Macroeconomic effects of the gender revolution

Bergholt, Drago; Fosso, Luca; Furlanetto, Francesco Working Paper Macroeconomic effects of the gender revolution Working Paper, No. 19/2024 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Bergholt, Drago; Fosso, Luca; Furlanetto, Francesco (2024) : Macroeconomic effects of the gender revolution, Working Paper, No. 19/2024, ISBN 978-82-8379-348-2, Norges Bank, Oslo, https://hdl.handle.net/11250/3177155 This Version is available at: https://hdl.handle.net/10419/322324 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-nd/4.0/ Working Paper Macroeconomic Effects of the Gender Revolution Norges Bank Research Authors: Drago Bergholt Luca Fosso Francesco Furlanetto Keywords: Economic growth, gender inequality, labor market, productivity, VAR with common trends 19 | 2024 Norges Bank Working Paper 1 Working papers fra Norges Bank, fra 1992/1 til 2009/2 kan bestilles på epost: [email protected] Fra 1999 og senere er publikasjonene tilgjengelige på http://www.norgesbank.no Working papers inneholder forskningsarbeider og utredninger som vanligvis ikke har fått sin endelige form. Hensikten er blant annet at forfatteren kan motta kommentarer fra kolleger og andre interesserte. Synspunkter og konklusjoner i arbeidene står for forfatternes regning. Working papers from Norges Bank, from 1992/1 to 2009/2 can be ordered by e-mail: [email protected] Working papers from 1999 onwards are available on www.norges-bank.no Norges Bank’s working papers present research projects and reports (not usually in their final form) and are intended inter alia to enable the author to benefit from the comments of colleagues and other interested parties. Views and conclusions expressed in working papers are the responsibility of the authors alone. ISSN 1502-8143 (online) ISBN 978-82-8379-348-2 (online) MACROECONOMIC EFFECTS OF THE GENDER REVOLUTION* DRAGO BERGHOLT†, LUCA FOSSO‡AND FRANCESCO FURLANETTO§ December 2024 Abstract: U.S. labor market data exhibit a major, secular decline in the employment and wage gaps between males and females. In this paper, we identify the underlying, structural forces and quantify the spillover from this gender convergence to the broader macroeconomy. A novel time series model maps empirical trends in data into (aggregate and gender-specific) structural trends. Identification is achieved with restrictions derived from a neoclassical model with gender-specific labor. Empirically, we find that secular changes in female-specific labor productivity account for approximately one-third of economic growth in the postwar U.S. economy, in addition to most of the observed gender convergence. Keywords: Economic growth, gender inequality, labor market, productivity, VAR with common trends. JEL Classification: C32, E10, E13, J2. *This work should not be reported as representing the views of the European Central Bank or Norges Bank. The views expressed are those of the authors and do not necessarily reflect those of the European Central Bank or Norges Bank. We thank for useful comments Stefania Albanesi, Jonas Arias, Philippe Andrade, Guido Ascari, Regis Barnichon, Martin Beraja, Paolo Bonomolo, Fabio Canova, Efrem Castelnuovo, Andrea Cerrato, Larry Christiano, Marco Del Negro, Martin Eichenbaum, Domenico Giannone, Luca Fornaro, Peter Karadi, Joseba Martinez, Leonardo Melosi, Karel Mertens, Silvia Miranda Agrippino, Evi Pappa, Giorgio Primiceri, Lukasz Rachel, Giuseppe Ragusa, Giovanni Ricco, Juan Rubio-Ramirez, Aysegul Sahin, Stefan Schiman, seminar participants at Northwestern University, Boston Fed, Boston College, Chicago Fed, Dallas Fed, Fed Board, University of Texas Austin, University of California Davis, Cleveland Fed, Washington University of Saint Louis, European Central Bank, Norges Bank, University of Aix-Marseille, University of Milano-Bicocca, 2021 Sailing the Macro Conference in Ventotene, 2021 conference in honor of Fabio Canova in Hydra, 2022 Padova Macro Talks, 2022 Barcelona Summer Forum, 2022 Dolomiti Macro Meetings in Selva Val Gardena, 2022 Forskermøtet in Stavanger, 2023 SNDE conference in Orlando, 2023 IAAE conference in Oslo, 2023 CEF conference in Nice, 2023 Dynare Conference in Malta, 2023 Fall Midwest Macro Meetings in Lubbock, 2024 Workshop in Empirical Macroeconomics King’s College London, 2024 RCEA International Conference in London, 2024 T2M Conference in Amsterdam and 2024 SMN Conference in Madrid. †Norges Bank. Email: [email protected] ‡European Central Bank. Email: [email protected] §Norges Bank. Email: [email protected]. Corresponding author. 1 INTRODUCTION Women’s increased labor market participation is arguably one of the fundamental changes observed in modern economies during the last century. Consider, for example, the U.S. labor market data in Figure 1: in the 1960s, the employment rate for females was less than half of the employment rate for males. But the female-to-male employment ratio increased steadily throughout the 1980s and 1990s, before converging to around 85 percent in recent decades. Gender differences in wages display a similar picture. Although women’s hourly wages stayed relatively flat at 60 percent of men’s wages until the mid 1970s (despite a substantial employment catch-up during that period), they have since outgrown male wages at about the same pace as the additional growth in female employment, resulting in a major wage convergence between the genders. In total, more than 60 percent of the female-to-male employment gap, and about half of the female-to-male wage gap, have disappeared in the last 5-6 decades. It is hardly a coincidence that Goldin (2006) refers to a “quiet revolution” when describing these trends. The goal of our paper is twofold: first, we want to quantify the consequences of this gender revolution for the U.S. macroeconomy. In particular, we estimate the spillover effects on economic growth in terms of U.S. GDP, employment, and productivity. An important part of our motivation is the concern that talent may be significantly misallocated when only a minority of women participate in paid work, as advocated by Hsieh, Hurst, Jones, and Klenow (2019). Second, we aim to shed light on the structural drivers behind the gender convergence in employment and wages. At first glance, the observation that these two trends co-move may suggest that labor demand factors have been dominant, as stressed by Aguiar and Hurst (2007) and Fukui, Nakamura, and Steinsson (2023). However, the aggregate time series shown in Figure 1 are silent about any reallocation across different skill segments of the labor force, as well as reallocation across sectors. Our aim is to appropriately disentangle labor demand and labor supply factors once we control for the fact that the large increase in employment of female workers was concentrated in the market for high-skilled workers and in the service sector. To identify the structural trends of interest in this paper, we develop a neoclassical model involving gender-specific labor which builds on Albanesi (2024) and Fukui et al. (2023).1The theoretical framework allows us to derive mutually exclusive identification restrictions on three gender-neutral macro trends and two gender-specific labor market trends. In turn, we impose these restrictions on a Structural Vector Autoregressive (SVAR) model fitted to relevant macro data, as well as to data on differences between females and males in wages and employment. The resulting econometric framework allows us to infer structural gender trends empirically, and to quantify their importance for the U.S. macroeconomy. As our key focus is on slow-moving, structural drivers that persist way beyond common business cycle horizons, we use a SVAR model with common trends, as in Del Negro, Giannone, Giannoni, and Tambalotti (2017) and Crump, Eusepi, Giannoni, and Sahin (2019).2The model can be seen as a multivariate, unobserved components model, in 1Albanesi (2024) estimates a real business cycle model with gender-specific labor to account for jobless recoveries. Fukui et al. (2023) extend the model with home production and open economy features, showing that women’s rising participation did not crowd out males’ participation, thus, being an expansionary factor for the macroeconomy. 2More recent extensions of the same model to explain inflation dynamics include Ascari and Fosso (2024), 2 Figure 1: Gender differences in employment and wages 1965 1975 1985 1995 2005 2015 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Notes: The employment (wage) gap is defined as the female-to-male ratio in employment (wage) rates. Based on data from The Current Population Survey, the United Census Bureau, the Bureau of Labor Statistics, and authors’ calculations. which the variables enter in levels, and transitory and permanent components in data are disentangled from each other. We seek to quantify the latter. As an example, let us consider GDP. Our model decomposes observed GDP dynamics into a cyclical and a permanent component. In turn, the permanent component—understood as the empirical or reduced-form GDP trend—is a function of underlying, structural drivers such as productivity and demographics (both of which can be gender-neutral or gender-specific). However, the mapping from these structural drivers to the empirical trend in GDP is unknown ex-ante. It is exactly this identification problem that we address with restrictions from economic theory. In particular, the theoretical framework presented here implies a log-linear mapping between trend GDP and (gender-neutral) technology shocks, (gender-neutral) automation shocks, (gender-neutral) labor supply shocks, as well as gender-specific labor demand and labor supply shocks. We disentangle these five structural forces based on their long-run impact on economic variables. This represents a key distinction from previous studies estimating VARs with stochastic trends. Del Negro et al. (2017) and Crump et al. (2019), for example, estimate selected common trends in data, but remain silent about the underlying structural sources. Our framework, instead, allows us to estimate the mapping from structural to empirical trends, and to instruct a Bayesian algorithm we deploy with prior information derived from theory. We estimate a baseline model where standard macro data for the U.S. economy are linked to data on aggregate gender differences in wages and employment. The idea is to Bianchi, Nicol` o, and Song (2023), Hasenzagl, Pellegrino, Reichlin, and Ricco (2022) and Maffei-Faccioli (2024). 3 infer the structural drivers of trends in wage and employment differences between females and males, and to quantify their importance for the U.S. macroeconomy. In a second step, following Dolado, Motyovszki, and Pappa (2021) we use information on individuals’ education and sector of employment contained in the Current Population Survey (CPS), and estimate extensions of the baseline model with data on gender differences in skills and sectors of employment. The results from these more granular extensions are then compared with those from the baseline. The idea is to separate “within-skill” and “within-sector” fixed effects from “within-gender” fixed effects. Thus, we can disentangle fundamental gender trends from trends in skills and sectoral composition, and gauge compositional effects that may bias our baseline results. Our main empirical result concerns the macroeconomic spillover from gender-specific labor market trends in our data: these trends, which capture the observed convergence between females and males, are also essential for macroeconomic growth in the postwar U.S. economy. During the 70s, 80s and 90s for example, they account for 30-50% of the overall trend increase in GDP, and for 20-40% of the overall trend increase in productivity. Moreover, they are responsible for a sizable share of the slowdown in trend GDP growth during the last 20 years, coinciding with a leveling-off of the gender convergence compared with earlier decades. In total, gender-specific factors explain almost one-third of GDP growth in the postwar U.S. economy, and they prevented an overall employment decline during this period. Importantly, these results are obtained when we control for aggregate macro trends such as the evolution of total factor productivity. In addition, structural factors that originate on the demand side of the labor market explain (i) most of the long-term gender convergence in employment and wages, (ii) the end of the gender convergence observed in the last 20-30 years, as well as (iii) almost all net spillovers from gender-specific labor market trends to the macroeconomy. From an econometric point of view, our empirical model suggests that the female-to-male employment ratio tends to increase permanently in periods when there is a permanent rise in the wages of females relative to the wages of males. This is consistent with a story about female-specific productivity growth, which shifts labor demand towards females. Supplyside explanations, by contrast, should have implied stagnant wage growth of females in periods with strong female employment growth. This is not what we typically see in our aggregate data. Finally, while the net spillover from labor supply factors to the macroeconomy is limited, we document an important role for skill-biased trends in females’ labor supply: when gender gap data by skill segments are considered, a supply-driven expansion of female labor in high-skill jobs emerges, and at the same time a contraction in the supply of female labor in low-skill jobs. In sum, these two forces have led to substantial reallocation of females from low-skill to high-skill jobs. Moreover, the compositional effects associated with these supply-driven labor flows across skill segments have likely contributed to the overall convergence in female and male wages. At the same time, they have largely counteracted each other in aggregate data on gender pay and employment gaps. This is partly why the net spillover effect from gender-specific labor supply factors to the macroeconomy are so limited in our baseline estimation. But trends in females’ labor supply are still important for the macroeconomy. Our paper speaks to a large literature studying the gender revolution. A useful distinction for our purposes is between papers discussing labor demand factors and labor supply 4 factors. Among the former, Galor and Weil (1996) emphasize technological factors that favored the demand for women in combination with an increase in the returns to intellectual skills (Beaudry and Lewis (2014), Rendall (2024)) and the rise of the service sector (Ngai and Petrongolo (2017), Buera, Kaboski, and Zhao (2019)), while Jones, Manuelli, and McGrattan (2015) and Hsieh et al. (2019) point to a reduction in gender discrimination and a reduction in barriers to schooling as important drivers of the convergence in wages. Among the latter, Albanesi and Olivetti (2016) and Goldin and Katz (2002) document the importance of advances in maternal health and contraception, Fern´ andez, Fogli, and Olivetti (2004) emphasize cultural factors developed during World War II, Attanasio, Low, and S´ anchez-Marcos (2008) point to the crucial role of availability and affordability of child care, while Greenwood, Seshadri, and Yorukoglu (2005) propose a model in which the emergence of home appliances favors females’ market production at the expense of home production. We contribute to this literature by proposing a horse race between labor supply and labor demand factors in the context of a macroeconomic timeseries model. While less detailed in terms of the underlying transmission mechanisms, our analysis provides a clear link between gender trends and macroeconomic outcomes. Within the large literature studying the gender convergence, our paper is closely related to two seminal papers quantifying the role of gender for macroeconomic dynamics in calibrated quantitative set-ups. Both Hsieh et al. (2019) and Heathcote, Storesletten, and Violante (2017) propose a decomposition of US macroeconomic growth in structural models with gender and find a major role for gender forces. Our results are in the same ballpark as theirs, but obtained using a simple time series model which is substantially less parameterized and estimated (rather than calibrated) on US quarterly data. Albanesi (2024) is also of particular interest. Using an estimated real business cycle model with a gender dimension, she finds that women’s relative labor supply and productivity have been important for changes in the business cycle, and more generally, for the economic performance in the US. The explicit use of theory to form priors for an estimated, empirical time series model relates our work to the well-known DSGE-VAR methodology proposed by Del Negro and Schorfheide (2004). As in their case, we are concerned about the potential misspecification induced by the tight cross-equation restrictions featured by a fully specified theoretical model. Therefore, we use theory only as a prior to inform the VAR with common trends. Differently from Del Negro and Schorfheide (2004), we focus on the variables’ permanent components and not on the cycle. Moreover, we form priors about specific structural elasticities, rather than the full covariance structure implied by theory. The rest of the paper is organized as follows: section 2 introduces the empirical trendcycle model that we fit to U.S. labor market data (aggregate and gender-specific). Section 3 describes the theoretical framework that disciplines our empirical assessment of trends, while section 4 derives the exact, theory-robust identification restrictions. Section 5 documents the paper’s main empirical results. Section 6 takes stock and puts our results in perspective. Section 7 relates the gender convergence in wages and employment to trends in the skills and the rise of services, while section 8 considers structural changes specific to male labor. Section 9 concludes. 5 2 A TIME SERIES MODEL WITH COMMON TRENDS The model that we estimate is a multivariate time series model with unobserved components, see Watson (1986), Stock and Watson (1988,2007), and Villani (2009). It decomposes a vector of observable data into two unobservable, stochastic components: the first component is characterized by cyclical but transitory fluctuations. The second component captures permanent changes, or trends, in data. One distinctive contribution of our paper is to map these trends into a vector of underlying, structural drivers. Importantly, the structural drivers may give rise to common trends in data, even if the structural drivers themselves are orthogonal to each other. To fix ideas, consider an n×1vector of data Yt, which is the sum of two unobserved states: Yt=ˆ Yt+¯ Yt,(1) ˆ Ytand ¯ Ytrepresent the cycles and trends in our data, respectively. As such, equation (1) is a purely statistical (reduced-form) decomposition of the data. The focus of our paper will be on the empirical trends in ¯ Ytand, more precisely, on the underlying causal drivers behind ¯ Yt. We suppose that ¯ Ytcan be decomposed into q≤nstructural trends, collected in the q×1vector Xt:¯ Yt=VXt(2) Here, Vis a n×qmatrix that maps the reduced-form trends into structural, economic factors. Similarly to Del Negro et al. (2017) and Crump et al. (2019), we treat the structural trends as separate random walk processes, potentially with a drift: Xt=c+Xt−1+ut, ut∼ N(0q,Σu)(3) Throughout we assume that the covariance matrix Σuis diagonal. This is standard given that utis a vector of structural trend shocks. Nevertheless, the presence of non-zero, off-diagonal elements in the matrix Vstill implies common, stochastic trends across the individual components of ¯ Yt. A vital part of our analysis will be to identify V, given that we are interested in the causal drivers of trends in data. Moreover, since our focus is on trends rather than on business cycle fluctuations, only a minimal set of restrictions is imposed on ˆ Yt. In particular, we model ˆ Ytas a vector autoregressive (VAR) process in reduced-form: Φ(L)ˆ Yt=et, et∼ N(0n,Σe)(4) Φ(L) = I−Φ1L−... −ΦpLpis an n×nmatrix of lag coefficients. Σeis freely estimated without any restrictions on the off-diagonal elements. However, we assume that permanent and transitory shocks are mutually uncorrelated, i.e. that cov(ut, et) = 0. In the parlance of Watson (1986), the model is an ”independent trend-cycle decomposition” and trends do not affect the cycle by construction.3 Equations (1)-(4) constitute the model that we confront with data. It is the combination of relevant (aggregate and gender-specific) labor market data, together with a proper identification of V, that allows us to infer gender-specific trends and quantify their importance for the U.S. macroeconomy. A first methodological contribution of this paper 3If anything, a violation of this assumption may bias estimates in the cyclical block given by (4). However, we view the assumption as rather innocuous, given that our sole interest is in secular trends and not in cyclical fluctuations. 6 4.2 REVISITING THE MAPPING TO EMPIRICAL TRENDS We are now in a position to specify a baseline, theory-consistent mapping Vfrom the structural trends Xtto trends in data ¯ Yt:       ¯ GDP t ¯ Wt ¯ Lt ¯wf,t ¯ lf,t       | {z } ¯ Yt =      1 1 1 ν14 ν15 1 0 0 ν24 ν25 0 1 ν33 ν34 ν35 0 0 0 1 −1 0 0 0 λ γ       | {z } V       At Ψt αt af,t ψf,t       | {z } Xt (23) The first column of Vimposes restrictions on the stochastic technology trend. We normalize it to have a unit effect on GDP and wages, and zero long-run effects on the remaining variables. The second column of Vgoverns the spillover from the aggregate labor supply trend, which is normalized to cause a unit increase in GDP and employment, while the third column in Vimposes that automation permanently increases GDP and lowers employment. The latter restriction implies that ν33 <0. Consistent with Figure 2 we impose zero-restrictions on the remaining elements in Vfor both labor supply and automation. Note that most zero-restrictions are based on the assumption that macro shocks do not permanently affect gender gaps in wages and employment. This assumption, which greatly facilitates identification of V, effectively rules out compositional effects of gender-neutral macro shocks. However, we inspect the possibility of compositional effects in section 7. The fourth and fifth columns in Vgovern the long-run implications of permanent changes in female-specific labor productivity and labor supply, respectively. We normalize both of these trends so that they have a unitary effect on the wage gap between females and males. That is, rather than estimating af,t and ψf,t, we identify ˜af,t ≡a γ−1 γ+λ f,t and ˜ ψf,t ≡ψ 1+λ γ+λ f,t . This is particularly convenient, as can be seen from the normalized, log-linear versions of (21)-(22): ˆwf,t =cw,f +ˆ ˜af,t −ˆ ˜ ψf,t (24) ˆ lf,t =cl,f +λˆ ˜af,t +γˆ ˜ ψf,t (25) A hat means that the variable is expressed in logarithms, with cw,f and cl,f being reducedform constants. Importantly, the gender substitution elasticities λand γ, two structural parameters of particular interest, enter directly as coefficients in (25). This means that λ=ν54 and γ=ν55 can be read directly from the estimated matrix V. The restrictions imposed on each column in Vare mutually exclusive, which is what we need to separately identify the five stochastic trends of interest. The aggregate technology trend, for example, is the only aggregate macro trend that makes the real wage co-move with GDP in the long-run and, consistent with the balanced growth path assumption, is the only one that has zero long-run effect on employment. Aggregate labor supply and automation can be disentangled in data because the former implies long-term co-movement GDP and employment, while the latter crowds out labor. Finally, femalespecific labor demand and labor supply are separable from aggregate macro trends because they are the only drivers of long-run wage and employment gaps between females 13 Table 1: Prior distributions and posterior estimates Prior Posterior Density Support Mean Mode 90% HPD ν14 af→¯ GDP Uniform [0,1] 0.91 0.98 (0.74, 0.99) ν24 af→¯ WUniform [0,1] 0.60 0.67 (0.33, 0.81) ν34 af→¯ LUniform [−0.5,0.5] 0.30 0.36 (0.09, 0.46) ν15 ψf→¯ GDP Uniform [0,2] 0.42 0.10 (0.02, 1.04) ν25 ψf→¯ WUniform [−2,0] -0.15 -0.02 (-0.48, -0.00) ν35 ψf→¯ LUniform [0,3] 0.54 0.29 (0.04, 1.23) −ν33 α→¯ LΓ(0.3,0.15) (0,∞)0.39 0.40 (0.21, 0.60) λ af→¯ lf,t Γ(1,0.5) (0,∞)1.53 1.63 (0.95, 2.18) γ ψf→¯ lf,t Γ(3,1.5) (0,∞)3.26 3.17 (2.44, 4.20) Notes: The posterior moments are generated from the last 10,000 of 50,000 draws generated from the RW Metropolis-Hastings algorithm. Γ(µ, σ2)refers to the Gamma distribution with mean µand variance σ2. and males. Moreover, they are uniquely identified because they imply opposite signs on the co-movement between gender gaps in wages and employment.5 4.3 PRIORS The last step is to specify prior shapes for the estimated parameters. Table 1 summarizes our choice of priors. We aim for an agnostic approach and use uniform priors for all elasticities governing the feedback from gender trends to the aggregate macroeconomy. The support of each uniform prior largely reflects the uncertainty bands computed during the Monte Carlo exercise, as shown in Figure 2. That is, consistent with theory, the femalespecific productivity shock behaves qualitatively as a technology shock in the aggregate given our priors, while the female-specific labor supply shock behaves as a gender-neutral labor supply shock for aggregate variables. The prior for the aggregate employment response to automation has a Gamma distribution, reflecting that automation (a decline in αt) crowds out employment. Note that we impose the prior on −ν33, since the Gamma distribution has a positive support. The final two parameters that we estimate are γand λ. The gender-specific labor demand elasticity γhas been quantified in a few existing studies. Weinberg (2000), for example, finds that γis around 2.4 in the US, while Acemoglu, Autor, and Lyle (2004) report a slightly higher value of 3. Albanesi (2024) and Fukui et al. (2023) have considered values between 4 and 5. Thus, we choose a Gamma-prior for γcentered around 3 with most of the probability mass located between 1 and 5. The evidence on λ, which captures complementarity between males’ and females’ leisure time, is more scant. Ngai 5Note that the zero restrictions on employment in response to a technology shock and on aggregate wages in response to automation and labor supply shocks are not crucial for identification. In Appendix D.2, we show that results change only marginally if we relax these assumptions. In addition, a more conservative set of priors tilted against macro effects of gender shocks is evaluated in Appendix D.1. 14 Figure 3: Estimated empirical trend of real GDP and aggregate employment 1965 1975 1985 1995 2005 2015 3.6 3.8 4 4.2 4.4 4.6 1965 1975 1985 1995 2005 2015 56 58 60 62 64 Notes: observed data (red solid line), median trend estimate (blue solid line), and 68% coverage bands (blue shaded areas). The grey areas represent NBER recessions. and Petrongolo (2017) use a value of 0.19 based on micro-evidence from Goux, Maurin, and Petrongolo (2014).6Thus, we choose a Gamma prior for λwith about 60% of the probability mass below one, and where the estimate from Goux et al. (2014) is covered by the 90% credible prior bands (even though much higher values are allowed as well during estimation). A defining feature of our priors is that firms can switch between female and male labor more easily than households (λ<γ) at the prior mode, a feature that seems highly reasonable. 5 EMPIRICAL RESULTS In this section, we present the main empirical results based on the estimated time series model described in section 2. Given the theoretical restrictions derived in section 3 and section 4, the vector of observable variables Ytincludes: (i) real GDP, (ii) real aggregate wages, (iii) the aggregate employment-to-population ratio, (iv) the ratio of female-to-male wages, and finally (v) the ratio of female-to-male employment. All variables enter the system in log-levels. The model is estimated over the sample period 1960:Q1-2019:Q4, and we choose p= 4 lags in the system given that data are observed at a quarterly frequency. We use Bayesian methods to estimate the model. In particular, a Gibbs sampler is designed to generate 50,000 draws, where the first 80% of the draws are discarded as a burn-in sample and the last 20% serve to generate posterior moments. The algorithm includes a Metropolis-Hastings step that draws from the posterior of the elasticities in V. Details on the estimation steps, data sources and construction are laid out in Appendixes A and B, respectively.7 6Goux et al. (2014) exploit a workweek reduction policy in France to obtain an estimate that reflects pure cross-hour effects across partners and not income effects. 7All our data are published by the Bureau of Economic Analysis and the Bureau of Labor Statistics, and can be downloaded from the FRED website. In the baseline specification, data on employment and wages include both single and married individuals. In Appendix D.3, we restrict our attention to married couples only. Olsson (2024) and Albanesi and Prados (2022) model explicitly the heterogeneity in marital status. 15 Figure 4: Posterior distributions of coefficients λand γ. 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 5.1 PERMANENT AND TRANSITORY COMPONENTS The first set of results is related to the decomposition of observable data into variablespecific, permanent and transitory components. Figure 3 reports the estimated permanent components of aggregate output and employment. Our model-implied trend estimates seem to be largely consistent with popular narratives for trends in US macro data: The GDP trend, for example, has displayed lower average growth in the last 20 years of data, and it leveled off completely during (and just after) the 2008 financial crisis. Moreover, the model assigns a large share of the observed employment decline in the last 15 years to permanent factors: it essentially concludes that the employment rate was back to its own trend by the end of our sample, despite being 3-4 percentage points lower than before the financial crisis. A secular decline in trend employment starting in the late 1990s is important for this result. Interestingly, our model-implied estimate of the output gap— defined as the difference between observed GDP and the inferred permanent counterpart (see Figure C.2 in the appendix)—exhibits a correlation coefficient of 0.88 with the output gap reported by the Congressional Budget Office (CBO). The latter constitutes a classic benchmark in the literature. Such a high correlation is neither obvious nor targeted, as our SVAR is not informed by data on the CBO estimates. We conclude that our model offers a reasonable description of trends and cycles in GDP in the postwar US economy, and that it can be used as a laboratory to investigate the structural drivers of trends in data.8 5.2 ESTIMATED PARAMETERS IN V Posteriors for the nine estimated parameters in Vare summarized in Table 1, while Figure 4 plots the full posterior distributions of λand γ. Despite using a prior with substantial mass below 1, we obtain a posterior density for λthat is centered around 1.5. This implies that a 1 percent increase in the wage of females relative to that of males, when caused by permanent productivity improvements among females, is associated with a 1.5 percent increase in the employment rate of females relative to that of males. Moreover, most of the posterior probability mass for λis located between 1 and 2. The posterior mean value for 8An equivalent decomposition is provided in Appendix C for the remaining variables. 16 γ, in contrast, is 3.3. Thus, a permanent rise in females’ labor supply, scaled to reduce the relative wage of females by 1 percent, increases the employment rate of females relative to males by more than 3 percent at the posterior mean. This number is close to, but somewhat higher than those obtained by Acemoglu et al. (2004) and Weinberg (2000) (they report values of 3 and 2.5, respectively). However, the posterior distribution for γ in Figure 4 covers well both of these estimates. When it comes to the remaining coefficients in Table 1, the posterior distributions for ν14,ν24 and ν34 are shifted further away from zero compared with the priors. These elasticities govern the sensitivity of aggregate output, wages and employment to a given change in females’ productivity. The shift is particularly pronounced for output, where most of the posterior mass is located close to the upper bound of the uniform prior. The estimated feedback elasticities for female-specific labor supply, ν15,ν25 and ν35, reveal a different pattern. Here, they all move from the prior towards zero in absolute values, reflecting that a given change in females’ labor supply has smaller effects on the macroeconomy in data than what our priors would indicate. However, to gauge the quantitative role of females-specific labor supply shocks, we would have to take into account the movement in ψf,t as well. This is done below. Finally, the employment sensitivity to a given change in automation, ν33, has a posterior centered around -0.4. Taking the normalization of this trend into account, a back-of-the-envelope calculation suggests that most of the labor productivity improvements arising from automation can be attributed to higher output rather than to job destruction.9 5.3 TREND DECOMPOSITIONS Estimated contributions of the different structural factors to each empirical trend in our data are documented in Figure 5. Let us first consider the trends in female-to-male employment and wage ratios, which we decompose into female-specific labor demand (green) and female-specific labor supply (light blue). Recall that these are the only structural factors that affect gender gaps in wages and employment in our framework. We find it instructive to distinguish between three separate phases in our sample: in the first 15-20 years of data, both female-specific demand and female-specific supply contributed significantly to a secular increase in the employment rate of females relative to that of males. However, the prominent role of women’s labor supply also kept their wage growth relatively modest, explaining why the wage gap between women and men remained somewhat stagnant during this period. Then, starting around 1980, the relative labor supply increase among female workers ceased to take place, causing women’s wages to significantly outgrow the wage of males. The final phase in our data started in the late 1990s. Since then, the wage and employment growth among females have been much more modest and more in line with what we observe for men. While some convergence has taken place, the gender gaps have been much more stable in the last 20-25 years of our data. Importantly, our model largely attributes this observation to lower, female-specific productivity growth. The second and third rows of Figure 5 document how gender-specific labor market factors have affected aggregate trends in the post-war US macroeconomy. Female-specific 9A unit innovation to our normalized automation shock raises output by 1 percent and lowers employment 0.4 percent. Thus, labor productivity increases by 1.4 percent, and more than two-thirds of this is due to higher output. 17 Figure 5: A structural decomposition of the empirical trends 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 1965 1975 1985 1995 2005 2015 0 0.1 0.2 0.3 0.4 0.5 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 1965 1975 1985 1995 2005 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1965 1975 1985 1995 2005 2015 -0.1 -0.05 0 0.05 0.1 0.15 0.2 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 Notes: Empirical trends (black lines) vs. the contributions of individual structural trends (colored bars). Vertical axes represent log deviations from initial values, horizontal axes measures time in quarters. Pointwise median estimates are reported. productivity growth (green), in particular, accounts for a sizable share of the overall increase in trend GDP and real wages. Moreover, female-specific productivity growth explains most of the postwar rise in employment rates, which would have fallen on average 18 Table 2: Growth factors in the US economy A. Baseline account (1) GDP (2) Labor productivity Total AΨα afψfTotal A α afψf 1960-1969 2.8 1.8 0.1 0.3 0.5 0.1 2.5 1.8 0.4 0.3 0.0 1970-1979 2.1 1.1 0.0 0.1 0.8 0.1 1.7 1.1 0.1 0.5 0.0 1980-1989 2.1 0.7 0.2 0.2 1.0 0.0 1.7 0.7 0.3 0.7 0.0 1990-1999 2.0 0.8 0.2 0.4 0.6 0.0 1.8 0.8 0.6 0.4 0.0 2000-2009 1.2 0.3 0.0 0.6 0.3 0.0 1.4 0.3 0.9 0.2 0.0 2010-2019 1.1 0.5 0.0 0.4 0.2 0.0 1.2 0.5 0.6 0.1 0.0 (3) Aggregate employment (4) Aggregate wages Total Ψα afψfTotal A afψf 1960-1969 0.3 0.1 -0.1 0.2 0.1 2.1 1.8 0.3 0.0 1970-1979 0.4 0.0 0.0 0.3 0.1 1.6 1.1 0.5 0.0 1980-1989 0.5 0.2 -0.1 0.4 0.0 1.4 0.7 0.7 0.0 1990-1999 0.2 0.2 -0.2 0.2 0.0 1.2 0.8 0.4 0.0 2000-2009 -0.1 0.0 -0.2 0.1 0.0 0.5 0.3 0.2 0.0 2010-2019 -0.1 0.0 -0.2 0.1 0.0 0.6 0.5 0.1 0.0 B. Counterfactual: no gender trends (1) GDP (2) Labor productivity Total AΨα afψfTotal A α afψf 1960-1969 2.7 2.1 0.3 0.3 – – 2.5 2.1 0.4 – – 1970-1979 2.1 1.7 0.3 0.1 – – 1.8 1.7 0.1 – – 1980-1989 2.1 1.4 0.4 0.3 – – 1.8 1.4 0.4 – – 1990-1999 2.0 1.2 0.3 0.5 – – 1.8 1.2 0.6 – – 2000-2009 1.0 0.4 0.0 0.6 – – 1.3 0.4 0.9 – – 2010-2019 1.1 0.6 0.1 0.4 – – 1.2 0.6 0.6 – – (3) Aggregate employment (4) Aggregate wages Total Ψα afψfTotal A afψf 1960-1969 0.2 0.3 -0.1 – – 2.1 2.1 – – 1970-1979 0.3 0.3 0.0 – – 1.6 1.6 – – 1980-1989 0.3 0.4 -0.1 – – 1.4 1.4 – – 1990-1999 0.1 0.3 -0.2 – – 1.2 1.2 – – 2000-2009 -0.2 0.0 -0.2 – – 0.5 0.5 – – 2010-2019 0.0 0.1 -0.1 – – 0.6 0.6 – – Note: Structural decompositions of the average, annual trend growth rates of (1) real GDP, (2) labor productivity, (3) aggregate employment and (4) aggregate wages (by decade), into total factor productivity A, automation α, gender-neutral labor supply Ψ, female-specific productivity af, and female-specific labor supply ψf. Panel A: baseline model with gender-specific trend shocks. Panel B: restricted model with dogmatic zero-priors on the elasticities governing feedback from gender-specific shocks to the macroeconomy. All numbers are point-wise median estimates. since 1960 in the absence of this secular trend. This suggests a rather muted crowding out of men when women enter the labor market, a point which we return to in section 8. 19 Female-specific productivity is also an important driver of aggregate labor productivity— here measured as output per worker—suggesting that women entering the labor market caused productivity gains beyond the mere scale effects of having a greater labor force. Finally, we note that while the trend in female-specific labor supply (light blue) has contributed to aggregate employment, it has played only a minimal role for GDP, real wages, and aggregate labor productivity. This finding suggests that the periods with extraordinary growth in females’ labor supply did not coincide with extraordinary increases in trend GDP, causing the posterior for ν15 to shift towards zero (compared with their respective prior distributions). When it comes to aggregate, gender-neutral factors, total factor productivity (blue) seems to be the quantitatively most important driver of trend GDP, wages and labor productivity in our data. Aggregate labor supply (purple) never plays an important role for GDP, but has stimulated total employment. Finally, labor-displacing automation (yellow) has not only contributed significantly to higher GDP and labor productivity over time, but also to lower employment. In particular, together with the slowdown of gender-specific trends, automation appears to be the main cause of the secular decline in aggregate employment rates over the recent decades. Automation also explains more or less the entire disconnect between real wages and labor productivity in our sample.10 Table 2 summarizes our account of economic growth in the postwar US economy. For each decade, we decompose the average, annual growth rates in the trend components of GDP and labor productivity into the estimated contributions of the five structural drivers. For now, we restrict attention to Panel A, which documents the growth accounting implied by the baseline model. Importantly, both GDP growth and labor productivity growth have declined substantially in our data, from about 2.8and 2.5percentage points per year in the 1960s, to 1.1-1.4percentage points in the last 20 years. The decline has mostly taken place in two waves, between the 1960s and 1970s, and at the beginning of the 2000s. Also the growth rate of wages has fallen consistently in our sample, while employment started to stagnate in the 1990s. The slow-down in economic growth (as well as the timing of the two waves) is well-documented and has motivated a large literature on the possible causes, see, e.g. Syverson (2017) for a review. For the sample as a whole, we find that most of the slow-down is attributed to total factor productivity, which in annual growth terms fell by more than 50% over the first two decades.11 However, up until the 1980s, lower total factor productivity growth was substantially counteracted by a secular increase in female-specific productivity: its contribution to trend GDP growth went from 0.5percentage points per year in the 1960s to 1percentage point in the 1980s. For labor productivity, its contribution increased from 0.3to 0.7percentage points per year. Thus, female-specific labor productivity doubled its annual growth rate—and as a result—its contribution to trend growth, between the 1960s and the end of the Cold War. This picture has changed fundamentally in the last 30 years of data: not only has the overall growth rate of gender-neutral macro trends continued to decline between the 10This result implies that the automation trend is the dominant driver of the labor share decline, as in Bergholt et al. (2022). This is visually confirmed in Figure C.3 in the appendix, where we compute and structurally decompose the model-implied trend in the US labor income share. 11The annual growth rate in total factor productivity fell from 1.8percentage points in the 1960s to 0.7 percentage points in the 1980s. 20 1990s and the 2010s, but also two-thirds of female-specific labor productivity growth has disappeared during this period. The latter result is imperative for growth accounting in recent decades according to our model: between the 1990s and 2010s, the slow-down in female-specific labor productivity is responsible for about half of the overall decline in the growth rates of trend GDP and labor productivity. Overall, we arrive at three main takeaways from the estimation of our empirical model: first, gender-specific labor market trends are quantitatively important for the US macroeconomy. For example, they account for almost one-third of the overall postwar increase in trend GDP, and more than one-fifth of the postwar increase in labor productivity. Second, about fifty percent of the slowdown in economic growth observed in the last 25 years is attributed to a slowdown in the employment convergence between females and males. Third, the catch-up of females’ employment observed in the last 60 years is mainly, if not entirely, a consequence of labor demand factors. Notably, all these results are confirmed also when using a prior specification tilted against macro effects of gender shocks, as shown in Appendix D.1. 6 TAKING STOCK The important impact of gender-specific labor market trends for macroeconomic growth documented in this paper is consistent with the analyses by Hsieh et al. (2019) and Heathcote et al. (2017) in the context of calibrated models. Hsieh et al. (2019) develop a Roy model of occupational choice and conclude that declining obstacles to human capital accumulation, as well as reduced discrimination, may explain around half of GDP per-capita growth between 1960 and 2010. Heathcote, Storesletten, and Violante (2010) study a neoclassical growth model with incomplete markets and overlapping generations, and find that female-specific demand factors explain most of the increase in females’ labor supply, which in turn drove half of the growth in earnings per capita between 1967 and 2002 (Heathcote et al.,2017). While the conclusions reached in these papers seem largely consistent with ours, we take a quite different approach by estimating an empirical time series model with substantially fewer cross-equation restrictions. 6.1 WHY IS FEMALE-SPECIFIC PRODUCTIVITY IMPORTANT? To better understand why the data prefer such an important role for female-specific productivity growth, we find it informative to confront the implications of trend shocks in our estimated model with empirical patterns in data. The first important piece of information is the observed, common rise of women’s wages and employment relative to males. Recall that, by construction, none of the genderneutral macro shocks can account for this empirical feature. Moreover, equations (24)- (25) demonstrate that the co-movement between wage and employment gaps naturally arises from gender-specific labor productivity af,t, as opposed to gender-specific labor supply ψf,t. Notably, the pre-1975 period represents an exception. Then the wage gap was rather stagnant compared with the employment gap, suggesting that female-specific labor supply must have played an important role as well. This is indeed captured in our trend estimates, as illustrated in Figure 5. 21 To appreciate why the estimation procedure also chooses female-specific productivity as a driver of macroeconomic variables such as GDP, it is important to understand how this trend departs from the gender-neutral alternatives. A first natural comparison is with total factor productivity, the largest contributor to macroeconomic growth according to our model. This is the only macroeconomic driver that can jointly capture the prominent upward trends of GDP, labor productivity, and aggregate wages in data. Automation cannot account for the trend in wages, while neither the wage trend nor the labor productivity trend can be accounted for by gender-neutral labor supply. However, total factor productivity implies a balanced growth path in our framework, where GDP, labor productivity and wages all grow at the same rate.12 In US macro data, instead, GDP and labor productivity have tended to outgrow aggregate wages. The only other macroeconomic driver that can capture this phenomenon is automation. However, automation by itself has limited explanatory power because it implies (i) an extreme wage disconnect where wages do not grow at all, and (ii) an overall decline in aggregate employment. While automation seems likely to be important for GDP and labor productivity in periods with falling employment rates, both of these implications are on average at odds with the last 60 years of US labor market data. Female-specific productivity speaks to all of these observations. Qualitatively, femalespecific productivity behaves similarly to total factor productivity in the sense that both GDP, labor productivity and wages rise in response to the shock. But quantitatively we allow female productivity to have different effects across these three variables, in contrast to total factor productivity. The special case of balanced growth following changes in females’ productivity, can be captured in our empirical framework with the parametric restrictions ν14 =ν24 and ν34 = 0. While these restrictions are satisfied at the prior mean in our baseline specification13 (see Table 1), the posterior parameters are updated to explain trends in data. Table 1 summarizes how the model chooses to quantitatively match female-specific productivity with our macroeconomic time series: at the posterior mean, a shock to women’s productivity—normalized to increase the relative wage of women by 1percent—is associated with a 0.9percent increase in GDP, a 0.6percent increase in the aggregate real wage, and a 0.3percent increase in the employment rate. Thus, higher female-specific productivity does not only account for the joint rise of these three variables over the full sample, but also for the slow-down in wage growth compared with GDP, as well as the increase in trend employment. Importantly, these effects are quantitatively relevant in our estimation because larger-than-normal growth in the wage of women tends to coincide with larger-than-normal growth in GDP, labor productivity, and aggregate wages, as well as a rise in aggregate employment. In the same vein, the flattening of the wage gap trend in the early 2000s coincides with lower income growth and a reversal of the aggregate employment rate. Finally, we note that ν14 −ν34 ≈ν24 at the posterior mean, implying that femalespecific productivity causes labor productivity and wages to respond similarly. In turn, this means that female-specific productivity has a negligible effect on the overall labor 12Critically, we consider deviations from balanced growth in Appendix D.2. The role of gender forces is even larger in that specification. 13Thus, at the prior mean, female-specific productivity shocks are separately identified from total factor productivity solely because the latter comes with zero-restrictions on the responses of gender gap variables. 22 Now we relax the above-mentioned assumption and allow for the presence of malespecific trend shocks Am,t and Ψm,t, in addition to the female-specific shocks Af,t and Ψf,t. Contractionary, male-specific shocks may, for example, capture the impact of video gaming and recreational computing on the labor supply of young men, as discussed in Aguiar, Bils, Charles, and Hurst (2021). The presence of male-specific trends implies that the wage and employment gaps wf,t and lf,t are driven by four gender trends in total, and we need to impose additional identification restrictions on the system. To this end, we augment the baseline empirical model with two more observables—the male employment rate and male wages—both observed in log-levels. The joint restrictions that we impose on female-to-male employment and wage gaps, as well as on the levels of males’ employment and wages, allow us to identify all of the gender-specific trends in the system. Details about the identification strategy and the theory-consistent priors are discussed in Appendix E where we present also impulse responses to male-specific shocks in the theoretical model which is once again our reference to set the priors in the empirical model. Results are presented in Figure 7. The first row plots the decomposition of trend employment and wage gaps between females and males. Both gaps are driven almost exclusively by female-specific shocks. Thus, ignoring male-specific shocks was largely inconsequential. The second row in Figure 7 reveals that female-specific shocks also remains an important driver of GDP and aggregate employment. The two male trends, in contrast, are never important at the aggregate level. Finally, the third row plots the decomposition of trends in the levels of female and male employment. A few comments are in place: first, the disappearance of aggregate employment growth observed in the last 20 years is entirely driven by female labor which stopped growing around 2000. The employment rate of males, instead, has declined every decade since 1960 up until the financial crisis, and has since then been relatively flat. Second, our account of trends in the female employment rate is similar to that of aggregate employment, albeit with a relatively larger role for female-specific productivity and labor supply (as opposed to the gender-neutral shocks). This is not surprising, given that these shocks have a smaller weight in aggregate employment. Third, the specification with males can shed light on whether secular increases in female employment crowd out male labor. The crowding-out elasticity is a key statistic in Fukui et al. (2023). On average over the sample, an increase in female employment of 1 percentage point, when driven by female-specific demand, leads to a decline in male employment of around 0.30 percentage points according to our estimates. Such a muted crowding out elasticity implies relatively large effects on aggregate employment and economic activity when women enter the labor market. The corresponding crowding out elasticity conditional on female-specific labor supply is even smaller, around 0.1. Part of the difference could be that the gender-specific productivity shocks imply stronger income effects on spouses’ labor supply because they have a greater impact on the family’s total income. This would be in line with our theoretical model, as shown in Figure E.1 in the Appendix. By comparison, Fukui et al. (2023) who do not make an explicit distinction between demand and supply-driven forces in their empirical section, report a value of 0.18. However, they refer to “relative” crowding out elasticities across regions with different exposure to gender trends, making a comparison less straightforward. Overall, we conclude that i) male shocks play a minor role, ii) our model estimates a 29 rather small degree of crowding out, consistently with the large macro effects of gender shocks and iii) estimates of the degree of crowding out are shock-specific, a point that to the best of our knowledge is novel. 9 CONCLUSION In this paper, we investigate and quantify the implications of gender-specific labor market trends for the U.S. macroeconomy. Using a SVAR model with common trends, we document the importance of gender-specific structural forces not only for the reduction of gender inequality (gender convergence) in the labor market, but also for economic growth. In particular, we show that gender-specific labor market trends account for up to 50% of trend growth in GDP over the period 1960-1990. Furthermore, the flattening of the gender convergence which started in the 1990s is key for the marked slowdown in trend growth observed over the last 25 years. Importantly, we document that gender differences matter for the macroeconomy using a pure “macro” approach: an empirical time series model is disciplined by neoclassical theory and estimated on selected macroeconomic variables. In that sense, our “let the data speak” approach is complementary yet very different from the more heavily parametrized structural models, such as those put forward by Hsieh et al. (2019), Heathcote et al. (2010) and Heathcote et al. (2017). Interestingly, and somewhat unexpectedly, our flexible setup reaches very similar conclusions on quite granular quantitative results, like the implications of gender convergence for growth and productivity. Regarding future growth prospects, one possible concern is that the muted growth since the 2000s represents a new normal unless the labor market participation among women starts accelerating again. Is this likely to happen? On one side, gender differences in the US labor market are still sizable, and experiences from other countries (see Albanesi, Olivetti, and Petrongolo (2023) for an international comparison) suggest that ample pockets of growth may still be available if the right institutional features are put in place.18 However, further growth could also prove more difficult than in earlier decades given that females’ employment rates are much higher now, and given the much smaller gap between females and males compared with the past. After all, policies cannot improve economic performance unboundedly, as the labor force participation rate has a natural upward bound of 100%. The extent to which labor market participation among women could start to rise again ultimately depends on why it stopped in mid-2000s, and the jury is still out on this question.19 18In addition, one can imagine that other long-lasting sources of growth can be exploited by addressing other forms of misallocation. For example, wage and employment gaps between native and migrants are still far from closed and there is substantial evidence of skill downgrading of migrants (Dustmann, Frattini, and Preston (2013)). 19A slowdown in gender convergence can be associated with cultural factors (Fogli and Veldkamp,2011; Fern´ andez,2013), lack of family-friendly policies (Blau and Kahn,2013), and increased income inequality inducing negative income effects on women married to high-earning husbands (Albanesi and Prados, 2022). Goldin (2014) and Goldin (2021) argue that the gender wage gap would be reduced further if firms did not disproportionately reward workers for long hours and duties difficult to plan in advance. Erosa, Fuster, Kambourov, and Rogerson (2022) validate this view in a Roy model with occupation-specific non-convex earnings functions. 30 While we believe that the application to the gender convergence is particularly interesting, our methodology can be applied to an array of questions concerning other secular trends as well. Examples include demographics, climate change, sectoral trends, immigration, as well as linkages between growth and inequality. In addition, our framework can be used to study gender differences at business cycle frequencies (cf. Albanesi (2024) and Albanesi and S¸ahin (2018)). We plan to investigate some of these topics in future research. 31 REFERENCES Acemoglu, D., D. H. Autor, and D. Lyle (2004). 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Industrial and Labor Relations Review 53(2), 290–308. 35 APPENDIX A A BVAR WITH COMMON TRENDS This section discusses the prior assumptions we formulate on the free parameters of the model presented in section 2 – including the assumptions on the prior volatilities of the structural trends’ shocks – and the instructions to estimate the model with a Gibbs sampling algorithm. A.1 PRIOR ASSUMPTIONS The initial conditions of the structural trends are distributed according to X0∼ N(X0, Iq). In principle, we do not have information about X0. However, we can use the information on ¯ Y0– the initial conditions of the empirical trends20 – as well as on the prior coefficients in V. Then, one can retrieve X0by solving the system in eq. (2), provided that the number of structural trends q=n, as it is the case in our model. The initial conditions of the cycles are distributed according to ˆ Y0∼ N(0n, In). This assumption implies that cycles fluctuate symmetrically around a zero mean. Finally, the priors for the remainder model’s coefficients are distributed according to: Σe∼ IW(κe,(κu+n+ 1)Σe)(A.1) ˜ Φ|Σe∼ N(˜ Φ,Σe⊗Ω)I(˜ Φ) (A.2) Σu∼ IW(κu,(κu+n+ 1)Σu)(A.3) where ˜ Φ = vec(Φ) and I(˜ Φ) is an indicator function that is equal to one, when the VAR of the cycle block is stationary, zero otherwise. The prior on the lag coefficients is standard Minnesota with mean zero and overall tightness hyperparameter equal to 0.2 (Giannone, Lenza, and Primiceri (2015)). IW is the Inverse-Wishart distribution with κdegrees of freedom and mode Σ. The prior on the transitory innovations is rather loose, with degrees of freedom κe=n+ 2 to ensure the existence of the mean and the prior mode Σe=I. Next, the prior on the trends’ shocks Σuis distributed according to an Inverse-Wishart, as well. The prior is rather tight, as we set the degrees of freedom κu= 100. Finally, the prior mode Σuis assumed to be diagonal. One non-trivial task is to come up with reasonable priors for the elements of σ2 u= [σ2 Aσ2 Ψσ2 ασ2 afσ2 ψf]′, the vector stacking the shocks’ volatilities of the structural trends in Xt. The reason is because the structural trends are unobservable in the first place. However, it is still possible to form fairly nonjudgmental priors on these structural volatilities by combining two pieces of information we already possess, namely: (i) the data and (ii) the theory-based prior beliefs on the free parameters in V(ν). To fix ideas, recall that empirical and structural trends are linked by the linear relationship ¯ Yt=VXtand that Xt=c+Xt−1+ut. Without loss of generality, one can express the empirical trends in their growth rates, as follows: ∆¯ Yt=V(c+ut) This equation implies that the covariance matrix of the empirical trends in growth rates is denoted by Σ∆¯ Y=V′ΣuV. Then, provided that the covariance matrix Σuis diagonal, the 20Specifically, we set ¯ Y0equal to the average of the HP-filter trend growth rate from the pre-sample data. 36 following linear relations apply: σ2¯ GDP =σ2 A+σ2 Ψ+σ2 α+ν2 14σ2 af+ν2 15σ2 ψfσ2 ¯ W=σ2 A+ν2 24σ2 af+ν2 25σ2 ψf σ2 ¯ E=σ2 Ψ+ν2 33σ2 α+ν2 34σ2 af+ν2 35σ2 ψfσ2 ¯ Ef−m=ν2 44σ2 af+ν2 45σ2 ψf σ2 ¯ Wf−m=σ2 af+σ2 ψf On the left-hand side of each equation, there are the volatilities of the empirical trends in growth rates, while on the right-hand side, there are the coefficients of Vand volatilities of the structural shocks. The empirical volatilities are available in the data and the parameters νij are simply the values around which the prior density of the long-run elasticities is centered. The only unknowns are the structural volatilities. It turns out that is straightforward to retrieve the structural volatilities in σ2 u, as they are the unknowns of a linear system of 5 equations in 5 unknowns and, therefore, there always exists a unique solution to the system. Consistently, this is how we proceed in practice. First, back out the empirical volatilities from the HP-filter trend growth rates of the endogenous variables using pre-sample training. Second, plug the empirical volatilities and the prior means of the parameters in V. Solve the system for the unknown volatilities and use them to center the prior density of the structural volatilities. Finally, notice that the very same reasoning applies when forming priors for the initial conditions and the drifts of the structural trends. Accordingly, the initial conditions X0 should be centered around X0=V¯ Y0, with ¯ Y0being the last period’s empirical trend in levels (last period in the training sample). As for the drifts, the constants cshould be centered around c=VE(∆¯ Yt), with E(∆ ¯ Yt)being the average of the empirical trends in growth rates (in the training sample). A.2 ESTIMATION OF THE STATE SPACE WITH GIBBS SAMPLING Consider the unobserved states of the model in section 2 in the following stacked formulation: VXt ˆ Yt=Vc 0+I0 0AVXt−1 ˆ Yt−1+I0 0IVut et(A.4) and the Covariance matrix of the model is given by Σ: Σ = V′ΣuV0 0 Σe(A.5) Then, the model samples 50000 draws and retains the last 10000 draws from a Gibbs algorithm, according to the following steps: 1. Draw from the joint distribution X0:T,ˆ Y−p+1:T, ν |c, A, Σu,Σe, Y1:T, which is given by the product of the marginal posterior of νvector of free parameters in Vconditional on the other parameters ν|c, A, Σu,Σe, Y1:Tand the distribution of the unobserved states conditional on νand the other parameters X0:T,ˆ Y−p+1:T| ν, c, A, Σu,Σe, Y1:T. 37 (a) p(ν|c, A, Σu,Σe, Y1:T)∝ L(Y1:T|ν, c, A, Σu,Σe)p(ν), where L(Y1:T|ν, c, A, Σu,Σe)is the likelihood of the data obtained from the Kalman filter applied to the state space of the model. The posterior of νis estimated by introducing a Metropolis-Hastings step. (b) Draws from p(X0:T,ˆ Y−p+1:T, c |ν, A, Σu,Σe, Y1:T)are obtained implementing Durbin and Koopman (2002) simulation smoothing algorithm. 2. Draw from the joint distribution A, Σu,Σe|X0:T,ˆ Y−p+1:T, Y1:T. The estimation of the remaining parameters is relatively straightforward, provided that the unobserved states follow rather standard vector autoregressive laws of motion. (a) TREND BLOCK. the posterior distribution of Σuis given by: p(Σu|X0:T) = IW(Σu+ T X t=1 (Xt−Xt−1)(Xt−Xt−1)′ | {z } Su , κu+T) (b) CYCLE BLOCK. The posterior distributions of the lag coefficients in Aand the covariance matrix Σeof the stationary VAR are standard: p(Σe|ˆ Y0:T) = IW(Σe+Se, κe+T) p(A|Σe,ˆ Y0:T) = N vec(A),Σe⊗T X t=1 ˆ Ztˆ Z′ t+ Ω−1−1! where ˆ Zt= (ˆ Y′ t−1,..., ˆ Y′ t−p), A=PT t=1 ˆ Ztˆ Z′ t+ Ω−1−1PT t=1 ˆ Ztˆ Y′ t+ Ω−1A, Se=PT t=1 ete′ t+ (A − A)′Ω−1(A − A) 38 As documented in Table D.1, the elasticities governing feedback from female-specific productivity shift substantially away from zero even when we put low prior weight on such an outcome. The posteriors for ν14 and ν24, which govern the feedback from femalespecific productivity to GDP and aggregate wages, are centered around 1.4and 0.75, respectively. Given that the estimates of λand γare relatively similar to those in the baseline, suggesting that also the in-sample estimates of realized productivity growth for female labor is rather similar across specifications, it follows that the high posterior values for ν14 and ν24 do indeed reflect a major role for female-specific productivity (instead of simply capturing lower estimates of female-specific productivity growth). The remaining parameters that govern macroeconomic feedback move less, especially those that determine the feedback from female-specific labor supply. Figure D.1 demonstrates that female-specific productivity remains important for the US macroeconomy even in a setting with very conservative priors against this outcome. Notably, the effect on labor productivity is even stronger in this specification since employment is slightly less affected by females’ productivity compared with the baseline. Figure D.1: Structural decomposition with priors stacked against macroeconomic feedback 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 1965 1975 1985 1995 2005 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1965 1975 1985 1995 2005 2015 -0.05 0 0.05 0.1 0.15 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 Notes: The colored bars display the point-wise median evolution of the empirical trends attributable to each structural trend. D.2 REVISITING THE LONG-RUN RESTRICTIONS ON EMPLOYMENT AND WAGES In this exercise, we relax the balanced-growth assumption which implies that employment is invariant to changes in TFP. We set a prior on the employment that allows for negative wealth effects on labor supply but does not rule out positive effects. We use a Normal 45 distribution centered around -0.1, a value broadly based on Boppart and Krusell (2020). In addition, we also relax the zero long-run effects of automation and labor supply on aggregate wages. In particular, we restrict the effects to be negative with most of the prior density concentrated around zero. As shown in Figure D.2, the data favor a small but non negligible negative effect of TFP shocks on employment. This implies that gender shocks play an even larger role in driving employment up, especially in the first part of our sample. The key role of gender shocks for GDP and productivity are confirmed also in this specification. Finally, we do not find evidence of a non-negligible effect of either automation or labor supply on aggregate wages. Figure D.2: Structural drivers of empirical trends – relaxing balanced growth path 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 1965 1975 1985 1995 2005 2015 0 0.1 0.2 0.3 0.4 0.5 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 1965 1975 1985 1995 2005 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1965 1975 1985 1995 2005 2015 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 Notes: The colored bars display the point-wise median evolution of the empirical trends attributable to each structural trend. 46 D.3 WHAT IF WE CONSIDER ONLY MARRIED INDIVIDUALS? In this exercise, we construct employment and wage gender gaps based only on married individuals to better reflect the counterpart in the theoretical model where decisions are taken at the household level. As shown in Figure D.3, all the main results are confirmed although the role of gender-specific labor supply shocks is further reduced in this exercise. Figure D.3: Structural drivers of empirical trends – model with data on married individuals 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 1965 1975 1985 1995 2005 2015 0 0.1 0.2 0.3 0.4 0.5 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 1965 1975 1985 1995 2005 2015 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1965 1975 1985 1995 2005 2015 -0.1 -0.05 0 0.05 0.1 0.15 0.2 1965 1975 1985 1995 2005 2015 0 0.2 0.4 0.6 0.8 1 Notes: The colored bars display the point-wise median evolution of the empirical trends attributable to each structural trend. 47 E A MODEL WITH MALE-SPECIFIC SHOCKS This section outlines the prior assumptions of the model specification that jointly identifies female-specific and male-specific structural trends, as presented in section 8. We augment the baseline specification with male-specific data on employment and wage rate levels. This enables to identify a male-specific labor demand trend and a male-specific labor supply trend in addition to female-specific trends. Vis modified accordingly:           ¯ GDP t ¯ Wt ¯ Et ¯ Wf−m,t ¯ Ef−m,t ¯ Wm,t ¯ Em,t           | {z } ¯ Yt =           1 1 1 ν14 ν15 ν16 ν17 1 0 0 ν24 ν25 ν26 ν27 0 1 ν33 ν34 ν35 ν36 ν37 0 0 0 −1 1 1 −1 0 0 0 γ λ −γ−λ 1 0 0 ν64 ν65 ν66 ν67 0 1 ν73 ν74 ν75 ν76 ν77           | {z } V           At Ψt αt ψf,t af,t ψm,t am,t           | {z } Xt (E.1) As in the baseline, the first three columns define the long-run effects of aggregate macro trends. Restrictions on GDP, wages, employment and the gender gaps are identical to the baseline. In addition, we assume that the long-run effect of technology, automation and labor supply on the level of males employment and wages is identical to their aggregate counterparts. This implies, for example, that the long-run feedback of automation to aggregate employment and males employment is identical – i.e., ν33 =ν73. Together with the zero long-run restrictions on the gender differentials, such assumption preserves the long-run gender neutrality of macro trends. The remainder columns identify the four gender-specific trends. A few remarks are in place. First, similarly to the baseline, female(male)-specific labor demand is separable from female(male)-specific labour supply because the former implies the same co-movement between gender gaps, while the latter implies a negative sign on the comovement between gender gaps. Consistent with the results from the theoretical model in Figure E.1, female-specific and male-specific shocks are assumed to have non-negative long-run effect on GDP and are mutually exclusive via the opposite sign effect on the employment gap. The uniform priors for the remainder gender-specific feedback to macro are formulated using the impulse responses in Figure E.1 as a reference point. Furthermore, both the female-specific and the male-specific shocks are normalized to have unit long-run effects on the wage gap, so that the feedbacks to the employment gap can be interpreted in terms of γand λ, as in the baseline. Furthermore, we assume that femalespecific and male-specific shocks have symmetric effects on both the gender gaps and macro aggregates. This implies that the elasticities of macro aggregate with respect to male-specific shocks span the same uniform boundaries of the macro feedbacks to femalespecific shocks. In this way, we remain agnostic about the relative strength of femaleand male-specific shocks. Finally, we also estimate the effects of female-specific shocks to males employment and wages – i.e., ν64,ν65,ν74,ν75. As discussed in section 8, this is particularly useful because we can make inference on the crowding out effect, conditional on either a female-specific shocks. These effects are captured by ν74 and ν75. The prior on both these elasticities is rather loose: it uniformly spans the probability set [-1,0]. This allows the likelihood to visit both regions with large and small crowding out effects. The 48 prior and posterior estimates are summarized in table E.1. Table E.1: Prior distributions and posterior estimates Prior Posterior Density Support Mean Mode 90% HPD ν14 ψf→¯ GDP Uniform [0,2] 1.61 1.39 (1.23, 1.92) ν24 ψf→¯ WUniform [−2,0] -0.24 -0.30 (-0.52, -0.03) ν34 ψf→¯ EUniform [0,3] 2.27 1.94 (1.89, 2.69) ν64 ψf→¯ WmUniform [0,1] 1.60 1.69 (1.07, 2.03) ν74 ψf→¯ EmUniform [−1,0] 2.73 2.07 (1.98, -3.53) ν15 af→¯ GDP Uniform [0,1] 0.94 0.99 (0.80, 0.99) ν25 af→¯ WUniform [0,1] 0.91 0.99 (0.77, 0.99) ν35 af→¯ EUniform [−0.5,0.5] 0.26 0.31 (-0.04, 0.46) ν65 af→¯ WmUniform [0,1] -0.01 0.00 (-0.21, 0.13) ν75 af→¯ EmUniform [−1,0] -0.33 0.41 (-0.63, -0.10) ν16 ψm→¯ GDP Uniform [0,2] 0.62 0.57 (0.13, 1.13) ν26 ψm→¯ WUniform [−2,0] -1.51 -1.52 (-1.97, -0.72) ν36 ψm→¯ EUniform [0,3] 0.81 0.91 (0.30, 1.25) ν66 ψm→¯ WmUniform [−1,0] -0.40 -0.20 (-0.89, -0.06) ν76 ψm→¯ EmUniform [0,10] 6.10 5.80 (5.30, 7.29) ν17 am→¯ GDP Uniform [0,1] 0.63 0.98 (0.09, 0.95) ν27 am→¯ WUniform [0,1] 0.72 0.82 (0.26, 0.94) ν37 am→¯ EUniform [−0.5,0.5] 0.15 0.20 (-0.29, 0.44) ν67 am→¯ WmUniform [0,2] 1.68 1.95 (1.18, 1.96) ν67 am→¯ EmUniform [0,1] 0.54 0.68 (0.07, 0.92) −ν33 α→¯ EΓ(0.3, .15) (0,∞)0.39 0.41 (0.20, 0.54) λ ai={f,m}→¯ Ef−m,t Γ(1, .5) (0,∞)1.31 1.57 (0.81, 1.61) γ ψi={f,m}→¯ Ef−m,t Γ(3,1.5) (0,∞)6.28 6.7 (5.48, 6.81) Notes: The posterior moments are generated from the last 10,000 of 50,000 draws generated from the RW Metropolis-Hastings algorithm. Γ(µ, σ2)refers to the Gamma prior density with mean µand variance σ2. 49 Figure E.1: Additional impulse responses: male vs female shocks Notes: Impulse response functions from simulations of the theoretical model. Pointwise median, 90% and 68% bands based on 1,000 independent draws from the parameter distributions. The y-axes measure responses in percent, the x-axes represent time in quarters. 50