Estimating large-dimensional connectedness tables: The great moderation through the lens of sectoral spillovers

Brunner, Felix; Hipp, Ruben Article Estimating large-dimensional connectedness tables: The great moderation through the lens of sectoral spillovers Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Brunner, Felix; Hipp, Ruben (2023) : Estimating large-dimensional connectedness tables: The great moderation through the lens of sectoral spillovers, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 14, Iss. 3, pp. 1021-1058, https://doi.org/10.3982/QE1947 This Version is available at: https://hdl.handle.net/10419/296344 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/ Quantitative Economics 14 (2023), 1021–1058 1759-7331/20231021 Estimating large-dimensional connectedness tables: The great moderation through the lens of sectoral spillovers Felix Brunner Nova School of Business and Economics Ruben Hipp Financial Stability Department, Bank of Canada We estimate sectoral spillovers around the Great Moderation with the help of forecast error variance decomposition tables. Obtaining such tables in high dimensions is challenging because they are functions of the estimated vector autoregressive coefficients and the residual covariance matrix. In a simulation study, we compare various regularization methods on both and conduct a comprehensive analysis of their performance. We show that standard estimators of large connectedness tables lead to biased results and high estimation uncertainty, both of which are mitigated by regularization. To explore possible causes for the Great Moderation, we apply a cross-validated estimator on sectoral spillovers of industrial production in the US from 1972 to 2019. We find that the spillover network has considerably weakened, which hints at structural change, for example, through improved inventory management, as a critical explanation for the Great Moderation. Keywords. VAR models, shrinkage, networks, industrial production. JEL classification. C32, C52, E23, E27. 1. Introduction With the onset of the Great Moderation, around 1984, key macroeconomic time series exhibit sharp decreases in growth rate volatility. Whether this shift in fluctuations is due to a structural change in the economy, improved economic policies, or just good luck has been extensively studied using a variety of approaches. In particular, for industrial production (IP), the literature provides manifold narratives, often using contemporaneous correlations between sectors to approximate dependencies. Recent advancements further allow econometricians to describe directional dependencies in the form of forecast error variance decompositions (FEVDs) (see Diebold and Yılmaz (2014)). Namely, FEVDs measure how much the variation of one sector can explain the variation of anFelix Brunner: [email protected] Ruben Hipp: [email protected] We are grateful for comments from Jason Allen, Tatjana Dahlhaus, Thibaut Duprey, Soojin Jo, Paulo M.M. Rodrigues, various seminar participants, and two anonymous referees. This work was supported by the Fundação para a Ciência e Tecnologia, Portugal. A special thanks goes to Matteo Barigozzi and Christian Brownlees, without whom this work would not exist. ©2023 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE1947 1022 Brunner and Hipp Quantitative Economics 14 (2023) other. They do so by condensing contemporaneous and lagged dependencies into a single connectedness table based on vector autoregressions (VARs). These tables, however, inherit the VAR’s estimation uncertainty in high dimensions: the estimation error blows up when the number of variables (N)approachesthenumberoftimeobservations(T). The goal of this paper is to examine sectoral spillovers as a potential driver of the decline in volatility during the Great Moderation. It consists of two parts: First, we explore regularization methods for estimating large networks from time-series observations and detail the properties of different approaches in an extensive Monte Carlo (MC) simulation. Second, we investigate the structural change hypothesis of the Great Moderation by applying this methodology to sectoral spillovers of IP. Econometricians typically face a high-dimensional setup when estimating FEVDs. For one thing, networks must include all relevant variables to have a unified and precise interpretation; that is, Nis large. In addition, time variation in the parameters—for example, a structural break or rolling windows—reduces the number of observations; that is, Tis often small. With large Nand small T, standard estimation methods produce poor estimates and bad forecasts due to overfitting. Regularization methods for regressions and covariance matrices counteract these ramifications. Yet, it remains unclear which ones to choose. Thus, we provide an extensive MC study of regularization techniques combined with FEVDs to guide researchers in large-dimensional network estimations. While there are successful applications of FEVDs in high dimensions (e.g., see Demirer, Diebold, Liu, and Yilmaz (2017)), two questions remain unanswered. First, how does estimation uncertainty affect the overall results of FEVDs? Second, would an additional regularization of the innovation covariance matrix improve the results? Our MC simulation results demonstrate the performance gain of each regularization step in the estimation of the overall FEVD network. We show that regularization of both the coefficients and the covariance matrix not only trade off bias for variance by reducing the estimation uncertainty of the FEVD, but also mitigates a positive bias in the entries. To the best of our knowledge, this result is novel in the literature and highlights the importance of regularization in the context of FEVDs. Perhaps surprisingly, a horse race of regularization methods applied to FEVDs yields similar results among estimators, with the winner being conditional on the setting. In our application, we investigate changes to the sectoral dependency structure of IP around the Great Moderation. The central question is whether the large volatility of the aggregate IP index originated from amplifications of sectoral fluctuations by the network in the spirit of Acemoglu, Carvalho, Ozdaglar, and Tahbaz-Salehi (2012). That is, we address the gap in the literature on how structural change in sectoral interconnectedness affects aggregate index volatility. The directed nature of FEVDs allows us to investigate linkages that were previously hidden under the rationale of correlations as a result of common exposure to aggregate shocks. Thus, we motivate our application by the fact that a strong intersectoral network of idiosyncratic shocks is observationally equivalent to the prevalence of aggregate shocks, if connectedness is ignored. We apply FEVDs to scrutinize IP spillovers between 88 sectors in the US from 1972 to 2019. The estimation is challenging since the split into preand post-Great Moderation periods reduces the effective sample size. We use cross-validated regularization to Quantitative Economics 14 (2023) Estimating large-dimensional connectedness tables 1023 tackle this challenge. The estimates provide time-varying spillover networks that uncover the corresponding directed dependency structure among sectors. We find that sectors that were initially influential decreased their outgoing links after 1984, reducing the importance of spillover effects overall. An analysis of contributions to the aggregate index suggests that inventory-heavy sectors added considerably to the high volatility pre-Great Moderation via their spillovers. We connect individual changes in spillovers to decreases in inventories-to-shipments ratio, which supports a narrative of structural change through improved inventory and supply chain management. After 2007, we observe again higher levels of aggregate volatility, but without out-sized contributions of some sectors. This observation insinuates no reversal of the Great Moderation’s structural change. Our comparative MC study connects to the literature on the regularization of regressions and covariance matrices. Regularizations have become popular not only because the increasing availability of data makes variable selection more critical, but also because technological advances make the application of high-dimensional estimators increasingly feasible. For the regression step, we consider shrinkage estimators, such as ridge regression by Hoerl and Kennard (1970), variable selection methods, such as LASSO by Tibshirani (1996), and combinations of the two, such as adaptive elastic-net by Zou and Zhang (2009). Whereas these methods find application primarily in crosssectional contexts, the time-series literature has succeeded in using regularization, for example, in the general case of VARs in Kascha and Trenkler (2015) and in the setting of FEVDs in Demirer et al. (2017). For the regularization of covariance matrices, we examine variable selection methods for the partial correlation matrix, as in the graphical LASSO by Friedman, Hastie, and Tibshirani (2008), optimal shrinkage estimators, as in Ledoit and Wolf (2004), and sample covariance thresholding, as in Bickel and Levina (2008), Rothman, Levina, and Zhu (2009), and Cai and Liu (2011). We contribute to this diverse literature by comparing the regularized estimators’ performances in the context of FEVDs. Empirically, we contribute to the understanding of the decline in IP volatility that took place during the Great Moderation. A prevalent hypothesis in the context of industrial production is the decline in aggregate shocks, as described by Foerster, Sarte, and Watson (2011). This contrasts with the results of Gabaix (2011), who considers overweight index constituents as a central driver of aggregate fluctuations in industrial production. Similarly, Carvalho and Gabaix (2013) argue that idiosyncratic shocks on the sectoral level can account for the shift in macroeconomic volatility. Acemoglu et al. (2012) connect to this idea and stress that strong production networks propagate sectoral productivity shocks to the rest of the economy and hinder diversification in the aggregate index. In contrast to the fast-expanding literature that uses input–output tables as network proxies (for a review, see Carvalho and Tahbaz-Salehi (2019)), our analysis directly sheds light on the network implied by sectoral IP correlations. Thus, we offer a unifying view on the two opposing explanations of Foerster, Sarte, and Watson (2011) and Gabaix (2011) by highlighting the transition from an economy with strong intersectoral spillovers from a few sectors to an economy with a less transmitting network. 1024 Brunner and Hipp Quantitative Economics 14 (2023) Our empirical results show that the intersectoral network has changed structurally with the Great Moderation, making idiosyncratic shocks less likely to impact the aggregate. In other words, before the Great Moderation, granular shocks were able to propagate through the network such that they amplified to strong aggregate fluctuations. This finding offers a new perspective on the observation that correlations between sectors have significantly changed with the Great Moderation, which is often attributed to the existence of a strong factor structure, that is, shocks to multiple sectors at once. Similar to Galí and Gambetti (2009), our findings support the notion that structural change has abated shock transmission, for example, through improvements to supply chain and inventory management as argued in Kahn, McConnell, and Perez-Quiros (2002), Summers et al. (2005), and Davis and Kahn (2008). The rest of the paper is organized as follows. In Section 2, we introduce the concept of FEVDs and provide an overview of various regularization methods. We assess their performance in a simulation study in Section 3.Section4applies the regularization of FEVDs to the IP setup to answer the question of sectoral spillovers. Finally, Section 5 concludes. The Supplementary Material (Brunner and Hipp (2023)), such as mathematical details, practical illustrations, and complementary empirical graphs can be found in the Online Appendix. 2. Methodology This section provides a general overview of FEVDs and introduces suitable regularization methodologies to mitigate estimation uncertainty in large-dimensional applications. We start with an N-dimensional stable VAR(1) process, yt=ν+Ayt−1+ut,ut∼N(0, ),∀t=1, ,T.(1) Following Pesaran and Shin (1998), we obtain the FEVD as a function of the VAR coefficient matrix Aand the innovation covariance matrix .1A detailed derivation is presented in Section 2.1. As in many structurally motivated economic models, we want to include a broad set of variables. However, considering many variables entails the curse of dimensionality. Thus, to estimate high-dimensional FEVDs, we have to estimate the VAR coefficient matrix Aand the innovation covariance matrix in large dimensions. For that purpose, we assess the performance of various estimation techniques by comparing different regu1Note that any VAR(p) translates into a VAR(1); hence, our specification provides the companion form to higher-order lag numbers without loss of generality. For details, see (Lütkepohl (2005, p. 15)). Quantitative Economics 14 (2023) Estimating large-dimensional connectedness tables 1025 Figure 1. Estimation overview for high-dimensional forecast error variance decompositions (FEVDs). The estimates ˆ Aand ˆ are required for the estimation of FEVDs. The lists describe the considered regularization methods in this paper. Note that although the VAR residuals are fed into the estimation of the covariance matrix, properties such as the degree of shrinkage and sparsity structure are determined independently by the estimators. larization approaches for large VARs.2Figure 1gives an overview of the regularization methods considered in this paper. The statistical learning literature contains two types of regularization techniques that are beneficial for FEVDs: the regularization of coefficients and the regularization of covariance matrices. Concerning the former, LASSO techniques tend to perform well in autoregressive setups. Such techniques apply regularization to the coefficient matrix Abut do not imply any regularization for the covariance matrix . The combination of Aand in the FEVD, however, suggests that poor estimation of the innovation covariance matrix renders the overall estimate noisy. That is, the estimation of suffers from a similar uncertainty induced by high dimensionality. Hence, we resort to covariance shrinkage estimators as an alternative to the sample covariance matrix. Our approach, therefore, is to combine regularization methods for the unknown regression coefficient and covariance matrix to achieve the best possible estimate of the FEVD with respect to the estimation error. We describe each method in detail in Section 2.2. 2We address the most prominent examples, but we are aware that there are additional approaches that are beyond the scope of this paper. L0 penalties in high-dimensional regressions are computationally infeasible for classical machines: the nonconvexity of the L0 penalty creates a combinatorial problem, which is NP-hard. Notable examples of variable selection are the information criteria AIC/BIC and the Variational Garotte in Kappen and Gómez (2014). We experimented with the latter approach in small scale simulations, which turned out to be computationally too expensive and did not yield any improvement. The covariance matrix estimation literature also deals with the regularization of the eigenvalues; for example, in Lam et al. (2016). 1026 Brunner and Hipp Quantitative Economics 14 (2023) 2.1 Generalized forecast error variance decompositions Due to the stability assumption, the VAR process in (1) can be written in moving average (MA) representation yt=μ+∞  k=0 kut−k,(2) where the MA parameters are defined as k=Ak,∀k≥0, and μrepresents the mean term. Note that the components of utare generally not orthogonal such that structural interpretations are economically meaningless. Koop, Hashem Pesaran, and Potter (1996) and Pesaran and Shin (1998) define the (unscaled) generalized impulse response (IR) function at horizon hto an impulse jon the jth entry of the reduced form innovation ut. They do so by integrating out the effect of all of the remaining impulses in the innovation vector: IR(h,j,j)=E[yt+h|uj,t=j]−E[yt+h|uj,t=0].(3) Under a Gaussian assumption, we can use E[ut|uj,t=j]=(σ1j,,σNj )σ−1 jj j=ejσ−1 jj j, with σij being the ij th entry in and ejas the jth column of the identity matrix. Substituting this expression in (3), the impulse response function can be rewritten as IR(h,j,j)=hejσ−1 jj j. It is customary to set j=√σjj , which yields the scaled generalized impulse response functions IR(h,√σjj ,j). We assemble the scaled IRsinthe(N×N)matrix g(h)=ψg ij (h)=IR(h,√σ11,1 ),,IR(h,√σNN,N)=hdiag()−1 2,(4) where diag(M)denotes a diagonal matrix with the diagonal values of the square matrix M. Analogously to standard impulse response analysis, we gain further insights rewriting ytas a vector-valued impulse response function multiplied by an innovation vector. Let P =diag()−1 2and define the generalized shock ug tas ug t:=P−1ut∼N(0N×1,), where =diag()1 2−1diag()1 2. Then it is possible to express (2)as yt=μ+∞  k=0 g(k)ug t−k.(5) Henceforth, we can interpret ug tas the innovation vector in the generalized impulse response analysis, where a single element receives a unity shock and all others remain at Quantitative Economics 14 (2023) Estimating large-dimensional connectedness tables 1027 zero. Implicitly, we assume that ug tis an orthogonal and exogenous shock vector. However, due to the distribution of ug twith the inverse correlation matrix , this step results in an approximation error. The approximation error can be linked to the partial correlation matrix of ut, and it is generally bigger when more entries are partially correlated.3 Note that the h-period forecast error variance of variable iexplained by innovations in variable jis (ψg ij (h))2. Then the effect of a generalized impulse of variable jat time t on the H-step-ahead forecast error variance of variable iis MSEyi,t+H−1|ug t+h=ejH−1 h=0= H−1  h=0ψg ij (h)2.(6) The H-step-ahead forecast error variance contributions from all variables to iis just its mean squared error (MSE): MSE[yi,t+H−1]=H−1  h=0h hii .(7) Pesaran and Shin (1998) divide (6)by(7) and get a table showing the contributions from innovations in variable jto the H-step-ahead forecast error variance of variable i. Like Diebold and Yılmaz (2014), we denote the H-step-ahead generalized FEVD as DgH = [dgH ij ]with entries dgH ij =MSEyi,t+H−1|ug t+h=ejH−1 h=0 MSE[yi,t+H−1].(8) Note that the numerator implicitly shocks single entries of ug tand the denominator shocks single entries of ut. In other words, the “generalized” FEVD approximates shocks with ug tand is only accurate if =IN;thatis,ifis diagonal. Optionally, Diebold and Yılmaz (2014) row-normalize these tables for a cleaner network interpretation. However, row normalization distorts the entries and further complicates estimation errors. Thus, if not explicitly stated, we do not perform this step. At this point, it is worth mentioning that for clean calculations of variance decompositions, we require the model in (1) to be identified. The literature on structural VARs contains many identification approaches that are suitable for meeting this objective in various circumstances. Yet, most restrictions such as exclusion restrictions (see Sims (1980)) dissent from the motivation of detecting links between variables. Some other schemes, such as heteroskedasticity identification (see Rigobon and Sack (2003)), are less restrictive on the directionality of effects and also have been successfully applied to FEVDs (see Hipp (2020)). However, in the high-dimensional context, such identification schemes are practically infeasible. Thus, we opt for the generalized version with imperfectly orthogonalized shocks to avoid stricter assumptions about the process, allowing us to interpret results more neutrally. Nevertheless, we acknowledge the drawbacks of using “generalized identification,” and address the resulting imprecision by testing deviations from orthogonality in the empirical section. 3For a discussion of ,seeRaveh (1985). 1028 Brunner and Hipp Quantitative Economics 14 (2023) 2.2 Estimating large forecast error variance decompositions 2.2.1 Regularizing vector autoregressive coefficients First, we follow the notation of Kascha and Trenkler (2015)forp=1 lags and transform the VAR(1) such that the coefficient matrix Acan be estimated in vector form. That is, y=Z⊗INβ+u,(9) where y=vec([y1,,yT]),Zt−1=(1, y t−1),Z=[Z0,,ZT−1],β=vec([ν,A]) and u=vec([u1,,uT]).WesetX:=(Z⊗IN)to obtain the general regression form. The ordinary least squares (unpenalized) estimator for the general regression form reads ˆ βOLS =argmin βy−Xβ2. Here, ·2denotes the square of the Frobenius norm. Based on this objective function, we aim to regularize the coefficient matrix A. On this account, we consider elastic-net regularization, which comprises the extreme cases of LASSO and ridge regression. Additionally, we introduce a new regularization target that enforces sparsity on the long-run dependencies. (Adaptive) elastic-net, LASSO, and ridge regression We outline the most general concept following Zou and Zhang’s (2009) adaptive elastic-net. This penalized estimator is a compound of the general concepts of elastic-net and adaptive LASSO. In particular, it simultaneously shrinks and selects entries in the coefficient matrices and, moreover, has the oracle property, which ensures optimal large-sample performance. A comprising definition of the adaptive elastic-net estimator class is ˆ βAEnet =argmin βy−Xβ2+λNET N2+N  i=1 wiα|βi|+(1−α)1 2β2 i, (10) where wi=|ˆ βi,ini|−γis an initial guess with γ>0andλNET is a tuning parameter that controls the strength of the elastic-net penalty, and must be chosen by the researcher. Note that the original paper proposed to only use the weights on the LASSO penalty. However, similar to Demirer et al. (2017), we put the weight before the shrinkage penalty and use the glmnet routine from Friedman, Hastie, and Tibshirani (2010).4 The regression in (10) is an enhanced version of the penalty regression, and thus, generalizes a family of regularized estimators. For example, the elastic-net penalty with α∈(0, 1)combines the LASSO and the ridge estimator and inherits the desirable properties of both; for example, it removes the degeneracy of the LASSO estimator caused by extreme correlations while still performing variable selection. Moreover, the absolute penalty term automatically selects variables while the quadratic penalty shrinks entries and stabilizes the solution paths (see Zou and Zhang (2009)). 4The glmnet routine is available for many programming languages. We employ the implementation in MATLAB by Qian, Hastie, Friedman, Tibshirani, and Simon (2013). Quantitative Economics 14 (2023) Estimating large-dimensional connectedness tables 1035 Table 1. Simulation results for the regularization of Apaired with the sample covariance matrix estimate of . Values represent the relative Frobenius norm of the FEVD estimates to the true generalized connectedness tables, when compared to the OLS estimates as  DgH reg − DgH/ DgH OLS −DgH. This number states the relative sizes of the estimation errors when compared to the OLS estimate. For example, a value of 20% signifies that the estimator makes only 20% of the error of the OLS estimates. We present such values for various NsandTs with 500 Monte Carlo repetitions. DGP 3 has 25 different random realizations of Aand . Ridge denotes the ridge regression and AENET denotes the adaptive elastic-net. Ridge LASSO AENET Geometric N\T75 175 500 75 175 500 75 175 500 75 175 500 DGP1 50 21.5% 22.5% 22.6% 21.6% 22.7% 22.6% 21.5% 22.6% 22.6% 22.4% 23.1% 22.8% 150 12.3% 13.8% 12.3% 13.8% 12.3% 13.8% 12.6% 13.9% 250 10.8% 10.8% 10.7% 10.9% DGP2 50 23.6% 23% 23.9% 23.3% 22.9% 25.0% 23.6% 23.0% 24.5% 29.8% 29.9% 35.4% 150 13.7% 14.5% 13% 14.1% 13.6% 14.2% 17.2% 19.2% 250 11.7% 11.1% 11.2% 15.2% DGP3 50 27.9% 36% 49.1% 27.8% 35.8% 49.7% 27.6% 35.6% 48.9% 33.2% 45.5% 77.5% 150 14.2% 18.6% 14% 18.4% 13.9% 18.5% 15.5% 22.7% 250 12.7% 12.6% 12.6% 14.1% DGP4 50 30.7% 39.2% 55.7% 30.5% 39.6% 57.8% 30.3% 39.3% 57.5% 43.3% 62.7% 111.5% 150 16.7% 24.7% 16.4% 24.9% 16.4% 24.8% 24.5% 45.3% 250 12.4% 12.3% 12.3% 15.1% We compare the performance gains over N={50, 150, 250}and T={75, 175, 500}.Note that OLS breaks down for N>Tsuch that we are not able to calculate any value for these cases.11 Since the estimation is a two-step procedure, we split the simulation into two parts. First, we regularize Apaired with the sample covariance for .Wecompare the ridge, LASSO, adaptive elastic-net, and geometric long-run regularization. The latter uses the LASSO penalty to perform variable selection. We choose the penalty parameters λNET such that they minimize the respective norm λ∗=argminλ DgH reg (λ)−DgH. Thus, the values show the best possible performance gain. Table 1contains the simulation results for the regularization of Afor 500 Monte Carlo repetitions. First, it is evident that regularization achieves a large overall efficiency gain. The largest performance gain for all DGPs and regularization methods is at N=250. However, for N=50 and T=500 we still observe a remarkable efficiency gain. That is, the best regularized estimators achieve reductions in the norms to the true values to 10.7−23.9% for DGPs 1-2,andto12.3−55.7% for DGPs 3-4. Surprisingly, ridge, LASSO and adaptive elastic-net perform similarly well. There is no clear winner among the estimators since the differences in performance are marginal. If at all, the variable 11In general, regularization methods are not limited to N<T. 1036 Brunner and Hipp Quantitative Economics 14 (2023) Table 2. Simulation results for the regularization of , paired with the best-performing adaptive elastic-net estimator for A. Values report the relative Frobenius norm of the FEVD estimates to the true connectedness tables, when compared to the OLS estimator with the sample covariance matrix as  DgH reg −DgH/ DgH OLS −DgH. This number states the relative sizes of the estimation errors when compared to the OLS and sample covariance estimate. For example, a value of 20% signifies that the estimates make only 20% of the error of the unregularized estimates. We present such values for various NsandTs with 500 Monte Carlo repetitions. DGP 3 has 25 different random realizations of Aand . Sample-Cov Threshold Ledoit–Wolf GLASSO N\T75 175 500 75 175 500 75 175 500 75 175 500 DGP1 50 26.1% 18.7% 15.4% 5.2% 4.7% 0.5% 6.9% 4.9% 0.6% 5.3% 4.7% 13.2% 150 14.9% 10.8% 3.8% 1.0% 4.3% 1% 3.8% 1.0% 250 9.5% 1.1% 1.1% 1.1% DGP2 50 24.3% 23.1% 27.4% 3.4% 8.4% 19.8% 3.6% 8.5% 19.9% 3.4% 8.6% 25.6% 150 14.4% 14.8% 2.8% 7.7% 2.8% 7.7% 2.8% 7.7% 250 11.5% 4.6% 4.6% 4.6% DGP3 50 28.7% 39.2% 59.4% 14.6% 29.7% 59.2% 14.1% 29.9% 62.4% 14.5% 28.8% 59.8% 150 14.4% 20.4% 4.8% 13.4% 4.6% 13.9% 4.7% 13.5% 250 13.2% 6.1% 6.1% 6.1% DGP4 50 32.3% 41.5% 61.9% 23.8% 38.4% 60.0% 25.5% 46.9% 84.6% 23.4% 38.4% 62.3% 150 18.1% 28.8% 12.5% 26.4% 14.5% 34.9% 12.1% 26.3% 250 12.8% 5.9% 6.2% 5.8% selection capabilities of LASSO and adaptive elastic-net provide a slight performance advantage over the pure shrinkage of ridge regression for simulations where Tis close to N. Finally, the geometric regularization of long-run effects underperforms other regularization methods for all DGPs, and does worse than the unregularized estimation for N=50 and T=500. As a second step, we compare regularization methods for . That is, we calculate the residuals using the adaptive elastic-net estimate and construct the FEVD with the (regularized) estimate of . We choose the adaptive elastic-net penalty parameter to minimize the Frobenius norm of the difference between the estimate matrix ˆ Aand the true parameter matrix A:λNET∗=argminλˆ A(λ)−A.Additionally,wesettherespective penalty parameter for the covariance regularization such that δ∗=argminδ DgH reg (δ)− DgH. Table 2shows the simulation results for the different regularizations. The first estimator is the sample covariance matrix and sets the benchmark. Again, we measure the performance of the regularization methods with the norm of the estimated FEVD to the true values relative to the unregularized estimates. For DGP1 with the identity as the data generating covariance matrix, the regularization methods provide a near perfect approximation of the FEVD as they favor diagonal entries. When DGP 2 adds autocorre- Quantitative Economics 14 (2023) Estimating large-dimensional connectedness tables 1037 lation, all regularization methods perform similarly, with the exception of the GLASSO estimator for N=50 and T=500. In contrast, DGPs 3 and 4have nondiagonal covariance matrices. In DGP 3, the estimators perform similarly, with each of them outperforming others for two combinations. We believe this result is due to chance, and thus, we do not declare a winner. Finally, the GLASSO estimator outperforms the other regularization techniques in DGP 4 by a small margin. The only exception is for N=50 and T=500, where only the adaptive-threshold estimator surpasses the sample covariance. 3.3 Bias, variance, and edge detection Next, we investigate the mean of the entries and the norms for regularized and unregularized FEVDs. For this purpose, we simulate time series with DGP 4,N=100 and T∈[100, 500]. The analysis compares OLS and the sample covariance to the adaptive elastic-net in combination with the sample covariance matrix, GLASSO, and the adaptive threshold estimator, respectively. We select the hyperparameters based on the best performance (minimization of the Frobenius norm to the true FEVD). Note that this procedure requires knowledge about the true parameters, such that it is applicable only in a simulation exercise. The left-hand panel in Figure 2shows the average mean distance, N−2N i=1N j=1mean( DgH ij −DgH ij ), which reflects the biases of the FEVD estimates. The right-hand panel of Figure 2then shows the average variance N−2N i=1N j=1var( DgH ij ). Note that this analysis relates to the “bias-variance” trade-off for estimators as regularization methods generally sacrifice unbiasedness to achieve a lower variance and, therefore, a reduction in prediction errors overall. The left-hand panel depicts the magnitude of the bias for small T(approaching N=100 from the right). We see that on average all estimators are positively biased, with the nonregularized estimator (solid line) profoundly overestimating entries in the FEVDs. That is, this estimator faces a strong positive bias the closer Tis to N.Perhaps surprisingly, the adaptive elastic-net (dotted curve) already diminishes the bias for small Tby a margin (almost by a factor of 100 for T=100). While we expected that regularization methods trade off bias for variance in FEVD tables, adaptive elastic-net also improves with respect to the bias for T<200. Similarly, when using regularizations on the covariance matrix, we see a consistent improvement with respect to the bias. The threshold estimate improves strongly and the GLASSO estimate improves slightly over its sample covariance counterpart, even for larger T. The right-hand panel plots each estimator’s variance, indicating the precision of the estimation. The nonregularized version not only faces heavy inaccuracies following from its bias but also shows an extremely large variance for T<150. Its variance lessens with increasing Tbut still underperforms compared to the regularized versions. For all regularization methods, we see gains in the variance, likely stemming from the “bias-variance” trade-off. The combination of coefficient and covariance regularization therefore dominates for all Ts. Pairing this finding with the findings of the left-hand panel, there appears to be no trade-off for T<200 but an overall improvement in bias and variance. Summing up, it is advisable to combine the regularizations for the coefficient and the covariance matrices as this combination provides a lower variance, and for most sample sizes, also comes with a lower bias. 1038 Brunner and Hipp Quantitative Economics 14 (2023) Figure 2. Simulation results for 500 Monte Carlo repetitions of DGP 4 and N=100. The left-hand panel shows the average mean difference of the estimates to the true values N−2N i=1N j=1mean( DgH ij −DgH ij ). The right-hand panel shows the respective average variance N−2N i=1N j=1var( DgH ij ). The sample size Tis on the x-axis. Next, in the context of networks, we explicitly care about the diagnostic ability of the estimator. That is, we are interested in how well the classification into zero and nonzero entries performs in the network matrix. For that purpose, we summarize the performance in terms of the correct detection of zero and nonzero entries in the three panels of Figure 3. A value is considered true positive (TP) in the case of a hit and false positive (FP) in the case of a false alarm (Type I error). Likewise, a true negative (TN) is given if the FEVD is correctly estimated to be sparse at a given edge, and there is a falsenegative (FN) when an existing edge is not found (Type II error). First, the probability of correct classification is summarized by the accuracy metric. Accuracy is defined as the fraction of correct predictions: accuracy =(TP +TN)/N2. We compare this metric of the estimators for increasing Tin Figure 3. The left panel shows that using the adaptive elastic-net estimator instead of OLS unlocks a big improvement in small samples. A small but additional gain can be achieved through the usage of GLASSO instead of the sample covariance matrix, while the adaptive threshold estimator does not show any improvements. The receiver operating characteristic (ROC) plots the false–positive rate FPR = FP/(FP +TN)against the true-positive rate TPR =TP/(TP +FN)while varying the discrimination threshold of setting values to zero. More precisely, the threshold varies from the lowest to the highest entry in the FEVD, and thus, sets increasingly more values to zero. For each threshold, the ROC plots the respective FPR and TPR in a diagram ranging from 0 to 1. A perfect estimator—that is, one that correctly classifies all edges no matter the threshold—would result in a line starting at (x,y)=(0, 1)and ending at Quantitative Economics 14 (2023) Estimating large-dimensional connectedness tables 1039 Figure 3. Simulation results for 500 Monte Carlo repetitions of DGP 4 and N=100. The first panel shows the accuracy for increasing sample size Ton the x-axis. The center panel shows the receiver operating characteristic (ROC) for T=200, which has FPR on the x-axis and TPR on the y-axis. The diagonal thin black line is the equivalent of a random estimate. The right-hand panel shows the area under the ROC (AUROC) for increasing sample size T. (x,y)=(1, 1). Conversely, a completely random guess would be shown as a 45◦diagonal line. We plot the ROC curve for T=150 in the center Panel of Figure 3. Again, the regularized estimators clearly improve the performance when it comes to classification, with the combination of adaptive elastic-net and GLASSO providing the best diagnostic capabilities. Finally, the right panel plots the area under the ROC curve (AUROC). The values range from 0.5 to 1, where 0.5 is a completely random guess and 1 is the perfect classification. This metric summarizes an estimate’s performance in detecting edges in a single number and lets us compare estimators for different Ts. The regularized estimators display high confidence in their classifications and achieve better performance, in particular for Tclose to N. While the gains over the unregularized version mainly originate from the regression stage of the estimation, the results improve even further after regularizing the covariance matrix with GLASSO. To sum up, our simulations demonstrate that the regularization of both estimation steps leads to a substantial improvement in the estimation of FEVDs. This is evident when comparing estimation errors as well as when using entrywise classification metrics. For large Nand small T, our results indicate that it is critical to regularize not only the coefficient but also the covariance matrix. Perhaps surprisingly, there are no tradeoffs when Napproaches Twith performance gains in all metrics. However, the use of regularization also shows improvements when estimating large-dimensional networks from large-T data sets and, therefore, we advise researchers to apply regularization techniques when faced with large-dimensional estimation problems. Comparing the covariance estimators, the adaptive threshold estimator appears to best reduce the estimation bias, while GLASSO mainly excels in its diagnostic ability. Thus, we advise researchers to pick the method based on the desired properties for the specific research question. Alternatively, model selection techniques such as out-of-sample CV or in-sample information criteria may help to validate the methods’ performances on specific data. 1040 Brunner and Hipp Quantitative Economics 14 (2023) 4. Empirical application:Production volatility spillovers and the great moderation The period known as the Great Moderation, starting in the mid-1980s, is characterized as a period with reduced fluctuations in many macroeconomic time series, such as real growth rates, industrial production (IP), and unemployment. The question whether the Great Moderation happened due to good luck, better monetary policy or any other structural change is vital for policymakers as they need to understand the impact of their actions on macroeconomic volatility.12 The large volatility of IP is particularly puzzling since the aggregated IP index sums over many weighted sector-level shocks. Thus, Foerster, Sarte, and Watson (2011) investigate this shift by decomposing IP into sectoral and common shocks and conclude that a decline in the volatility of common shocks induced most of the break in IP’s volatility. Conversely, Carvalho and Gabaix (2013)provide evidence that the shift in macroeconomic volatility could have originated from idiosyncratic microeconomic shocks on the sectoral level, offering a contrasting explanation that leaves the question of the origins of the Great Moderation all but conclusively answered. The estimation of sectoral connectedness tables allows us to gain a new perspective on the source of the decline in the volatility of aggregate IP. While the findings in Foerster, Sarte, and Watson (2011) are in accordance with much research that attributes the Great Moderation to declines in the shock volatility of common exogenous factors such as monetary policy (e.g., Leduc and Sill (2007), Justiniano and Primiceri (2008)), factor productivity (e.g., Arias, Hansen, and Ohanian (2007)), or oil supply (e.g., Nakov and Pescatori (2010)), an alternative explanation has emerged from the granular hypothesis of Gabaix (2011). The latter paper proposes that a small number of constituents can explain aggregate shocks if the index weight distribution is fat-tailed, an idea the findings of Foerster, Sarte, and Watson (2011) refute. In our view, one needs to consider not only sectoral weights but also directional dependencies to analyze the influence of a particular industry on the aggregate. That is, we entertain the possibility that correlations between sectors originate from a strong spillover network of sector-level innovations rather than common shocks as in Foerster, Sarte, and Watson (2011). Acemoglu et al. (2012) establish this idea based on input–output linkages and show that the diversification argument does not apply in the presence of strong network structures. More recently, Foerster, Hornstein, Sarte, and Watson (2022) give additional leeway to this hypothesis by attributing much of trend GDP growth rates to sector-specific factors. In a nutshell, we investigate if structural change of such a network has been an important driver of the Great Moderation, which is in line with supply-chain and inventory-based explanations presented in Kahn, McConnell, and Perez-Quiros (2002), Summers et al. (2005)andDavis and Kahn (2008). For insights into sectoral interconnectedness, we obtain estimates of the directional dependencies. In contrast to a fast-expanding literature on production networks that 12For early discussions of the potential drivers of the Great Moderation, see Stock and Watson (2003) and Bernanke (2004). Quantitative Economics 14 (2023) Estimating large-dimensional connectedness tables 1041 exploits input–output tables as network proxies (for a review, see Carvalho and TahbazSalehi (2019)), our methodology directly sheds light on the network implied by sectoral IP correlations. Our approach is in line with the finding of Carvalho (2014) showing that pairwise correlations between two sectors are also bigger for stronger input–output links. An advantage of inferring the network directly from the output data is the ability to capture not only the supply–demand relationships of economic sectors but also alternative channels of transmission. For example, the propagation of production volatility between two sectors goes beyond existing input–output relations if they compete for the same resources, if their outputs are substitutes or complements, or if poor inventory management amplifies demand shocks and their propagation up the supply chain. Thus, by estimating FEVD tables, our application is agnostic about the channel of propagation and adds to the understanding of sectoral spillovers and their contribution to aggregate fluctuations. We examine preand post-1984 periods in detail to give a comparison of how spillovers have changed. To address the possibility of exogenous macroeconomic shocks, we analyze the spillover networks while factoring out macroeconomic variation. Our findings provide novel empirical evidence and offer a unifying view on the otherwise opposing narratives of Foerster, Sarte, and Watson (2011), Gabaix (2011)and Acemoglu et al. (2012). We conjecture that sizeable spillovers from a handful of sectors initially generated strong sectoral comovements, which appear in other analysis as large aggregate shocks. With the Great Moderation, the structure of the spillover network changed such that sectoral shocks are less likely to amplify into aggregate volatility. Hence, our findings raise an alternative explanation in which structural change largely contributed to the Great Moderation. Finally, an analysis of the increase in production volatility around the Great Recession does not reveal a reversal of the initial structural change, but rather attributes a growing contribution share to macroeconomic factors. 4.1 Data Similar to Foerster, Sarte, and Watson (2011), we use sectoral data on IP throughout the period 1972–2020. We mainly analyze the three-digit industry classification of the North American Industry Classification System (NAICS) with N=88 sectors, whereas the data spans up to N=138 sectors corresponding to the five-digit industry classification. The sectoral indices are available on a monthly basis. Since the preand post-Great Moderation periods have different sample sizes, they face distinct degrees of estimation uncertainty. Hence, we split the whole sample into four equally sized subsamples of T=144 months. The subsamples span from 03/1972 to 02/1984, from 03/1984 to 02/1996, from 03/1996 to 02/2008, and from 03/2008 to 02/2020. In our analysis, the boundary between the first and second samples marks the onset of the Great Moderation. For simplicity, we label these samples as 1972–1983, 1984–1995, 1996–2007, and 2008–2019, respectively. Let IPi,tdenote the value of IP of sector iat date t. We take monthly growth rates and annualize the respective percentage points, gi,t=1200×ln(IPi,t/IPi,t−1). The aggregate level of IP growth is the weighted average over the sectors, gt=N i=1wi,tgi,t,with 1042 Brunner and Hipp Quantitative Economics 14 (2023) weights wi,t. Our first subsample, from 1972 to 1983, coincides with the pre-Great Moderation period and the other three subsamples are post-Great Moderation. The subsample IP volatilities of 11.67%, 5.85%, 6.36%, and 8.79% illustrate that the average monthly volatility of aggregate IP diminished with the Great Moderation and stayed fairly constant thereafter with the exception of the Great Recession of 2007–2009.13 To account for the possibility of exogenous factors affecting all sectors simultaneously and thereby inducing the illusion of a strong spillover network, we condition on three contemporaneous and lagged factors that the literature has brought forward as possible explanations for the Great Moderation. First, we include the monthly monetary policy shock series gMP tof Wieland and Yang (2020) based on Romer and Romer (2004) for the first three subsamples and a similar version of Bauer and Swanson (2022)forthe most recent subsample. Second, we consider the possibility of exogenous productivity shocks through the inclusion of the updated interpolated utilization-adjusted TFP series gTFP tof Fernald (2014). And third, we account for commodity supply shocks by adding percentage changes of the S&P GSCI Commodity Index gCOM tto our data. 4.2 Estimation We interpret the spillover constituent in the data as a VAR(1) model and infer connectedness from monthly cross-autocorrelations of gi,t. The higher frequency allows setting the forecast horizon to 3 months, which corresponds to the (undirected) covariance matrix of the quarterly data. This connection between monthly and quarterly frequencies provides insights into the contagion within a quarter. In particular, Foerster, Sarte, and Watson’s (2011) average pairwise correlations and aggregate shocks may be better understood if we break up quarterly volatilities into three serially correlated monthly volatilities. Hence, the central regression specification is as follows: yt=μ+Ayt−1+ 12  l=0 Blxt−l+ut,∀t=1, ,T, yt=[g1,t,,gN,t], xt=gMP t,gTFP t,gCOM t, ut∼N(0, ). While Foerster, Sarte, and Watson (2011) test hypotheses about correlations through common factors, our framework aims to see correlations through the lens of intersectoral spillovers in Aand . Nevertheless, we include the factors xtin our analysis to account for the original authors’ results concerning common variation. As control variables, they serve as a means to rule out that the estimated connectedness tables falsely ground on common innovations that the literature has investigated as explanatory for the Great Moderation.14 13See Figure D.1 in Appendix D for a visual display of the growth rates of IP on an aggregate level. 14Note that confirmatory factor analysis with observable factors aids us in answering our main research question, whereas latent factor techniques like principal components would not allow to distinguish between factor narratives and the granular network story. Quantitative Economics 14 (2023) Estimating large-dimensional connectedness tables 1043 Table 3. Summary of the model selection and with cross-validated hyperparameters. MSE is the out-of-sample CV loss, denoted as a percentage of the respective unregularized candidate model (OLS and sample covariance). AIC is the Akaike information criterion for the dynamic model fit with scale 105. Panel A reports values for candidate models of the coefficient matrix and using the sample covariance matrix as the innovation covariance matrix. OLS overfits, such that the log-determinant of the residual covariance matrix is 0, yielding infinity in the AIC column. Panel B contains the same comparison for the residual covariance matrix estimators. MSE AIC 1972–1983 1984–1995 1996–2007 2008–2019 1972–2019 Panel A: Model selection for regression OLS 100% 100% 100% 100% Inf Elastic-Net 7.87% 7.40% 7.51% 6.74% 4.863 Adaptive Elastic-Net (OLS init.) 7.71% 7.34% 7.36% 6.58% 4.956 Adaptive Elastic-Net (ENet init.) 6.13% 6.12% 6.22% 5.33% 4.649 Panel B: Model selection for covariance matrix Sample Covariance 100% 100% 100% 100% 4.649 GLASSO 99.70% 99.49% 99.56% 99.55% 4.592 Ledoit–Wolf 99.81% 99.85% 99.85% 99.80% 4.812 Adaptive Threshold 99.73% 99.51% 99.57% 99.58% 4.613 We regularize A,{Bl}12 l=0,andwith the techniques mentioned in the previous sections. Since the simulations did not point toward a consistent winner throughout all settings, we validate all regularization methods on the data. Namely, we run a 12-fold CV as in Section 2.3 to select the hyperparameters αand λNET in the regularized regressions, and δand ηin the covariance estimation. Here, the best estimate of Afrom the regression stage serves to obtain the residuals for the estimation of . Note that although the regression residuals are fed into the estimation of the covariance matrix, the two estimation steps are independent, such that sparsity can emerge at different positions in the coefficient and covariance matrix. Then, similar to the model comparisons of Sims and Zha (2006), we choose among the optimized candidate models based on Akaike’s information criterion (AIC) over the full sample.15 Table 3shows the out-of-sample MSEs and in-sample AICs for all optimized candidate estimators. First, Panel A reports the results for the regression stage of the estimation. Unsurprisingly, the adaptive elastic-net with initial elastic-net weights dominates all other estimation schemes. In our large-N-small-T setting, the OLS estimator even overfits the data to an extend that the log-likelihood is not defined with the residual covariance matrix being noninvertible. Second, Panel B reports the same statistics for the estimation of the residual covariance matrix after the first-step model selection. Here, regularized estimation methods again consistently outperform the sample covariance matrix in terms of MSE, while GLASSO provides the best model fit in terms of AIC 15As pointed out in Hurvich and Tsai (1989), AIC needs a second-order correction for small samples, that is, the large-N-small-T setting we operate in. We explored robustness with respect to outcomes of a corrected AIC as in Bedrick and Tsai (1994). More precisely, the correction term in the corrected AIC is bigger for dense models such that the model selection criterion more clearly favors sparse models. 1044 Brunner and Hipp Quantitative Economics 14 (2023) Table 4. Comparison between static and dynamic model choices evaluated for the Akaike information criterion (AIC). Dynamic refers to the sample being split into four equally-sized subsamples 1972–1983, 1984–1995, 1996–2007, and 2008–2019, each of size T=144. Static refers to the full sample 1972–2019 with size T=576. All modeling choices are 12-fold cross-validated and the best predictor model is selected according to the methods described in the main text. Values have scale 105. A AIC Static Dynamic static 4.6900 4.6417 dynamic 4.6444 4.5920 throughout. For all subsequent analysis, we focus only on the selected estimator, that is, the one that achieves the lowest AIC.16 Similar to Sims and Zha (2006), we additionally investigate whether the break in the sample is due to the coefficient matrix Aor to the contemporaneous matrix in Table 4. That is, we compare specifications that use static matrices estimated for the whole sample from 1972 to 2019 instead of dynamically allowing them to change per subsample. Naturally, dynamic specifications achieve better goodness-of-fit, but AIC penalizes them more strongly because of the higher use of degrees of freedom. Nevertheless, dynamic specifications for both Aand indeed improve the model quality. We see this finding as support for a change in the coefficient as well as the covariance matrix with the onset of the Great Moderation. As structural identification of two-way causality in utproves difficult in the presence of high dimensionality, we rely on “generalized identification” to offer neutral insight into the connectedness of variables. With the inclusion of all production sectors in yt and potential sources of common variation in xt, the covariance of the innovation vector utbecomes more sparse. Thus, the generalized approach gets more precise at approximating exogenous variability such that the generalized-shock covariance matrix  is indeed close to the identity. To support the hypothesis that we are not missing out on significant sources of exogenous variations, we will additionally conduct a formal test for being statistically distinguishable from the identity matrix. Finally, taking the estimates for Aand as inputs, we calculate the FEVDs with the forecast horizon H=3. We row-normalize DgH in (8) to show the percentage contribution to the variance. Additionally, we present key figures related to the network literature. In particular, we use the same measures as Diebold and Yılmaz (2014): in-, out-, and average connectedness. These measures are defined as the row sum, column sum, and the average row sum without the diagonal entries, respectively,17 Ci←·DgH= j=i dgH ij (in-connectedness to i), 16Stone (1977) establishes the asymptotic equivalence of CV and AIC in terms of model choice. 17To facilitate intuition, we slightly deviate from the original authors’ terminology and use the terms inand out-connectedness for what they call fromand to-connectedness. Quantitative Economics 14 (2023) Estimating large-dimensional connectedness tables 1051 Table 6. Decomposition of variance contributions by channels for H=12. 1denotes the one vector of respective size. Aggregate responses provide an approximation of the variance of the aggregate IP index via innovation response variances. Factor innovations and granular innovations have direct/intrinsic effects through their own variance and indirect/extrinsic effects through intersectoral linkages on index constituents. The contemporaneous spillover contribution comprises all variations within the same period, and the lagged component consists of all responses happening via the autoregressive coefficients. 1972–1983 1984–1995 1996–2007 2008–2019 Data variance vIP =var(gt)148.1 34.7 36.8 87.4 Aggregate response vagg =vf+vgran 149.1 23.5 25.9 85.0 Factor Contributions vf=¯ w·2FRV111.7 1.1 −0.1 17.8 of which direct vfDir =¯ w·2BfB12.7 1.0 −0.1 10.9 of which indirect vfInd =vf−vfDir 9.0 0.1 0.0 7.0 Granular Contributions vgran =¯ w·2IRV1137.4 22.4 26.0 67.1 of which intrinsic vIC =¯ w·2diag(IRV)116.7 9.4 12.1 17.1 of which extrinsic vEC =vgran −vIC 120.7 13.0 13.8 50.0 of which contemp. vcont. =¯ w·2154.5 19.7 21.0 33.7 of which lagged vlag =vgran −vcont. 82.9 2.6 5.0 33.4 Extrinsic Share vEC/vagg 0.81 0.55 0.53 0.59 Lagged Share vlag/vagg 0.56 0.11 0.19 0.39 the weakening of the spillover network as the main driver of the decline in granular contributions. Similarly, strong changes happen in the lagged transmission channel as opposed to contemporaneous reactions, refuting the narrative of aggregate shocks as the main explanation for the Great Moderation. On the other hand, returning to the narratives of Kahn, McConnell, and Perez-Quiros (2002), Summers et al. (2005), and Davis and Kahn (2008), the decline in extrinsic and lagged contributions is consistent with the idea that advances in transportation, supply and distribution networks, inventory management, and sales forecasting related to new information technologies have streamlined intersectoral dependencies, and thus, have reduced the volatility of individual sectors. Overall, there is a considerable share of the decline in aggregate volatility that our model attributes to the weakening of the spillover network, remarkably after adjusting for common factor-based explanations. Hence, our findings suggest that structural change of the network contributed to the Great Moderation. Lastly, we care about single-sector contributions via the IRV in (17). To do so, Figure 7tracks the sectors’ contributions to aggregate variations over the four subsamples. We place the sectors with the highest contributions at the bottom and vice versa. In brackets, we show the percentage of the sectors’ contributions via the extrinsic channel as in Table 6: vgran ·→IP =¯ w·2diagIRV(H)   intrinsic component +¯ w·2IRV(H)−diagIRV(H)   extrinsic component . (18) For example, a value of 73% signifies that 73% of the estimated contributions come via estimated spillovers; or in other words, the network noticeably amplifies the sector’s rel- 1052 Brunner and Hipp Quantitative Economics 14 (2023) Figure 7. Bump chart of sectoral contributions to the variation of the aggregate IP index. Contributions are calculated as in (17). Data includes IP indices for 88 three-digit-level sectors. Adjacent sectors are in similar colors. The top contributors per subsample are labeled. Values in brackets correspond to the extrinsic component of the contribution. For example, the value of the Motor Vehicles sector in the first subsample states that 73% of this sector’s contribution is due to spillovers (extrinsic) to other sectors and 27% is due to its intrinsic component. evance in the aggregate index. Complementarily, Appendix D presents figures for the intrinsic (Figure D.10) and extrinsic (Figure D.9) contributions separately. First, note that with the Great Moderation, total contributions decrease sharply and relative contributions of individual sectors vary in importance. While the Motor Vehicles sector is the strongest contributor due to its size and outgoing linkages, the Coal Mining sector’s effect on the aggregate is relatively small due to lower spillovers of its large intrinsic variation documented in Figure D.10.20 In contrast to Foerster, Sarte, and Watson’s (2011) narrative that metal-related industries are well explained by common factors, our analysis shows that downstream sectors related to the metal industry exhibit strong contributions in the form of spillovers. This finding is consistent with the idea of shocks propagating upstream, for example, the amplification of demand shocks through excessive inventory holdings. Considering that heavy industries are still among the most influential sectors after 1984 (compare Figure 7) with nearly unchanged intrinsic contributions, we reckon that macroeconomic shocks and sectoral innovations alone do not suffice to explain the decline in aggregate volatility. Instead, structural change in the form of improvements to inventory management, supply chains, and information technology offers a plausible narrative for the less volatile industrial output. Despite the 20High volatility before the Great Moderation in the Coal Mining sector mainly stems from the strikes in 1974, 1977–1978, and 1981. Quantitative Economics 14 (2023) Estimating large-dimensional connectedness tables 1053 partial increase in spillover contributions in the 2008 to 2019 subsample (compare Figure D.9 in Appendix D), there is no strong indication for a reversal of the Great Moderation’s structural change. High intrinsic and common-factor contributions surrounding the Great Recession tell a more convincing story than a return of the strong spillovers. Throughout the application, two principal findings stand out and repeat in various analyses. First, we document a sharp decline in network connectedness after 1984 in Table 5, which also shows in many visualizations such as Figures 4,6,and7. As illustrated in Figure 5, our evidence suggests that outgoing links have weakened considerably. A decomposition of aggregate variance in Table 6shows that the importance of spillover effects has particularly faded with the Great Moderation. Second, we find that sectors demanding inputs from other sectors are among those that reduced their outgoing links the most, which is visible in Figures 6and 7. We see this result as indication that supply chain improvements have played an important role with the onset of the Great Moderation. More precisely, improvements to inventory management offer a potential explanation as they deter demand shocks from propagating upstream (see Figure D.8). Finally, we find no substantial evidence that a revival of sectoral spillovers drove the temporary increase in production volatility around the Great Recession. Our results provide novel insights into the changes observable during the Great Moderation. Foerster, Sarte, and Watson (2011) emphasize that the decrease in aggregate variance is not due to some sectors but is rooted instead in the change of aggregate shocks; that is, shocks to multiple sectors at once. In contrast, our analysis reveals that changes in the propagation of sector-specific shocks may have contributed largely to the decline in aggregate variance. Put differently, a sector contributes to the aggregate variance not only via its index weight, but also via the spillover effects it has on other constituents. Such a reduction in spillovers supports the granular hypothesis of Gabaix (2011) and its network refinement in Acemoglu et al. (2012). Further, we conjecture that improved inventory management structurally changed the spillover network such that its amplifying effect subsided. Thus, our findings offer a unifying perspective to the opposing explanations of Foerster, Sarte, and Watson (2011)andGabaix (2011) by connecting their narratives to the inventory-based explanations of Kahn, McConnell, and Perez-Quiros (2002), Summers et al. (2005), and Davis and Kahn (2008). 5. Conclusions In this paper, we investigate the estimation of high-dimensional connectedness tables via vector autoregressive models. In a simulation study, we compare different regularization methods for the coefficient and the covariance matrix. We evaluate their performance in the estimation of FEVDs and find that the regularization of both matrices improves bias and variance. Since there is no one-fits-all estimator in our simulations, we suggest validating the estimators through cross-validation in practice. In an application to US industrial production, we are able to uncover changes in the structure of the intersectoral spillover network around the Great Moderation. Our results support the notion that receding network effects have largely aided the decline in 1054 Brunner and Hipp Quantitative Economics 14 (2023) aggregate variance by decreasing the influence of granular innovations. Specifically, the outgoing spillovers of downstream sectors before the Great Moderation was unmatched thereafter, suggesting that supply chains and inventory management may be important drivers of the structural change behind the Great Moderation. Finally, we would like to attempt to extrapolate our findings to the current macroeconomic environment. With the COVID-19 pandemic, disruptions in the production process due to supply chain bottlenecks and rapid changes in the consumer product demand have rendered lean inventory management less effective, such that many firms adapted safeguard inventories. 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