝜃𝛼𝛼𝛼𝛼(𝑠𝑠𝐺𝐺)�𝜒𝜒𝐾𝐾𝑖𝑖𝜂𝜂+(1−𝜒𝜒)𝐵𝐵𝑖𝑖𝜂𝜂�1𝜂𝜂 �𝐿𝐿𝑍𝑍 𝑍𝑍+𝛾𝛾�(1 −𝜒𝜒)𝐾𝐾𝑖𝑖𝛿𝛿+𝜒𝜒𝐵𝐵𝑖𝑖𝛿𝛿�1𝛿𝛿 �𝐿𝐿𝐺𝐺 𝐷𝐷+(1−𝜃𝜃−𝛾𝛾)𝑙𝑙 � 2.2 Speci�ication Issues and the empirical Model To get a regression form we have to specify the func�on 𝛼𝛼(𝑠𝑠𝐺𝐺). We use a polynomial of the third degree because of its flexibility: 𝛼𝛼 = 𝜉𝜉0+𝜉𝜉1𝑠𝑠𝐺𝐺+𝜉𝜉2𝑠𝑠𝐺𝐺 2+𝜉𝜉3𝑠𝑠𝐺𝐺 3 (4) The construct simplifies as in the HLP model if the squared and cubic terms are sta�s�cally insignificant, but they may be helpful in the es�ma�on especially if the linear part is sta�s�cally insignificant or has an unexpected sign. HLP use 𝑠𝑠𝐺𝐺,𝑎𝑎=𝐿𝐿𝐺𝐺/𝐿𝐿𝐿𝐿, where LF is the labour force. The interac�on HLP have in mind then is that of 𝐿𝐿𝐺𝐺 with 𝐿𝐿𝑍𝑍 in (2) and (2’). For the private-public interac�on we explore two addi�onal ideas. First, personnel-interac�on, s(LG*LZ) is modelled as an econometric interac�on term specified as 𝑠𝑠𝐺𝐺,𝑏𝑏=𝐿𝐿𝑍𝑍𝐿𝐿𝐺𝐺 The essence of the exercise is ge�ng 𝜉𝜉i to capture the curvature of f(sG)(1-sG) in (16) of HLP (2023), which is a version of (2) which is modified by inclusion of othr parts of the model. Second, 𝑠𝑠𝐺𝐺 could be defined more broadly as money going from the government to firms, which we take as percentage of BERD flows financed by government, GB. The theory of HLP (2023) can be interpreted as seeing a log-log effect of a func�on of GB = S/BERD, from BERD being split up into BERD = R+S (private and government money sources of BERD), on the change or growth rate of BERDST in (2) and (2’): 𝑠𝑠𝐺𝐺,𝑐𝑐=𝐺𝐺𝐵𝐵,𝑙𝑙𝑜𝑜 𝑙𝑙𝑙𝑙𝑙𝑙𝐺𝐺𝐵𝐵, or GB squared If, instead, researchers would go from the government to the firms as during projects of universi�es executed for private business then money goes the opposite direc�on from firms to governmental research ins�tu�ons, leaving open which sign is more plausible, and resul�ng in the risk of sta�s�cal insignificance. 4 Using versions of GB or LG*LZ/(LF)2 never yields interes�ng results. We set 𝐾𝐾𝑖𝑖=𝑍𝑍/𝑙𝑙 and 𝐵𝐵𝑖𝑖=𝐷𝐷/𝑙𝑙 in the spillover functions of (2’) and (3’) to bring the idea of domain speci�ic knowledge in HLP to the macro level. Knowledge in turn is measured as accumulated private and public R&D capital stock. Finally, we add the rate of depreciation of 0.15 to the growth rates on the left-hand side as in the process of making stock data explained in the next section. Taking natural logarithms his leads to the following system of equations with 𝑙𝑙𝑍𝑍,𝑙𝑙𝑁𝑁 as growth rates of Z and N . log(𝑙𝑙𝑍𝑍+ 0.15)= 𝑙𝑙𝑙𝑙𝑙𝑙𝛼𝛼+log [𝛼𝛼(𝑠𝑠𝐺𝐺)] + �1𝜂𝜂 � �𝑙𝑙𝑙𝑙𝑙𝑙�𝜒𝜒�𝑍𝑍 𝑁𝑁�𝜂𝜂+(1−𝜒𝜒)�𝐷𝐷 𝑁𝑁�𝜂𝜂�+𝜏𝜏𝑙𝑙𝑙𝑙𝑙𝑙𝐿𝐿𝑍𝑍−𝛽𝛽𝑙𝑙𝑙𝑙𝑙𝑙𝑍𝑍 (2’’) log(𝑙𝑙𝐷𝐷+ 0.15)=�1𝛿𝛿 ��𝑙𝑙𝑙𝑙𝑙𝑙�(1 −𝜒𝜒)�𝑍𝑍 𝑁𝑁�𝛿𝛿+𝜒𝜒�𝐷𝐷 𝑁𝑁�𝛿𝛿�+𝜑𝜑𝑙𝑙𝑙𝑙𝑙𝑙𝐿𝐿𝐺𝐺−𝜀𝜀𝑙𝑙𝑙𝑙𝑙𝑙𝐷𝐷 (3’’) Allowing for non-unit coef�icients of logZ and logD suggests that these terms do not just come in through the division by Z and D in (2’) and (3’), but rather not only labour is a factor of production but also Z and D , similar to a Cobb-Douglas case but without the constraint that logZ and logD have the same coef�icients as labour up to the sign. The es�mated results for f(sG) and sG can be used in the short-run general equilibrium growth rate equa�on (16) in HLP (2023) for private R&D to calculate the factor f(1-sG) and perhaps compare its values to the possible short-run maximum. 3. Data Transla�ng it to the macro level we use log Z = LBERDST and log D = LPUBST, where L stands for the natural logarithm and ST for stock, BERD is business R&D, and PUB is the flow of GERD minus BERD both taken from OECD MSTI un�l 2017. More recent years in OECD MSTI are incomplete or characterized as provisional or es�mated. Stocks are constructed using the perpetual inventory method with a deprecia�on rate of 0.15 (see Hall et al. 2010 for a survey).3 Labour produc�vity data, LTH07, have been constructed by Ziesemer (2023a) for alterna�ve elas�ci�es of subs�tu�on of CES produc�on func�ons including human capital. We use values calculated under the assump�on of a CES = 0.7. These data typically have more than forty yearly data points per country. In the dynamic R&D equations, LZ denotes business enterprise researchers (FTE); LG government researchers (FTE); sG is either (i) LG / LF , with labour force data from WDI, which start only in 1990, or (ii) the (log of) the percentage of BERD �inanced by government from OECD-MSTI, sG = GB , with values between 0.8 and 32 percent and entered as GB/100 in the regressions, or (iii), focusing on the idea of personnelinteraction, sG =LG*LZ , which has a panel maximum of 9.77E+10; therefore we divide 3We are grateful to ANONYMOUS for providing the R&D data. 5 LG*LZ by 1E+11 = 100 billion (one billion = 1E+9) to make sure that 1sG >0. When using GB , we drop Austria and Sweden from the set of 17 OECD contries; when using the labour interaction we drop Austria, Finland, and USA because they both have a very low number of observations. For GB, LZ, LG we have then fourty data points per country with some gaps, and a total of unbalanced observation for GB = 590, LZ = 573, LG = 569 . For the number of �irms, N in (1’) and its growth rate 𝑙𝑙𝑁𝑁 in (2’’), we use ‘Number of enterprises’ from ‘OECD SSIS: Structural Sta�s�cs of Industry and Services, 05_82_LESS_K: Business economy, except financial and insurance ac�vi�es’. These data typically have 19 yearly data points with a total of balanced observa�on of 250 when no lags reduce it. Table 1: Growth rate of number of enterprises for 17 OECD countries AUT 0.0157 FIN 0.0106 NLD 0.0798 BEL 0.0449 FRA 0.0261 NOR 0.0141 CAN 0.0069 GBR 0.0235 PRT -0.0095 DEU 0.0318 IRL 0.0875 SWE 0.0232 DNK 0.0031 ITA -0.0054 USA -0.0011 ESP -0.0050 JPN -0.0182 Average 0.0193 Source: OECD SSIS: Structural Statistics of Industry and Services, 05_82_LESS_K: Business economy, except financial and insurance activities. The unweighted average over the countries’ growth rates of the number of firms in Table 1 is 0.0193, with significantly nega�ve rates for Spain and Italy, slightly nega�ve growth rates for USA and Japan, and slightly posi�ve rates for Denmark and Norway. Ireland and the Netherlands have the highest growth rates. Table 1 shows more details on the number of firms. Table 2 provides a data descrip�on for all variables. Table 2 Data descrip�on for 17 OECD countries: Coefficients (standard errors) Variable Business researchers LZ (FTE) Government researchers LG (FTE) Number of enterprises ETP LTH07 (c) Private R&D LBERDST Public R&D LPUBST Labour force (a) Panel average 89551 (7182) 55996 (2618) 1316163 (76508) 1.9 (0.006) 10.02 (0.012) 9.74 (0.0133) 24,54mio Std. dev. 171925 62459 1209696 0.339 1.894 1.573 35.9 mio Maximum 1201000 295864 4326720 2.93 14.42 13.5 1.67E+08 Minimum 607 FTE 1507 81264 0.658 4.658 5.8 1.44 mio Average trend (b) 0.047 (0.0026) 0.035 (.0016) 0.015 (0.0026) 0.014 (8E-3) 0.05 (0.003) 0.04 (.0016) 0.0077 (0.0006) Periods 40, 1981-2020 40, 1981-2020 19, 2002-2020 55, 1963-2017 55, 19632017 55, 1963 -2017 31, 19902020 Unb. obs. 573 569 250 935 911 902 527 (a) From regression of variable on a constant. Standard error of es�ma�on in parentheses. (b) From fixed effects regression with cross-sec�on weights of log of variable on a trend. Standard error of es�ma�on in parentheses. (c) Labour augmen�ng technical change, log level. 6 As the number of enterprises has the lowest number of observa�ons, we first conduct analyses where they are not necessary or ignored as in earlier research that was not based on endogenous growth theory. When using the number of enterprises in preliminary country-specific regressions for the system (2’’) and (3’’) together with the numbers of researchers, we get informa�on on the number of observa�ons available per country. For most countries we have 13 observa�ons per equa�on or 26 total observa�ons, leaving us with a system total of 414 observa�ons or 212 per equa�on, which is in the order of magnitude of the number of observa�ons for firms, which is 250. We have 16 (32) observa�ons for Canada; 8 (16) for the USA, 6 (12) for Japan, and 11 (22) for Norway. We form three panel data set; one including all 17 countries and two not including those with small numbers of observa�ons: in the second dropping Japan, Norway, and USA, and in a third one including Norway, but not Japan and the USA. 4. Econometric Aspects The data for LTH07, LBERDST, and LPUBST have panel (near) unit roots (see Table A.1). Therefore, we use cointegra�on methods for the es�ma�on for 17 OECD countries4. We use three different es�ma�on methods, which have been developed to deal with nonsta�onarity and endogeneity: group-mean versions of FMOLS and DOLS, which are consistent es�mators (Pedroni 2001), as well as PMG/ARDL (pooled mean group es�mator) (Pesaran et al. 1999). As the PMG/ARDL method yields different results than FMOLS and DOLS when assuming one cointegra�ng equa�on for the three variables of (1), we get somewhat suspicious in regard to this assump�on. In the �me-series literature, the ARDL method underlying the PMG for each country requires having only one cointegra�ng equa�on (Pesaran and Shin 1999) and two cointegra�ng rela�ons of two variables are generally held to be more informa�ve than one of three variables (Kilian and Lütkepohl 2017). Therefore, we also test for the number of cointegra�ng rela�ons using the JohansenFisher panel cointegra�on tests for the cointegra�on rank. As the number of firms has a small number of observa�ons, we cannot use the Fisher-Johansen test and FMOLS es�ma�on. For es�ma�on we lean on DOLS, and PMG/ARDL. For cointegra�on tes�ng we use the Pesaran CIPS5 and the Bai and Ng (2004) PANIC6 test (with MQF meaning to allow for a VAR(p) with p>1 in the tes�ng for the number of (non)sta�onary common factors found according to the average of the Bai/Ng criteria), and in connec�on with PMG/ARDL es�mates, also the bounds test. Results for the spillover parameters in the dynamic R&D func�ons (2’), (3’) should be in the unit interval. Therefore, we set 𝜒𝜒=𝑒𝑒𝑐𝑐 1+𝑒𝑒𝑐𝑐 or 𝜒𝜒=1 𝑒𝑒−𝑐𝑐+1. For any es�mated value of the parameter c this yields a value of χ in the unit interval. Without this specifica�on we find values outside the unit interval in country-specific explora�ons (not shown). 4 AUT,BEL,CAN,DEU,DNK,ESP,FIN,FRA,GBR,IRL,ITA,JPN,NLD,NOR,PRT,SWE,USA. 5 Cross-sec�onally Augmented IPS. 6 ‘PANIC’ abbreviates ‘Panel Analysis of Nonsta�onarity in Idiosyncra�c and Common Components’. 13 Table 6 Regression coefficients of dynamic private and public R&D equa�ons with government R&D-labour share and CES constant-returns-to-scale spillover func�ons (a) Panel, model → Variables ↓ OECD 17, iden�cal OECD17, ident., resid. augm. OECD14 (b) ident. OECD14 (b) ident., resid augm OECD 14 (b), asym. OECD 14 (b), asym., Resid augm. α = c1 0.042 (3.56) 0.24 (16.3) 3222(2.73) 0.036 (14.3) 0.726 (3.91) 0.76 (15.0) LG/LF , c2 23.06 (2.31) 15.49 (3.8) - - 27.0 (2.51) 17.7 (3.57) (L G /LF)3, c 7 - - 757747.5 (4.32) 560011.7 (6.77) - - 𝑐𝑐14 in 𝜒𝜒1=𝑒𝑒 𝑐𝑐(14) 1+𝑒𝑒𝑐𝑐(14) 0.72 (9.75) 0.79 (-8.2) 0.63(4.58) 0.67 (4.95) -2.03 (-3.26) -2.42 (-8.7) 𝜒𝜒1 0.67 0.69 0.65 0.662 0.116 O.082 𝑐𝑐18 in 𝜒𝜒2=𝑒𝑒 𝑐𝑐(18) 1+𝑒𝑒 𝑐𝑐(18) 0.72 (9.75) 0.79 (-8.2) 0.63 (4.58) 0.67 (4.95) 2.986 (5.97) 33.46(216) 𝜒𝜒2 0.67 0.69 0.65 0.662 0.952 1 η, c13 -2.93 (-5.75) -3.65 (-10.8) -5.95 (-4.1) -6.89 (-7.8) 0.52 (0.49) 0.95 (1.89) δ, c17 -2.93 (-5.75) -3.65 (-10.8) -5.95 (-4.1) -6.89 (-7.8) -4.61 (-2.24) -70.4 (-11.7) σ 1 =1/(1-η) 0.254 0.215 0.144 0.127 2.085 18.02 σ2 = 1/(1-δ) 0.254 0.215 0.144 0.127 0.178 0.014 LOG(LZ), c8 0.140 (4.09) 0.112 (7.28) 0.086 (2.4) 0.067 (3.4) 0.106 (3.05) 0.093 (4.77) LBERDST(-1),c 10 -0.419 (-11.9) -0.40 (-30.14) -0.326 (-7.057) -0.312 (-18.2) -0.31 (-8.45) -0.302 (-17.85) LOG(L G ), c 9 0.126 (4.73) 0.123 (12.27) 0.148 (5.25) 0.136 (11.0) 0.09 (3.35) 0.076 (6.10) LPUBST(-1), c 11 -0.395 (-12.1) -0.396 (-45.93) -0.36 (-9.76) -0.351 (-29.3) -0.286 (-8.84) -0.276 (-22.68) Intercept 1 (c) 7.51 (1131.6) 5.887 (2239.9) -4.003 (-166.46) 7.6 (2857.7) 4.01 (243.0 4.05 (894.2) Resid 1 (-1) - 0.953 (95.2) - 0.95 (72.3) - 0.955 (79.6) Intercept 2 4.24 (17.08) 4.28 (90.6) 3.72 (58.1) 3.76 (68.3) 3.603 (14.9) 3.66 (69.1) Resid 2 (-1) - 0.946 (97.2) 0.93 (11.4) 0.94 (94.8) - 0.950 (93.8) Obs per equa�on (f) 207 2002-17 189 2003 -17 185 2002-17 171 2003-17 185 2002-17 171 2003-17 Itera�ons to convg. 59 66 59 69 56 83 Log likelihood 703.9 1107.8 642.63 983.62 665.96 995.2 DW sta�s�c (d) 1st, 2nd equa�on 0.026 0.03 1.65 1.94 0.0297 0.034 1.64 1.92 0.03 0.038 1.656 1.957 Pesaran CD: p-val. (e) (resid of 1 st , 2 nd eq) 0.044, 0.0002 Na, na 0.3068, 0.0234 0.0000, 0.1832 0.3777, 0.0173 0.0000, 0.0720 Unit root LLC t* (resid of 1 st , 2nd eq) 0.0996 0.2579 0.0000 0.0000 0.1551 0.3049 0.0000 0.0000 0.2042 0.3290 0.0000 0.0000 ADF-Fisher Chi sq. (resid of 1st, 2nd eq) 0.1615 0.0288 0.0000 0.0000 0.1443 0.0141 0.0000 0.0000 0.2341 0.0228 0.0000 0.0000 PP-Fisher Chi square (resid of 1st, 2nd eq) 0.0655 0.0183 0.0000 0.0000 0.0448 0.0036 0.0000 0.0000 0.1613 0.0096 0.0000 0.0000 (a) Es�ma�on Method: Full Informa�on Maximum Likelihood (BFGS /Marquardt steps); coefficient covariance computed using the Huber-White method; z-values (= coefficient/ std. error) in parentheses. Results depend on ini�al values. Iden�cal or asymmetric CES and spillover parameters in the private and public R&D produc�on func�ons. (b) Excluding Japan, Norway, USA. (c) In the first regression we have log(c1) + c24, which is a combina�on of a non-linear and a linear specifica�on of an intercept allowed only under ML es�ma�on (Greene 2012). (d) The Durbin-Watson sta�s�c is used only as descrip�ve informa�on, not as a test (see Epple and McCallum 2006). (e) Null: No cross sect.dep. (f) �me span determined by CAN with most observa�on; loss of observa�on per dropped country is only 6 or 7. 14 Japan, Norway, and USA. In columns 2, 4, and 6 we add the residuals of the previous regressions in order to correct for serial correla�on. In columns 1 to 4 for iden�cal spillover func�ons we find values for the linear own-spillover parameters χ between 0.65 and 0.69, which is close to the value of 0.7 assumed in the calibra�on of HLP, and it implies a cross-spillover, 1χ, from private to public R&D and vice versa of 0.35 to 0.31. The elas�city of subs�tu�on between private and public R&D capital is between 0.12 and 0.26, calculated from the es�ma�on of the CES parameter between -2.9 and -6.9. The labour produc�on terms have the expected posi�ve sign and the R&D capital terms have the expected nega�ve sign. The government labour share has a posi�ve sign in all columns. The func�on f = 1+ξ(sG)α with α = 1 or 3 runs up (in the data range going to sG = 0.0074) to 1.17 for a coefficient of ξ = 23.06 in column 1, to 1.115 for a coefficient of ξ = 15.49 in column 2, to about 1.3 for column 3, and 1.25 for column 4. As sG < 0.0074, the expression f(1sG) in formula (16) of HLP also is at almost the same values as those for f just indicated. By implica�on, the share of government researchers clearly enhances growth also at this level of the analysis. This confirms that HLP make reasonable assump�ons regarding the CES spillover and labour interac�on terms in the dynamic R&D func�on. In column 5 and 6 we now allow the CES spillover func�ons to differ between private and public. The private own spillover goes to about 0.1 implying a cross-spillover from public to private R&D of 0.9. The public own spillover goes to 0.95 or even 1, implying a cross-spillover from private to public of 0.05 or even zero. The elas�ci�es of subs�tu�on go to remarkably high values for private R&D and very low values for public R&D. The labour share interac�on effect is about the same as in column 1 and 2. There is obviously some panel heterogeneity (comparing columns 1 and 2 with 3 and 4, and an impact from serial correla�on correc�on comparing even and odd numbered columns. The presence of panel heterogeneity suggests es�ma�on on a country by country basis if there is an interest in ge�ng all the parameters more exactly. The fourth last row of Table 6 tries to indicate the p-values for the hypothesis of no cross-sec�on dependence (csd). These are ‘non-available’ or close to this again. In each case one of the equa�ons has csd at the five percent level and the other has not. Residual augmenta�on turns around which one has (no) csd. Periods are too short to allow for panel unit root analysis considering csd. Given the overrejec�on of the null of independence by the Pesaran CD test (see Pesaran and Xie 2023), we may have more independence than indicated. Panel unit root tests without considering cross-sec�on dependence for the residuals are shown in the last three lines. In the equa�ons without serial correla�on correc�on there are common and individual unit roots according to the LLC and the ADF Fisher Chi square test, and, less likely, according to the PP Fisher Chi square test. This would suggest having no cointegra�on. However, in equa�ons with serial correla�on correc�on, unit roots vanish with the serial correla�on correc�on. For other well-known tests considering csd we do not get any test output because the data series is too short. Therefore, our conclusion on cointegra�on is more intui�ve than sta�s�cally exact. Next, we replace the labour share of government researchers by the labour interac�on term, sG =LG*LZ /E+11. Table 7, column 1 and 2, shows results for 17 OECD countries and in 15 column 3 and 4 for 14 OECD countries leaving out Japan, Norway, and the USA because of the small number of observa�ons. In column 1 and 3 we assume that private and public R&D equa�ons both have the same spillover parameters, 𝜒𝜒1=𝜒𝜒2, and the same CES parameters of the spillover func�on, δ = η, and in columns 2 and 4 we allow these parameters to differ for the two equa�ons. In Table 8 we add the serial correla�on correc�on to these es�mates. In column 1, we have χ = 0.69, and the CES parameter is δ = η = -3.199 leading to an elas�city of subs�tu�on for private and public R&D in the spillover func�on of σ = 0.238 for the 17 OECD countries. The parameters for the f func�on in private R&D dynamics, c2, c6, c7, imply the possibility that labour interac�on goes (within the data range) to almost 65% beyond the model without interac�on as shown in Figure 1. Table 7 Regression coefficients of dynamic private and public R&D equa�ons with R&Dlabour-interac�on terms and CES spillover func�ons with constant returns to scale (a) Panel → Variables ↓ OECD 17 iden�cal OECD 17(e) asymmetric OECD 14 (b) iden�cal OECD 14 (b) asymm. α = c1 0.554 (2.29) 0.303 (2.55) 0.464 (2.19) 0.443 (2.02) LG*LZ , c2 1.481 (4.44) 0.963 (3.15) -0.573 (-0.552) 0.93 (0.76) (LG*LZ)2 , c6 -1.045 (-3.47) -0.887 (-2.97) 14.73 (2.60) 4.92 (0.78) (L G *L Z )3 , c 7 0.22 (2.835) 0.221 (2.79) -28.74 (-2.95) -13.49 (-1.31) 𝑐𝑐14 in 𝜒𝜒1=𝑒𝑒 𝑐𝑐(14) 1+𝑒𝑒 𝑐𝑐(14) 0.81 (9.38) -1.61 (-4.59) 0.7 (4.56) -2.55 (-5.54) 𝜒𝜒1 0.69 0.166 0.668 0.072 𝑐𝑐18 in 𝜒𝜒2=𝑒𝑒 𝑐𝑐(18) 1+𝑒𝑒 𝑐𝑐(18) 0.81 (9.38) 3.29 (3.96) 0.7 (4.56) 4.07 (2.82) 𝜒𝜒2 0.69 0.964 0. 668 0.983 η, c13 -3.199 (-5.985) 1.22 (1.95) -7.61 (-3.14) 1.428 (2.265) δ, c17 -3.199 (-5.985) -3.64 (-3.06) -7.61 (-3.14) -3.42 (-3.278) σ 1 =1/(1-η) 0.238 -4.545 0.116 -2.336 σ2 = 1/(1-δ) 0.238 0.2155 0.116 0.226 LOG(LZ), c8 0.091 (3.054) 0.113 (3.82) 0.0715 (1.96) 0.08 (2.25) LBERDST(-1),c 10 -0.481 (-13.46) -0.444 (-14.09) -0.379 (-8.55) -0.381 (-9.85) LOG(LG), c9 0.1135 (04.54) 0.08 (3.34) 0.1 (3.34) 0.107 (3.375) LPUBST(-1), c11 -0.386 (14.1) -0.347 (-13.5) -0.303 (-8.895) -0.281 (-9.235) Intercept 1 (c) 6.629 (583.3) 6.58 (687.8) 6.03 (438.0) 5.83 (383.7) Trend 1 -0.01 (1.364) -0.0078 (-1.13) -0.00878 (-1.09) -0.0048 (0.64) Intercept 2 4.84 (15.9) 4.756 (16.78) 4.143 (12.43) 3.966 (11.82) Trend 2 -0.011 (1.76) -0.00956 (-1.58) -0.01 (1.483) -0.0072 (7.83) Obs per equa�on 207 207 185 185 Log likelihood 727.2485 757.1327 656.6719 677.3284 Pesaran CD: p-val. (d) 0.9752, 0.5146 0.9577, 0.9867 0.8336, 0.3693 0.7967, 0.9078 Unit root LLC t* (e) 0.1930, 0.1680 0.4067, 0.1317 0.3615, 0.1564 0.3870, 0.2404 ADF-Fisher Chi sq. (e) 0.0261, 0.0066 0.4010, 0.0292 0.2854, 0.0252 0.2924, 0.0541 PP-Fisher Chi sq. (e) 0.1459, 0.0041 0.3505, 0.0095 0.1344, 0.0122 0.1759, 0.0603 (a) Es�ma�on Method: Full Informa�on Maximum Likelihood (BFGS /Marquardt steps); z-values (= coefficient/ std. error) in parentheses. Iden�cal or asymmetric CES spillover parameters in the private and public R&D produc�on func�ons. (b) Excluding Japan, Norway, USA. (c) In the first regression we have log(c1) + c24, which is a combina�on of a non-linear and a linear specifica�on of an intercept, allowed only under ML es�ma�on (Greene 2012). (d) Tests applied to residuals of 1st and 2nd eq; null: No cross sect. dep. (e) Null: unit root. 16 In column 2, we allow the parameters to be different in private and public R&D dynamics. The own spillover parameter is now low in the private R&D func�on, χ1 = 0.166, implying a high public-to-private R&D cross spillover 1-χ1 = 0.834, and it is high in public R&D own spillovers, χ2 = 0.964 implying a low private-to-public cross spillover of 1χ2 = 0.036. From the CES parameters we get σ1 = -4.545, sign change indica�ng complementarity in private R&D dynamics, and σ2 = 0.2155 indica�ng low subs�tutability in public R&D dynamics. The f func�on (similar to Figure 1 and not shown) would go to almost 1.4 in the maximum at LZLG = 0.756. In column 3, imposing iden�cal func�ons again, now for 14 OECD countries, the own spillover parameter now is χ = 0.67, a bit smaller than for 17 countries, and the elas�city of subs�tu�on is 0.116, which is about half of that for 17 countries, both together indica�ng panel heterogeneity. The f func�on (similar to Figure 1 and not shown) again remains near 1.4 in its maximum, which is located again slightly below LZLG = 0.8. In column 4 for 14 OECD countries, we again allow both func�ons to have different parameters. Again, own spillovers in private R&D are low at χ1 = 0.072 and high in public R&D χ2 = 0.983. Elas�ci�es of subs�tu�on change sign again in private R&D dynamics and are low in public R&D, at σ1 = -2.336 and σ2 = 0.226 repec�vely. The f func�on peaks with a maximum below 1.3 at a value of LL slightly larger than 0.3, which is much lower than under iden�cal parameters and lower than for 17 countries in column 2. Figure 1 The impact of private-public research-labour interac�on on private R&D growth In spirit and even numerically, the results of Table 7 are close to those of Table 6 using the government research labour share. The fourth but last row shows high probabili�es for cross-sec�on independence in the residuals of all equa�ons. The panel unit root tests for the residuals show high probabili�es for unit roots sugges�ng lack of cointegra�on. In column 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0 1.1 1.2 1.3 1.4 1.5 1.6 LZLG_E-11 f 17 to 4 we therefore have very low Durbin-Watson sta�s�cs near zero (not shown). We have saved the residuals of the equa�ons and added their lags to the regressions. Table 8 Regression coefficients of dynamic private and public R&D equa�ons with R&Dlabour-interac�on terms, CES spillover func�ons with constant returns to scale, and residual augmenta�on (a) Panel → Variables ↓ OECD 17, iden�cal OECD 17, asymmetric OECD 14 (b), iden�cal OECD 14 (b) asymmetric α = c1 0.234 (4.22) 1.877 (3.35) 0.567 (6.07) 0.475 (7.9) LG*LZ , c2 1.787 (8.79) 1.114 (6.56) 0.456 (0.814) 1.549 (2.45) (LG*LZ)2 , c6 -1.252 (-5.045) -0.897 (-4.47) 10.56 (3.365) 1.955 (0.558) (LG*LZ)3 , c7 0.26 (3.40) 0.203 (3.34) -21.3 (-3.998) -7.82 (-1.41) 𝑐𝑐14 in 𝜒𝜒1=𝑒𝑒 𝑐𝑐(14) 1+𝑒𝑒 𝑐𝑐(14) 0.954 (8.075) -1.604 (-8.57) 1.087 (3.77) -2.278 (-6.413) 𝜒𝜒1 0.722 0.167 0.748 0.093 𝑐𝑐18 in 𝜒𝜒2=𝑒𝑒 𝑐𝑐(18) 1+𝑒𝑒 𝑐𝑐(18) 0.954 (8.075) 33.3 (184.1) 1.087 (3.77) 31.68 (52.65) 𝜒𝜒2 0.722 1 0.748 1 η, c13 -4.31 (-7.977) 0.881 (2.38) -12.04 (-3.51) 0.597 (0.815) δ, c17 -4.31 (-7.977) -63.69 (-15.73) -12.04 (-3.51) -54.2 (-15.15) σ1 =1/(1-η) 0.188 8.41 0.077 2.48 σ2 = 1/(1-δ) 0.188 0.0155 0.077 0.018 LOG(LZ), c8 0.054 (2.76) 0.083 (4.41) 0.029 (1.367) 0.049 (2.50) LBERDST(-1),c10 -0.469 (-28.05) -0.433 (-26.28) -0.377 (-23.69) -0.376 (-24.5) LOG(LG), c9 0.136 (9.646) 0.093 (6.74) 0.13 (9.84) 0.124, (10.76) LPUBST(-1), c11 -0.363 (-33.5) -0.408 (-30.1) -0.33 (-26.3) -0.323 (-30.6) Intercept 1 (c) 7.73 (41.06) 4.94 (18.6) 6.21 (1226.35) 6.01 (1478.0) Trend 1 -0.0097 (-4.90) -0.0075 (-3.84) -0.008 (-5.669) -0.0047 (-3.399) Resid 1 (-1) 0.952 (78.64) 0.962 (72.3) 0.945 (77.26) 0.962 (78.6) Intercept 2 4.83 (46.56) 4.80 (46.8) 4.132 (58.1) 0.124 (10.76) Trend 2 -0.01 (-6.046) -0.0111 (-5.47) -0.01 (-8.65) -0.008 (-6.669) Resid 2 (-1) 0.934 (67.9) 0.95 (68.84) 0.928 (86.77) 0.949 (88.2) Obs per equa�on 189 189 171 171 Log likelihood 1105.047 1114.284 985.5202 996.8732 Adj R-sq 0.379, 0.252 0.39, 0.256 0.374, 0.239 0.387, 0.245 DW sta�s�c (d) 1.66, 1.89 1.691, 1.951 1.644, 1.871 1.677, 1.93 Pesaran CD: p-val. (e) Na, na Na, na 0.0000, 0.1184 0.0000, 0.0845 Unit root LLC t*(f) 0.0000, 0.0000 0.0000, 0.0000 0.0000, 0.0000 0.0000, 0.0000 ADF-Fisher Chi sq. (f) 0.0000, 0.0000 0.0000, 0.0000 0.0000, 0.0000 0.0000, 0.0000 PP-Fisher Chi sq. (f) 0.0000, 0.0000 0.0000, 0.0000 0.0000, 0.0000 0.0000, 0.0000 (a) Es�ma�on Method: Full Informa�on Maximum Likelihood (BFGS /Marquardt steps); z-values (= coefficient/ std. error) in parentheses. Results depend on ini�al values; mostly we find the highest likelihood when ini�al values from the calibra�on are used. Iden�cal or asymmetric CES and spillover parameters in the private and public R&D produc�on func�ons. (b) Excluding Japan, Norway, USA. (c) In the first regression we have log(c1) + c24, which is a combina�on of a non-linear and a linear specifica�on of an intercept allowed only under ML es�ma�on (Greene 2012). (d) The DurbinWatson sta�s�c is used only as descrip�ve informa�on, not as a test (see Epple and McCallum 2006). (e) Tests applied to residuals of 1st and 2nd eq; null: No cross sect. dep. (f) Null: unit root. 18 Therefore, we report both results in order to make the sensi�vity visible. The results for residual augmenta�on are shown in Table 8.9 The result for the residual augmenta�on of Table 7, column 1, is shown in column 1 of Table 8. The spillover parameter is now χ = 0.72, which is almost iden�cal to the calibra�on of HLP (2023) and similar to that of Table 7, column 1. The elas�city of subs�tu�on is now about 0.188, which is below that from the CES parameter assump�ons of δ = η = 0.287 in the calibra�on of HLP, which leads to a CES above unity. The f func�on (not shown) goes to about 1.8, which is slightly higher than without added residuals, and has a maximum at LZLGE-11 = 1.07. The result for the residual augmenta�on of column 2 of Table 7 is shown in column 2 of Table 8. The own-spillover parameters are now χ1 = 0.167 and χ2 = 1, implying again high(er) public-to-private spillovers and no private-to-public spillovers, which is more extreme than in the earlier results above. The elas�ci�es of subs�tu�on are now 8.41 and 0.0155, high(er) in private R&D and low(er) in public R&D. The f func�on goes slightly beyond 1.4 and reaches a maximum at LZLGE-11 = 0.888. The result for the augmenta�on of column 3 of Table 7 is shown in column 3 of Table 8. The spillover parameter is now χ = 0.748, slightly higher than the result in column 1 and the calibra�on of the theore�cal model of HLP. The elas�city of subs�tu�on is now 0.077, which is very low. The f func�on goes to about 1.5, which is slightly higher than without added residuals, and has a maximum at LZLG = 1.07 as in column 1. The result for the residual augmenta�on of column 4 of Table 7 is shown in column 4 of Table 8. Spillover parameters again are very low for private, χ1 = 0.093, and very high for public R&D func�ons, χ2 = 1, implying the opposite for the cross spillovers: a strong effect from public to private and none for private to public. The labour interac�on func�on f again peaks around 0.35 with a maximum of about 1.4. Spillover parameters are near the calibra�on values of 0.7 in HLP(2023) in all equa�ons of Table 7 and 8 when imposing that the spillover func�ons are iden�cal for private and public R&D capital growth processes, and they get very low for private and very high for public R&D when we allow them to differ between private and public R&D processes, implying that public R&D has strong spillovers to private R&D but private R&D has (almost) no spillovers to public R&D. Elas�ci�es of subs�tu�on do not change signs anymore when lagged residuals are added in Table 8. They are low when assuming iden�cal ones for private and public R&D. They get higher for private R&D and even lower for public R&D when the spillover func�ons are allowed to differ. Residual augmenta�on improves the DW sta�s�c and makes unit roots vanish and therefore reduces the suspicion of non-cointegra�on. This could change under cross-sec�on dependence, which remains present in Table 8. However, the test is (close to) unavailable when countries with less (more) observa�ons are included and it rejects the null of independence too o�en (Pesaran and Xie 2023). 9 One plausible reason is the omission of foreign R&D because the mo�va�ng model by HLP (2023) is a closed economy model. Its inclusion can be done in future research. 19 In all equa�ons of Table 7 and 8 we see a coefficient for labour much below unity and for R&D capital much less nega�ve than minus one. This may indicate that the linear labour part of the func�on, LZ /Z or LD/D, could be replaced by a Cobb-Douglas func�on. The parameters would suggest that they have decreasing returns. In addi�on, we have small nega�ve �me trends roughly between one percent or a half, which may not only be mere detrending but rather indicate produc�vity decreases in dynamic R&D capital produc�on func�ons. The appearance of decreasing returns together with technical change in R&D produc�on func�ons is similar to and familiar from classical economics and has entered neoclassical economics in agriculture-industry models in the 1960s (Jorgenson 1961) and later for the case of posi�ve technical change. The idea that high elas�ci�es of subs�tu�on (or even complementari�es) can have the same effect as technical change has been discussed more recently by Klump and De La Grandville (2000) and the related literature (see Ziesemer 2023a). This idea may be even more important if technical change is nega�ve at the macrolevel, perhaps through structural shi�s to sectors with low produc�vity growth. Decreasing returns, technical change and the elas�city of subs�tu�on are therefore closely related, and it is hardly surprising that they all appear in R&D produc�on func�ons again. It remains to be clarified in future research whether the assumed linearity and the calibrated subs�tu�on in the spillovers func�on in the model of HLP (2023) is more or less exactly equivalent to the empirically low subs�tu�on with decreasing returns and nega�ve �me trends shown above. Perhaps simula�on analysis can help show in the future how similar they are. 7. Panel data analysis: Country fixed effects and autoregressive processes In this sec�on we try to add country-specific fixed effects in the LSDV form. Only the nonlinear, itera�ve least squares es�ma�on method for the case of iden�cal CES spillover func�ons for both processes give a regression output, which is presented in Table 9. Again, we need to start with the calibra�on values of the theore�cal growth model. For Japan and the USA, we have an insufficient number of observa�ons. Therefore, we include the other 15 countries, for which we typically have 13 observa�ons per equa�on of each country (11 for Norway, 16 for Canada). When we add autoregressive (ar) processes for the residuals, we lose an observa�on per lag in each equa�on for each country leaving us with only eleven observa�ons for most of the equa�ons. Results are shown in Table 9 for varying assump�ons on ar processes. Because of the small number of observa�ons and fixed effects absorbing degrees of freedom, many coefficients are sta�s�cally insignificant. CES parameter η is sta�s�cally significant at the five or one percent level with values between -0.64 and -0.7 leading to an elas�city of subs�tu�on of about σ = 0.6. Again, this provides support for the idea of a CES spillover func�on. The spillover parameter c14 in the e-func�ons are between 0.375 and 0.86, leading to χ between 0.59 and 0.7. However, they are sta�s�cally insignificant. Se�ng c to zero yields χ = 𝑒𝑒0 1+𝑒𝑒0 = 0.5. In column 2 and 3, in the private R&D equa�on an ar(2) process 20 is significant, and in the public R&D equa�on an ar(1) process is significant at all standard levels, and an ar(2) process is significant at the 10 percent level. Introducing the ar processes turns the unexpected nega�ve sign of the government labour term in the public R&D equa�on (column 1) into a posi�ve one (column 2 and 3). Unlike Table 7 and 8, the coefficients of the labour-interac�on terms in the private R&D equa�on are sta�s�cally insignificant. This does not change when taking one or two of them out, which is a major difference with the previous tables, raising the ques�on whether the interac�on specifica�on or the collabora�on idea are false, or significance would come about with more observa�ons and degrees of freedom. When ar processes are included the func�ons peak at f = 1.875 and 1.89 both at LL = 0.41, and a minimum with f < 1 at LL = 0.047. The corresponding func�on f(1-s) is above unity for the range 0.17439 < s < 0.45360 for the es�mate of column 2, and the range 0.17369 < s < 0.45680 for the es�mate of column 3; both ranges are within the data range for s, (0,1). A posi�ve minimum of interac�on is required to get a nonnega�ve effect. We were unable to get results when the government labour share was used instead of an interac�on term. Table 9 Regression coefficients of dynamic private and public R&D equa�ons with R&Dlabour-interac�on terms, fixed effects, CES spillover func�ons with constant returns to scale, and autoregressive-processes (a) Panel → Variables ↓ OECD 15 OECD 15 OECD 15 LG*LZ , c2 -5.69 (1.54) -2.277 (-0.335) -2.373 (-0.36) (LG*LZ)2 , c6 29.6 (1.91) 26.84 (0.849) 27.6 (0.892) (LG*LZ)3 , c7 -36.1 (-1.77) -39.2 (-0.89) -40.28 (-0.934) 𝑐𝑐14 in 𝜒𝜒=𝑒𝑒 𝑐𝑐(14) 1+𝑒𝑒 𝑐𝑐(14) 0.375 (0.38) 0.858 (0.481) 0.606 (0.347) 𝜒𝜒 0.593 0.7 0.647 η, c13 -0.643 (-2.77) -0.70 (-2.019) -0.66 (-3.14) σ =1/(1-η) 0.61 0.588 0.602 LOG(LZ), c8 0.496 (4.58) 0.525 (3.87) 0.525 (3.966) LBERDST(-1),c10 -1.43 (-5.70) -2.654 (-5.54) -2.616 (-5.43) ar 1st eq: ar(2) - 0.243 (2.83) 0.240 (2.84) LOG(LG), c9 -0.082 (-0.664) 0.213 (1.845) 0.214 (1.788) LPUBST(-1), c11 -0.277 (-0.91) -1.877 (-2.439) -2.36 (-3.13) Trend per country (b) yes yes yes ar 2nd eq: ar(1); ar(2) - 0.597 (c) (7.34) 0.6; -0.148 (6.32) (-1.68) Period (d) 2002-2017 2003-2017 2004-2017 Obs (e) 392 347 332 (a) Es�ma�on Method: Itera�ve Least Squares; t-values (= coefficient/std. error) in parentheses. Iden�cal CES and spillover parameters in the private and public R&D produc�on func�ons. Excluding Japan, USA. Country specific intercepts and trends in both equa�ons; slope homogeneity for ar processes. (b) Results are shown in Table 10. (c) ar(1) only. (d) 13 observa�ons per country and equa�on, 11 for NOR, 16 for CAN; periods indicated for CAN, all others three less, NOR 5 less. (e) 15 obs lost per ar lag. 21 For the three equa�ons in Table 9, the results for country-specific �me trends per country are shown in Table 10. Averaging over the countries, only public R&D has a nega�ve trend in public research produc�vity in column 2 of Table 10 belonging to column 1 of Table 9, where we have no autoregressive processes. When ar processes are introduced produc�vity trends become posi�ve on average for these 15 OECD countries for data of this millennium un�l 2017. However, �me trends may also simply detrend variables on the right-hand side of the es�mated equa�ons. Unfortunately, we do not get results with fixed effects and ar-terms when allowing for different parameters in the two R&D func�ons or when using different es�ma�on methods. Moreover, using lags as instrumental variables has not led to any regression output, perhaps because of losing one more year of observa�ons when using lagged instrumental variables. Table 10 Research produc�vity trends per country: Coefficients of �me trend per equa�on with fixed effects Specification (a) OECD 15 no ar process OECD 15 1st eq.: ar(2); 2 nd eq. ar(1) OECD 15, 1st eq.:ar(2);2 nd eq.ar(1), ar(2) Equation → Country↓ DLBERDST DLPUBST DLBERDST DLPUBST DLBERDST DLPUBST AUT 0.0140 -0.0159 0.0711 0.03177 0.0700 0.0467 BEL 0.0441 0.0250 0.0735 0.0722 0.0720 0.0842 CAN - 0.0109 -0.0268 - 1.7531e-05 0.0103 - 0.0012 0.0236 DEU - 0.0109 0.0171 0.0307 0.0473 0.0290 0.0661 DNK - 0.01305 -0.0153 0.0185 0.0295 0.0169 0.0534 ESP - 0.033 -0.07374 - 0.00105 -0.01516 - 0.0023 0.0025 FIN - 0.0241 -0.0306 - 0.0330 -0.00269 - 0.0337 0.0102 FRA + 0.0074 +0.01323 0.0150 0.0210 0.0140 0.0294 GBR + 0.0163 0.0055 0.0241 0.02045 0.0232 0.0235 IRL 0.0723 0.0174 0.0685 0.0092 0.0680 0.0169 ITA 0.0012 -0.0320 0.0177 -0.0347 0.0179 -0.0257 NLD 0.0625 0.0603 0.08735 0.0789 0.0869 0.0856 NOR 0.0199 -0.00134 0.0550 0.0419 0.0538 0.0549 PRT - 0.0560 -0.0517 - 0.0181 -0.0370 - 0.0170 -0.0512 SWE - 0.00166 0.0021 - 0.0070 0.0388 - 0.0084 0.0563 Average 0.0059 -0.0071 0.027 0.021 0.0259 0.0318 (a) Corresponding to the three equations and columns of Table 9. 8. A VAR model in growth rates for 14 OECD countries with cuts in public R&D 8.1 The panel VAR model In this sec�on we compare the dynamic proper�es of the HLP model with those of a VAR (vector autoregressive) model in growth rates for the 14 OECD countries with sufficiently many observa�ons. We consider the growth rates in terms of log differences (dL) of technical progress TH07, private R&D stock, BERDST, public R&D stocks, PUBST, number of enterprises, ETP, number of business researchers, BR, and number of government researchers, GR. VARs in log-levels of pooled data with lag length eight or less are all unstable. The corresponding VECMs are mostly unstable, but for some constella�ons VECMs 22 are stable for some combina�ons of lag length and number of long-term rela�ons. But then the equa�ons turn out to have fixed effects as in Hsiao (2022), chapter 5.2.1.1, special case iv. The recommended procedure then is to take differences of the underlying VAR to get rid of the fixed effects. This leads us to a VAR in growth rates, for which use of maximum likelihood es�ma�on, or GMM with lagged regressors as instrumental variables is recommended. It turns out that all equa�ons have fixed effects again. The reason for this may be that differencing leads to country-specific coefficients stemming from the �me trend, which in Hsaio’s textbook model has slope homogeneity. Finding these fixed effects suggests that �me trends do have slope heterogeneity as in the single-country VECM es�mates of Soete et al. (2022). To get rid of the fixed effects, we may take differences again, leading to a VAR in changes of growth rates without constants that have vanished through the differencing. A steady state then requires zero changes of growth rates. The maximum number of lags according to Schwert’s formula is eight. VARs with eight or seven lags are unstable. Therefore, we try using a VAR with lag length six, which is stable and recommended by standard length criteria AIC (preferred by Kilian and Lütkepohl 2017), HQ, LR, and FPE, whereas SIC suggests 2 lags, leaving probably more (risk of) serial correla�on in the model. However, while stable in changes of growth rates the model generates growth rates some of which get exorbitantly large when simula�ng forward to 2100, also for any other lag length. Therefore, we have to go back to a VAR in growth rates, for which we have two op�ons: (i) under the assump�on that the bias from fixed effects is small as coefficients from trends cannot differ strongly, we can ignore the fixed effects and es�mate a VAR in growth rates using maximum likelihood es�ma�on; (ii) we can take fixed effects into account, leading us to panel VAR with equa�ons es�mated separately using the orthogonal devia�on method of Arellano and Bover (1995). Both versions generate comparable results suppor�ng the HLP model. For the VAR in growth rates of pooled data, among the stable models with lag length up to 7, we choose the model with four lags, which is suggested by the Hannan-Quinn criterion (when the maximum number of lags is 6), which is known to be consistent.10 We then get rid of cross-sec�on dependence. Models with lags three or four have no nega�ve growth rates in the forward simula�on un�l 2100; allowing for more lags, leads to several nega�ve growth rates in the long run, which is unrealis�c as the corresponding variables would run to nega�ve values in the long run. We report results from an ML es�mate of a VAR in growth rates with four lags (ignoring fixed effects) in an appendix. For the VAR with GMM-OD (orthogonal devia�ons) there are two approaches. Abrigo and Love (2016) have developed a VAR version for STATA where the complete system is es�mated simultaneously. Alterna�vely, we can es�mate the equa�ons separately and then import them into a simultaneous equa�on system (Chu et al. 2021) and es�mate the intercepts using the seemingly unrelated regression (SUR) method. This implies a loss of efficiency. However, we use four lags in line with the lag length tests from the VAR based on 10 When allowing for a maximum of seven lags, HQ suggests three lags and we get cross-sec�on dependence in the residuals. 29 (iii) The growth of private knowledge stocks decreases under a public R&D cut, 𝜕𝜕𝑍𝑍 � 𝜕𝜕𝑠𝑠𝐺𝐺> 0. These theore�cal results are meant to hold for the steady state values in HLP. The corresponding empirical results from our impulse response analysis using the GMM-OD VAR model are shown in Figure 3 for the growth of labour produc�vity and of BERDST. The result (i) for technical change is confirmed in Figure 3. The result (ii) for business researchers per firm is confirmed in Figure 4. This plot compares to HLP’s Fig. 5, upper-le� panel, which has more business researchers in the phase shortly a�er the shock, but then is comparable to the falling patern shown here. The result (iii) for business R&D knowledge stock growth holds empirically in the long run of shown in Figure 3. Result (iii) has therefore confirma�on from our empirical GMM-OD VAR. 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 AUT - 63 - AUT - 63 - BEL - 25 - CAN - 87 - CAN - 87 - DEU - 49 - DNK - 11 - ESP - 73 - ESP - 73 - FIN - 35 - FRA - 97 - FRA - 97 - GBR - 59 - IRL - 21 - ITA - 83 - ITA - 83 - NLD - 45 - PRT - 07 - SWE - 69 - SWE - 69 - (BR_1M/ETP_1M)/(BR_0M/ETP_0M) Figure 4: Effects of public R&D cut on firm-level R&D, 2010-2100. 0M indicates baseline expected panel average value; 1M indicates expected panel average value from shock scenario. Each line represents one country. The growth reduc�on of firm knowledge shown in Figure 3 is also present in the transition phase of the theore�cal model (HLP, sec�on 4.2). This holds also for technical change (a�er a theore�cally unclear impact effect), which is also approximately visible in Figure 3. Overall, we consider this to be convincing evidence in favour of proper�es (i) to (iii) of the steady state of the HLG model; transi�onal results are also confirmed so far. Results upon 30 impact are hard to compare because the GMM-OD VAR has lags which the theore�cal growth model does not have. The dynamic relation between public and private R&D In the HLP model, firm level R&D, 𝐿𝐿𝑍𝑍(𝑡𝑡)𝑙𝑙(𝑡𝑡) ⁄, reacts to the nega�ve R&D shock on public R&D researchers, as far as the interac�on effect is concerned, with an ambiguous impact effect, and then falls to its new lower steady state. The corresponding GMM-OD VAR result is shown in Figure 4. There is no impact or early effect on average; the simula�on for the shock goes below that of the baseline scenario as in the theory. In par�cular, private R&D per firm in Figure 4 and public R&D in terms of researchers in Figure 2 go down together as compliments. The availability of more data allowing for (panel) �me-series analysis and the progress in econometrics in this millennium has lead to strong evidence for this complementarity (David et al. 2000; Becker 2015; Ziesemer 2021b). ‘The cut yields the percentage change in the steady-state mass of firms per capita’ (HLP, p.15). This increase is visible a�er some periods in Figure 2 for an unaffected popula�on growth path. Having shown in Figure 4 that under a public R&D cut firm-level R&D goes down and in Figure 2 that the number of firms goes up, the ques�on is how exactly private R&D (not per firm) goes together with public R&D, both in terms of number of researchers. HLP suggests that this is a hump shape form because firm level R&D goes together with public R&D but the number of firms doing R&D goes the opposite way. 31 Figure 5 Decreasing change of business researchers through more government researchers. We take both variables as a share of the exogenous labour force data. Figure 5 shows the rela�on between government researchers per person in the labour force, gr/lf, and business researchers as a share of the labour force, br/lf, in terms of the nearest-neighbour or lo(w)ess fit, using 60% of all data to generate a point, atribu�ng it to the middle observa�on, and then shi�ing one point further to repeat the procedure. Using 90% or 30% of the data per regression would lead to a similar graph. The upper-le� part of Figure 5 shows the decreasing slope between gr/lf and br/lf in terms of data. The upper-right part shows this also in terms of points of the baseline simula�on of the GMM-OD VAR (indicated as 0m). The lower-right part shows the decreasing slope for the public-R&D-cut scenario (indicated as 1m). The lower-le� shows their change from the baseline to the public R&D cut scenario in the VAR model (indicated as 1_0m), where business R&D is reduced much less than public R&D; this is ‘a �ny amount’ in the steady state in HLP, and here between - 0.0000 and -0.0006 for the period 2013-2020. All figures suggest that business R&D reacts less strongly to public R&D at higher values of public R&D. In all four graphs of Figure 5 the data points most far away from the origin are those of Finland; they spoil the idea of an 32 inverse u-shaped curve by bending the downward line upward. The outlier posi�on of Finland and perhaps other countries suggests the use of fixed effects methods for Figure 5. To underpin these results with more sophis�catedly tes�ng regression methods than the loess fit of Figure 5, we use pooled and weighted DOLS, because the variables have panel unit roots according to all tests (see Table A.2) and are sta�onary a�er differencing (not shown). The long-term rela�on is (with constants digested in the modified variables and not appearing in the regression output;14 p-values in parentheses) 𝐵𝐵𝐵𝐵/𝐿𝐿𝐿𝐿 = 274.1(𝐺𝐺𝐵𝐵/𝐿𝐿𝐿𝐿)2 − 18615.84(𝐺𝐺𝐵𝐵/𝐿𝐿𝐿𝐿)3 (0.0000) (0.0625) Figure 6: The HLP hump shape rela�on. Plot of DOLS regression rela�on between government and business researchers (% labour force, lf). Plo�ng the equa�on leads to Figure 6 with data in the range of the upward sloping part as can be seen from comparison with the data in Figure 5, where they are below 0.008, in the upward sloping part as noted by HLP. This provides some support for the idea of a humpshape form of HLP. However, we cannot test the shape in connec�on with all desirable proper�es of mean-group es�mators with the limited number of observa�ons available.15 14 Sample (adjusted): 1982-2020. Periods included: 39. Cross-sec�ons included 14. Total panel (unbalanced) observa�ons: 420. Panel method: Weighted es�ma�on. Cointegra�ng equa�on determinis�cs: C (no �me trend as variables have to stay in the unit interval). Automa�c leads and lags specifica�on (based on AIC criterion, max=*). Long-run variance weights (Prewhitening with lags from AIC maxlags = -1, Bartlet kernel, Newey-West automa�c bandwidth, NW automa�c lag length). Adjusted R-squared: 0.84. S.E. of regression: 0.001010. 15 The related (pooled) mean group es�mators of DOLS or PMG/ARDL require more lags than the data allow here. A PMG/ARDL es�mate, which looks similar to Figure 6 is BR/LF = 397.8780(GR/LF)2 -29927.32 (GR/LF)3. This is the best available PMG/ARDL es�mate. It uses only observa�ons 2007-2020; the required similarity with the mean group es�mator has p = 0.43. The selected model then is PMG(3,2,2) with 3 as the maximum number of differenced terms of the dependent variable indica�ng that more lags might be desirable, but they are not feasible with the limited number of observa�ons. Moreover, we have no constants, neither restricted nor 0.002 0.004 0.006 0.008 0.010 0.012 0.014 -0.004 -0.002 0.000 0.002 0.004 0.006 0.008 0.010 gr/lf br/lf 33 Figure 7 The growth rate of the number of firms is lower in the shock scenario than in the baseline scenario in the first year a�er the shock and higher later. In the model of HLP, the growth rate of the number of firms (% labour force) jumps down a�er the public R&D cut and then quickly increases beyond its baseline of zero to which it returns in the long run (see HLP, Figure 6). In Figure 7, differing from Figure 3 through the use of a Kernel fit, which goes more to the ini�al and final observa�ons, we show a similar result: the growth rate of the number of firms (% labour force) starts at zero in 2006, is lower in 2011, directly at the public R&D cut in 2010, and higher in all periods a�erwards. In the HLP model, impact effects are strong compared to our VAR model in which the lags in principle smooth the early effects, but here only by one period. The adjustment process of the complete HLP model is conducted in terms of the number-offirms/labour ra�o and the public/private knowledge ra�o. In Figure 8 we show that the public/private knowledge stock ra�o is going down rela�ve to baseline (un�l 2100 indeed) as in the HLP analysis. The number-of-firms/labour ra�o evolves as in the theore�cal analysis of HLP, first going slightly down and then up. unrestricted, also because of the limited number of observa�ons, and no cointegra�on according to the bounds test and the insignificance of the adjustment coefficient. Cointegra�on perhaps requires linking to more variables, which might also repair the other proper�es and preserve the shape, which is the major point of interest here. 34 Figure 8: The adjustment process a�er a public R&D cut. Upper part: The number of firms (% labour force) first goes down and then goes up, in the data period and also un�l 2100. Lower part: the public-private knowledge ra�o is decreasing. Although Figure 3 shows that the growth rate of the public knowledge stock goes up a�er the shock and that of private knowledge goes down, Figure 8 implies that the growth rate of private R&D remains higher than that of public R&D. 9. Summary and conclusion We have derived a Cobb-Douglas func�on for technical progress depending on public and private R&D knowledge stocks from two cointegra�ng equa�ons for each of three es�ma�on methods. Standard results from the literature are shown to hold also when including the number of firms as in the HLP model. 35 For the dynamic R&D capital growth func�ons of the HLP model, results obtained through full informa�on maximum likelihood (FIML) es�ma�on without fixed effects are summarized as follows. The research produc�vity of the private R&D process is enhanced through collabora�on of business and government researchers by 17-80% above baseline. The CES spillover func�ons in their R&D growth func�ons have linear own (cross) spillover distribu�on parameters between 0.65 and 0.75 (0.35-0.25) if private and public R&D processes are restricted to have the same CES spillover func�on as assumed in HLP’s calibra�on of the theore�cal model. Elas�ci�es of subs�tu�on between private and public R&D stocks in the spillover func�on are low under this assump�on. When spillover func�ons are allowed to be different between private and public R&D, (i) own spillover (distribu�on) parameters get lower for private R&D processes and close to unity in public R&D func�ons, implying almost no spillovers from private to public R&D and strong spillovers from publicto-private R&D; (ii) CES parameters for spillover func�ons get posi�ve for private R&D func�ons and more nega�ve for public R&D func�ons. Generaliza�on of the linear labour argument in the R&D growth func�ons to a Cobb-Douglas func�on in labour and R&D capital shows decreasing returns to labour and R&D capital. A nega�ve research produc�vity trend in both R&D growth func�ons is found when the R&D labour interac�on term is modelled as product of government and business researcher, but not if it is modelled as government researchers as a share of the labour force as in HLP. Fixed effects in combina�on with autoregressive processes in nonlinear, itera�ve least squares es�ma�on for the case of iden�cal CES spillover func�ons in dynamic R&D func�ons yields a spillover parameter range of 0.59-0.7 and a CES of 0.6. A VAR model turns out to have not only fixed effects as in Hsiao’s (2022) textbook model but also country-specific �me trends which turn up as fixed effects a�er differencing. We have assumed that these fixed effects are small rela�ve to each other and do cause only negligible biases in results shown in an appendix. In contrast, considering fixed effects using the orthogonal devia�on version of system GMM we find the slopes for all equa�ons separately and import them into a simultaneous equa�on model, leading to a GMM-OD VAR. The results from a public R&D cut in the VAR are compared to those in the theore�cal model of HLP. All steady-state and adjustment proper�es of the theore�cal HLP model are in line with those of the VAR model. The VAR ignoring fixed effects in the appendix has comparable results. A sugges�on for further research for theory with calibra�on is to allow for asymmetric private and public R&D dynamics, with strong public-to-private spillovers and weak privateto-public spillovers in the CES spillover func�on. The HLP model is the ideal basis for the analysis of the consequences of asymmetric CES spillover func�ons. Overall, we provide an empirical inves�ga�on that stays close to the endogenous growth model and confirms the assump�ons and results of the HLP model. 36 References Arellano, M., and O. Bover (1995) Another look at the instrumental variable es�ma�on of error-components models. Journal of Econometrics 68: 29–51. Abrigo, M. R., & Love, I. (2016). Es�ma�on of panel vector autoregression in Stata. The Stata Journal, 16(3), 778-804. Bai, Jushan, and Serena Ng (2004). A PANIC atack on unit roots and cointegra�on. Econometrica, 72, 1127–1177. 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In the change from the baseline to the public R&D cut scenario in the VAR model (indicated as 1_0m), business R&D is reduced much less than public R&D; this is ‘a �ny amount’ in the steady state in HLP, and here between -0.0000 and -0.00025 for the period 2013-2020, which is smaller than the effect for the GMM-OD VAR in the main text. 46 Figure A.7 The growth rate of the number of firms is lower in the shock scenario than in the baseline scenario in the early phase and higher later. Figure A.8: The adjustment process a�er a public R&D cut: the number of firms (% labour force) first goes down and then goes up; the public-private knowledge ra�o is decreasing. The last falling part of Figure A.8 for the number of firms would hold for a theore�cal constella�on crossing the isoclines two �mes more o�en than drawn in their Fig.1. This would imply a falling phase for the public/private R&D capital ra�o though, which we do not find empirically in our VAR. However, adding the labour force to the VAR, which we have not done in line with the theore�cal model, the shock would yield non-decreasing growth rates of the number of firms, and sta�s�cally significantly decreasing growth rates of the labour force. This would imply that the number of firms would always increase rela�ve to the labour force. This constella�on would be obtained when the moment a�er the shock has both variables below the isoclines. The UNU-MERIT WORKING Paper Series 2024-01 The green transformation as a new direction for techno-economic development by Rasmus Lema and Carlota Perez 2024-02 Evidence on an endogenous growth model with public R&D by Thomas H.W. Ziesemer