The innovation long-run risk component

Franceschini, Fabio Working Paper The innovation long-run risk component Quaderni - Working Paper DSE, No. 1215 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Franceschini, Fabio (2025) : The innovation long-run risk component, Quaderni - Working Paper DSE, No. 1215, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/8613 This Version is available at: https://hdl.handle.net/10419/331532 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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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ ISSN 2282-6483 The Innovation Long-Run Risk Component Fabio Franceschini Quaderni - Working Paper DSE N° 1215 The Innovation Long-Run Risk Component Fabio Franceschini ∗ 5th November 2025 [click here for the latest version] Abstract This paper provides robust empirical evidence that shocks to aggregate Research and Development (R&D) have persistent effects on macroeconomic dynamics and represent a significant risk for investors, as predicted by the ‘long-run risk’ literature. The analysis focuses on a single variable, ‘effective R&D’, which captures the entire contribution of R&D to productivity growth, flexibly accounting for knowledge spillovers and product proliferation effects. Deviations of effective R&D from its equilibrium level can be empirically identified leveraging the error correction term in the cointegration relationship among R&D, total factor productivity, and the labor force. In US data, structural effective R&D shocks affect productivity and consumption growth rates beyond business cycle horizons and are associated with a significant risk premium in a cross section of stock and bond portfolios (around 2% annually), with cash-flow sensitivities proving a key determinant. Keywords: R&D, Long-run risk, Asset Pricing, Cointegration JEL Codes: E32, E44, G12, O30 ∗Department of Economics, University of Bologna (Italy); francesc[email protected]. This paper was a chapter of my Ph.D. thesis and was funded by the European Union - NextGenerationEU, in the framework of the “GRINS -Growing Resilient, INclusive and Sustainable project” (PNRR - M4C2 - I1.3 - PE00000018 - CUP J33C22002910001). Special thanks to my advisors Max Croce and Martín Gonzalez-Eiras for many helpful discussions. I also thank Emanuele Bacchiocchi, Frederico Belo, Jeremy Boccanfuso, Laura Bottazzi, Marco Brianti, Svetlana Bryzgalova, Giuseppe Cavaliere, Enzo Dia, Luca Fanelli, Luca Gemmi, Howard Kung, Filippo Massari, Antonio Minniti, Elisa Ossola, Oliviero Pallanch, Andrea Renzetti and Alwyn Young as well as seminar participants at the University of Bologna, the University of Milano-Bicocca, the INSPIRE seminar series, and the 27 th INFER Annual Conference for comments and discussions from which I greatly benefited. The views and opinions expressed are solely those of the author and do not necessarily reflect those of the European Union, nor can the European Union be held responsible for them. 1 Non-technical summary This paper demonstrates that fluctuations in aggregate Research and Development (R&D) have persistent effects on the broader economy with significant implications for asset prices. Specifically, as R&D shocks increase expected growth in productivity and consumption at horizons beyond the business cycle, investors demand higher compensation from assets whose values are more sensitive to R&D dynamics, as these assets amplify uncertainty about future consumption streams. The analysis centers on a transformation of R&D called ‘effective R&D’, which expresses R&D in units of productivity gains while accounting for spillovers from past innovations and product proliferation. Constructed using quarterly US data from 1947 to 2024, shocks to this measure – isolated from productivity growth independent shocks – are shown to affect productivity and consumption growth for up to 20 years. These shocks are then tested as a risk factor. First, using 183 stock and bond portfolios with a methodology that extracts all latent risk factors to address omitted variable bias, the paper provides robust evidence that effective R&D shocks represent a priced risk factor. Second, examining how assets’ cash flows respond to these shocks reveals that this sensitivity is a crucial channel through which the risk premium emerges. This analysis also yields descriptive evidence supporting several insights: returns of firm-level R&D are higher when economy-wide R&D investment is elevated, and firm characteristics related to funding capacity and growth opportunities systematically relate to sensitivity to aggregate R&D. Overall, this paper makes four main contributions. It establishes the empirical significance of an R&D-based factor identified by macroeconomic theory (a twofold contribution), flexibly connects asset pricing to endogenous growth models through the ‘effective R&D’ measure and provides an econometric framework to recover it, offers descriptive evidence on how firm characteristics affect sensitivity to aggregate R&D, and supports the persistence amplification mechanism of R&D widely adopted in recent macroeconomic literature. 2 1 Introduction Investments in Research and Development (R&D) have a profound and well-documented impact on the economy. At the aggregate level, R&D has been linked to slow-moving fluctuations in macroeconomic quantities such as productivity and consumption (Comin and Gertler 2006; Evans et al. 1998), and theory has shown that this nexus carries substantial implications for asset prices (Kung and Schmid 2015). The underlying mechanism is that of the ‘Long-Run Risk’ (LRR) framework from Bansal and Yaron (2004), who first demonstrated how persistence in consumption growth enables macroeconomic models to account for the large equity risk premium observed in financial markets, a premium that far exceeds predictions based on observed consumption growth volatility alone (Mehra and Prescott 1985). This framework has since become a central reference in macro-finance research, 1 but has also faced sustained criticism,2making rigorous empirical validation of models built on it essential. This paper provides robust empirical evidence that shocks to US aggregate R&D have persistent effects on productivity and consumption growth, and carry substantial implications for asset prices. Specifically, I focus on a theoretical measure of aggregate R&D – ‘effective R&D’ – that uniquely summarizes the total impact of R&D on productivity growth, accounting for virtually any degree of spillover and product proliferation effects. I then propose an empirical framework to recover its time series and the associated structural shocks, illustrating their persistent effects on the economy. Finally, I show that structural shocks to effective R&D serve as a risk factor with significant power in pricing the cross-section of financial assets. This constitutes a twofold novelty, as it introduces the first empirically significant risk factor that is (1) based on aggregate R&D, and (2) constructed from structural shocks in a theoretically identified system. In the long-run risk framework, agents have recursive preferences as in Epstein and Zin (1989), which are characterized by an aversion to swings in forecasts of distant future consumption growth. Persistent shocks can induce such fluctuations with very small variance, making them a powerful theoretical feature by allowing models to account for sizable risk premia with little added complexity, but also an elusive one to validate empirically, since low-frequency processes are challenging to identify in finite samples. A growing literature has tackled the empirical challenge directly, either by devising refined empirical strategies (Bryzgalova et al. 2025; Dew-Becker and Giglio 2016; Gourieroux and Jasiak 2024; Ortu et al. 2013; Schorfheide et al. 2018) or by constructing new data (Liu and Matthies 2022). Another strand has derived the LRR pivotal conditions as reduced forms of richer structural models, yielding additional testable implications (e.g. Kaltenbrunner and Lochstoer 2010). Within this framework, Croce (2014) showed that the predictable component of productivity growth is persistent and does transmit to consumption growth. He termed this component the ‘productivity long-run risk component’, which Kung and Schmid (2015) later rationalized 1Applications include, among others, exchange rate dynamics (Colacito and Croce 2011), climate change asset pricing (Bansal et al. 2021), term structure models (Ai et al. 2018), and oil price dynamics (Ready 2018). 2 Most notably Constantinides and Ghosh (2011), Beeler and Campbell (2012), and Epstein et al. (2014). 3 Figure 1: Consumption growth is US real total consumption per capita from BEA; ‘TFP’ is the utilization-adjusted, excluding R&D capital, TFP from Fernald (2012). Of both is plotted the 6 th component of Ortu et al. (2013) decomposition, which filter the fluctuations with half-life between 8 and 16 years. Excess effective R&D is in levels, not growth rates. as originating from persistent fluctuations in R&D investment. This paper provides direct empirical evidence on the macroeconomic mechanism and financial predictions developed in this theoretical framework, while contributing a novel econometric procedure, grounded in macroeconomic theory, to identify a process that captures revisions to long-run expectations of innovation – an ‘innovation long-run risk component’. The results in this study depart from the predictions in Kung and Schmid (2015) only regarding the specific mechanics behind R&D shock propagation, which empirically does not appear to arise from the persistence of effective R&D itself, but rather from its interaction with external macroeconomic factors. Figure 1displays the estimated effective R&D, along with the processes it is expected to drive, i.e., the components with half-lives between 8 and 16 years of consumption and productivity growth rates, whose comovement is apparent. Effective R&D expresses aggregate R&D in units of expected productivity growth, as prescribed by endogenous growth theory. It is a log-linear combination of aggregate R&D, the stock of ideas and a product variety measure, capturing the nonlinear impact of R&D on productivity growth. Its definition is formalized within a theoretical framework that illustrates the minimal conditions linking aggregate R&D to asset prices, while intentionally abstracting from the optimal choices driving R&D dynamics, leaving the latter to empirical analysis. This theoretical framework relies primarily on a production function of ideas that nests Kung and Schmid (2015) as a special case, while more flexibly accommodating both fully and semi-endogenous growth mechanisms. This greater flexibility, together with a closer alignment of the econometric approach and the underlying theory, allows the resulting 4 effective R&D measure to address some undesirable statistical properties of the empirical measure in Kung and Schmid (2015). 3 In particular, I show that their effective R&D measure exhibits persistence that, while consistent with traditional calibrations of consumption LRR, is substantially higher than that of the productivity component it is intended to drive. Moreover, its proximity to a unit root raises concerns of spurious inference in standard econometric applications. By contrast, the effective R&D measure developed here generates highly persistent effects while remaining highly stationary, enabling more reliable application of established asset pricing cross-sectional tests. Another related measure is that of Kogan et al. (2017), which offers more granular information and complements effective R&D by focusing on the outcome of the innovation process – successful innovations – rather than its input. The empirical identification of effective R&D relies on the widely supported assumption of productivity growth stationarity, which implies that the linear combination of trending variables constituting effective R&D forms a cointegration relationship that can be estimated. For technical reasons related to the timing of the variables, this relationship is estimated using a single-equation approach (Phillips and Hansen 1990). For robustness, the cointegration model is estimated using productivity levels instead of the stock of ideas, thereby introducing productivity drivers external to R&D into the error-correction term, which forms a ‘gross’ effective R&D measure. Identification of net effective R&D dynamics is pursued in several ways: (1) linearity of both the mapping from gross to net effective R&D and the VAR model representing system dynamics ensures structural shocks are not distorted; (2) all estimates are performed using both the gross measure and a net measure, recovered relying on a recursive formula based on a one-step productivity forecast regression with robust controls that have been shown in previous work to capture productivity growth dynamics; (3) the dynamic implications from the VAR are tested and extended to consumption growth by employing the effective R&D structural shocks from the VAR in a local projection exercise (Jordà 2005; Montiel Olea and Plagborg-Møller 2021). A crucial result is that, although effective R&D is highly stationary, its shocks consistently affect productivity and consumption growth well beyond the business cycle, likely over horizons of ten years. These are significantly longer horizons compared with the evidence in Kung and Schmid (2015), while controlling for external factors, using methods robust to small-sample bias, and allowing for richer multivariate dynamics. The persistence of effects from R&D fluctuations has also been studied by Anzoategui et al. (2019), who emphasized the role of gradual technology adoption in generating persistence, and many other studies which have leveraged it as a channel for propagating business cycle shocks. 4 Relative to these studies, this paper extends the analysis by focusing on the financial implications of R&D persistency and by pivoting on the unique R&D measure that reflects the time-varying efficiency of R&D. The cross-sectional asset pricing tests rely on the effective R&D structural shocks from 3They refer to ‘effective R&D’ as ‘R&D intensity’. 4 For example, Benigno and Fornaro (2018) illustrate its role in propagating negative shocks, Antolin-Diaz and Surico (2025) for military spending shocks, Beqiraj et al. (2025) for monetary shocks, Aksoy et al. (2019) for demographic changes. 5 the macroeconomic analysis, which are orthogonalized to productivity growth shocks. This ensures that R&D shocks are isolated from the main confounding factor in these reduced-form tests. Combined with the procedure from Giglio and Xiu (2021), which exploits a wide cross-section of assets to mitigate concerns about omitted risk factor bias, this yields highly reliable estimates of the risk premia associated with effective R&D shocks. The premia are insignificant for contemporaneous shocks but highly significant and consistent for rolling sums over multi-year horizons, yielding roughly 2% annually – consistent with investors’ underreaction to R&D news documented in Eberhart et al. (2004). Furthermore, a key feature of long-run risk models is that risk is transmitted through asset cash flows. This is tested following Bansal et al. (2005) using stock portfolios sorted by firm characteristics previously linked to R&D investment, as well as by industry portfolios. These results also provide descriptive evidence bridging the corporate finance literature on R&D investment and the asset pricing literature on firm-specific R&D risk premia. Beyond Kung and Schmid (2015), only Hsu (2009) link aggregate R&D and financial markets. This paper extends both studies by providing novel explicit cross-sectional risk premium estimates and by employing an R&D measure with a tighter mapping to theory. From a technical perspective, the theoretical definition of effective R&D is grounded only in a definition of Total Factor Productivity (TFP) contributors and a ‘lab-equipment’ production function of ideas inspired by Jones (1999), making the results broadly applicable. 5 While the framework allows R&D efficiency in generating new ideas to be diluted by both decreasing returns to past ideas and an expanding variety of products, it does not explicitly assess their relative contribution to fitting the data, with product variety significance depending on which measure of R&D expenditure is employed. On the empirical side, cointegration models have been widely applied to study the relationship between R&D and technological progress in macroeconomic studies (Bottazzi and Peri 2007; Ha and Howitt 2007; Herzer 2022b; Kruse-Andersen 2023; Madsen 2008). However, these works mainly focus on foreign spillovers and the distinction between fullyand semi-endogenous economies, whereas this paper emphasizes the dynamic properties of R&D and its financial implications. Moreover, none of these studies employ a single-equation approach, which, on the other hand, is frequently used in the empirical macro-finance literature (e.g., Lettau and Ludvigson 2001; Melone 2021) and is adopted here. The paper proceeds as follows. Section 2outlines the theoretical framework, defining effective R&D and its key macroeconomic and financial predictions. Section 3describes the econometric framework proposed to identify effective R&D and test the associated predictions. Section 4presents the cointegration results and forecasting regressions that yield the effective R&D measures. Section 5documents the impact of effective R&D on macroeconomic quantities, and Section 6reports the cross-sectional asset pricing results. Section 7concludes. 5 The ‘lab-equipment’ class of models, introduced in Romer (1987), uses final output goods to produce ideas, in contrast to labor-based models à la Romer (1990). 6 2 Theoretical framework 2.1 Persistent macroeconomic shocks and financial markets In a discrete-time, arbitrage-free economy, the expected excess return of any asset 𝑖 over the risk-free rate 𝑅𝑓 𝑡 is proportional to the covariance between its return 𝑅𝑖𝑡 and the Stochastic Discount Factor (SDF) 𝑀𝑡:6 E𝑡[𝑅𝑖𝑡+1]−𝑅𝑓 𝑡=−𝑅𝑓 𝑡⋅Cov𝑡[𝑀𝑡+1,𝑅𝑖𝑡+1]. (1) To discipline and better understand the dynamics of the SDF, asset pricing theory often relates the SDF to the intertemporal marginal rate of substitution (IMRS) of a representative agent with preferences over consumption streams. Shocks to state variables that matter for investor consumption become the key drivers of asset valuations. As illustrated by Bansal and Yaron (2004), persistent macroeconomic quantities strongly affect the IMRS and, consequently, asset valuations, when the representative agent has recursive preferences as in Epstein and Zin (1989). These preferences separate the intertemporal elasticity of substitution (IES) from risk aversion, and imply that the agent is sensitive both to contemporaneous consumption shocks 𝜀𝑐,𝑡+1 =ln 𝐶𝑡+1−E𝑡[ln𝐶𝑡+1], (2) and to shocks to long-run consumption prospects 𝜀𝑥,𝑡+1 ={E𝑡+1−E𝑡}(∞ ∑ 𝑗=1(𝜅𝑥)𝑗⋅Δln 𝐶𝑡+1+𝑗), (3) as reflected in the log SDF, ln𝑀𝑡+1 =E𝑡[ln 𝑀𝑡+1]−𝑏𝑐𝜀𝑐,𝑡+1−𝑏𝑥𝜀𝑥,𝑡+1,(4) where 𝜅𝑥∈(0,1) is a function of the equilibrium consumption–wealth ratio, and 𝑏𝑐>0 and 𝑏𝑥>0 are the loadings on contemporaneous and long-run consumption shocks, respectively. The persistence of macroeconomic shocks plays a central role in this framework because the more persistently a shock affects consumption growth, the larger the fluctuations it generates in 𝜀𝑥,𝑡+1, and hence the stronger its impact on the IMRS.7 Assuming that 𝜀𝑐,𝑡+1 and 𝜀𝑥,𝑡+1 are also the key stochastic determinants of asset return 6A standard reference is Cochrane (2005). 7 For instance, if consumption growth simply followed Δln 𝐶𝑡+1 =𝜌𝑐Δln 𝐶𝑡+ 𝜀𝑡+1 with |𝜌𝑐|<1 and  𝜀𝑡+1 i.i.d., then the long-run shock would amount to 𝜀𝑥,𝑡+1 =1 1−𝜅𝑥𝜌𝑐 𝜀𝑡+1. 7 Nevertheless, the influence of the initial condition decays exponentially through the weights 𝜅𝑡𝑠 , which decline rapidly over time under the baseline specification. Furthermore, as illustrated below, the main analysis can be conducted equivalently using either 𝑠𝑡 or 𝑠𝑡 , and both measures are employed to ensure robustness of the findings. Section A.3 provides a detailed discussion of the recovery’s accuracy and precision, deriving analytical expressions for the standard errors of the recovered series that account for both the unobservability of the initial condition and sampling variability. To obtain a reliable estimate of 𝛾1 , two assumptions are made. First, it is assumed that a set of pervasive macroeconomic factors 𝐟𝑡 spans the non-innovation component of TFP growth, so that 𝑎𝑡=𝐜′𝐟𝑡 for some vector 𝐜 . This assumption is supported by the extensive literature on the predictable part of TFP growth (Ai et al. 2018; Croce 2014), of which 𝑎𝑡 represents the non-idea component; accordingly, it should already be captured by the information set of the predictors previously used. Second, the effects of the first lag of effective R&D on the external component 𝑎𝑡 are negligible in magnitude. This assumption is conceptually motivated by the external component’s likely dependence on numerous aggregate economic factors, and is strongly corroborated by the empirical findings presented in subsequent analysis. Therefore, while feedback mechanisms may eventually amplify the contribution of effective R&D to the fluctuations of 𝑎𝑡 , the immediate impact is expected to be negligible. Under these assumptions, 𝛾1is identified by 𝑏𝑠in the estimation of (8) as Δln𝑍𝑡+1 =𝑏0+𝑏𝑠𝑠𝑡+𝐛′𝑓𝐟𝑡+𝑢𝑡+1.(20) Since this estimation focuses solely on the parameter 𝑏𝑠 rather than the full dynamics of the R&D impact, this forecasting regression is also estimated as a single equation rather than within a system. Relative to a multivariate approach, this method trades a modest increase in estimation variance for a substantial reduction in bias, incorporating numerous control variables in 𝐟𝑡 , including lagged values of the dependent variables, with lag lengths selected using standard information criteria. An alternative approach to recover 𝑠 is to estimate the cointegration between ln𝑆𝑡 , ln𝑍𝑡 , ln𝐿𝑡 , and the factors 𝐟𝑡 directly. However, this approach requires estimating a long-run covariance matrix with 144 elements using fewer than 280 observations, which is challenging and may lead to imprecise and unstable cointegrating parameter estimates. 3.2 Innovation shocks and long-run dynamics Following the estimation of effective R&D, the dynamics of 𝑎𝑡 , Δln𝑍𝑡 , and 𝑠𝑡 (or 𝑠𝑡 ) are studied jointly, as a system, to achieve two objectives. First, this approach enables an integrated assessment of the long-run impact of effective R&D, accounting for dynamic feedbacks and contemporaneous correlations that univariate models would overlook. Second, it allows identification of structural shocks to the innovation component. The identification strategy, detailed below, isolates shocks to innovation efforts that are orthogonal to fluctuations in other macroeconomic variables of the system, most importantly productivity growth. This 14 separation is crucial to minimize contamination from other sources of risk in estimates of the risk premium from reduced-form asset pricing tests using effective R&D as a risk factor. The stochastic processes from Section 2are then extended to incorporate arbitrary persistence in 𝑎𝑡 (controlled by 𝜌𝑎 ) and feedback effects between 𝑎𝑡 and 𝑠𝑡 (governed by 𝜃𝑠 and 𝜃𝑎 ). Importantly, the shocks to effective R&D that are not determined by external factors, 𝜀𝑠,𝑡 , are assumed to have no effect on the contemporaneous level of the non-idea component. This is a common assumption in the literature (see, for instance, Moran and Queralto (2018)), since the innovation process rarely has outcomes within such a short time frame to be considered contemporaneous.14 The resulting structural system is 𝑎𝑡+1 =𝜃𝑠𝑠𝑡+𝜌𝑎𝑎𝑡+𝑏𝑎𝑎𝜀𝑎,𝑡+1 (21a) Δln𝑍𝑡+1 =(𝛾1+𝜃𝑠) 𝑠𝑡+(𝜌𝑎−1)𝑎𝑡+𝑏𝑎𝑎𝜀𝑎,𝑡+1 (21b) 𝑠𝑡+1 =𝜌𝑠𝑠𝑡+𝜃𝑎𝑎𝑡+𝑏𝑎𝑠𝜀𝑎,𝑡+1+𝑏𝑠𝑠𝜀𝑠,𝑡+1,(21c) where 𝜀𝑎,𝑡+1 and 𝜀𝑠,𝑡+1 are i.i.d. shocks from a standardized normal distribution, and 𝑏𝑎𝑎 , 𝑏𝑎𝑠 , and 𝑏𝑠𝑠 are free parameters controlling the volatility of the shocks and their crosseffects. A similar system can be written for 𝑠𝑡 by substituting 𝑠𝑡= 𝑠𝑡+𝛼𝑍𝑎𝑡 into the above equations. Notably, each variable continues to be driven by the same underlying structural shocks whether 𝑠 or 𝑠 is used; only the reduced-form coefficients differ, allowing the same identification scheme to recover the same structural shocks. More details of the system are provided in Appendix A.2. In this work, the estimation of (21) is approached by focusing on parsimonious 2variable Vector Autoregression (VAR) models that approximate the VAR–moving-average representation obtained from substituting the external factor out of the 3-variable systems: Δln𝑍𝑡+1 =(𝛾1+𝜃𝑠) 𝑠𝑡+𝜌𝑎Δln𝑍𝑡−(𝜃𝑠+𝜌𝑎𝛾1) 𝑠𝑡−1+𝑏𝑎𝑎𝜀𝑎,𝑡+1−𝑏𝑎𝑎𝜀𝑎,𝑡 (22a) 𝑠𝑡+1 =𝜌𝑠𝑠𝑡−𝜃𝑎𝜌𝑎 1−𝜌𝑎Δln𝑍𝑡+(𝜃𝑎𝜃𝑠+𝜃𝑎𝜌𝑎(𝛾1+𝜃𝑠) 1−𝜌𝑎) 𝑠𝑡−1+ 𝑏𝑎𝑠𝜀𝑎,𝑡+1+𝑏𝑠𝑠𝜀𝑠,𝑡+1+𝜃𝑎𝑏𝑎𝑎 1−𝜌𝑎𝜀𝑎,𝑡,(22b) where the number of lags is chosen based on standard information criteria. A Cholesky decomposition of the reduced-form residuals’ covariance matrix, with 𝑠𝑡 ordered last, identifies the structural shocks to effective R&D, 𝜀𝑠,𝑡 , from the estimates of (22) . As anticipated, the same holds if the system is expressed in terms of 𝑠𝑡 instead of 𝑠𝑡 , since only the coefficients change. Crucially, this approach leaves the external component unobserved, which may introduce serial correlation in the reduced-form residuals if its dynamics are poorly reflected in the two-variable system, potentially invalidating structural identification. However, the adequacy of this specification can be assessed ex post through standard diagnostic tests. 14 The implausibility of contemporaneous outcomes is further supported by the significant lag in the diffusion of new technologies, as argued in Rotemberg (2003) and Anzoategui et al. (2019). 15 A natural way to incorporate a wider information set in the system could be approaching the system (21) with a Factor-Augmented VAR in which the pervasive macroeconomic factors 𝐟𝑡 that have been previosuly argued to span the productivity external component were to be included among the endogenous variables in place of 𝑎𝑡 . However, as with the cointegration problem, the rapidly increasing number of parameters in this approach leads to substantial costs in terms of estimation noise (11 endogenous variables imply 121 coefficients per lag in the VAR) and a high risk of overfitting. Robustness of the impulse response functions (IRFs) is ensured by the use of recursive residual bootstrap standard errors of VAR estimates (Lütkepohl 2005), complemented with local projection (LP) estimates following Jordà (2005) and Montiel Olea and Plagborg-Møller (2021). LPs exploit lags of both the macroeconomic factors and the dependent variable, providing results that are generally more robust to misspecification though more volatile than those from the VAR. Following the standard approach, to analyze the dynamic responses of productivity and consumption growth rates to effective R&D shocks with local projections, cumulative responses are estimated using the following specification: ℎ ∑ 𝑗=1Δ𝑦𝑡+𝑗 =𝑏𝑦,ℎ,0+𝑏𝑦,ℎ,𝑠⋅𝜀𝑠,𝑡+𝑘 ∑ 𝑙=0(′𝑦,ℎ,𝑓,𝑙𝑡−𝑙+𝑏𝑦𝑦,ℎ,𝑙⋅𝑦𝑡−𝑙), (23) where 𝑘 is set to five, exceeding the annual data frequency to bolster robustness; the results are robust to alternative nearby choices of 𝑘 . Conveniently, the coefficients 𝑏𝑦,ℎ,𝑠 precisely reflect the impact of R&D shocks on the dynamics of long-run expectations, truncated at horizon ℎ ; that is, they approximately indicate how strongly these shocks affect the long-run risk factor 𝜀𝑥,𝑡, as defined in (3). 3.3 Asset pricing tests The key prediction from the theoretical framework in Section 2is that if shocks to effective R&D influence the long-run dynamics of consumption growth (and agents have recursive preferences), then 𝜆𝑥 in (5) should be positive and significant. Although the covariance-based nature of asset pricing would, in principle, allow effective R&D to serve as the long-run risk factor in levels – as is common in the macro-finance literature, since estimation procedures net out predictable components – this analysis focuses on structural shocks to effective R&D as the risk factor, while also reporting results based on the levels. This choice provides a more stringent test of the theory and ensures consistency with the tests of the macroeconomic predictions. The key concern in estimating factor models such as (5) is omitted variable bias, which arises when the estimation model fails to capture all priced sources of risk in the economy. This concern is particularly relevant here because the theoretical framework underlying this work, like most asset-pricing models, is deliberately stylized and does not explicitly account for all systematic risks. Recent work by Giglio and Xiu (2021) addresses this issue by proposing a methodology that improves the robustness of risk premia estimates through 16 explicit control for omitted variables. To further investigate the sources of innovation long-run risk premia, this study replicates the analysis of Bansal et al. (2005). Their methodology, specifically designed for the LongRun Risks framework, focuses on stocks’ cash-flow risks rather than total return variation, isolating the channel through which long-run risk factors theoretically generate equity risk premia. While this approach is less stringent than the Giglio and Xiu (2021) methodology, it provides complementary evidence that facilitates comparison with existing studies and contributes to broader discussions on the economics of R&D. The remainder of this section presents the two methodological approaches in detail, while the next sub-section describes the test assets. A robust approach to risk premia estimation In the context of a standard factor structure for returns such as 𝐑𝑡−𝑅𝑓 𝑡−1 =𝜷𝝀+𝜷𝐯𝑡+𝐮𝑡,(24) when the number of observations and test assets go to infinity, the ‘true’ risk factors in the economy 𝐯𝑡 can be recovered by the Principal Component Analysis up to an arbitrary rotation 𝐯𝑡=𝐻𝐯𝑡 , where 𝐻 is a full-rank matrix. Then, Giglio and Xiu (2021) focus on an observable factor 𝑥𝑡 of interest that is affine in the ‘true’ factors, with measurement error 𝑤𝑡 , 𝑥𝑡=𝜁0+𝜻𝑣𝐯𝑡+𝑤𝑡,(25) and show that the risk premia associated with it can be effectively estimated without bias. In this framework, the risk premium associated with 𝑥𝑡 amounts to 𝜻𝑣𝝀 , which corresponds to the expected excess return of an asset with a beta of one with respect to 𝑥𝑡 and zero with respect to any other independent factor. The key to its estimation is that 𝜻𝑣𝐻−1 can be obtained by regressing 𝑥𝑡 on 𝐯𝑡 , while 𝐻𝝀 can be obtained by regressing 𝐑𝑡 on 𝜷𝐻−1 , the latter being estimated by regressing returns on the ‘true’ factors. This delivers all the necessary elements to recover 𝜻𝑣𝝀, since 𝜻𝑣𝐻−1𝐻𝝀=𝜻𝑣𝝀. (26) In this setting, a crucial modeling choice concerns the number of principal components treated as the ‘true’ risk factors of the economy. This analysis considers multiple specifications for the number of factors, guided by the standard approach of Bai and Ng (2002) and the criteria proposed in Alessi et al. (2010). 15 Equally important is the breadth of the test assets: the wider the span of economic states they represent, the more robust the control for omitted risk factors will be. Moreover, to allow the information contained in the shocks to be incorporated into prices gradually (Eberhart et al. 2004), the innovation long-run risk 15The latter consists of two criteria and is implemented by taking the median across 300 repetitions. 17 factor is examined both as contemporaneous shocks to effective R&D 𝜀𝑠,𝑡 and as rolling sums of these shocks. Specifically, results are reported for rolling sums with horizons of 1 quarter (i.e., contemporaneous shocks), two years (matching the analysis on cash flows), and four years (corresponding to the typical length of a business cycle). A traditional estimation approach Bansal et al. (2005) applies the standard procedure of Fama and Macbeth (1973) with two modification. First, they ignore the risk associated with short-term fluctuations in consumption growth, based on the empirical finding that such fluctuations account for a negligible portion of the equity risk premium (Mehra and Prescott 1985). This observation is precisely what motivates the focus on long-run risks, whose premia are predicted to be larger by orders of magnitude. Second, they measure asset risk by the sensitivity of cash-flow growth to the risk factors rather than by return sensitivities. This choice aligns more closely with the theoretical formulation of Long-Run Risk and other consumption-based models, in which return betas are endogenously determined by the sensitivity of cash flows to the risk factors. The model they test is: E𝑡[𝑅𝑖𝑡+1]−𝑅𝑓 𝑡=𝜆𝑥𝛽𝑖 𝑥,𝐷,(27) where the dividend-beta, 𝛽𝑖 𝑥,𝐷, is estimated from the univariate time-series regression Δln𝐷𝑖𝑡=𝛽𝑖 0,𝐷+𝛽𝑖 𝑥,𝐷⋅1 𝐻𝐻 ∑ 𝑙=1𝜀𝑠,𝑡−𝑙+𝑢𝑖𝑡.(28) Bansal et al. (2005) uses the raw series of consumption growth as regressor, so the moving average primarily serves to filter out high-frequency fluctuations and identify shifts in longrun consumption prospects. In contrast, since shocks to effective R&D are shown in the macroeconometric analysis to have persistent effects on consumption growth, they capture changes in long-run consumption prospects directly, without the need for filtering. To assess this, the sensitivity of the results to the aggregation horizon is examined, reporting outcomes both for the traditional horizon in Bansal et al. (2005) ( 𝐻=8 quarters) and for 𝐻=1 quarter. Further details on this framework are provided in Appendix A.4. 3.4 Data Macroeconomic data The baseline measure of R&D in this study is the real US quarterly private R&D expenditures (chained 2017 dollars), consistent with closely related studies (Beqiraj et al. 2025; Kung and Schmid 2015; Moran and Queralto 2018). Private R&D reflects the profit-maximizing innovation decisions at the core of most endogenous growth models more directly than total R&D, since government expenditures operate through distinct institutional mechanisms, 18 likely materializing in a fundamentally different knowledge production function. The total R&D series is used as a robustness check. Both series are provided by the Bureau of Economic Analysis (BEA) via the Federal Reserve Economic Data (FRED) online database and span 1947 Q1 to 2025 Q2.16 Total Factor Productivity is obtained following Fernald (2012). The baseline series is the quarterly TFP growth adjusted for utilization and excluding R&D from capital, while robustness checks use the raw TFP series. The utilization adjustment is preferred because removing cyclical utilization dynamics – like any non-idea-related factor – enhances the signal-to-noise ratio of the underlying technological component of productivity, thereby improving the precision of the estimates; the R&D capital is excluded because its construction through simple cumulation and depreciation of R&D flows is inconsistent with the knowledge production function studied in this work. Differences between the series mainly arise from the utilization adjustment, with minimal impact from the exclusion of R&D from capital. The series span 1947 Q2 to 2025 Q1, and levels are obtained by cumulating the growth rates.17 The labor force is measured by the total employment level, with non-farm employment used as a robustness check. Both series are provided monthly by the Bureau of Labor Statistics via FRED. The quarterly series is constructed by taking the last value of each quarter, spanning 1948 Q1 to 2025 Q2.18 The predicting factors consist of two distinct sets, previously employed to forecast TFP growth in Ai et al. (2018): (1) nine identified factors, extending those used in Bansal and Shaliastovich (2013); and (2) nine non-identified factors directly from Ludvigson and Ng (2009). These are referred to with the shorthands ‘BS’ and ‘LN’, respectively. The BS factors set originally comprised the US Cycle Adjusted Price Earnings (CAPE) ratio, the 3-month Treasury-bill yield, and the 3and 5-year Treasury bond yields. Ai et al. (2018) later expanded this set to include the US stock market integrated daily volatility. This work additionally incorporates the 10-year Treasury bond yield, real US corporate profits, real US nonfinancial corporate liquid assets, and labor input growth, in order to better capture macroeconomic dynamics at longer horizons, as well as aspects related to financing conditions, profitability, and product proliferation — all known to influence productivity and R&D decisions. These controls span 1951 Q4 to 2025 Q1. 19 The LN factors set instead is formed by the principal components of a wide array of macroeconomic and financial variables. These series have monthly frequency and are aggregated as quarterly averages, yielding a time span from 1960 Q1 to 2025 Q2. 20 As illustrated in Appendix C, all BS variables appear to exhibit 16 The baseline real series is obtained by deflating the nominal R&D series Y006RC1Q027SBEA with the deflator Y006RG3Q086SBEA. The total series has ID Y694RX1Q020SBEA. 17 The data are provided online by the author and also include the utilization adjustment and changes in capital excluding R&D. 18IDs: CE16OV and LNS12035019. 19 CAPE is from R. Shiller’s website; bill and bond yields are from Finaeon until the first availability of DGS3MO , DGS3 , DGS5 , and DGS10 on FRED; daily stock market data is from CRSP. US corporate profits and liquid assets are the series CPROFIT and BOGZ1FL104001005Q on FRED, deflated by GDPDEF from FRED. 20Publicly available on S. Ludvigson’s website. 19 unit-root behaviour, whereas there is only weak evidence of non-stationarity among the LN factors. Therefore, while the LN factors are used in levels, the BS factors are included in first differences to mitigate the risk of spurious regression. Finally, consumption is measured as real total personal consumption expenditures per capita, in chained 2017 US dollars. The series is provided by the BEA via FRED and span 1947 Q1 to 2025 Q2.21 More details on the data can be found in Appendix B.1. Test assets data The selection of test assets for the robust risk premia estimation is guided by the need to cover as broad a portion of the economic state space as possible, ensuring that the resulting estimates are generalizable and robust. Expanding on the approach of Bryzgalova et al. (2025), this work employs 153 anomaly stock portfolios from Jensen et al. (2021), 17 industry-sorted stock portfolios from K. French’s database, and 13 bond portfolios. The bond portfolios are constructed from the zero-coupon yield data of Gürkaynak et al. (2007) by fitting Nelson-Siegel-Svensson curves and subtracting the return on a three-month Treasury bill. These portfolios span maturities of 6 months and 1, 2, 3, 4, 5, 6, 7, 10, 15, 20, 25, and 30 years. The final dataset covers the period from 1971 Q4 to 2024 Q4. The test assets employed for the Bansal et al. (2005) exercise include only stock portfolios, as in the original study: 10 sorted by size, 10 by book-to-market equity, and 10 by past-year returns. This set is referred to as the ‘legacy pool’. It is expanded with 2-by-3 portfolios jointly sorted by size (larger vs. smaller market capitalization relative to the median) and various firm characteristics related to R&D investment to form the ‘extended pool’, and then further complemented by 17 industry portfolios to form the ‘wide pool’. The firm characteristics considered in the extended pool capture (i) the intensity of innovative efforts (firm-specific R&D ratio to market capitalization), (ii) financing capacity (leverage, turnover, and profitability), and (iii) growth opportunities (assets growth and Tobin’s 𝑞 ). These dimensions have been associated with dispersion in cross-sectional risk premium and different forms of sensitivity to innovation dynamics, therefore are likely to generate heterogeneity in exposure to the long-run risk carried by aggregate R&D. Specifically, variation in firms’ R&D intensity affects the spillovers and ‘fishing out’ effects experienced by a firm, which Jiang et al. (2016) showed to be priced in financial markets, while financing capacity and growth opportunities interact with aggregate R&D investment by affecting firms’ ability to react to innovation shocks (Aghion et al. 2012; Brown et al. 2009; Hall 2002; Hall et al. 2010; Li 2011; Maletic 2018; Zhang 2014). Their inclusion further provides descriptive statistics that contribute to the discussion on the cross-sectional variation in returns with firm-specific R&D intensity, which, since its first documentation in Chan et al. (2001), remains debated (Ahmed et al. 2025; Leung et al. 2020). The payout series is constructed following Bansal et al. (2005) and Hansen et al. (2005), with details provided in Appendix B.2. Monthly 21ID: A794RX0Q048SBEA. 20 stock data come from CRSP, and annual accounting data from the Compustat Fundamentals dataset. Monthly returns are compounded to obtain quarterly figures and then deflated using the consumption deflator. The portfolio return and cash-flow growth rate series begin at different dates: the long-pool and industry-sorted portfolios start in 1967 Q1, while the others begin in 1975 Q1; all series end in 2022 Q4. Key statistics of the formed series are reported in Appendix B.2. 4 The empirical innovation component 4.1 The gross effective R&D The estimation results for the long-run relation in (17) are shown in Table 1. The first column of the table shows the baseline specification, with the columns to the right substituting one variable at a time with the alternative robustness measure. The last column uses the baseline data but employs the IM-OLS method instead. First, the 𝛼𝑍 estimates have the expected sign and are always significantly different from zero. Simply put, this means that R&D expenditure and TFP levels increase together; raw R&D rises with the scale of the economy, as expected. A similarly expected coefficient is that of labor, 𝛼𝐿 , which suggests some dilution in R&D’s power to advance the technological frontier and implies a mediating effect on how R&D expenditure relates to the technological frontier to obtain a meaningful measure of effective R&D. The only exception occurs when total R&D is used, highlighting the possibility that public R&D may operate through a different production function than private R&D. 22 Further, the use of nonfarm employment does not result in discernible changes, while the use of raw TFP, although it does not materially affect the estimates, alters the short-term fluctuations in the resulting error correction term, as can be seen from the plots of all ECT time series in Appendix C(Figure 10). Visual inspection of the time series also reveals the considerable instability of the IM-OLS estimates, as reflected by the large condition number of the coefficients’ covariance matrix 𝜅 , even though the estimates essentialy confirm the baseline results. The lower part of Table 1reports descriptive statistics of the error correction term resulting from the estimation, i.e., the time series of 𝑠𝑡 . All estimated gross effective R&D series are found to be stationary according to both the ADF and KPSS tests, with none exhibiting a time trend or a squared time trend. The persistency of the series, as measured by an AR(1) fit, is in line with the persistent component of productivity growth that is transmitted to consumption growth, which Ortu et al. (2013) and Croce (2014) has shown having a half-life between 2 and 16 years. As shown in Appendix C(Table 10), correlations among ECTs from different specifications never drop below 86%, consistent with the close estimation results, except for the IM estimation. 22Unreported estimates are in fact significantly negative when only public R&D is used. 21 Table 1: Cointegration results. Standard errors in parentheses. 𝑇 is the number of observations; 𝜅 is the condition number of the coefficients’ covariance matrix. Statistics in the bottom part of the table refer to the error correction term 𝑠 : 𝜎𝑠 is its standard deviation; 𝑡𝑡 and 𝑡𝑡2 are the time trend and squared time trend coefficients (with HAC standard errors); ADF and KPSS are stationarity tests in levels; AR(1) is the coefficient from an AR(1) fit; HL are the lower and upper bounds, at 95% confidence, of the half-life implied by the AR(1) estimate, in years. Baseline 𝑆: Tot. R&D 𝑍: Raw TFP 𝑄: N.F. Empl. Est. Meth.: IM 𝛼𝑍3.526*** 4.197*** 3.655*** 3.349*** 2.821*** (0.439) (0.516) (0.490) (0.464) (0.552) 𝛼𝐿0.909*** −0.354 0.956*** 0.953*** 1.387*** (0.336) (0.395) (0.356) (0.325) (0.398) T309 309 309 309 309 𝜅 3.4×1063.4×1063.3×1063.5×1061.1×108 𝑠𝑡 𝜎𝑠 0.130 0.144 0.128 0.129 0.253 tt 0.00 0.00 0.00 0.00 0.00 tt20.00 0.00 0.00 0.00 0.00 ADF −2.57** −2.45** −2.92*** −2.45** −9.18*** KPSS 0.09 0.09 0.09 0.10 0.29 AR(1) 0.96 0.96 0.95 0.97 0.15 HL low 2.6 2.7 2.1 2.7 0.1 HL high 21.0 23.6 12.0 25.1 0.1 * p <0.1, ** p <0.05, *** p <0.01 4.2 The effective R&D The estimated 𝑠𝑡 series allows for the estimation of equation (20) , which yields 𝛾1 , the missing parameter required to recover the effective R&D series. Information criteria (Appendix C, Figure 8) select one lag for specifications based on the adjusted TFP measure and two lags for those based on the raw TFP measure. The corresponding results are reported in Table 2. The estimated R&D coefficient is consistently positive and statistically significant, confirming a strong empirical relationship between innovation intensity and productivity growth. Its magnitude closely matches the empirical estimates in Kung and Schmid (2015) and implies an annualized increase of about 0.8% in TFP growth following a one-standard-deviation increase in 𝑠𝑡 . Control variables are always jointly significant, and specifications using LN-based controls consistently achieve higher explanatory power, as reflected in higher R 2 values and lower information criteria. Table 2also reports the estimated implied 𝜅𝑠 and the recovered 𝑠𝑡 series. Specifically, 𝑠𝑡 is obtained by applying equation (19) for each 𝑡 in the sample, using parameter estimates of 𝛼𝑍 from Table 1and 𝑏𝑠 from Table 2. The series are trimmed at the beginning of the sample, up to the first 𝑡 such that 𝜅𝑡𝑠<0.01 , ensuring that the influence of the unobserved initial condition is negligible, as the omitted portion corresponds to only one-hundredth of 22 Table 2: One-step productivity growth forecast results. HAC standard errors in parentheses. 𝑇 is the number of observations; 𝑅2 is the goodness-of-fit; 𝑘 is the number of control variables among the regressors; 𝑊(𝑘) is the Wald statistic for their joint significance. Details on 𝜅𝑠 and its standard errors are in Section A.3. The lower panel reports statistics for the recovered time series 𝑠 (see Section 3.1): 𝜎𝑠 is its standard deviation; 𝑡𝑡 and 𝑡𝑡2 are the coefficients of the linear and quadratic time trends (with HAC significance levels); ADF and KPSS are stationarity tests in levels; AR(1) is the autoregressive coefficient from a first-order fit; HL are the 95% confidence bounds of the implied AR(1) half-life, in years. Baseline 𝑆: Tot. R&D 𝑍: Raw TFP 𝑄: N.F. Empl. BS LN BS LN BS LN BS LN 𝑏𝑠(%) 1.558*** 1.549*** 1.066*** 1.223*** 0.997*** 0.794** 1.507*** 1.520*** (0.429) (0.285) (0.318) (0.254) (0.355) (0.337) (0.418) (0.289) 𝑇 292 261 292 261 291 260 292 261 R2(%) 9.5 12.4 7.4 11.8 21.8 41.7 9.1 12.2 𝑘 10 10 10 10 20 20 10 10 𝑊(𝑘) 79.97*** 61.00*** 64.56*** 50.27*** 287.53*** 2775.10*** 73.09*** 61.99*** 𝜅𝑎0.964 0.971 0.950 0.949 0.945 0.945 0.955 0.949 (0.014) (0.013) (0.016) (0.012) (0.017) (0.012) (0.014) (0.012) 𝑠𝑡(𝜅𝑡𝑎<0.01) 𝑇𝑠 226 225 207 220 183 151 219 219 𝜎𝑠 0.058 0.057 0.068 0.065 0.060 0.062 0.055 0.055 𝑡𝑡 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 𝑡𝑡20.00 0.00 0.00 0.00 0.00*** 0.00*** 0.00 0.00 ADF −3.91*** −3.95*** −3.17*** −3.49*** −3.95*** −3.50*** −3.56*** −3.56*** KPSS 0.09 0.09 0.18 0.13 0.22 0.19 0.14 0.14 AR(1) 0.71 0.70 0.72 0.68 0.68 0.70 0.70 0.70 HL low 0.4 0.3 0.4 0.3 0.3 0.3 0.3 0.3 HL high 0.8 0.8 0.8 0.7 0.7 0.8 0.7 0.7 * p <0.1, ** p <0.05, *** p <0.01 𝑠0 . Appendix A.3 provides a detailed discussion of the uncertainty in this recovery, arising both from the initial condition approximation and the estimation noise, and it also includes details on the computation of the reported standard errors. Figure 11 in Appendix Cshows all recovered series with confidence intervals, illustrating that precision greatly varies across specifications. The series from the baseline specification, in particular, exhibit the smallest uncertainty, primarily reflecting the lower uncertainty in the estimates of 𝑏𝑠 . Descriptive statistics reported in the lower panel of Table 2indicate broadly consistent time series volatility across specifications. The recovered series display no trend, are stationary as confirmed by ADF and KPSS tests, and exhibit substantially lower persistence than the 𝑠𝑡 series. Notably, both the persistence of 𝑠 and the magnitude of 𝑏𝑠 are significantly lower than those assumed in the theoretical model of Kung and Schmid (2015). This discrepancy is not problematic, as the econometric framework developed in this work accounts for feedback effects between innovation and external components – a feature absent in that model but 23 Table 4: Risk premia estimation following Giglio and Xiu (2021). Each column reports the risk premia associated with the respective specification of effective R&D structural shocks, based on the corresponding estimations in Table 3. t-statistics are reported in square brackets. The test assets are 183 portfolios spanning 1971 Q4 to 2023 Q4. The number of factors (6, 14, and 22) corresponds to optimal selections per Alessi et al. (2010) and Bai and Ng (2002), with 22 being the smallest optimal number from the latter method. Baseline 𝑆: Tot. R&D 𝑍: Raw TFP 𝑄: N.F. Empl. 𝑠 Horizon: 1 quarter 6 Factors 0.01 0.02 0.02 0.02 0.02 [0.93] [0.90] [1.22] [1.03] [1.28] 14 Factors 0.04 0.03 0.05 0.04 0.04 [1.32] [0.74] [1.35] [1.12] [1.08] 22 Factors −0.01 −0.09 0.01 −0.02 −0.04 [−0.21] [−1.30] [0.09] [−0.38] [−0.55] Horizon: 2 years 6 Factors 0.06 0.04 0.08*0.06 0.10 [1.63] [0.91] [1.87] [1.63] [1.44] 14 Factors 0.22** 0.12 0.27*** 0.19** 0.29* [2.35] [1.24] [2.92] [2.05] [1.86] 22 Factors 0.15 0.11 0.23*0.10 0.18 [1.12] [0.73] [1.81] [0.74] [0.84] Horizon: 4 years 6 Factors 0.08 0.07 0.09 0.08 0.11 [1.33] [1.11] [1.33] [1.36] [0.97] 14 Factors 0.48*** 0.34** 0.52*** 0.45*** 0.69*** [3.28] [2.50] [3.52] [3.11] [2.75] 22 Factors 0.54*** 0.43** 0.61*** 0.48** 0.80** [2.80] [2.23] [3.17] [2.52] [2.28] Num.Obs. 213 213 213 213 213 * p <0.1, ** p <0.05, *** p <0.01 long-run risk specifically, these sensitivities are all negative. Second, dividend-betas are negative across most firm-specific R&D-intensity levels. The exception is small, highly R&Dintensive firms, which exhibit a distinctive response depending on the nature of aggregate R&D changes: their payouts raise when increases are unexpected, but fall when predictable components are included (i.e. with respect to overall effective R&D changes). Third, among small firms, variables related to financing capacity – turnover, profitability, and leverage – correlate positively with sensitivities, whereas those capturing investment opportunities show the opposite pattern, with Tobin’s q exhibiting a particularly strong negative relation with payout betas. A full structural interpretation of the observed patterns in dividend-betas is beyond the scope of this work, but the evidence appears broadly consistent with: (i) firms benefiting from 30 Table 5: Dividend betas for additional test assets from the extended pool at a 1-quarter and 2-year horizons (1975 Q1–2022 Q4). Stocks are double-sorted into 2×3 portfolios by size (NYSE median) and accounting characteristics: RD (R&D/market cap), To (sales/assets), Prof (gross profits/assets), Lvg (debt/assets), AG (asset growth), and TQ (Tobin’s 𝑄 ). Risk factors: Cons. (consumption growth), Raw/Adj. TFP (raw/adjusted total factor productivity), 𝑠 : shock/level (effective R&D shock/level). Portfolio Cons. Raw TFP Adj. TFP 𝑠: shock 𝑠: level Horizon 1 8 1 8 1 8 1 8 1 8 RD(1-small) 0.09 0.18 0.05 0.39 −0.07 0.03 0.01 0.00 −0.59 −0.58 RD(2-small) 0.06 0.04 0.02 0.46 −0.11 0.19 0.04 −0.02 −1.16 −0.73 RD(3-small) 0.75 −0.68 0.59 1.17 −0.35 0.45 0.54 1.09 −1.55 −3.37 RD(1-big) 0.01 0.03 0.00 0.08 −0.02 −0.01 0.00 −0.03 −0.27 −0.29 RD(2-big) 0.05 0.12 0.01 0.23 −0.03 0.00 0.03 −0.07 −0.36 −0.28 RD(3-big) 0.03 0.09 −0.01 0.24 −0.07 −0.09 −0.02 −0.17 −1.20 −1.20 To(1-small) 0.05 0.13 0.05 0.27 −0.01 0.03 0.02 0.06 −0.25 −0.76 To(2-small) 0.10 0.29 −0.01 0.81 −0.13 0.22 0.01 0.28 0.20 0.99 To(3-small) 0.38 1.06 0.23 1.85 −0.15 0.08 0.12 0.75 2.64 3.53 To(1-big) 0.01 0.06 −0.01 0.08 −0.02 −0.02 −0.01 −0.05 −0.31 −0.40 To(2-big) 0.03 0.09 0.00 0.17 −0.03 −0.03 0.00 −0.07 −0.45 −0.32 To(3-big) 0.04 0.11 0.03 0.29 −0.05 0.02 0.03 0.02 −0.21 −0.24 Prof(1-small) 0.00 −0.01 −0.02 0.20 −0.07 0.11 0.01 −0.18 −1.15 −0.82 Prof(2-small) 0.23 0.58 0.11 0.97 −0.13 −0.12 0.06 0.42 1.00 0.56 Prof(3-small) 0.38 1.34 0.19 2.29 −0.15 0.35 0.09 0.71 3.98 6.16 Prof(1-big) 0.01 0.05 0.00 0.09 −0.02 −0.02 −0.01 −0.02 −0.25 −0.22 Prof(2-big) 0.03 0.08 0.00 0.16 −0.03 −0.09 0.00 −0.12 −0.64 −0.67 Prof(3-big) 0.05 0.13 0.02 0.34 −0.06 0.14 0.03 −0.05 −0.41 −0.07 Lvg(1-small) 0.01 0.15 −0.01 0.20 −0.05 −0.05 0.02 −0.09 −0.73 −0.68 Lvg(2-small) 0.18 0.47 0.08 1.03 −0.19 0.22 0.06 0.17 −0.86 0.00 Lvg(3-small) 0.10 0.30 0.04 0.75 −0.06 0.07 0.01 0.24 1.21 1.54 Lvg(1-big) 0.03 0.09 0.01 0.19 −0.04 −0.01 0.00 −0.07 −0.61 −0.52 Lvg(2-big) 0.02 0.08 0.00 0.14 −0.03 −0.06 0.00 −0.08 −0.36 −0.41 Lvg(3-big) 0.02 0.04 0.00 0.10 −0.02 0.02 0.00 −0.03 −0.19 −0.08 AG(1-small) 0.19 0.82 0.01 1.66 −0.13 0.18 0.04 0.67 2.84 3.12 AG(2-small) 0.18 0.43 0.08 1.02 −0.20 0.16 0.05 −0.07 −1.57 −0.27 AG(3-small) 0.09 0.19 0.05 0.38 −0.05 0.04 0.03 0.09 0.29 0.47 AG(1-big) 0.06 0.14 0.00 0.29 −0.09 −0.05 0.01 −0.14 −1.02 −0.84 AG(2-big) 0.02 0.05 0.01 0.21 −0.01 0.01 0.00 −0.06 −0.30 −0.49 AG(3-big) 0.01 0.07 0.00 0.08 −0.03 −0.02 0.00 −0.04 −0.29 −0.01 TQ(1-small) 0.35 0.87 0.05 2.43 −0.50 0.78 0.18 0.27 −1.51 1.92 TQ(2-small) 0.14 0.65 0.01 1.10 −0.12 −0.15 0.06 0.20 0.36 0.63 TQ(3-small) 0.06 0.14 0.05 0.24 −0.01 0.00 −0.01 −0.06 −0.13 −0.37 TQ(1-big) 0.03 0.15 −0.01 0.25 −0.06 −0.09 −0.02 −0.06 −0.51 −0.47 TQ(2-big) 0.01 0.09 −0.01 0.15 −0.03 −0.07 0.01 −0.05 −0.43 −0.52 TQ(3-big) 0.03 0.03 0.03 0.14 −0.03 0.10 0.01 −0.07 −0.35 0.03 31 peers’ higher R&D investment, unless they simultaneously raise their own R&D investment in response to predictable aggregate R&D increases; (ii) a greater ability to generate internal cash flows – higher turnover and profitability – improving the ability to react to aggregate innovation news, particularly in smaller, typically more constrained firms, although evidence from leverage-sorted portfolios is ambiguous; 24 (iii) aggregate innovation tends to reduce payout growth of ‘leading’ firms, i.e. the largest and the fastest-growing ones. Resulting estimates of risk premia from the cross-sectional step are reported in Table 6. Notably, the premia associated with consumption cash-flow risk differ markedly across horizons: they are highly significant at the 1-quarter horizon but only barely significant in the wide pool at the 2-year horizon. This decline is even more apparent in the cross-sectional R 2 , which falls from roughly 20–30% to a range of 2–50% across pools. The magnitude of the premia, at the horizon common to Bansal et al. (2005), exceeds theirs but remains well within the confidence intervals. Risk premia associated with productivity long-run risk, as measured based on moving averages of raw TFP, are much more consistent: premia are significant for two of the three test-asset pools at both horizons and explanatory power increases at the 2-year horizon. By contrast, estimates based on adjusted TFP are less stable across horizons, exhibiting negative premia at the 1-quarter horizon and positive premia at the 2-year horizon, with R 2 comparable to those for raw TFP, except in the legacy pool, where R2drops by over 50%. Like productivity, the premia associated with innovation long-run risk, based on the structural shocks, are significant for the two widest of the three test-asset pools and remain so across horizons. By contrast, forming the risk factor from effective R&D levels yields premia that are smaller in magnitude and never significant at the 5% level, highlighting the substantial role of the predictable component of effective R&D in pricing. Nonetheless, three of the six premia estimated for effective R&D in levels have t-statistics above 1.2, indicating that some risk is still captured. Overall, these results support the notion that a substantial portion of the premia for holding long-run risk derives from cash-flow risk, consistent with the standard long-run risk framework. 24 In general, higher leverage implies reduced financial slack or even distress. This is difficult to reconcile with the finding that high-leverage portfolios do not exhibit the strongest consumption sensitivities. A more consistent interpretation is that leverage partly proxies for better access to external finance rather than distress, possibly reflecting omitted firm characteristics correlated with leverage—an issue likely exacerbated by the coarse two-by-three sorting used here. 32 Table 6: Cash-flow risk premia. 𝑡 -statistics in square brackets. Risk factors: Cons. (consumption growth), Raw/Adj. TFP (raw/adjusted total factor productivity), 𝑠 : shock/level (effective R&D shock/level). The legacy pool comprises 16 test assets (224 observations, 1967 Q1–2022 Q4); the extended pool, 51 test assets (192 observations, 1975 Q1–2022 Q4); and the wide pool, 68 test assets (192 observations, 1975 Q1–2022 Q4). Cons. Raw TFP Adj. TFP 𝑠: shock 𝑠: level Horizon: 1 quarter Legacy pool 0.68 0.18 −1.47 1.16 0.02 [1.60] [0.84] [−1.54] [1.34] [0.66] R2(%) 30.99 0.87 58.96 38.69 19.84 MAPE (%) 0.19 0.25 0.12 0.18 0.23 Ext. pool 1.72** 1.97*** −2.47** 2.28** 0.06* [2.58] [3.14] [−2.12] [2.31] [1.68] R2(%) 24.85 20.44 17.95 16.77 4.89 MAPE (%) 0.41 0.44 0.43 0.46 0.48 Wide pool 1.45*** 1.38** −1.09*** 2.00** 0.03 [3.11] [2.39] [−3.48] [2.51] [0.97] R2(%) 18.05 14.16 4.38 14.97 4.06 MAPE (%) 0.45 0.47 0.49 0.47 0.48 Horizon: 2 years Legacy pool 0.26 0.20 0.27 0.35 0.01 [1.42] [1.48] [0.67] [1.40] [0.70] R2(%) 54.21 59.62 6.35 55.46 21.37 MAPE (%) 0.15 0.12 0.25 0.15 0.23 Ext. pool 0.32 0.40** 1.17** 0.71** 0.04 [1.65] [2.15] [2.15] [2.14] [1.63] R2(%) 3.06 23.66 29.24 20.32 4.58 MAPE (%) 0.48 0.41 0.41 0.44 0.48 Wide pool 0.28*0.36** 0.42*0.60*0.03 [1.93] [2.53] [1.71] [1.92] [1.26] R2(%) 2.17 18.61 12.23 17.33 4.96 MAPE (%) 0.49 0.45 0.47 0.46 0.48 * p <0.1, ** p <0.05, *** p <0.01 33 7 Conclusion This paper focuses on a theoretical measure of aggregate R&D that is designed to reflect the contribution of R&D investments to productivity growth dynamics. This measure – termed ‘effective R&D’ or the ‘innovation component’ – accounts the mediating role of idea spillovers and product proliferation in the effect of R&D on productivity growth, consistent with both fullyand semi-endogenous growth mechanisms. A univariate empirical framework is introduced to recover fluctuations in two versions of this measure: a gross effective R&D series, derived from the cointegration relationship among R&D, TFP, and labor force; and a net measure, constructed, recursively, relying a one-period TFP growth forecast. Both series are stationary in quarterly U.S. data, though they differ markedly in persistence: the gross measure exhibits half-lives of 3 to 21 years, while the net measure displays a half-life of less than one year. Embedding either series in a VAR with productivity growth shows that innovation shocks generate persistent movements in productivity growth, propagating over horizons of a decade. Structural identification ensures that shocks to effective R&D are orthogonal to contemporaneous productivity growth, while local projection exercises confirm these dynamics and extend them to consumption growth, which responds to innovation shocks at horizons well beyond the business cycle, possibly up to 15 years. The paper’s primary contribution is demonstrating that shocks to the innovation component constitute a significant cross-sectional risk factor, consistent with long-run risk asset pricing theories, associated to a substantial premium, in the order of 2% annually. This finding is particularly robust, leveraging structural VAR identification to eliminate spurious correlations with other productivity-related factors, employing recent estimation techniques to control for omitted risk factors, and drawing on a broad pool of 183 stockand bond-based test assets. The analysis further highlights the importance of the cash-flow channel in innovation long-run risk and reveals heterogeneous exposures across firm characteristics linked to R&D. In particular, payouts of small, R&D-intensive firms increase with aggregate R&D shocks, while those of other R&D-sorted portfolios decline. Measures of internal financing capacity (turnover, profitability) correlate positively with payout sensitivities, whereas proxies for investment opportunities (asset growth, Tobin’s q) correlate negatively. Taken together, these findings establish aggregate innovation as a fundamental macroeconomic risk factor with distinct asset pricing implications. Beyond this contribution, the empirical methodology developed here provides a platform for future work exploring international evidence, firm-level responses, and the differential roles of alternative types of R&D, among others. 34 References Aghion, Philippe et al. 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For 𝛼𝐿 , the slope is not significantly different from zero at the 10% level when all specifications are included ( 𝑡=1.71 ), but equals −8.63 with a 𝑝 -value below 1% when restricted to specifications where 𝛼𝐿is significantly different from zero. the parameter estimates in the observable component, i.e. the feasible recovery estimator 𝑠𝑡 : Var(E[𝑠𝑡∣ 𝜽,𝑋]∣𝑋)=Var ( 𝛼𝑍(𝑡−1 ∑ 𝑗=0 𝜅𝑠𝑗⋅Δln 𝑍𝑡−𝑗)+𝑡−1 ∑ 𝑗=0 𝜅𝑠𝑗Δ𝑠𝑡−𝑗∣𝑋) (65) =Var(𝑡−1 ∑ 𝑗=0 𝜅𝑠𝑗⋅Δln 𝑆𝑡−𝑗− 𝛼𝐿(𝑡−1 ∑ 𝑗=0 𝜅𝑠𝑗Δln𝐿𝑡−𝑗)∣𝑋), (66) where the first equality follows from 𝑠0 being independent of both the parameter estimates and the data sample, with zero mean by definition, while the second equality results from explicitly expressing the first differences of the gross effective R&D. Following the Delta method, this is estimated as  Var(E[𝑠𝑡∣ 𝜽,𝑋]∣𝑋)=∇ 𝑠𝑡( 𝛼𝑍, 𝛼𝐿, 𝑏𝑠)⊤⎡ ⎢ ⎢ ⎣  𝜎2𝛼𝑍𝜎 𝛼𝑍, 𝛼𝐿0 𝜎 𝛼𝑍, 𝛼𝐿 𝜎2𝛼𝐿0 0 0  𝜎2 𝑏𝑠 ⎤ ⎥ ⎥ ⎦∇𝑠𝑡( 𝛼𝑍, 𝛼𝐿, 𝑏𝑠), (67) where the covariances between 𝑏𝑠 and 𝛼• are conservatively assumed to be zero, following earlier arguments and the evidence in Figure 6, 26 and the gradient evaluated at the point estimates is ∇𝑠𝑡( 𝛼𝑍, 𝛼𝐿, 𝑏𝑠)=⎡ ⎢ ⎢ ⎣ ∑𝑡−1 𝑗=0− 𝑏𝑠⋅𝑗⋅ 𝜅𝑠𝑗−1(Δln𝑆𝑡−𝑗− 𝛼𝐿Δln𝐿𝑡−𝑗) −∑𝑡−1 𝑗=0 𝜅𝑠𝑗Δln𝐿𝑡−𝑗 ∑𝑡−1 𝑗=0− 𝛼𝑍⋅𝑗⋅ 𝜅𝑠𝑗−1(Δln 𝑆𝑡−𝑗− 𝛼𝐿Δln𝐿𝑡−𝑗)⎤ ⎥ ⎥ ⎦.(68) 26 This assumption is more conservative with respect to 𝛼𝑍 than to 𝛼𝐿 , although the evidence does not provide a strong case for a positive covariance in either case. 46 The second term, on the other hand, captures the uncertainty arising from the unobservability of the initial effective R&D, 𝑠0 , which, conditional on the parameter estimates, is the only remaining random quantity: E[Var(𝑠𝑡∣ 𝜽,𝑋)∣𝑋]=E[ 𝜅𝑠2𝑡⋅Var( 𝑠0∣ 𝜽,𝑋)∣𝑋] (69) =E[ 𝜅𝑠2𝑡∣𝑋]Var( 𝑠0|𝑋)+Cov [ 𝜅𝑠2𝑡,Var( 𝑠0∣ 𝜽,𝑋)∣𝑋]. (70) Under stationarity of 𝑠𝑡 and a consistent recovery provided by 𝑠𝑡 , both Var( 𝑠0∣𝑋) and Var( 𝑠0∣ 𝜽,𝑋)can be consistently estimated by  𝜎2𝑠 =1 𝑇𝑇 ∑ 𝑡=1(𝑠𝑡)2(71) =1 𝑇𝑇 ∑ 𝑡=1{(Δln𝑆𝑡− 𝛼𝐿Δln 𝐿𝑡)2[𝑇−𝑡 ∑ 𝑙=0 𝜅𝑠2𝑙]}. (72) Then, approximating 𝜅𝑠2𝑡 using a second-order expansion and applying the Delta method to approximate the covariance term,  E[Var(𝑠𝑡∣ 𝜽,𝑋)∣𝑋]=( 𝜅𝑠2𝑡+𝑡(2𝑡−1) 𝜅𝑠2(𝑡−1)⋅ Var( 𝜅𝑠|𝑋)) 𝜎2𝑠 +∇𝜅2𝑡( 𝜅𝑠) Var( 𝜅𝑠|𝑋)∇ 𝜎2𝑠( 𝜅𝑠), (73) where ∇𝜅2𝑡( 𝜅𝑠)=2𝑡⋅ 𝜅𝑠2𝑡−1 (74) ∇ 𝜎2𝑠( 𝜅𝑠)=1 𝑇𝑇 ∑ 𝑡=1{(Δln𝑆𝑡− 𝛼𝐿Δln 𝐿𝑡)2[𝑇−𝑡 ∑ 𝑙=0𝑙⋅ 𝜅𝑠2𝑙−1]}. (75) The total variance of the effective R&D recovery is thus quantified by  Var[𝑠𝑡∣𝑋]=  Var(E[𝑠𝑡∣ 𝜽,𝑋]∣𝑋)+  E[Var(𝑠𝑡∣ 𝜽,𝑋)∣𝑋]. (76) A.4 The cash-flow-based asset pricing framework According to the argument presented in the main text, Bansal et al. (2005) adopt the simpler cross-sectional pricing condition E𝑡[𝑅𝑖𝑡+1]−𝑅𝑓 𝑡=𝜆𝑥𝛽𝑖 𝑥(77) as a reasonable approximation of the theoretical long-run risk pricing equation. The pricing equation in (27) then follows from applying the Campbell (1996) decomposition, which expresses unexpected returns as the sum of news about future cash-flow growth and discount 47 rates: ln𝑅𝑖𝑡+1−E𝑡[ln𝑅𝑖𝑡+1]≈𝛿𝑖 𝐷,𝑡+1−𝛿𝑖 𝑅,𝑡+1 (78) where 𝛿𝑖 𝐷,𝑡 ={E𝑡−E𝑡−1}[∞ ∑ 𝑗=0 𝜅𝑗Δln𝐷𝑖𝑡+𝑗], 𝛿𝑖 𝑅,𝑡 ={E𝑡−E𝑡−1}[∞ ∑ 𝑗=1 𝜅𝑗ln𝑅𝑖𝑡+𝑗]. (79) This decomposition implies that any return beta can be approximated as the difference between a dividend beta, 𝛽𝑖 𝑥,𝐷, and a discount-rate beta, 𝛽𝑖 𝑥,𝑅:27 𝛽𝑖 𝑥=Cov [𝑅𝑖𝑡,𝜀𝑥,𝑡] Var [𝜀𝑥,𝑡]≈Cov[𝛿𝑖 𝐷,𝑡,𝜀𝑥,𝑡] Var [𝜀𝑥,𝑡]−Cov[𝛿𝑖 𝑅,𝑡,𝜀𝑥,𝑡] Var [𝜀𝑥,𝑡]=𝛽𝑖 𝑥,𝐷−𝛽𝑖 𝑥,𝑅,(80) and focusing on the dividend beta alone isolates the component of risk stemming from assets’ fundamentals, abstracting from that arising through the discount-rate channel. The beta estimates from (28) are asymptotically equivalent to those from 1 𝐻𝐻 ∑ 𝑙=1Δln 𝐷𝑖𝑡+𝑙 = 𝛽𝑖 0,𝐷+ 𝛽𝑖 𝑥,𝐷𝜀𝑠,𝑡+ 𝑢𝑖𝑡+1.(81) The latter formulation makes the interpretation of the sensitivity as the ‘long-lasting impact on cash-flow growth’ more explicit, thereby better clarifying the mapping between the sensitivity estimates and the theoretical parameter 𝛽𝑖 𝑥,𝐷 from (80) . However, as illustrated by Hodrick (1992), the former offers inferential advantages in small samples. The premium 𝜆𝑥is then estimated by regressing the dividend-betas on the assets’ returns. B Details on the data B.1 Macroeconomic data This section presents the macroeconomic data, showing the series and descriptive statistics. 27 The approximation follows from both (78) and 𝑅𝑖 𝑡≈1+ln 𝑅𝑖 𝑡 . See Campbell and Vuolteenaho (2004) for a systematic application. 48 Figure 7: Raw macroeconomic data. 𝑠 denotes the panel showing data used to estimate effective R&D, while BS and LN indicate the respective factor sets. Table 7: Descriptive statistics: effective R&D data. N. Obs. is the number of observations. 𝑡𝑡 and 𝑡𝑡2 are the time trend and squared time trend coefficients (with HAC standard errors); ADF and KPSS are stationarity tests in levels; AR(1) is the coefficient from an AR(1) fit. Priv. R&D Tot. R&D Raw TFP Adj. TFP Tot. Empl. N.F. Empl. 𝑡𝑡 0.017*** 0.019** 0.018*** 0.018*** 0.018*** 0.017*** 𝑡𝑡20.000 0.000 0.000*** 0.000 0.000*** 0.000*** AR(1) 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** ADF −1.482 −2.027** −2.346** −2.891*** −1.353 −0.948 KPSS 2.004*** 1.924*** 1.976*** 1.978*** 2.001*** 2.011*** N. Obs. 314 314 314 314 310 310 * p <0.1, ** p <0.05, *** p <0.01 49 Table 8: Descriptive statistics: BS factors. N. Obs. is the number of observations. 𝑡𝑡 and 𝑡𝑡2 are the time trend and squared time trend coefficients (with HAC standard errors); ADF and KPSS are stationarity tests in levels; AR(1) is the coefficient from an AR(1) fit. CAPE 10Y yield 3M yield 3Y yield 5Y yield Int. Vol Corp. Profits N.F. Liq. Assets 𝑡𝑡 −0.001 −0.004*−0.003 −0.004*−0.004*0.002 0.002 0.005* 𝑡𝑡20.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 AR(1) −0.334*** −0.053 −0.187*** −0.117*−0.102*−0.297*** 0.031 0.068 ADF −13.638*** −11.188*** −13.206*** −11.844*** −11.734*** −15.514*** −10.569*** −10.688*** KPSS 0.078*0.184*0.099*0.144*0.170*0.259*0.052*0.097* N. Obs. 270 270 270 270 270 270 270 269 * p <0.1, ** p <0.05, *** p <0.01 Table 9: Descriptive statistics: LN factors. N. Obs. is the number of observations. 𝑡𝑡 and 𝑡𝑡2 are the time trend and squared time trend coefficients (with HAC standard errors); ADF and KPSS are stationarity tests in levels; AR(1) is the coefficient from an AR(1) fit. f1 f2 f3 f4 f5 f6 f7 f8 f9 𝑡𝑡 0.005 0.005 0.002 −0.004 −0.001 0.006 −0.001 0.017*** 0.003 𝑡𝑡20.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000*** 0.000 AR(1) 0.725*** 0.360*** 0.749*** 0.426*** 0.703*** 0.620*** 0.138** 0.513*** 0.329*** ADF −6.467*** −7.252*** −4.577*** −6.929*** −4.911*** −5.758*** −10.204*** −7.415*** −9.526*** KPSS 0.324*0.514** 0.525** 0.373*0.224*0.278*0.190*0.603** 0.062* N. Obs. 262 262 262 262 262 262 262 262 262 * p <0.1, ** p <0.05, *** p <0.01 50 B.2 Financial data Cash-flow growth rates First, a measure ℎ𝑖,𝑡 of capital gain is computed for each stock by adjusting CRSP ex-dividend returns RETX for share repurchases as ℎ𝑖,𝑡 =(𝑃𝑖,𝑡+1 𝑃𝑖,𝑡 )⋅min(𝑛𝑖,𝑡+1 𝑛𝑖,𝑡 ,1). (82) For each portfolio, these stock-level measures are aggregated using market-capitalization weights to obtain a portfolio capital gain series ℎ𝑝,𝑡 . From each portfolio series, the value of one dollar invested at the beginning of the sample is computed recursively as 𝑉𝑝,𝑡+1 =ℎ𝑝,𝑡+1𝑉𝑝,𝑡 with 𝑉𝑝,0 =1 . Payouts are then given by 𝐷𝑝,𝑡+1 =𝑦𝑝,𝑡+1𝑉𝑝,𝑡 , where 𝑦𝑝,𝑡+1 is the portfolio dividend yield, obtained from 𝑅𝑝,𝑡 =ℎ𝑝,𝑡+𝑦𝑝,𝑡 . Capital gains are less than proportional to price appreciation when equivalent shares outstanding decline, typically due to share repurchases, which are a form of payout not recorded in dividend data. Quarterly dividend series are obtained by summing monthly values and deflating them with the implicit price deflator for nondurable and services consumption, constructed as in Bureau of Economic Analysis (2024). The series are log-transformed and, following Bansal et al. (2005), deseasoned with a 4-quarter rolling mean to remove residual seasonality. Finally, cash-flow growth rates are computed as first differences of the log-transformed, de-seasoned real quarterly payouts. Portfolio Formation To mitigate liquidity concerns, the sample is restricted to common stocks with market capitalization above the 1st percentile of the monthly NYSE distribution and share prices above $2, removing less than 0.4% of total market capitalization at any point in time. Firms must also have at least twelve consecutive monthly observations to enter the final sample. Portfolio construction follows standard methodology: portfolios are formed at the end of June, value-weighted, and held until the following June. Sorting variables are Market capitalization (Size), Book-to-Market (BM), Momentum (Mom), firm-specific R&D intensity (RD), Turnover (To), Profitability (Prof), Leverage (Lvg), Asset growth (AG), Tobin’s Q (TQ), and Industry (Ind). Most variables are sorted using a 2 × 3 framework, where stocks are first split by whether market capitalization is above or below the NYSE median and then divided into terciles within each size group. Exceptions are Size, BM, and Mom, which follow univariate sorts as in Bansal et al. (2005), and Industry, which is directly classified. All portfolios exhibit patterns consistent with established literature. Size portfolios All firms are assigned to quintiles based on market capitalization relative to NYSE breakpoints. Both returns and cash-flow growth decrease with size. 51 BM portfolios All non-financial firms (SIC outside 6000–6999) are assigned to quintiles based on book equity in fiscal year 𝑡−1 to market capitalization at end of calendar year 𝑡−1 , relative to NYSE breakpoints. Both returns and cash-flow growth increase with the B/M ratio. Mom portfolios All firms are assigned to quintiles based on cumulative returns from month 𝑡−12to month 𝑡−1. Both returns and cash-flows increase with momentum. RD portfolios See Chan et al. (2001) and Lin (2011). All non-financial and non-utility firms (SIC outside 4000–4999, 6000–6999) are ranked by R&D expenditures from the previous fiscal year to market capitalization at end of calendar year 𝑡−1 . Returns and cash-flow growth increase with with R&D intensity in both small and big portfolio. To portfolios See Haugen and Baker (1996). All firms are assigned to quintiles based on the sales-to-assets ratio from the previous fiscal year. Returns and cash-flow growth increase with turnover for both small and big portfolios. Prof portfolios See Fama and French (2015), Hou et al. (2015), and Novy-Marx (2013). All non-financial and non-utility firms are sorted on gross profits over assets. Returns and cash-flow growth increase with profitability for both small and big portfolios, with stronger effects among small portfolios. Lvg portfolios See Bhandari (1988). All non-financial firms are sorted on debt-to-assets ratio, used instead of debt-to-market capitalization for consistency with corporate finance literature (e.g. Rajan and Zingales 1995) and with other sorts related to financing capabilities in this study. Returns and cash-flow growth increase with leverage among small portfolios, while cash-flow growth decreases among big portfolios, with returns showing no clear pattern. AG portfolios See Cooper et al. (2008) and Hou et al. (2015). All non-financial and non-utility firms are sorted on asset growth (first difference of assets over lagged value). Both cash-flow growth and returns decrease with asset growth for small and big portfolios. TQ portfolios See Hou et al. (2015). All firms are sorted on Tobin’s Q, defined as (assets - book equity + market capitalization) / assets as in Chung and Pruitt (1994). Cash-flow growth and returns clearly decrease with Tobin’s Q for small portfolios, with no evident pattern for big portfolios. 52 C Additional tables and figures C.1 Tables This subsection presents supplementary tables on correlations, R&D measures, and test asset portfolio statistics. Table 10: Correlation among error correction terms from the specifications in Table 1. Baseline 𝑆: Tot. R&D 𝑍: Raw TFP 𝑄: N.F. Empl. 𝑆: Tot. R&D 0.880 . . . 𝑍: Raw TFP 0.855 0.721 . . 𝑄: N.F. Empl. 0.997 0.858 0.859 . Est. Meth.: IM 0.511 0.504 0.377 0.503 Table 11: Correlation among estimates of 𝑠𝑡from the specifications in Table 2. Specification Baseline 𝑆: Tot. R&D 𝑍: Raw TFP 𝑄: N.F. Empl. BS LN BS LN BS LN BS LN Baseline-LN 1.000 . . . . . . . 𝑆: Tot. R&D-BS 0.777 0.777 . . . . . . 𝑆: Tot. R&D-LN 0.806 0.806 0.996 ..... 𝑍: Raw TFP-BS 0.858 0.859 0.640 0.641 .... 𝑍: Raw TFP-LN 0.853 0.853 0.644 0.636 0.994 ... 𝑄: N.F. Empl.-BS 0.998 0.998 0.752 0.781 0.868 0.862 . . 𝑄: N.F. Empl.-LN 0.998 0.998 0.752 0.781 0.868 0.861 1.000 . Table 12: Correlation among estimates of structural shocks from the VAR specifications in Table 3. Baseline 𝑆: Tot. R&D 𝑍: Raw TFP 𝑄: N.F. Empl. 𝑆: Tot. R&D 0.690 . . . 𝑍: Raw TFP 0.915 0.630 . . 𝑄: N.F. Empl. 0.991 0.699 0.901 . 𝑠 0.926 0.646 0.837 0.929 * p <0.1, ** p <0.05, *** p <0.01 53 Table 13: Summary statistics of the (updated) Kung and Schmid (2015) R&D intensity measure. Column 1 reports yearly R&D expenditure ( 𝑆 , NSF) and R&D stock ( 𝐼 , BLS) for 1963-2020 (yearly observations from 1986 onward); columns 2-3 show data from the baseline specifications in the main analysis. 𝜉 corresponds to commonly used labor-share values, following the original paper. 𝑇 denotes the number of observations; 𝜎 , the standard deviation; 𝑡𝑡 and 𝑡𝑡2 , linear and quadratic time trends; ADF and KPSS, respective stationarity tests; AR(1), the first-order autoregressive coefficient; and HL, the implied half-life based on AR(1). 𝑠𝑡:(ln𝑆𝑡−ln𝐼𝑡) (ln𝑆𝑡−1 𝜉ln𝑍𝑡) 1−𝜉:− 0.35 0.3 𝑇 62 314 314 𝜎 0.321 0.864 0.835 𝑡𝑡 −0.043 0.01 0.01 𝑡𝑡20.000 −0.00 −0.00 ADF −2.82∗3.80 3.72 KPSS 1.29∗∗∗ 0.82∗∗∗ 0.79∗∗∗ AR(1) 0.995∗∗∗ 1.000∗∗∗ 1.000∗∗∗ (0.063) (0.000) (0.000) HL low 40.2 ∞ ∞ HL high ∞ ∞ ∞ ∗∗∗𝑝<0.01,∗∗𝑝<0.05,∗𝑝<0.1 Table 14: Granger causality F-test 𝑝 -values from a VAR of effective R&D ( 𝑠 ) and the aggregated innovation measure of Kogan et al. (2017) (KPSS), both included as structural shocks orthogonal to productivity growth, estimated from a first-step bivariate VAR with productivity growth (as in Table 3). VAR lags are selected by Hannan–Quinn information criterion (max 10); 𝑘 denotes second-step VAR lags. Results are shown using standard, HC, and HAC covariance estimators. Baseline 𝑆: Tot. R&D 𝑍: Raw TFP 𝑄: N.F. Empl. 𝑠 𝑘 1 1 2 1 1 N. Obs. 298 298 298 298 216 GC(𝑠) p.v. 0.072 0.926 0.095 0.074 0.073 GC(KPSS) p.v. 0.786 0.395 0.667 0.662 0.154 GC(𝑠) HC p.v. 0.069 0.934 0.131 0.072 0.066 GC(KPSS) HC p.v. 0.838 0.436 0.699 0.745 0.345 GC(𝑠) HAC p.v. 0.011 0.921 0.013 0.008 0.031 GC(KPSS) HAC p.v. 0.824 0.497 0.636 0.719 0.288 * p <0.1, ** p <0.05, *** p <0.01 54 Table 15: Descriptive statistics of the legacy pool test assets used in Section 3.3. Quarterly returns and cash-flow growth rates are reported from 1967 Q1 to 2022 Q1; summary statistics include mean and standard deviation. Portfolio CF Growth Mean CF Growth SD Returns Mean Returns SD Size(1) 134.68 604.09 2.47 13.20 Size(2) 68.34 373.78 2.31 11.83 Size(3) 36.33 193.21 2.18 10.77 Size(4) 29.38 161.65 2.14 10.06 Size(5) 6.38 36.26 1.59 8.34 BM(1) 3.09 16.54 1.72 9.76 BM(2) 2.31 21.74 1.78 8.86 BM(3) 1.49 17.17 1.67 8.22 BM(4) 3.47 36.58 1.96 8.65 BM(5) 8.70 69.61 2.41 9.37 Mom(1) 1.11 20.96 1.47 12.61 Mom(2) 7.44 74.50 1.75 9.31 Mom(3) 8.34 58.26 1.78 8.47 Mom(4) 17.24 107.76 1.85 8.77 Mom(5) 34.09 280.68 1.98 11.66 55 Figure 11: 𝑠𝑡 from all specifications tested. Bands show 95% confidence intervals assuming normality, with variance computed as in (64) . Shaded areas indicate NBER recessions. 62 Figure 12: Structural shocks from VAR estimations; selected results are reported in Table 3. Shaded areas indicate NBER recessions. 63 Figure 13: Principal component scree plots of the test assets in Section 3.3. The left y-axis shows the variance explained by each factor (solid line), and the right y-axis shows the cumulative variance explained (dotted line). Vertical green lines indicate the optimal number of factors according to Alessi et al. (2010), while the vertical blue line marks the minimum optimal number of factors as in Bai and Ng (2002). The horizontal red line corresponds to the reciprocal of the number of test assets. 0 5 10 15 20 0 25 50 75 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Component(s) % Cumulative % Cumulative Variance Factor Variance Figure 14: Cross-sectional average returns: variance explained by the principal components used as risk factors in Section 3.3. The left y-axis shows the variance explained by each factor (solid line), and the right y-axis shows the cumulative variance explained (dotted line). Vertical green lines indicate the optimal number of factors according to Alessi et al. (2010), while the vertical blue line marks the minimum optimal number of factors as in Bai and Ng (2002). The horizontal red line corresponds to the reciprocal of the number of test assets. 64 Quest’opera è soggetta alla licenza Creative Commons