Cross-Sectional Dynamics Under Network Structure: Theory and Macroeconomic Applications

-0.5 in 1925Cross-Sectional Dynamics Under Network Structure: Theory and Macroeconomic Applications

{ {\bf JEL codes:} C32, E32, E37.}

{ {\bf Key words:} Vector Autoregression, Spatial Autoregression, Dynamic Network Effects, Input-Output Economy, Business Cycles, High-Dimensional Time Series, Local Factors}.

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Introduction

In economics, we often encounter a cross-section of units linked by a network of bilateral ties, such as sectors connected through supply chains or individuals acquainted to each other. A large theoretical and empirical literature documents that networks amplify idiosyncratic shocks and generate comovement in cross-sectional variables at a given point in time.\footnote{ See work surveyed in Carvalho-TahbazSalehi2019,BramoulleGaleottiRogers2016 and the following literature review. }

How network-induced comovements unfold over time, however, is less well understood. The literature predominantly assumes contemporaneous network interactions (e.g. BramoulleDjebbariFortin2009, AcemogluEtAl2012, ElliottGolubJackson2014), embodied by the Spatial Autoregressive (SAR) model. This implies that an innovation contemporaneously transmits along network connections of all orders. The resulting static framework is useful for “{networked}" steady state comparisons. Among the few studies that consider lagged network interactions, most posit that network effects materialize exactly one link per period (e.g. LongPlosser1983, GolubJackson2010, Elhorst2012). While useful for analyzing the qualitative implications of the laggedness of network interactions, this assumption is of limited use for empirical work concerned with “{networked}" transition dynamics. ZhuPanLiLiuWang2017 generalize the lag length, but the generality of their empirical framework and its relation to structurally motivated models of contemporaneous or single-lagged network interactions remain unclear.

I construct an econometric framework that derives the dynamics of cross-sectional variables from the lagged innovation transmission along bilateral links between cross-sectional units: the Network-VAR (NVAR). Like existing studies, I assume uni-directional transmission\footnote{ i.e. either downstream or upstream, the distinction being relevant only for directed networks.} and time-invariant links. Unlike existing studies, the model can accommodate general patterns on how innovations travel along links over time and, consequently, how network connections of higher order accumulate as time progresses. To obtain this generality, I distinguish between the frequency of network interactions and the frequency of observation. As the former diverges to infinity, the impulse-response to contemporaneous high-frequency disturbances and the networked covariance among observations approach their counterparts from the SAR model. The same impulse-response is also obtained when considering long-term effects of permanent innovations. By taking a stance on the timing of network effects, the NVAR goes beyond such steady state comparisons and characterizes transition dynamics.

In the NVAR, the interdependence of observations $y_{it}$ and $y_{j,t-h}$ arises as the interplay of the temporal distance between periods $t$ and $t-h$ and the cross-sectional distance between units $i$ and $j$ encoded by the network. Under a sparse network, this dependence is modeled parsimoniously; the dynamic comovement among all series is rationalized by the dynamic innovation transmission along a few bilateral links among units. This is akin to the assumption that longer-term dynamics are driven by a set of shorter-term dynamics, upheld by the general class of VARMA($p,q$) models. The dynamics under the NVAR can be represented by a Dynamic Factor Model (DFM) whereby the number of factors equals the rank of the network adjacency matrix. At the cost of restricting their dynamics, the NVAR naturally incorporates sparse factors as locally important nodes in the network.

The NVAR is useful in two distinct kinds of work with cross-sectional time series. By conducting inference on the timing of innovation transmission along bilateral links, it can be used to estimate dynamic network effects. Thereby, the network can be taken as given or it can be inferred from dynamic cross-correlations in the data, possibly aided by shrinking towards observed links. When both the network and effect-timing are estimated, the NVAR is also applicable as a dimensionality-reduction technique for modeling high-dimensional processes. Conditional on a network, inference on the timing of network effects boils down to a linear regression with covariates that summarize lagged observations using bilateral links. Joint inference is implemented by iterating on analytically available conditional estimators, with Bayesian as well as frequentist interpretations. I illustrate each of these two model uses with a respective application.

In the first application, I show that the NVAR approximates the process of sectoral output in a Real Business Cycle (RBC) input-output economy with lagged input-output conversion (IOC). In turn, I evaluate the potential of lagged IOC to generate endogenous business cycles, as suggested theoretically by LongPlosser1983. I generalize their one period-lagged IOC by assuming that inputs from the past $p$ periods are imperfectly substitutable in the production at any time $t$ -- a stand-in for a range of frictions that prevent just-in-time input-sourcing -- and by allowing the model frequency to differ from the observational frequency. I characterize impulse-responses under a difference-stationary TFP process with persistent aggregate and sectoral shocks. The analysis illustrates the differing implications of lagged IOC and persistence in exogenous shocks for the dynamics of sectoral and aggregate output; the former formalizes the idea that idiosyncratic changes in, say, energy production take time to feed through the networked economy, while the latter merely prolongs the effects of each round of transmission. Using input-output tables and monthly output growth across 23 manufacturing sectors in the US economy, I estimate their respective contributions to persistence in aggregate output growth.\todo{to the autocorrelation function of ...?}

The results suggest that lagged IOC explain 61% of the autocorrelation in aggregate output growth. With persistence in aggregate TFP, this number grows to 93%, leaving little room for persistence in idiosyncratic sectoral TFP. My analysis reveals that shocks to more central sectors not only affect aggregate output more strongly in the long run -- as established by AcemogluEtAl2012\todo{CITE MORE?} --, but they also tend to materialize more sluggishly. This relationship, however, is far from clear-cut; for example, owing to its position at the top of supply chains, a TFP shock in the primary metals sector materializes more slowly than a shock to the food and beveredge sector, despite similar long-term effects.\todo{relate results to Foerster et al. and Krishna Kamepalli et al.?}

In the second application, I investigate the merits of the NVAR as a dimensionality-reduction technique for forecasting high-dimensional (cross-sectional) processes. I consider monthly industrial production growth, monthly CPI inflation and quarterly GDP growth across OECD countries. Under a sparse network, the NVAR models the dynamics parsimoniously; the dynamic comovement among all series is rationalized by the dynamic innovation transmission along a small set of bilateral links among countries. In line with that, I estimate the network by shrinking links to zero using a Lasso-penalty. I also consider alternative specifications that apply a Ridge-penalty and shrink to observable links.\todo{LEFT TO DO}

In my setting, the NVAR reduces out-of-sample mean squared errors by 13%-68% relative to the ex-post best-performing DFM. This corroborates my equivalence result: the NVAR better forecasts these series, whose cross-country dynamics seem to be driven by many micro links rather than a few influential countries. This corresponds to a setting with numerous sparse factors and differing sets of non-zero loadings across units or, equivalently, a sparse, yet high-rank network adjacency matrix. \todo{add sentence on sparseness and interpretability of network I estimate}

\paragraph*{Contribution}

There is a large econometric literature on spatial and network-interactions Manski1993,KelejianPrucha2010JoE,Anselin2003,Lee2007,BramoulleDjebbariFortin2009,ShaliziThomas2011,KuersteinerPrucha2020,DePaula-Rasul-Souza2024. It is concerned with identifying spatial/peer-effects and accounting for heterogeneity in cross-sectional and panel data-regressions.\footnote{ A separate strand devises methods for dyadic data, i.e. the case when edges rather than nodes in the network are observed Graham2020. } It predominantly assumes contemporaneous network interactions, as exemplified by the canonical SAR model. Among the few studies that consider lagged interactions Elhorst2012,KnightNunesNason2016, ZhuPanLiLiuWang2017 cast them in an an explicit time series model, generalizing the lag length and discussing stationarity and large $T$-inference. I characterize Granger causality in their environment as the cross-sectional innovation transmission along network connections of different order, formalizing the idea that network effects materialize dynamically over time. In turn, I generalize this mapping between network connections and dynamics, which allows me to derive the SAR model as the limit of an underlying process driven by high-frequency lagged network interactions.\footnote{ BarigozziCavaliereMoramarco2025,Maung2022,ArmillottaFokianos2023 extend ZhuPanLiLiuWang2017 along different dimensions than I do. } Like PinkseSladeBrett2002,LamSouza2020,QuLeeYang2021,DePaula-Rasul-Souza2024 do in the static environment, I discuss inference on the network itself -- not only the effect-timing --, which institutes the NVAR as a dimensionality-reduction technique -- not only a tool for estimating dynamic network effects -- and motivates my equivalence result to factor models.

Many studies incorporate networks in time series analyses. Following DieboldYilmaz2009,DieboldYilmaz2014, some researchers interpret quantities of a given time series model -- typically a VAR -- using network analysis BillioGetmanskyLoPelizzon2012,AnufrievPanchenko2015,ChenLiLiLinton2025. Other studies restrict time series models using networks CarvalhoWest2007,ChudikPesaran2011,DavisZangZheng2016,AhelegbeyBillioCasarin2016JAE,ZhuPanLiLiuWang2017,BarigozziBrownlees2018,BillioCasarinRossini2019,CaporinErdemliogluNasini2021,MartinPassinoCucuringuLuati2025.\footnote{ HollyPetrella2012,DahlhausSchaumburgSekhposyan2021,DeGraeve-Schneider2023,ChodorowReich-Gabaix-Viviano2025 exploit networks for shock-identification in VARs. Bykhovskaya2023 models the temporal evolution of a weighted network, which belongs to the realm of dyadic regressions Graham2020 and (dynamic) network-formation rather than -effects. } Among them, the Global VAR (GVAR) PesaranSchuermannWeiner2004,ChudikPesaran2014 restricts static and dynamic interdependencies across units in a panel-VAR using exogenous network-weights -- e.g. bilateral trade --, enabling the estimation of many interconnected VAR models under a weak exogeneity assumption.\footnote{ See CanovaCiccarelli2013 for an overview of panel-VAR approaches. } The NVAR also belongs to this second category; it restricts innovations in a (cross-sectional) VAR to transmit along bilateral links among variables (units), which leads to rich patterns of multi-step causality as in DufourRenault1998. In contrast to the GVAR and most “{networked}" time series approaches above, I focus on a single variable per cross-sectional unit and a single type of connections among them, and I entertain the assumption that innovations transmit along bilateral links. This allows me to characterize a range of theoretical properties -- including the relation to static SAR models and DFMs --, to structurally motivate the NVAR with an RBC production economy, and to efficiently conduct inference by relying on analytical conditional estimators.

When the network is estimated parsimoniously, the NVAR also addresses the literature on dimensionality-reduction methods for time-series modeling. By reducing the number of parameters and applying shrinkage priors, it leverages both approaches available for addressing the large parameter problem in the Wold representation Geweke1984. Compared to standard shrinkage methods Litterman1986,Tibshirani1996, it applies shrinkage to links -- which in turn summarize the information in predictors at all lags -- rather than to predictors themselves. Compared to reduced rank regression VeluReinselWichern1986 or factor models Geweke1977,StockWatson2002, it finds the linear combination to summarize the information in predictors by relying on network connections of different order and, ultimately, on bilateral links among units. While many of the network-restricted time series approaches mentioned above appear to have a reduced-rank representation -- consider the GVAR's construction of weighted averages of other countries' variables --, the simple setup of the NVAR allows me to characterize this representation explicitly. Owing to the equivalence result that compares the NVAR to the DFM -- corroborated by their relative performance forecasting the three processes I consider --, the NVAR contributes to the long-standing literature on sparse factors BoivinNg2006,Onatski2012,BaiNg2019,FanMasiniMedeiros2022,AnatolyevMikusheva2022,Freyaldenhoven2025. Interestingly, in my application, a sparse network is preferred to a dense one, suggesting the presence of sparse rather than weak factors and running counter to the results of GiannoneLenzaPrimiceri2021.

With the other application of the NVAR, I contribute to the series of efforts to microfound aggregate dynamics using production networks (see Horvath1998,Horvath2000,Dupor1999,Shea2002,Carvalho2010,FoersterSarteWatson2011,AcemogluEtAl2012,DiGiovanni-Levchenko-Mejean2014 and survey of Carvalho-TahbazSalehi2019). These studies rely on contemporaneous IOC -- mirroring the prevalence of contemporaneous interactions in econometrics and in the related study of spatial economies DesmetParro2025 -- and show that the production network greatly contributes to aggregate volatility. A notable exception is LongPlosser1983, who show that lagged IOC endogenizes business cycles -- i.e. persistence in aggregates --, without relying on autocorrelation in exogenous shocks, non-rational expectations, or other frictions. Their analysis points to a more prominent role for production networks than as mere amplification devices of existing dynamics; it formalizes (qualitatively) the notion that sectoral shocks take time to feed through the economy, an idea well-accepted in the trade literature AlessandriaKaboskiMidrigan2010,LiuTsyvinski2024,AntrasTubdenov2025. CarvalhoReischer2021 characterize persistence in the LongPlosser1983-economy and show that its implied evolution due to observed changes in the US input-output network accounts well for empirical measures of the changing persistence of aggregate output growth. Their analysis suggests that the endogenization of business cycles through lagged IOC is not only theoretically attractive, but has empirical merit as well. I quantify the empirical salience of LongPlosser1983's hypothesis by taking an RBC economy with general lags in IOC to the data. While lagged IOC leads to the same steady state as under contemporaneous IOC, it decomposes the long-term responses to granular TFP shocks, with distinct transition dynamics from those generated by autocorrelated TFP processes. By conducting inference on the timing of IOC along with the autocorrelations and variance of exogenous TFP processes, I quantify the contribution of lagged IOC to the entire autocorrelation function of aggregate output growth, supplementing and extending earlier decompositions of aggregate volatility FoersterSarteWatson2011,DiGiovanni-Levchenko-Mejean2014,DeGraeve-Schneider2023.

\paragraph*{Outline}

The rest of this paper is structured as follows. The model and its properties are discussed in (ref). (ref) treats inference. In (ref), I study how input-output connections affect output dynamics in the US economy. In (ref), I illustrate the merits of the NVAR as a dimensionality-reduction technique. (ref) concludes.

Dynamics via Lagged Network Effects: Theory

After providing some basic background on networks in (ref), I present the NVAR and its main properties in (ref), and I discuss further properties in (ref). Details and proofs are in (ref). The discussion of inference is deferred to (ref).

Bilateral Connections in Networks

A network is represented by an $n \times n$ adjacency matrix $A$ with elements $a_{ij}$. I consider a directed, weighted and possibly signed network; $a_{ij} \in [-1,1]$ shows the sign and strength of the link from unit $i$ to unit $j$, with $a_{ij} \neq a_{ji}$ possibly. If $a_{ij}=0$, I say unit $i$ is not connected to unit $j$. Self-links are permitted: $a_{ii} \neq 0$. The set of bilateral links $\{a_{ij}\}_{i,j=1:n}$ give rise to a myriad of higher-order (hyper-dyadic) links among units, referred to as walks.\footnote{Whenever convenient to simplify notation, I write $a:b$ for the set of integers $\{a,a+1,...,b\}$, $a\leq b$.}

A walk is a product of bilateral links that lead from unit $i$ to unit $j$ over intermediary units. It is non-zero if all of these units are sequentially connected. Simple matrix algebra reveals that $[A^K]_{ij}$ contains the sum of walks from $i$ to $j$ of length $K$. I refer to this quantity as the $K$th-order connection from $i$ to $j$. Consider the following example:

Even though unit $3$ is not directly connected to unit $1$ ($a_{31}=0$), there exists a second-order connection via unit $2$ ($a_{32}a_{21}\neq 0$). In a production network, unit $1$ could be a supplier to unit $2$, who in turn is a supplier to unit $3$.

Lagged Innovation Transmission via Bilateral Links

Consider a stationary cross-sectional time series $y_t = (y_{1t}, ..., y_{nt})'$ with mean zero. Under an NVAR, the dynamics of $y_t$ are driven by the lagged transmission of innovations $u_{it}$ along bilateral links $a_{ij}$ among cross-sectional units $i,j=1:n$. Throughout, I assume that links are fixed over time and that transmission is uni-directional: the direct link from $i$ to $j$, $a_{ij}$, transmits innovations from $j$ to $i$. Innovations $u_t$ may be cross-sectionally correlated.

Single Lag in Innovation Transmission: NVAR(1,1)

Consider a VAR(1) with an autoregressive matrix proportional to the adjacency matrix $A$:

for $\alpha \in \mathbb{R}$. Under this process, the one step-ahead expectation of $y_{it}$ is proportional to the weighted sum of one period-lagged values of $y_{jt}$ for all units $j$ to which $i$ is directly linked: $\mathbb{E}_{t-1}[y_{it}] = \alpha \sum_{j=1}^n a_{ij} y_{j,t-1}$. For example, in the RBC input-output economy of LongPlosser1983, the expected output in sector $i$ tomorrow is a weighted average of the production in its supplier-sectors $j$ today. In GolubJackson2010, individual $i$'s expected opinion tomorrow is a weighted average of their friends' opinions today. For reasons clarified below, I dub this process NVAR(1,1).

Under (ref), dynamics of $y_t$ at horizon $h \geq 1$ are driven by $h$th-order network connections. More precisely, assuming $\alpha \neq 0$, $y_j$ Granger-causes $y_i$ at horizon $h$ iff there exists an $h$th-order connection from $i$ to $j$:\footnote{ In other words, given all other variables $y_{kt}$ for $k \neq j$, $y_{jt}$ is useful in forecasting $y_{i,t+h}$ iff there exists an $h$th-order connection from $i$ to $j$. }

(ref) illustrates these Granger-causality dynamics -- also referred to as Generalized Impulse-Response Functions (GIRFs)\footnote{ The GIRF is “{generalized}" because it disregards shock identification, but considers the propagation of reduced form errors $u_t$ over future time periods $t+h$, $h\geq 1$.} -- for the network from (ref) under $\alpha=1$. It points to two insights. First, lagged network interactions generate persistence; even white noise-innovations $\{u_{jt}\}_{j=1}^n$ lead to persistent reactions of $y_t$ because each $u_{jt}$ affects connected units $i \neq j$ with a lag.\footnote{Furthermore, each individual $y_{jt}$ reacts persistently to its own white noise-innovation $u_{jt}$ as long as there is a cycle in the network, i.e. a walk from $j$ to $j$. Notably, this can hold even without a self-link -- $a_{jj} = 0$.} This result is behind the endogenous business cycles in LongPlosser1983's economy with lagged input-output conversion. Second, under this type of persistence, network-connections affect not only the strength of impulse-responses, but also their timing. For example, while unit 2 is directly linked to unit 1 and therefore experiences the latter's innovation with a lag of one period, unit 3 only has a second-order connection to unit 1 and therefore reacts only after two periods. This result relates to DufourRenault1998, who point out that Granger-causality can take the form of chains.\footnote{ Specifically, even if a series $x_t$ does not Granger-cause a series $y_t$ at horizon 1, under the presence of a third series $z_t$, $x_t$ might Granger-cause $y_t$ at higher horizons as the causality could run from $x_t$ to $z_t$ to $y_t$. } In the case of a (cross-sectional) time series driven by lagged innovation transmission along bilateral links, their insight emerges naturally and their generally non-trivial conditions for Granger-(non)causality boil down to the presence or absence of network connections of relevant order between the relevant variables (units). As I show in the following, this holds even under a more general transmission-timing than considered in (ref).

While the process in (ref) reveals useful theoretical insights in both macro- LongPlosser1983 and microeconomics GolubJackson2010, it is too restrictive for most empirical work. It assumes that innovations transmit at the speed of one link per period and that transmission materializes completely within a single period. This implies, for instance, that in response to news gathered by a friend of a friend, $j$, an individual $i$ adjusts their opinion in two periods, not earlier, not later. Similarly, an innovation to a sector $j$'s output affects the output of other sectors $i$ located two positions downstream (“{customers of customers}") in exactly two periods. In either case, earlier responses are ruled out altogether, and later adjustments only occur if $i$ has a third- or higher-order connection to $j$. In the following, I generalize the timing of lagged network effects by extending the simple process above along two dimensions.

Multiple Lags in Innovation Transmission: NVAR($p,1$)

Let the cross-sectional time series $y_t$ evolve according to a VAR($p$), where each autoregressive matrix is proportional to the same network adjacency matrix $A$:

with $\alpha = (\alpha_1, ..., \alpha_p)' \in \mathbb{R}^{p}$. I dub this process NVAR($p,1$). It simplifies the process presented in ZhuPanLiLiuWang2017 by not distinguishing self- and cross-links, by abstracting from restrictions on the rows of $A$, and by disregarding the intercept and covariates. These simplifications prove useful for characterizing a range of proprties in this section, for conducting inference in (ref), and for taking a macroeconomic model of lagged input-output conversion to the data in (ref).

Compared to (ref), under the NVAR($p,1$) with $p>1$, dynamics at horizon $h$ are affected by lower-order connections:

\footnotetext{$ceil(x)$ rounds $x \in \mathbb{Q}$ up to the next integer.}

Under $\alpha_l > 0 \; \forall \; l$ and $a_{ij} \geq 0$, (ref) implies that $y_j$ Granger-causes $y_i$ at horizon $h$ iff there exists a connection from $i$ to $j$ of at least one order $k \in \{\underline{k}, \underline{k}+1, ..., h\}$. The proof of (ref) in (ref) establishes that the GIRF is of the form

Each coefficient $c^{h}_{k}$ for $k=\underline{k}:h$ is a polynomial of $\{\alpha_l\}_{l=1:p}$ and shows the importance of connection-order $k$ for the impulse-response at horizon $h$. If $i$ has a first- and no higher-order connections to $j$, then $\partial y_{i,t+h} / \partial u_{j,t} = \alpha_h a_{ij}$ for $h=1:p$ and zero otherwise. Hence, the NVAR($p,1$) specifies that dynamics are driven by the transmission of innovations along direct links lagged over $p$ periods. $\alpha$ determines how the transmission is spread out over these $p$ periods and, consequently, how transmission along higher-order connections accrues as time progresses. Note that the transmission is assumed to be the same for all unit pairs $(i,j)$ and invariant over time.

\todo{note that $\alpha \neq$ timing of network effects, as $q$ matters too!}

Multiple Rounds of Innovation Transmission: NVAR($p,q$)

To further generalize the timing of innovation transmission, suppose that we observe $\tilde{y}_\tau \sim$ NVAR($p,1$) every $q \in \mathbb{N}$ periods; let the observed series $y_t$ evolve according to the state space system

I dub this process NVAR($p,q$). $q$ indicates the relation between the frequency of network interactions, at which $\tilde{y}_\tau$ evolves, and the frequency at which data $\{y_t\}_{t=1:T}$ is observed. If $q=1$, the two coincide and, trivially, $y_t$ and $\tilde{y}_\tau$ are the same NVAR($p,1$) process. Instead, if $q>1$, then network interactions occur at higher frequency than data is observed. For example, under yearly observations, $q=4$ implies quarterly network interactions. As a result, multiple rounds of transmission occur in a single observational period and dynamics at horizon $h$ are affected by higher-order connections, $k>h$:

\footnotetext{ Assuming again $\alpha_l > 0 \; \forall \; l$ and $a_{ij} \geq 0$, (ref) implies that $\tilde{y}_j$ Granger-causes $\tilde{y}_i$ at horizon $h$ iff there exists a connection from $i$ to $j$ of at least one order $k \in \{\underline{k}, \underline{k}+1, ..., hq\}$. }

(ref) implicitly assumes that $\tilde{y}_\tau$ is a stock variable; every $q$ periods we observe a snapshot of it. To accommodate a flow variable $\tilde{y}_\tau$, we can model $y_t$ as

with similar properties as under (ref).\footnote{ See (ref) and (ref). Analogous calculations apply if $y_t=\tilde{y}_\tau + ... + \tilde{y}_{\tau-q+1}$.}

By restricting the NVAR($p,q$) in (ref), we can accommodate more intricate relations between the frequencies of network interactions and observation; for any $q \in \mathbb{Q}_{++}$, we can write $q=q_1q_2$ with $q_2, q_1^{-1} \in \mathbb{N}$, and we can model $y_t$ as an NVAR($p^*,q_2)$ with $p^* = p/q_1 \in \mathbb{N}$ and with restricted parameters $\{\alpha_l\}_{l=1:p^*}$ to comply with the stated interpretations of $p$ and $q$. For example, under monthly observations, $q=4/3$ signifies that network interactions occur (roughly) every three weeks. This amounts to observing every $4$ periods a snapshot of a weekly process that depends on its value in the past $p$ $3$-week-periods, i.e. on its value three weeks ago, six weeks ago, and so forth, until $p^*=3p$ weeks ago. (ref) elaborates.

Further Properties of the NVAR

\paragraph*{Stationarity}

In the NVAR, the interdependence of $y_{it}$ and $y_{j,t-h}$ is an interplay of the cross-sectional distance between units $i$ and $j$ -- encoded by the network -- and the temporal distance between periods $t$ and $t-h$. Correspondingly, (ref) characterizes stationarity in terms of eigenvalues of the network adjacency matrix $A$ and roots of an AR process shaped by the timing of innovation transmission along a single link, $\alpha$.\footnote{ (ref) follows from (ref). Intuitively, the conditions ensure that $\lim_{k\rightarrow \infty}\; c(\alpha,k)A^k = 0$, for any polynomial in $\alpha$ of order $k$, $c(\alpha,k)$. By (ref), this ensures that the effects of disturbances vanish for higher horizons. } Under $p>1$, this simplifies checking for stationarity, especially for large $n$. Note that the second statement requires the AR($p$) process $(1 - \phi_1 L - ... - \phi_p L^p)x_t = v_t$ with $\phi_l = \alpha_l \lambda_i$ to be stationary for each eigenvalue $\lambda_i$ of $A$, though AR-coefficients are typically required to be real-valued.

\paragraph*{Relation to SAR Model}

In all of economics, the literature overwhelmingly considers contemporaneous network interactions as embodied by the Spatial Autoregressive (SAR) model. If $x = (x_1, ..., x_n)'$ is the cross-sectional variable of interest, it posits

Provided that $|\lambda_i|<1$ for all eigenvalues $\lambda_i$ of $A$, we can write $x= (I-A)^{-1} v = (A + A^2 + A^3 + ...) v$. Hence, the implicit assumption is that connections of all order materialize in a single period. In line with that, (ref) establishes that the response of $x_i$ to an innovation $v_j$ in the SAR model from (ref) is equal to the long-run response of $y_{it}$ to a persistent innovation $\tilde{u}_{j\tau}$ in the NVAR($p,q$) from (ref) or (ref).\footnote{ (ref) follows from (ref). }

Both responses are given by the Leontief inverse $(I- a A)^{-1}$, which is a sufficient statistic for the long-term cross-sectional comovement. By taking a stance on the timing and frequency of network interactions, the NVAR shows how any such long-term effects materialize over time; it goes beyond steady state comparisons and characterizes transitional dynamics.\footnote{ Note the correspondence between the timing of transitional dynamics and the timing of responses to temporary innovations; as for any VAR, the long-term response to a permanent innovation is equal to the cumulative response to temporary innovations. } $\alpha$ temporally decomposes the spatial autoregressive parameter $a$.

The long-run is defined in terms of the frequency of network interactions. To justify the use of the SAR at the expense of the NVAR, then, we need this frequency to be high enough and the data frequency to be low enough. (ref) formalizes this idea; if we define the observed process $y_t$ as the sum of an underlying process $\tilde{y}_\tau$ evolving at the higher frequency of network interactions, then, as this frequency diverges to infinity, $y_t$'s response to a high-frequency innovation that occurs within observational period $t$ converges to the response of $x_i$ to an innovation $v_j$ in the SAR model from (ref).\footnote{ (ref) follows from (ref), applied for $\rho_j=0$. }

\paragraph*{Difference to Exogenous Persistence}

Instead of relying on lagged network interactions, one can also introduce transitional dynamics to the SAR model from (ref) through autocorrelated innovations $v$. This approach is predominant in macroeconomic studies of input-output-economies; under fairly standard assumptions (see e.g. Carvalho-TahbazSalehi2019), sectoral production and prices evolve according to (ref), whereby $v$ is Total Factor Productivity (TFP), which -- across all areas of macroeconomics -- is typically assumed to follow an autoregressive process.\footnote{ It can be split into an idiosyncratic and an aggregate component; see (ref). } \todo{and in SAR some allow for autocorr. errors?} \todo{also, cite some others in footnote on idiosync. vs. agg. component}

Lagged network interactions and autocorrelated innovations lead to different kinds of dynamics. As first illustrated in (ref) for the NVAR(1,1), and as discussed throughout this section for the general NVAR($p,q$), lagged network interactions relate impulse-responses to network-connections of different order, whereby the precise mapping is determined by $\alpha$ and $q$. As a consequence, how strongly $y_i$ reacts to an impulse to $y_j$ depends on the strength of network-connections from $i$ to $j$ of relevant order, and how fast it reacts depends on how many lower- as opposed to higher-order connections the two units share. For instance, in the environment of GolubJackson2010 this implies that an individual adjusts their opinion faster after news shared by closer than more distant friends. Similarly, in the context of LongPlosser1983 it means that a sector contracts its production faster after a negative productivity shock to more immediate suppliers than to ones located further upstream.

In contrast, autocorrelated innovations lead to exponentially decaying impulse-responses. This shape is preserved when they are coupled with contemporaneous network interactions; if $x_t = aA x_t + v_t$ and $(1-\rho_jL)v_{jt} = \epsilon_{jt}$, then

The strength of connections from $i$ to $j$ merely scales the exponentially decaying impulse-response; network connections merely amplify exogenous dynamics.

When the two sources of dynamics are combined, their qualitatively distinct contributions remain. Consider an NVAR($p,1$), and let $(1-\rho_jL)u_{jt} = \varepsilon_{jt}$. Then, using (ref),

for $\underline{k}(l) = ceil(l/p)$. Under $\rho_j = 0$, this expression simplifies to (ref). Under $p=1$ and $\alpha_1 = 1$, it simplifies to $\sum_{l=0}^h \left[ {A^h} \right]_{ij} \rho_j^{h-l}$. While $\alpha$ determines the timing of innovation transmission along direct links and, therefore, also the timing of transmission along all higher-order connections from $i$ to $j$, $\rho_j$ determines the persistence of $y_i$'s response after each round of transmission.

(ref) illustrates for an NVAR($2,1$) and $\rho_j=.6$, once under $\alpha = (.8,.2)'$ (top row), once under $\alpha = (.2,.8)'$ (bottom row). The left panel depicts how connection-orders accumulate over horizons for these $\alpha$, while the middle and right panels show the resulting impulse-responses for two sectors that share only a first- or only a second-order connection, respectively. Under $\alpha=(.8,.2)'$, the innovation travels faster through the network than under $\alpha = (.2,.8)'$; in the former case, the transmission along a direct link (darkest blue dots and bars) is strongest after one period, in the latter after two periods. Consequently, under $\alpha=(.8,.2)'$, the transmission along second-order connections (lighter blue dots and bars) is strongest in the second period, under $\alpha = (.2,.8)'$ in the fourth period. The exogenous persistence ensures that the fraction $\rho_j$ of any transmission at horizon $h$ is carried over to horizon $h+1$, a fraction $\rho_j^2$ to horizon $h+2$, etc. (ever lighter gray bars).

Even under autocorrelated innovations, lagged and contemporaneous network interactions yield the same long-run responses. This is shown by (ref).\footnote{ (ref) follows from (ref). In contrast to (ref), it considers the long-run rather than contemporaneous response of $x$, since the implicit assumption in the SAR model with autocorrelated innovations is that they evolve at observational frequency. }

\paragraph*{Networked Contemporaneous Correlation}

Trivially, contemporaneous network interactions lead to contemporaneous correlation among $\{x_i\}_{i=1:n}$; under (ref), we have $\mathbb{V}[x] = \mathbb{V}[(I-A)^{-1}v]$. In this case, $\text{Cov}(x_i,x_j)$ reflects the bilateral exposure between $i$ and $j$ and their mutual exposure to third units, whereby exposure is determined by network connections of all order.\footnote{ We have $\mathbb{V}[x]=\mathbb{V}[(I+A+A^2+...)v]$, provided that $|\lambda_i|<1$ for all eigenvalues $\lambda_i$ of $A$. Assuming that $\mathbb{V}[v] = \Sigma = diag(\sigma^2_1, ..., \sigma^2_n)$, we get $$ \text{Cov}(x_i,x_j) = \sigma^2_i \mathbf{1}\left\{ { i=j } \right\} + \sigma^2_j \sum_{h=1}^\infty \left[ {A^{h}} \right]_{ij} + \sigma^2_i \sum_{h=1}^\infty \left[ {A^{h}} \right]_{ji} + \sum_{o=1}^n \sigma^2_o \sum_{h_i=1}^\infty \sum_{h_j=1}^\infty \left[ {A^{h_i}} \right]_{io} \left[ {A^{h_j}} \right]_{jo} \; .$$ } DeGraeve-Schneider2023 exploit this insight for shock identification in the context of a production economy.

Networked contemporaneous correlation is obtained as soon as the network interaction-frequency exceeds the frequency of observation, even under finite values of the former. In an NVAR($p,q$), we can define the “{observable innovations}" $u_t = y_t - \mathbb{E}[y_t|\mathcal{F}_{t-1}]$, where $\mathcal{F}_{t-1} = \{\tilde{y}_{(t-1)q}, \tilde{y}_{(t-1)q-1}, ... \}$. Under $q=1$, $u_t = \tilde{u}_\tau$ inherits the variance of the exogenous innovations $\tilde{u}_\tau$. Under $q>1$, $u_t$ is a linear combination of past $\tilde{u}_\tau$, and its variance reflects units' bilateral exposure and mutual exposure to third units, whereby exposure is determined by network connections of orders $k \in 1:(q-1)$, along which the underlying innovations $\tilde{u}_\tau$ travel within a single period of observation. For example, if $y_t$ evolves as in (ref) for $q=2$, we have $u_t = \tilde{u}_\tau + (I+\alpha_1 A) \tilde{u}_{\tau-1}$. If $\tilde{u}_\tau \sim WN(0,\Sigma)$ with $\Sigma=diag(\sigma^2_1, ..., \sigma^2_n)$, then $$ \text{Cov}(u_{it},u_{jt}) = 2\sigma_i^2\mathbf{1}\left\{ { i=j } \right\} + \alpha_1 a_{ij}\sigma_j^2 + \alpha_1 a_{ji} \sigma_i^2+ \alpha_1^2 \sum_{k=1}^n a_{ik}a_{jk} \sigma^2_k \; . $$ Thus, $u_{it}$ and $u_{jt}$ comove based on $a_{ij}$, $a_{ji}$ and $\{a_{ik}a_{jk}\}_{k=1:n}$. (ref) state $\mathbb{V}[u_t]$ for the NVAR($p,q$) processes in (ref) and general $q>1$.

(ref) above establishes that an SAR for $x$ is obtained in the limit when writing $x$ as the sum of an underlying process driven by lagged network interactions that evolves at an ever higher frequency. In addition, by (ref), the Normality-assumption $v\sim N(0,\Sigma)$ in the SAR model can be justified by temporally independent high-frequency innovations $\tilde{u}_\tau$ with $\mathbb{V}[\tilde{u}_\tau] = q^{-1}\Sigma$. Concretely, for the NVAR($p,q$) from (ref) and for $a= \sum_{l=1}^p \alpha_l$, (ref) shows that $\sqrt{q}u_t \overset{d}{\rightarrow} N\left( { 0, \mathbb{V}[(I-aA)^{-1}\tilde{u}_\tau] } \right)$ as $q \rightarrow \infty$, provided the high-frequency innovations $\tilde{u}_\tau$ are strictly stationary.

\paragraph*{Relation to Dynamic Factor Model}

The NVAR is a restricted VAR in which innovations transmit across series along constant, bilateral links (in one direction). Relative to a VAR($p$) with $p>1$, the NVAR($p,1$) induces parsimony; e.g. for each $y_{it}$, the $n$ links $\{a_{ij}\}_{j=1:n}$ in $A_{i\cdot}$ summarize the information in the $np$-dimensional set $\{y_{j,t-l}\}_{j=1:n,l=1:p}$:

where $x_{it} = \sum_{j=1}^n a_{ij}y_{j,t-l}$ is the $i$th row of $X_t = [A y_{t-1}, ..., A y_{t-l}]$. If, in addition, the network $A$ is sparse -- as is the case across a wide range of applications\todo{cite some general ref.; Jackson-book? who else?} --, then the NVAR rationalizes the dynamic comovement among all $\{y_{it}\}_{i=1:n}$ by the dynamic innovation transmission along few bilateral links among $i=1:n$.

The Dynamic Factor Model (DFM) also reduces dimensionality by summarizing a large set of predictors by a few linear combinations.\todo{cite here; Ambrogio's paper, and look up original source?} (ref) shows that the NVAR($p,1$) can be written as a DFM, while a DFM with restricted factor dynamics can be written as an NVAR($p,1$) in the limit as $n \rightarrow \infty$.\footnote{ The second part of (ref) restricts $f_t \in \mathbb{R}^r$ to follow an NVAR($p,1$). This is less restrictive than it may seem; for $p=1$, it requires $f_t \sim VAR(p)$, while under $r=1$ it requires $f_t \sim AR(p)$. Also, note that, as usual, the factor representation is not unique, and we can re-scale $A$ and $\phi$ to ensure $a_{ij} \in [-1,1]$.} Together with (ref) above, it suggests that, at the cost of restricting factor dynamics, the NVAR allows the linear combinations of predictors to vary more flexibly across units, to the point that it naturally acommodates sparse factors as (linear combinations of) locally important units in the network.\footnote{ The DFM constructs $\{\mathbb{E}_{t-1}[y_{it}]\}_{i=1}^n$ as different ($\Lambda_{i\cdot}$) linear combinations of the same factors $f_t$, which in turn are linear combinations of observables. In contrast, the NVAR constructs $\{\mathbb{E}_{t-1}[y_{it}]\}_{i=1}^n$ as the same ($\alpha$) linear combination of different covariates $x_{it}$, which in turn are linear combinations of observables.}

Dynamics via Lagged Network Effects: Inference

The NVAR can be used to estimate dynamic (lagged) network effects, as determined by $\alpha$, the time profile of network effects. When the network $A$ is estimated as well, it can also be used as a dimensionality-reduction technique useful for forecasting (cross-sectional) processes. In (ref), I discuss inference on $\alpha$, treating $A$ as given. In (ref), I discuss joint inference on $(\alpha,A)$. Details are in (ref).

Timing of Network Effects

\paragraph*{NVAR($p,1$)}

The NVAR($p,1$) from (ref) can be written as the linear regression in (ref). Defining $\Sigma = \mathbb{V}[u_t]$, we obtain the Least Squares (LS) estimator for $\alpha$:

As usual, under $u_t\sim N(0,\Sigma)$, it is also the (conditional) Maximum Likelihood (ML) estimator and the posterior-mean and -mode under a Uniform prior for $\alpha|\Sigma$. Under $\Sigma=I$, it yields the OLS estimator, which takes a “{pooled}" form:

(ref) establishes consistency and asymptotic Normality of $\hat{\alpha}_{OLS}$ under large $T$, large $n$ and large $(n,T)$ asymptotics. The derivations under large $n$ assume that the observed network adjacency matrix $A_n$ converges to some limit $A_*$ so that, for example, $\frac{1}{n} \sum_{i=1}^n \left( { A_{n,i\cdot} y_{t-l}} \right)' u_{it} \overset{p}{\rightarrow} \mathbb{E}\left[ { \left( { A_{*,i\cdot} y_{t-l}} \right)' u_{it} } \right]$. \todo{verify; at least notation could be better: $y_t$ also depends on $n$; it's not $A$ taht converges, but network-statistics}

We can estimate $\Sigma$ standardly by $\hat\Sigma|\alpha = \frac{1}{T}\sum_{t=1}^T u_t u_t'$. The joint ML estimator $(\hat\alpha,\hat\Sigma)$ is obtained by iterating on $\hat{\alpha}|\Sigma$ and $\hat\Sigma|\alpha$ until convergence (see MengRubin1993).

\paragraph*{NVAR($p,q$)}

In case of an NVAR($p,q$), as defined in (ref), an estimator for $\alpha$ can be obtained by data augmentation.\footnote{ The discussion holds likewise for the NVAR($p,q$) from (ref). } However, point identification is not guaranteed. For example, under $q=2$ and $p=1$, the observed process follows

which suggests that $\alpha_1$ is identified only up to sign. The identification problem is akin to estimating an AR($p$) observed every $q$ periods, discussed in PalmNijman1984.\footnote{ It is also similar to estimating continuous time models using discrete time data (see e.g. Phillips1973).}\todo{add DSGE citation?} It is due to the fact that the mapping between the parameters $\alpha$ in the high-frequency process $\tilde{y}_\tau$ and the parameters in the observed process $y_t$ is generally not bijective. For a general AR($p$) and $q=2$, the vector $(\alpha_1, \alpha_3, ...)$ is identified (jointly) up to sign (see (ref)).

In the application in (ref), the restrictions on $\alpha$ imposed by economic theory render $\alpha$ point-identified, as suggested by a unique posterior mode under Uniform priors. In other cases, one could follow the suggestion of PalmNijman1984 and inform the estimation of $\alpha$ with a prior. This is facilitated by the clear interpretation of $\{\alpha_l\}_{l=1:p}$; it is the GIRF for units $i$ and $j$ that only share a first-order connection (see (ref)) and indicates how innovations transmit along a single link over time.

Conditioning on $\tilde{\Sigma}$, the posterior $p(\alpha|Y_{1:T})$ can be obtained using the Gibbs sampler of CarterKohn1994. Treating the unobserved data in $\tilde{Y}_{1:T_\tau}$ as parameters, it iteratively draws from $p(\tilde{Y}_{1:T_\tau}|\alpha,Y_{1:T})$ and $p(\alpha | \tilde{Y}_{1:T_\tau})$ to obtain a sample from $p(\alpha, \tilde{Y}_{1:T_\tau}|Y_{1:T})$.\footnote{The particular state space model where in some periods $\tau$ no data is observed implies that in these periods the updating-step of the Kalman filter is skipped and the updated distribution of states equals their predicted distribution. See (ref), and see SchorfheideSong2015 for a discussion of the analogous case of a mixed-frequency VAR.} Under a Uniform prior for $\alpha$, the resulting posterior mode of $\alpha$ converges to the ML estimator obtained using the Expectation-Maximization (EM) algorithm.\todo{citation? or show in appendix?!} To estimate $\tilde{\Sigma}$ as well, an additional iteration step is added to the Gibbs sampler. Using a uniform prior, we get an Inverse-Wishart conditional posterior for $\Sigma |\tilde{Y}_{1:T_\tau}, \alpha$ with mode $\frac{1}{T_\tau}\sum_{\tau=1}^{T_\tau} \tilde{u}_\tau \tilde{u}_\tau'$.

Joint Inference: Network & Effect-Timing

When interest lies in dynamic network effects, joint estimation of $(\alpha,A)$ is useful because network data may be difficult to collect or it may appear restrictive to condition the analysis on available network data. In addition, estimating $(\alpha,A)$ jointly enables us to use the NVAR as a dimensionality-reduction technique.

\paragraph*{NVAR($p,1$)}

The NVAR($p,1$) from (ref) can also be written as the linear regression

where $z_t = \sum_{l=1}^p \alpha_l y_{t-l} = \big[ y_{t-1},y_{t-2},...,y_{t-p} \big] \alpha $, and the $T \times n$ matrices $Y$, $Z$ and $U$ stack $y_t$, $z_t$ and $u_t$ along rows, respectively. To simplifty notation, I suppress the dependence of $z_t$ on $\alpha$ and that of $X_t$ on $A$.

To render $(\alpha,A)$ jointly identified, I normalize $\alpha_1 = 1$, with appropriate redefinitions of $\alpha$ as well as $y_t$, $z_t$ and $X_t$. Relative to the alternative of restricting the norm of $\alpha$, this normalization facilitates analytical calculations and asymptotic analysis, but it requires $\alpha_1 \neq 0$ in the true data generating process.\footnote{ In case $a_{ij} \geq 0$ is restricted, it requires $\alpha_1 > 0$. }

Suppose data on a network $B$ with elements $b_{ij}$ is available. Under independent priors $a_{ij} \sim N(b_{ij},\lambda_a^{-1})$, we obtain a matrix-variate Normal conditional posterior for $A$:

Its mean and mode, $\bar{A}'$, is the conditional optimizer for $A$ under a LS objective function with a Ridge-penalty:

As $\lambda_a \rightarrow \infty$, we impose $A=B$. As $\lambda_a \rightarrow 0$, we ignore $B$ and infer $A$ from the data alone. No domain restrictions on $A$ are imposed because any parameter value $(\alpha,A)$ can be rescaled to yield $a_{ij}\in [-1,1] \; \forall \; i,j$, so that $A$ can be interpreted as a network.\footnote{To enforce $a_{ij} \in [0,1]$ even under low $\lambda_a$, $a_{ij} \geq 0$ must be imposed. This leads to the high-dimensional Normal posterior being truncated to $\mathbb{R}^{n^2}_{+}$ and considerably complicates the analysis, as both computing the mode and drawing from this distribution is computationally intensive.}

Under a Laplace prior, the conditional posterior of $A$ and its mode -- the Lasso estimator -- is available analytically when imposing $a_{ij}\geq 0$ and shrinking to $b_{ij}=0$. We get

truncated to $\mathbb{R}_+^{n^2}$. Drawing from this distribution or computing its mode is computationally feasible only for a diagonal $\Sigma$, which renders the distribution of each row $i$ of $A$ independent across $i$. Under $\Sigma=I$, we get $$ A_{i\cdot} \mid (Y, \alpha, \Sigma=I, \lambda_a) \sim N \left( { (\bar{A}')_{i,\cdot}, \bar{U}_A} \right) \; , \quad \text{truncated to} \; \mathbb{R}_+^{n} \; . $$ A draw from this distribution is obtained using Gibbs sampling by iteratively drawing from the Normal conditional densities $a_{ij} \mid (A_{i,-j}, Y, \alpha, \Sigma=I, \lambda_a)$ for $j=1:n$. Its mode is computed by iterating on the latters' modes.\footnote{Taken together, these two results mean that we can draw from the distribution of $A \mid (Y, \alpha, \Sigma, \lambda_a)$ or compute its mode by iterating on each column of $A$ given all other columns.}

Given $\Sigma$ and $\lambda_a$, the joint posterior $p(\alpha,A|Y)$ is obtained by Gibbs sampling, iteratively drawing from the conditional posteriors $p(A | Y, \alpha, B, \Sigma, \lambda_a)$ and $p(\alpha|Y,A, \Sigma)$. To estimate $\Sigma$ as well, an additional step is added to draw from $p(\Sigma | \alpha, A, Y)$. Under uniform priors for $\alpha$ and $\Sigma$, the posterior mode of $p(\alpha,A,\Sigma|Y)$ is equal to the GLS estimator ($\hat{\alpha},\hat{A},\hat{\Sigma})$ of the objective function in (ref), obtained by iterating until convergence on the three respective conditional estimators. Fixing $\Sigma=I$, we obtain the OLS estimator of $(\alpha, A)$, for which consistency and asymptotic Normality under $T \rightarrow \infty$ are established in (ref). The choice of $\lambda_a$ for predictive purposes as well as the possibility to construct $B$ as a combination of multiple link-types is discussed in (ref).\todo{yes, do that!}

\paragraph*{NVAR($p,q$)}

As in the estimation of $\alpha|A$, if the network interaction frequency is higher than the observation frequency, joint inference on $(\alpha,A)$ can be conducted by relying on data augmentation. As before, identification is not guaranteed. In fact, relative to the estimation of $\alpha|A$ the problem is likely worsened, even if $A$ may be tightly shrunk to a known (or sparsely parameterized) network $B$. However, the estimation of $(\alpha,A)$ in the application in (ref) of this paper is judged based on forecasting performance, not its ability to deliver point identification.

Business Cycles by Lagged Input-Output Conversion

LongPlosser1983 show that time lags between the production of goods and their subsequent use as intermediaries for producing other goods can generate endogenous business cycles. In this application, I empirically quantify the importance of their proposed channel. I generalize their RBC economy with one-period lagged IOC by assuming that firms' production requires inputs produced in the past $p$ periods. This leads to sectoral output growth evolving at some model-frequency as an NVAR($p,1$), which translates into an NVAR($p,q$) for some $q\in \mathbb{N}$ at my monthly frequency of observation. Thereby, $A$ is the input-output matrix, $u_t$ contains sectoral productivity processes, while $\{\alpha_l\}_{l=1:p}$ show how input-sourcing is spread out over the $p$ periods. By estimating $\alpha$, $q$ and the persistence in $u_t$, I use the NVAR to quantify the extent to which business cycles in this framework are due to lagged IOC as opposed to persistence in exogenous productivity processes.

After theoretically motivating the analysis in (ref), I discuss the setup, data and estimation in (ref). (ref) presents the results. Details are in (ref).

Theory

The following analysis is based on Carvalho-TahbazSalehi2019 -- who discuss a static economy -- and the Appendix to AcemogluAkcigitKerr2016. Derivations are in (ref).

Assume there are $n$ sectors, in each of which a representative firm produces a differentiated good $i$ by combining labor services $l_{i\tau}$ and goods produced by other sectors $j$, $\{x_{ij\tau}\}_{j=1}^n$, using a constant returns to scale (CRS) Cobb-Douglas production function. Firms maximize profits, taking prices as given. The profits of firm $i$ in period $\tau$ are

where $b_i > 0$, $a_{ij} \geq 0$ and $b_i + \sum_{j=1}^n a_{ij} = 1$. $z_{i\tau}$ denotes Total Factor Productivity (TFP) in sector $i$, $\{p_{i\tau}\}_{i=1}^n$ are the prices of the $n$ different goods, and $w_\tau$ is the price of labor. $x_{ij\tau}$ is the amount of good $j$ used in the production at time $\tau$. As discussed below, it can differ from the amount of good $j$ purchased in period $\tau$, $x^{ij}_\tau$.

In this environment, prices are entirely determined by supply. To characterize output, I assume the presence of a representative household who supplies one unit of labor inelastically and exhibits log-preferences over the $n$ goods:

where $\sum_{i=1}^n \gamma_i = 1$. The first-order condition (FOC) yields $c_{i\tau} = \gamma_i \frac{w_\tau}{p_{i\tau}}$.\footnote{Hence, $\gamma_i$ is the share of good $i$ in households' expenditures.} This result holds even if households have access to a storage technology, as market clearing under representative households in a closed economy implies that households spend their whole period $\tau$ income, $w_\tau$, on consumption. \todo{see P009diary3, p.152 for dynamic HH problem}

Different assumptions on the timing of IOC lead to different dynamics of sectoral prices and output in this economy. Typically, it is assumed that inputs are converted into outputs in the same period when they are purchased, i.e. $x_{ij\tau} = x^{ij}_\tau$. Dropping time subscripts in this static environment, define $\tilde{y} = ln(y)$, $y= (y_1, ..., y_n)$ and analogously for $\tilde{z}$. In equilibrium, sectoral output satisfies

where $k^y$ is a vector of constants, and $a_{ij} = p_j x^{ij} / (p_i y_i)$ is value of good $j$ purchases by sector $i$ as a fraction of the value of sector $i$'s output. In this environment, while the network $A$ amplifies idiosyncratic TFP shocks and therefore affects the variance of $\tilde{y}$, any autocorrelation in $\tilde{y}$ is inherited from that in $\tilde{z}$ (see e.g. Carvalho2010,AcemogluEtAl2012).

To analyze dynamics under lagged IOC, I assume perfect foresight.\footnote{ As discussed in FanHongParro2023, this assumption is standard for modeling dynamic spatial economies, which are closely related to dynamic network economies.} If, as in LongPlosser1983, it takes one period to convert purchased inputs into output, then $x_{ij\tau} = x^{ij}_{\tau-1}$ and sectoral output approximately follows an NVAR($1,1$):

where the time variation in $k^{y1}_\tau$ is due to time variation in the numéraire $w_\tau$. So long as $\beta < 1$, in steady state, prices are higher and output is lower than in the economy with contemporaneous IOC. Also, the meaning of $a_{ij} = \beta^{-1}p_j x^{ij} / (p_i y_i)$ changes slightly. These differences vanish as $\beta \rightarrow 1$. More importantly, while the former, static economy is always in steady state, the economy with one period-lagged IOC features transition dynamics; after a disturbance to $\tilde{z}_\tau$, $\tilde{y}_\tau$ only asymptotically converges to steady state.\footnote{ Relatedly, as discussed in (ref), the response of $\tilde{y}_{(\tau)}$ to a change in $\tilde{z}_{(\tau)}$ in the static economy corresponds to the long-term response of $\tilde{y}_{\tau}$ to a permanent change in $\tilde{z}_{\tau}$ in this dynamic economy (disregarding the slightly changed meaning of $a_{ij}$).} It is due to these transition dynamics that this framework can generate endogenous business cycles, i.e. autocorrelation in $\tilde{y}_\tau$ even in absence of autocorrelation in $\tilde{z}_\tau$.

To take the economy with lagged IOC to the data, I generalize the lag length by assuming that firms require inputs produced in the past $p$ periods for production. For ease of exposition, let $p=2$. Let $x_{ij\tau}$ aggregate quantities of input $j$ purchased at different periods in the past using a Constant Elasticity of Substitution-aggregator:

where $x^{ij}_{\tau,\tau-h}$ denotes the use of good $j$ purchased at time $\tau-h$ in the production of good $i$ at time $\tau$.\footnote{ This means that a good perishes after two periods (with regard to its suitability as an input in production). Therefore, the amount of good $j$ purchased at time $\tau$ can be used in production at periods $\tau+1$ and $\tau+2$: $x^{ij}_\tau = x^{ij}_{\tau+1,\tau} + x^{ij}_{\tau+2,\tau}$.} This shortcut stands for frictions like delivery costs and -lags and storage capacity constraints KhanThomas2007, AlessandriaKaboskiMidrigan2010, LiuTsyvinski2024, AntrasTubdenov2025.\footnote{ As in the LongPlosser1983-economy above, the presumption is that storage is done by the buyer.} In the Cobb-Douglas case $r\rightarrow 0$, sectoral output approximately follows an NVAR(2,1):

where once again $k^{y2}_\tau$ varies over time only to the extent that the numéraire $w_\tau$ changes in value. Under a more general elasticity of substitution $r \in [0,1)$,\footnote{ This notably excludes complementarity ($r<0$) and perfect substitutability ($r=1$). } the analogous result is obtained by log-linearizing around the steady state (see (ref)). This specification nests the one period-lagged economy, which is obtained under $\alpha_2 = 0$. Relative to that case, $\alpha_2 >0$ increases prices and decreases output in steady state, provided that $\beta < 1$. Also, we have $a_{ij} = \left[ { \beta \alpha_1 + \beta^2 \alpha_2 } \right]^{-1} (p_j x^{ij})/(p_i y_i)$. As $\beta \rightarrow 1$, the links $a_{ij}$ retain their interpretation as the output shares of different inputs $j$ in the production of good $i$. The parameters $(\alpha_1,\alpha_2)$ show the shares of an input $j$ purchased at different periods in the past in the overall usage of input $j$ in the production of good $i$. Their homogeneity means that the time profile of input-sourcing is assumed to be constant over time and across input-output pairs $(i,j)$. The restrictions $\alpha_1, \alpha_2 \geq 0$, $\alpha_1 + \alpha_2 = 1$ and $\sum_{j=1}^n a_{ij} < 1 \; \forall \; i$ imply that $\tilde{y}_\tau$ is stationary as long as $\tilde{z}_\tau$ is.\footnote{ BermanPlemmons1994 show that for an element-wise nonnegative matrix with row sums strictly smaller than 1, the absolute value of the largest eigenvalue is strictly less than 1. Stationarity then follows by (ref). }

The theory above does not restrict the process of (log) sectoral TFP $\tilde{z}_{i\tau}$. In general, it may display secular growth and persistent shocks of both aggregate and idioyncratic nature. Consider a difference-stationary specification with aggregate and idiosyncratic TFP processes $e^a_\tau$ and $e_{i\tau}$ evolving as AR(1) processes as in FoersterSarteWatson2011: $$ \Delta \tilde{z}_{i\tau} = \gamma_i + \delta_i e^a_\tau + e_{i\tau} \; , $$ $(1- \rho_a L) e^a_{\tau} = \varepsilon^a_{\tau} \sim WN$ and $(1-\rho_i L) e_{i\tau} = \varepsilon_{i\tau} \sim WN$. Under contemporaneous IOC ((ref)), this leads to

We obtain IRFs $\partial \Delta \tilde{y}_{i,\tau+h}/ \partial \varepsilon_{j,\tau}$ and $\partial \Delta \tilde{y}_{i,\tau+h}/\partial \varepsilon^a_\tau$ akin to those in (ref) in (ref). This points to two sources of persistence in output growth: persistence in the aggregate TFP process $e^a_\tau$ and persistence in idiosyncratic TFP processes $\{e_{i\tau}\}_{i=1}^n$. The role of input-output links is limited to amplification; the response of output growth in sector $i$ to a TFP shock in sector $j$ is scaled by element $(i,j)$ of the Leontief-inverse $(I-A)^{-1}$, as in Carvalho2010. By summing up connections of all order from $i$ to $j$, the latter shows the importance of sector $j$ in sector $i$'s supply chain.

Lagged IOC can be an additional source of persistence. Under (ref), we get

which leads to IRFs $\partial \Delta \tilde{y}_{i,\tau+h}/ \partial \varepsilon_{j,\tau}$ and $\partial \Delta \tilde{y}_{i,\tau+h}/\partial \varepsilon^a_\tau$ akin to those in (ref) in (ref). As discussed in (ref), lagged IOC and TFP shocks' autocorrelation imply distinct dynamics; $\alpha$ relates the timing at which a sector's output is first impacted by a TFP shock in another sector and the strength of the ensuing response to network connections of different order, whereas $\rho_a$ and $\{\rho_j\}_{j=1:n}$ induce an exponentially decaying response after every round of networked transmission.

Under a difference-stationary specification for log sectoral TFP $\tilde{z}_\tau$, TFP shocks have temporary effects on output growth, but persistent effects on output levels. By (ref), the long-term response of (log) output to a TFP shock -- equal to the cumulative response of output growth -- is the same under contemporaneous and lagged IOC, in line with the fact that any TFP level yields the same steady state in both economies (disregarding differences in $A$ between the two).

To take the processes in (ref) and (ref) to the data, one has to take a stance on what a period in the theoretical models above signifies. Let $\{\Delta y_t \}_{t=1}^T$ be observed output growth. As it is a flow variable, the model-frequency must be either equal to the observational frequency or an integer-multiple thereof (see (ref)):

and for $q \in \mathbb{N}$. Other things equal, under a higher $q$ the economy approaches faster the new steady state level of output following a TFP shock. Relatedly, under lagged IOC, it also means that the IRFs at any single horizon increasingly depend on higher-order connections, and it leads to network-induced cross-sectional correlation in innovations at observational frequency, whereas for $q=1$ the correlation is entirely due to aggregate TFP shocks.\footnote{ See (ref) as well as (ref) for more detailed discussions of these points. In the limit as $q \rightarrow \infty$, the long-term response referenced in (ref) -- a function of all connection-orders -- materializes after a single observational period. } As the meaning of $q$ depends on the frequency of observation, its choice is discussed in (ref).

Application-Setup, Data & Estimation

\todo{need to discuss aggregation weights} \todo{need to do a more careful decomposition analysis; see latex comment}

I quantify the relative contributions of these two (three) drivers of aggregate persistence as seen through the lens of the theory of real business cycles with contemporaneous and lagged IOC. To do so, I estimate the state space models characterized by (ref) and (ref), respectively, based on industrial production growth across US manufacturing sectors, while calibrating the links $a_{ij}$ using input-output data. The analysis first seeks to determine whether there is a role for lagged IOC at all by comparing the data fit of specifications with lagged IOC to that with contemporaneous IOC based on model selection criteria. Presuming that one of the former is preferred, the role of lagged IOC can be quantified by computing the change in the autocorrelation implied by the estimated model with lagged IOC when the persistence in TFP shocks is set to zero.

I compute log differences of monthly industrial production (IP) indices across 23 manufacturing and mining sectors in the US economy provided by the Federal Reserve Board. The indices are available from January 2005 through August 2022. \todo{sure? not later?} To eliminate seasonal patterns, I regress each series on month-dummies, take the residuals and add back the mean. To construct $A$, I use annual data on input-output (IO) matrices provided by the Bureau of Economic Analysis (BEA). I take the input-output matrix for 2010. Following the theory in (ref), links $a_{ij}$ are calibrated as

where $sales_{j \longrightarrow i}$ is the total value of goods and services purchased by sector $i$ from sector $j$ as determined by the corresponding entry in the BEA's “{use}" table.\footnote{This is in line with the literature. See AcemogluAkcigitKerr2016 for example. Carvalho-TahbazSalehi2019 discuss the IO data in more detail.} The value of $a_{ij}$ shows how many dollars worth of output of sector $j$ sector $i$ needs to purchase in order to produce one dollar's worth of its own output. I abstract from the differences in the calibration of $a_{ij}$ between contemporaneous and lagged IOC, which renders the analysis valid as $\beta \rightarrow 1$. More importantly, the calibration assumes that firms' input shares reported for the course of a year are equal to those at higher frequency intervals. Details on the matching of IP and IO data is provided in (ref).

For the specification with lagged IOC, I consider $p=\{1,2,3,4,5,6\}$ and $q \in \{1,2,3\}$, implying network interaction frequencies of a month, two weeks and 10 days. Under contemporaneous IOC, I take $q=1$, as it already refers to the limit case of an infinitely high frequency of network interactions. In both cases, I consider Normal AR(1) processes for idiosyncratic and aggregate TFP: $(1-\rho_a L) e^a_{\tau} = \varepsilon^a_{\tau} \sim N(0,\sigma^2_a)$ and $(1-\rho_iL) e_{i\tau} = \varepsilon_{i\tau} \sim N(0,\sigma^2_i)$. To separately identify both TFP processes, I normalize $\sigma^2_a = 1$ and $\delta_1 = 1$, re-defining $\delta$. To accommodate the restrictions $\alpha_l \geq 0 \; \forall \; l$ and $\sum_{l=1}^p \alpha_l = 1$, I drop $\alpha_p$ from $\alpha$ and impose the domain restrictions $\alpha_l \in [0,1]$ for $l=1:p-1$ and $\sum_{l=1}^{p-1} \alpha_l \leq 1$. Under lagged IOC, this yields $p-1 + 4n$ unknown parameters: $\theta = (\alpha',\gamma',\delta',\rho_a,\rho', \sigma^{2\prime})'$, where $\rho$ and $\sigma^2$ stack $\{\rho_i\}_{i=1}^n$ and $\{\sigma^2_i\}_{i=1}^n$, respectively. Under contemporaneous IOC, $\alpha$ is dropped from $\theta$, leading to $4n$ parameters.\footnote{ With $n=116$ sectors and $T=211-p$ periods available for estimation, this yields an observations-to-parameters ratio of 52.5 under $p=1$ and 50.7 under $p=6$. }

The inference from (ref) is not applicable because of the autocorrelation and factor structure of TFP processes and due to the restrictions on $\alpha$ under lagged IOC. The likelihood of both models can be evaluated with a Kalman filter, as stated in (ref). I consider Bayesian inference on $\theta$ under uniform priors on the respective domains.\footnote{ Specifically, $\rho_i, \rho_a \in [0,1)$ and $\sigma^2_i, \sigma^2_a \in \mathbb{R}_{++}$ for $i=1:n$, $\delta \in \mathbb{R}^n$ and $\alpha \in [0,1]^{p-1} \cap \{\alpha: ||\alpha||_1 \leq 1\}$. } As a result, the posterior mode equals the Maximum Likelihood estimator. The posterior is obtained numerically using a Sequential Monte Carlo algorithm, which -- as a by-product -- estimates the marginal likelihood and therefore enables model selection. \todo{mention also that SMC traces well whole posterior and is used under possible bi-modality, and in our case there is no such thing (unique MLE, identified params)?}

Results

(ref) reports the Marginal Data Density (MDD) for different specifications of the model with lagged IOC. For comparison, its value under contemporaneous IOC is -11,590. This number is beaten by all but a few lagged IOC specifications with $q=1$. The most preferred specification features $q=2$ and $p=5$, i.e. bi-weekly network interactions where innovations travel 2.5 months along single input-output links. The subsequent analysis is based on this preferred specification.

\todo{compare my alpha and q to Ramey(1989)}

\todo{put IRFs on RHS top of each other}

\todo{put in footnote: this is like mediation analysis in Dufour & Wang (2023); they do it by variable, I do it by connection order.}

(ref) illustrates the composition of impulse responses, analogously as (ref) does for two units that share only a first- or second-order connection. The top-left panel shows the estimated temporal propagation of TFP shocks along supply chain linkages of different order. Remarkably, within the first two months after the shock, the impact is mostly limited to direct customer-sectors. After that, the shock spreads somewhat more quickly to higher-order connections.

The top-right panel shows the strength of network connections of different order from the sector “{Fabricated Metal Products}" to the sectors “{Mining (except oil and gas)}" and “{Chemical Products}", respectively. Firms producing fabricated metals depend on chemical products directly as well as indirectly in their supply chain. In contrast, they rely on mining products only indirectly, though this higher-order dependence is of a similar magnitude.

The lower panels of (ref) illustrate the resulting impulse responses to a respective one standard deviation idiosyncratic TFP shock to Chemical Products and Mining. As a result of its stronger direct reliance on chemicals, the response of the growth in the production of fabricated metals to a TFP shock in the chemical sector materializes much faster than does the response to a TFP shock in mining. Yet, the two are of a similar magnitue at their peaks.

The responses refer to percentage point increases in sectoral output growth. To interpret the magnitudes, recall that the data is in monthly frequency and that the (persistent) response of the level of sectoral output is obtained by summing up the illustrated (transitory) response of output growth. Expectedly, the magnitudes are rather small, as they refer to responses of sectoral output to idioyncratic TFP shocks in a single other sector.

Larger responses are obtained when considering aggregate TFP shocks. On top of their direct effect on the output growth of all sectors, the latter have an indirect effect, as the supply chain network amplifies initial effects. (ref) shows the respones of output growth in the sectors “{Oil and Gas Extraction}" and “{Machinery}" to a one standard deviation shock to aggregate TFP. The oil and gas extraction sector is positioned rather at the top of supply chains. Due to its weak reliance on other sectors as suppliers, its response to aggregate TFP shocks is only weakly amplified by supply chain connections. In contrast, after a similar initial response, the machinery sector experiences a hump-shaped response due to second-order transmission operating via its supplier-sectors. The relevant network-quantity for these IRFs is a sector's weighted reliance on all other sectors as suppliers, with weights given by their exposures to the aggregate TFP shock, $\delta$ (see (ref)).

A similar reasoning explains the differing IRFs of aggregate output growth to sectoral TFP shocks. (ref) shows these responses to TFP shocks in “{Mining (except oil and gas)}" and “{Chemical Products}", respectively. As the chemical sector sits on the top of supply chains, it leads to a much more persistent increase in aggregate industrial production than does an increase in the TFP in the mining industry. The relevant network-quantity for these IRFs is all sectors' weighted reliance on the particular sector in question as a supplier, with weights given by sectors' contribution to aggregate output growth.

Existing studies use a static framework of contemporaneous input-output conversion to show that the effects of sectoral TFP shocks on aggregate output are stronger for sectors with more central positions in the supply chain network. By (ref), the present analysis leads to the same long-run effects, but it sheds light on the transition dynamics. The left panel of (ref) shows the time profiles of the response of aggregate industrial production to TFP shocks in different sectors. It illustrates that, due to different positions in the supply chain network, under lagged IOC sectors differ not only in terms of the strength of their impact on aggregate output, but also by its timing. Sectors at the bottom of supply chains, such as the food and beverage sector, have a much more immediate effect on aggregate output than sectors that act as important suppliers to other sectors in the economy.

Although stronger effects tend to take more time to realize, there is no clear relationship between the strength and timing of the response of aggregate output to sectoral TFP shocks. This is illustrated by the right panel of (ref). For example, a TFP shock in the food and beverage sector has a similar long-term effect on aggregate output as a shock to primary metals, yet the latter materializes much more sluggishly. One month after the TFP shock in the primary metals sector, a similar fraction of the long-term effect on aggregate output has materialized as in the case of a TFP shock to mining support activities, yet the latter are estimated to lead to a stronger long-term effect.

The autocorrelation of aggregate output growth at the posterior mean is estimated to be 0.389. In a hypothetical environment without persistence in exogenous shocks, this number drops to 0.237. As a result, lagged IOC can account for about two thirds of aggregate persistence. Together with persistence in the aggregate TFP process, this number increases to 0.365. In contrast, if only persistence in sectoral TFP processes is added, the autocorrelation increases only to 0.276. Overall, these results indicate that lagged IOC and a single, persistent aggregate TFP process can account wel for business cycles in this RBC environment.

Dimensionality-Reduction by Innovation Transmission through Parsimonious Networks

In (ref), a process $y_t$ is driven by an observed network, and we quantify how network effects materialize over time. In this section, I forecast a rather high-dimensional series $y_t$ and, for the most part, I assume that no network data is available.\todo{"for the most part"...} When $(\alpha,A)$ are jointly estimated and the estimation of $A$ is regularized, the NVAR becomes useful as a dimensionality-reduction technique. It rationalizes the dynamic comovement among all $\{y_{it}\}_{i=1:n}$ by the dynamic innovation transmission along a few bilateral links among units $i=1:n$.

In (ref), I discuss the potential of the NVAR to reduce dimensionality, building on (ref). (ref) then sets up the application to forecast macroeconomic aggregates across countries, and (ref) presents the results. Details are in (ref).

NVAR-Estimation for Dimensionality-Reduction

Even for intermediate $n$, an unrestricted VAR($p$) poorly forecasts $y_t \in \mathbb{R}^n$.\todo{cite..} Relative to it, under $p>1$, the NVAR($p,1$) reduces the number of parameters in the autoregressive matrices from $pn^2$ to $n^2 + p-1$, owing to the assumption that innovations transmit cross-sectionally only via bilateral links; rather than freely relating $y_{it}$ to $\{y_{j,t-l}\}_{j=1:n,l=1:p}$, it uses the $n$ links $\{a_{ij}\}_{i=1:n}$ to compress this information to $x_{i,t-l} = \sum_{j=1}^n a_{ij}y_{j,t-l}$, and in turn the $p-1$ free parameters in $\alpha$ determine the importance of different lags of $x_{it}$ for the dynamics of $y_{it}$.\footnote{ Even though $\mathbb{E}_{t-1}[y_{it}]$ contains the same linear combinations of $\{y_{j,t-l}\}_{j=1:n}$ at all lags $l=1:p$, dynamics at higher horizons $h$ are driven by different linear combinations, since higher-order network connections accumulate over time (see (ref)). } If, in addition, $A$ is estimated parsimoniously, then the NVAR leverages both approaches available to address the large parameter problem in the Wold representation Geweke1984: it reduces the number of parameters and applies shrinkage priors (regularization). Regularization of $A$ is motivated by the sparseness of real world-networks across a wide range of applications\todo{cite} and by the fact that the dynamic comovement of two series $y_{it}$ and $y_{jt}$ can be captured with higher-order connections between $i$ and $j$ without a direct link between them (see (ref)).

(ref) discussed the estimation of $A$ under a Normal prior (L2-penalty) and under an Exponential prior (L1-penalty and restricting $a_{ij}\geq 0$). In the following, these two approaches are labeled NVAR-R and NVAR-L. The joint posterior of $(\alpha,A)$ is obtained by iteratively drawing from the conditional posteriors of $\alpha|A$ and $A|\alpha$, the joint posterior mode -- the Ridge-/Lasso-regularized LS-/ML-estimator -- by iterating on the conditional posterior modes.

When the NVAR-R and NVAR-L are applied for predictive purposes, selecting the hyperparameter $\lambda_a$ -- i.e. the degree of shrinkage -- becomes particularly important. Following GiannoneLenzaPrimiceri2015, setting a hyperparameter to its marginal posterior mode (MPM) under a Uniform hyperprior maximizes the marginal data density (MDD) and, hence, the one-step ahead predictive ability.\todo{optimizes ... under a quadratic loss function?} Under NVAR-R and NVAR-L, respectively, we obtain the conditional posteriors

We get the joint posterior of $(\alpha, A, \lambda_a)$ by augmenting the Gibbs sampler from (ref) with a step to draw $\lambda_a$ from this conditional posterior. In turn, we arrive at the marginal posterior of $\lambda_a$ by simply considering its posterior draws in isolation.

While well-grounded in theory, the approach of GiannoneLenzaPrimiceri2015 is computationally expensive in the present environment, as it requires obtaining the full posterior.\footnote{ This is particularly disadvantageous if only the posterior mode estimator is of interest. Even if the full posterior of $(\alpha, A)$ evaluated at the optimal $\lambda_a$ is desired, however, it is a drawback, as it requires conducting posterior sampling twice. The disadvantage disappears only if one is interested in the marginal posterior of $(\alpha, A)$ under a Uniform prior for $\lambda_a$ (model-averaging rather than -selection). } For NVAR-L, a heuristic approach is to maximize the Bayesian Information Criterion (BIC) suggested by ZouHastieTibshirani2007 for other Lasso-applications. This involves counting the number of non-zero elements in $\hat{A}(\lambda_a)$. For NVAR-R, an analogous criterion is the conditional MDD $p(Y|\lambda_a,B,\alpha,\Sigma)$, which is derived in (ref) and can be maximized when evaluated at $\hat\alpha$ (and $\hat\Sigma$).

\todo{see commented-out text on interpretability and other}

Forecasting-Setup

To validate the NVAR's merit as a dimensionality-reduction technique, I forecast a range of macroeconomic time series across countries and compare its performance to that of the Dynamic Factor Model (DFM) of GewekeZhou1996:

where $u_t\sim N(0,I_n)$, $ \eta_t \sim N(0,I_r)$ and $\Lambda_{1:r,\cdot}$ is lower-triangular with positive diagonal elements. The equivalence result in (ref) suggests that, relative to the DFM, the NVAR restricts factor dynamics, but allows the linear combinations of predictors to vary more flexibly across units and acommodates sparse factors as (linear combinations of) locally important units in the network. Therefore, the NVAR is expected to capture cross-sectional dynamics in finite samples better than the factor model in the presence of many sparse factors or, equivalently, in case of a sparse, yet close-to-full-rank network adjacency matrix. Intuitively, this is the case of dynamics driven by many bilateral links rather than a few influential units. More generally, (ref) suggests the NVAR could improve upon the poor forecasting performance of factor models under a high dispersion of factor loadings across series (see BoivinNg2006).\footnote{ This notably includes the case of sparse factors, i.e. loading-vectors with zero and non-zero entries. As BoivinNg2006 point out, selecting the number of factors separately for each series is a poor remedy: series that depend on less dominant factors are still poorly forecasted, as including more estimated factors adds noise to the forecasts.}

\todo{Can we show that in results? note that it holds not unit-by unit, as I thoguht initially (see commented-out text here)}

To investigate these hypotheses, I use the NVAR and DFM to forecast monthly industrial production (IP) growth, monthly CPI inflation and quarterly real GDP growth across OECD-member countries, -applicants and -partners.\footnote{ As of August 2024, this includes 45 countries. On top of 34 OECD members, there are 8 applicants (Argentina, Brazil, Bulgaria, Croatia, Indonesia, Peru, Romania and Thailand) and 3 partners (China, India, South Africa).} The series are obtained from the IMF's IFS database, and YoY growth rates are used. I limit attention to the pre-COVID periods 2001:M1 - 2019:M12 and 2001:Q1 - 2019:Q4, respectively, and I delete countries with missings or more than two consecutive constants. This yields datasets with $n=42$, $n=34$ and $n=39$ observations. The series are de-seasonalized by subtracting fitted values from a regression on period-dummies, and the resulting series are standardized.

Comparison is based on mean squared errors (MSEs) at different forecasting horizons, whereby the mean is taken over countries as well as forecasting origins. I consider expanding windows with 24 and 16 origins, respectively, ranging from 2017:M12 to 2019:M11 and from 2015:Q4 to 2019:Q3. To reduce the computational burden, the models are fully estimated only at the first origin, after which parameters are fixed and, if required, hidden states are re-estimated by conditioning on these initially estimated parameters. At the first origin, we have $T=203$ and $T=60$ observations, respectively.

I limit the analysis to point forecasts obtained under the posterior mode of $(\alpha,A)$ in the NVAR-R and NVAR-L as presented in (ref) -- the Ridge-/Lasso-regularized ML-/LS-estimator --, and the posterior mode of $(\Lambda,\Phi)$, $\Phi = [\Phi_1, ..., \Phi_p]'$ under a Uniform prior in the DFM -- the ML-/LS-estimator (see (ref)).\todo{and orthogonal factors!} For the DFM, I consider $p=1:4$ lags and $r=10$ factors. In turn, I select the ex-post best-performing specification that minimizes the cumulative MSE over the first three horizons, and I compare all NVAR specifications against this benchmark. This renders the assessment of the NVAR's suitability for forecasting independent of methods to select the number of factors ex-ante. For the NVAR, I also consider $p=1:4$ lags, and I mostly focus on $q=1$ -- the case referred to by (ref) \todo{mostly...}. In both NVAR models, I set $\Sigma = I$, and, unless otherwise stated, I shrink $A$ to $B=0$. The degree of shrinkage, as embodied by $\lambda_a$, is chosen based on two methods. The first takes the MPM and, therefore, maximizes the MDD exactly, while the second approximates this choice by using grid-search to maximize the BIC (in case of the NVAR-L) or the conditional MDD (for the NVAR-R).

Results

(ref) illustrates the results for IP growth and NVARs with $q=1$. The ex-post best DFM features $r=4$ factors and $p=1$ lags. As shown by the black line, it reduces the MSE of one-step ahead forecasts by 13% relative to forecasting the unconditonal mean of zero. As expected, this improvement vanishes for higher horizons. The NVAR-R (blue lines) further reduces the MSE at the first horizon, amounting to -27% relative to the same benchmark (-16% relative to the DFM), whereby the noticeable improvement relative to the DFM persists for the first two horizons. Thereby, both methods to select $\lambda_a$ yield similar answers -- $\lambda_{a,MPM} = 160$ (solid line) and $\lambda_{a,MDD} = 141$ (dashed line) -- and indistinguishable forecasting performances. They both lead to $p=2$ as the ex-post best-performing model. For NVAR-L (red lines), smaller values and differing lag-lengths are selected: $\lambda_{a,MPM} = 31$ with $p=1$ (solid line) and $\lambda_{a,BIC} = 8$ with $p=3$ (dashed line). Nevertheless, the forecasting performances are again similar. For the first horizon, they yield MSE reductions of -42% and -40% relative to the unconditional mean (-33% and -30% relative to the DFM). The improvement is long-lasting, reverting to levels obtained under the DFM and NVAR-R only for six-period ahead forecasts.\footnote{The models' performances under different different choices of $p$ and $r$ are shown in (ref) and (ref).}

The results in (ref) are obtained using the respective posterior mode, i.e. frequentist point estimator. As shown in (ref), the factor models yield similar performances even for forecasts at the posterior mean or the posterior mean forecasts. Selecting the best DFM using these alternative forecasting-types does not change any of the above numbers by more than two percentage points. For NVAR-L, the normalization $\alpha_1=1$ is applied, whereas for NVAR-R, $||\alpha||_1 = 1$ is imposed.\footnote{ Under the former normalization, NVAR-R yields poor performance, as the free parameters in $\alpha$ are increased to extremely high values, while all elements in $A$ are shrunk accordingly. This is likely because, for highly correlated series with $\mathbb{E}[y_t y_{t-1}']\approx \mathbb{E}[y_t y_{t-2}']$, $\alpha_{-1}$ is weakly identified, as e.g. $A y_{t-1} + \alpha_2 A y_{t-2} \approx (1+\alpha_2)Ay_{t-1}$. This issue does not occur for NVAR-L, likely because it shrinks numerous elements of $A$ all the way to zero. The issue does not occur either when estimating the full posterior of NVAR-R or NVAR-L.} The computational time needed to obtain the posterior modes of the DFM and the NVAR for a given $\lambda_a$ are of a similar order of magnitude and amount to 5-10 seconds. However, this burden is considerably increased by the need to re-compute the posterior mode for many different $\lambda_a$ in the search for the value that maximizes MDD or BIC and the need to compute the full posterior distribution in the search for the MPM of $\lambda_a$.\footnote{For NVAR-R, this yields about 50-100s in the former case and about 10min in the latter case. These times are doubled for NVAR-L.}

Qualitatively, the same conclusions apply when forecasting monthly CPI inflation. The left panel of (ref) shows that, in this case, the DFM reduces the one-step ahead MSE by 54% relative to the unconditional mean forecast, with a noticeable improvement persisting throughout the first six months. The NVAR-R improves slightly upon this, yielding -67% and -64% (-27% and -22% relative to the DFM). The NVAR-L delivers again the best performance: -86% and -85% (-68% relative to DFM).

When forecasting quarterly GDP growth, the conclusions change. The performance of the best DFM (-30%) is beaten only slightly and only at the first horizon. The {NVAR-R} models yield -37% and -35% (-10% and -7% relative to DFM), and the NVAR-L with $\lambda_a$-selection according to BIC yields -38% (-12% relative to DFM). At longer horizons, the NVAR-R yields a similar performance as the DFM, beating the unconditional mean by about 10-15%, while the reduction by the mentioned NVAR-L model reverts faster to zero. The NVAR-L with $\lambda_a$ selected by MPM yields -22% and thereby underperforms the DFM by 12%.

\todo{see commented-out section on $B$-selection here and whole appendix-section}

\todo{see commented-out section on $q\neq 1$ here}

Conclusion

In this paper, I develop the Network-VAR (NVAR) -- an econometric framework that rationalizes the dynamics of a cross-sectional variable by the dynamic innovation transmission along bilateral links among cross-sectional units -- and I consider two applications. First, I estimate the contribution of lagged input-output conversion to business cycles through the lens of an RBC economy, thereby providing a structural underpinning of the NVAR. Second, I use the NVAR as a dimensionality-reduction technique for forecasting cross-country macroeconomic aggregates.

More work is needed to explore the extent to which the NVAR is useful to address sparse factor-issues and improve upon the forecasting performance of alternative dimensionality-reduction techniques. Its application for forecasting very high-dimensional processes would benefit from refinements of the crude shrinkage priors used in this paper.

Furthermore, rather than assuming time-invariant links, an important methodological step forward would be to jointly study dynamic network effects and -formation.\todo{cite Bykhovskaya, P007 and also KuersteinerPrucha2020} \todo{mention macro studies on growth via network changes?} Rather than assuming stationarity, an interesting avenue for further research would be to study networked cointegration. Finally, the NVAR could be augmented to accommodate heterogeneous propagation patterns across units or over time, which in the context of a production economy amounts to endogenizing firms' inventory management. I leave these directions for future research.

\todo{mention paper with Xin? Mention also papers with Wayne (ID as $q \rightarrow \infty$) and Frank/Mansi (nowcasting)? Mention them also/only in main text?}

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