Dynamic Networks in Large Financial and Economic Systems
DDynamic Networks in Large Financial and
Economic Systems
Jozef
Barun´ık ‡ and Michael Ellington † July 16, 2020
Abstract
We propose new measures to characterize dynamic network connections in largefinancial and economic systems. In doing so, our measures allow one to describeand understand causal network structures that evolve throughout time and overhorizons using variance decomposition matrices from time-varying parameter VAR(TVP VAR) models. These methods allow researchers and practitioners to examinenetwork connections over any horizon of interest whilst also being applicable to awide range of economic and financial data. Our empirical application redefinesthe meaning of “big” in big data, in the context of TVP VAR models, and trackdynamic connections among illiquidity ratios of all S&P500 constituents. We thenstudy the information content of these measures for the market return and realeconomy.
JEL Classifications: C10, C40, C55, C58, G00Keywords: Dynamic networks, Network connections, Time-varying parameter VAR, Liq-uidity ‡ Institute of Economic Studies, Charles University, Opletalova 26, 110 00, and The CzechAcademy of Sciences, IITA, Pod Vod´arenskou Vˇeˇz´ı 4, 182 08, Prague, Czech Republic [email protected]†
University of Liverpool Management School, Chatham Building, Liverpool, L69 7ZH, UK [email protected] a r X i v : . [ ec on . E M ] J u l Introduction
Economic agents create links among one another. These relationships can be seen througha variety of data such as: the co-movement of economic variables like consumption growth(e.g. Richmond, 2019); financial variables including stock return volatilities (e.g. Her-skovic et al., 2020); and firm characteristics describing the supply chain (e.g. Garveyet al., 2015). A natural way to describe and understand these connections is to viewthem as a network. Viewing linkages in this manner allows one to characterize and trackconnections through entities of interest, and also contributes to understanding how shockspropagate throughout a system.However, connections evolve over time and shocks creating network linkages pos-sess differences in persistence. The implication here is that network connections, andshock propagation throughout these systems, are dynamic over time and across horizons.Adding to this, the emergence of big data sources means that the dimensions of networkstructures that researchers and practitioners wish to understand are growing rapidly.Despite these two issues, dynamic networks remain ill-defined and poorly understood.Researchers and practitioners are bound to thinking in terms of correlation-based mea-sures and methods that permit time-variation into the modelling process either beinginfeasible or computationally inefficient.In this paper, we propose dynamic network measures that stem from time-varyingparameter VAR (TVP VAR) models. Our measures not only allow users to describe andunderstand network structures that evolve throughout time and over horizons, but alsoare readily available in the presence of big data. Adding to this, they have direct causalinterpretation that permits the understanding of how shocks with different persistencecreate dynamic networks among economic and financial variables. Our methods areapplicable to a wide range of economic and financial data. We illustrate by characterizingdynamic network connections among illiquidity proxies of N =496 financial assets. Wethen demonstrate its usefulness in two applications. First we relate dynamic networkmeasures the market return, and then to the real economy and financial stress indicators.Typically, the network literature in economics and finance study static networks (seee.g. Elliott et al., 2014; Glasserman and Young, 2016). To illustrate this, Figure 1 (a)shows a star topology where variable 1 is central and its shocks propagate to the remain-ing N variables. In this case, variable 1 is creating a directed, or asymmetric network inwhich the nodes are connected with the same strength. Figure 1 (b) is an N star networktopology where the links among the N central variables have weighted strengths. While1he network is more complex, it is missing key ingredients to be able to describe dynamicnetwork in the economic and financial systems. First, these networks cannot adequatelycapture evolving relationships among variables. Second, the network describes aggregateconnections across different horizons, while short-term, medium-term, and long-term con-nections are hidden. x x x x x x N (a) Asymmetric network x x x x N (b) Weighted asymmetric network
Figure 1: Emergence of Network:
The sub-figure (a) presents a network containing x , . . . , x N variables represented as nodes. Here node x influences all other x , . . . , x N nodes.Nodes x , . . . , x N are not connected to each other and also do not influence node x . The sub-figure (b) presents an N -star network where nodes x , . . . , x N are connected to a set of nodesexclusively, and also to one another. Arrows denote the direction of the connection and thedensity of the line denotes the strength of the connections. Our approach provides a potential resolution to the account for the two key ingre-dients prior work misses. To illustrate, Figure 2 introduces a dynamic network withmultiple layers that represent distinct connections among variables. We interpret theselinks as dynamic horizon specific network connections where the layers distinguish short-term and long-term linkages. Without loss of generality, the curves connecting the N central variables allow for the possibility of connections across layers. We represent timedynamics of network connections at periods t = 1 , . . . , T on a time line.For simplicity we assume that the N central variables are the same across each layerbut will have connections of different strengths over time and horizon. Specifically, net-work connections are more intense in the long run during the first period relative to theshort run meaning that relationships are more persistent and a shock to the central vari-ables will create linkages with a long-run strengths. During the second period long-termnetwork connections lose their strength and network connections are stronger in the shortrun. 2 h o r t H o r i z o n x x x x N L o n g H o r i z o n x x x x N Sh o r t H o r i z o n x x x x N L o n g H o r i z o n x x x x N t j ↑ t k ↑ T − T Figure 2: A Dynamic Horizon Specific Network:
This figure presents a multi-layerN-star network with snapshot of two time and two horizon specific network layers. Arrowsdenote directions of connections and the line density denotes strength. The curves from the N central nodes allow for connections to spillover across layers. We interpret each layer as anetwork specific to a horizon of interest; for example short-term depicted by light blue colorand long-term depicted by light red color. Our main objective is to provide a framework for tracking and understanding thecausal nature of dynamic networks that form within a potentially large system of vari-ables. Current studies almost exclusively examine static networks that mimic time dy-namics using an approximating window (see e.g. Demirer et al., 2018) . In doing so, weemploy a locally stationary TVP VAR that allows us to estimate the adjacency matrix fora network at each point in time using the variance decomposition matrix. We decomposethis into horizon specific components that allow us to disentangle short-term, medium-term, and long-term network connections. Our methods are general enough to permitone to examine any horizon of interest, such as over the business cycle for macroeconomicapplications, or even on a daily basis for financial applications.Variance decompositions are a natural way to characterize a network working withcausal linkages. Specifically, variance decompositions tell us much of the future variance ofvariable j is due to shocks in variable k , and under mild assumptions have a direct causalinterpretation (Rambachan and Shephard, 2019). The variance decomposition matrix Geraci and Gnabo (2018) estimates multiple pairwise time varying parameter models in an attemptto characterize a network of financial stocks using autoregressive coefficients. j to k link is not necessarily the same as the k to j link, and hence, the adjacencymatrix is not symmetric. Therefore we can define weighted, directed versions of networkconnectedness statistics readily that include degrees, degree distributions, distances anddiameters.We consider horizon specific dynamics of network connections by using the spectralrepresentation of the variance decomposition matrix. A shock with a strong long-termeffect will have high power at low frequencies, and in case it transmits to other variables,it creates long-term connectedness. Long-term connectedness of economic variables maybe attributable to permanent changes in expectations of economic agents. Conversely,short-term connections may appear when changes are transitory. Barun´ık and Kˇrehl´ık(2018) define notion of the horizon specific connectedness measures for a simple VARthat we further generalize to a locally stationary processes here. To capture horizon-specific dynamic network connections, we propose a general framework for decomposingthe aggregate dynamic network to any frequency, or horizon specific, band of interest.Our empirical applications show the usefulness of dynamic network connections thatform on individual stock illiquidities for the market return, and the real economy . Re-garding the former, we uncover a strong link between the market return and illiquiditynetwork connections that intensifies during periods of financial stress. We link this to thereinforcing mechanism between market and funding liquidity in Brunnermeier and Ped-ersen (2009) and argue that our measures may be of use for portfolio formation and/orrebalancing. Concerning the latter, we document a negative (positive) link with real ac-tivity (financial stress indicators). We relate this finding to Kiyotaki and Moore (2019)and suggest that our illiquidity connectedness measures act as indices that track horizon There is a growing literature that focuses on illiquidity. For example some focus on the pricingimplications of liquidity risk (see e.g. P´astor and Stambaugh (2003), Acharya and Pedersen (2005),Bekaert et al. (2007)). Meanwhile others concentrate on the dynamics of market illiquidity and marketreturns (Amihud, 2002; Chen et al., 2018); as well as the link between market illiquidity and the realeconomy (Næs et al., 2011; Ellington, 2018)
Our measures rely on locally stationary processes because this assumes the process isapproximately stationary over a short time interval. Intuitively, this allows us to incor-porate time-variation and establish our measures of horizon specific time-varying networkconnections. Consider a doubly indexed N -variate time series ( X t,T ) ≤ t ≤ T,T ∈ N with com-ponents X t,T = ( X t,T , . . . , X Nt,T ) > that describe all variables in an economy. Here t refersto a discrete time index and T is an additional index indicating the sharpness of the localapproximation of the time series ( X t,T ) ≤ t ≤ T,T ∈ N by a stationary one. Coarsely speaking,we can consider ( X t,T ) ≤ t ≤ T,T ∈ N to be a weakly locally stationary process if, for a large T , given a set S T of sample indices such that t/T ≈ u over t ∈ S T , the sample ( X t,T ) t ∈ S T approximates the sample of a weakly stationary time series depending on the rescaledlocation u . Note that u is a continuous time parameter referred to as the rescaled timeindex, and T is interpreted as the number of available observation, hence 1 ≤ t ≤ T and u ∈ [0 , p as X t,T = Φ ( t/T ) X t − ,T + . . . + Φ p ( t/T ) X t − p,T + (cid:15) t,T , (1)where (cid:15) t,T = Σ − / ( t/T ) η t,T with η t,T ∼ N ID (0 , I M ) and Φ ( t/T ) = ( Φ ( t/T ) , . . . , Φ p ( t/T )) > are the time varying autoregressive coefficients. In a neighborhood of a fixed time point u = t /T , we approximate the process X t,T by a stationary process f X t ( u ) as f X t ( u ) = Φ ( u ) f X t − ( u ) + . . . + Φ p ( u ) f X t − p ( u ) + (cid:15) t , (2)with t ∈ Z and under suitable regularity conditions | X t,T − f X t ( u ) | = O p ( | t/T − u | +1 /T )5hich justifies the notation “locally stationary process.” Crucially, the process has timevarying VMA( ∞ ) representation (Dahlhaus et al., 2009; Roueff and Sanchez-Perez, 2016) X t,T = ∞ X h = −∞ Ψ t,T ( h ) (cid:15) t − h (3)where Ψ t,T ( h ) ≈ Ψ ( t/T, h ) is a stochastic process satisfying sup ‘ || Ψ t − Ψ ‘ || = O p ( h/t ) for1 ≤ h ≤ t as t → ∞ . Since Ψ t,T ( h ) contains an infinite number of lags, we approximatethe the moving average coefficients at h = 1 , . . . , H horizons. The connectedness measuresrely on variance decompositions, which are transformations of the Ψ t,T ( h ) and will allowthe measurement of the contribution of shocks to the system.Since a shock to a variable in the model does not necessarily appear alone, i.e. or-thogonally to shocks to other variables, an identification scheme is crucial in calculatingvariance decompositions. We will adapt a generalized identification proposed by Pesaranand Shin (1998) to a locally stationary processes. An important feature of the gener-alized impulse responses is its direct causal interpretation. Under the mild conditions(Rambachan and Shephard, 2019), our measures will be causal measures.Moreover, as argued by Barun´ık and Kˇrehl´ık (2018) a natural way to describe horizonspecific dynamics (the long-term, medium-term, or short-term) of the connectedness inthe network is to consider a spectral representation of the approximating model. Henceinstead of impulse responses in the system, we will focus on the frequency response of ashock that in addition will be local. As a building block of the measures, we consider atime-varying frequency response function Ψ t/T e − iω = P h e − iωh Ψ t,T ( h ) which we obtainfrom a Fourier transform of the coefficients with i = √− X t,T at frequency ω which is locally thesame as the spectral density of f X t ( u ) at u = t/T can be defined as a Fourier transformof VMA( ∞ ) filtered series as S X ( u, ω ) = ∞ X h = −∞ E h f X t + h ( u ) f X > t ( u ) i e − iωh = n Ψ ( u ) e − iω o Σ ( u ) n Ψ ( u ) e + iω o > . (4)The time-varying spectral density is a key quantity for understanding frequency dy-namics, since it describes how the variance of the time varying covariance of X t,T isdistributed over the frequency components ω . Using the spectral representation for thelocal covariance that is associated with local spectral density, E h f X t + h ( u ) f X > t ( u ) i = Z π − π S X ( u, ω ) e iωh dω (5)6e can naturally introduce time varying frequency domain counterparts of variance de-compositions.The following proposition establishes time varying spectral representation of the vari-ance decomposition of shocks from asset j to asset k , and it is central to the developmentof the network connectedness measures in the time-frequency domain. Proposition 1 (Dynamic Adjacency Matrix) . Suppose X t , T is a weakly locally stationaryprocess with σ − kk ∞ X h =0 (cid:12)(cid:12)(cid:12)(cid:12)h Ψ ( u, h ) Σ ( u ) i j,k (cid:12)(cid:12)(cid:12)(cid:12) < + ∞ , ∀ j, k. Then the time-frequency variance decompositions of the j th variable at a rescaledtime u = t /T due to shocks in the k th variable on the frequency band d = ( a, b ) : a, b ∈ ( − π, π ) , a < b forming a dynamic adjacency matrix is defined as h θ ( u, d ) i j,k = σ − kk Z ba (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Ψ ( u ) e − iω Σ ( u ) (cid:21) j,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω Z π − π "n Ψ ( u ) e − iω o Σ ( u ) n Ψ ( u ) e + iω o > j,j dω (6) where Ψ ( u ) e − iω = P h e − iωh Ψ ( u, h ) is local impulse transfer function or frequency re-sponse function computed as the Fourier transform of the local impulse response Ψ( u, h ) Proof.
See Appendix A.It is important to note that h θ ( u, d ) i j,k is a natural dissagregation of traditional vari-ance decompositions to time-varying frequency bands, since portion of the local errorvariance of the j th variable at a given frequency band due to shocks in the k th variableis scaled by the variance of the j th variable. Note that while the Fourier transform ofthe impulse response is generally a complex valued quantity, the quantity introduced byproposition (1) is the squared modulus of the weighted complex numbers, thus producinga real quantity.This relationship is an identity which means the integral is a linear operator, sum-ming over disjoint intervals covering the entire range ( − π, π ) recovers the time domaincounterpart of the local variance decomposition. The following remark formalizes thisfact. Note to notation: [ A ] j,k denotes the j th row and k th column of matrix A denoted in bold. [ A ] j, · denotes the full j th row; this is similar for the columns. A P A , where A is a matrix that denotes thesum of all elements of the matrix A . emark 1 (Aggregation of Adjacency Matrix) . Denote by d s an interval on the realline from the set of intervals D that form a partition of the interval ( − π, π ) , such that ∩ d s ∈ D d s = ∅ , and ∪ d s ∈ D d s = ( − π, π ) . Due to the linearity of integral and the constructionof d s , we have h θ ( u, ∞ ) i j,k = X d s ∈ D h θ ( u, d s ) i j,k . Remark (1) is important as it establishes the aggregation of frequency specific net-work connectedness measures to its time domain, total counterpart. Hence one can eas-ily obtain short-term, medium-term, and long-term time varying network connectednesscharacteristics that will always sum up to an aggregate time domain counterpart.As the rows of the time-frequency network connectedness do not necessarily sum toone, we normalize the element in each by the corresponding row sum h e θ ( u, d ) i j,k = h θ ( u, d ) i j,k , N X k =1 h θ ( u, ∞ ) i j,k (7)Our notion that we can approximate well the process X t,T , by a stationary process f X t ( u ) in a neighborhood of a fixed time point u = t/T , means that all associated localquantities approximate well their time varying counterparts. Following the argumentsin Dahlhaus (1996) and using mild assumptions one can easily see that local variancedecompositions at a frequency band e θ ( u, d ) approximate well the time-varying variancedecompositions of the process X t,T .Note that local generalized variance decompositions form a dynamic network adja-cency matrix defining a time-varying network at a given frequency band. Hence we canuse these measures directly as a time-varying network connectedness that contains richerinformation in comparison to typical network analysis. In a typical network, adjacencymatrix is filled with zero and one entries depending on the node being linked or not. Inour notion, variance decompositions can be viewed as weighted links showing strengths ofthe connections. In addition, the links are directional, meaning that the j to k link is notnecessarily the same as the k to j link, and hence the adjacency matrix is asymmetric.Even more important, using our notion above, the adjacency matrix is time-varying andfrequency specific that allows the study of time-varying network connectedness at variousfrequency bands of the user’s choice.Now we can define network connectedness measures that characterize a time varyinghorizon specific network. We define local network connectedness measures at a given8requency band as the ratio of the off-diagonal elements to the sum of the entire matrix C ( u, d ) = 100 × N X j,k =1 j = k h e θ ( u, d ) i j,k , N X j,k =1 h e θ ( u, ∞ ) i j,k (8)This measures the contribution of forecast error variance attributable to all shocks in thesystem, minus the contribution of own shocks. Similar to the local network aggregateconnectedness measure that infers the system-wide connectedness, we can define measuresthat will reveal when an individual variable in the economy is transmitter or receiver ofshocks. The local directional connectedness that measures how much of each variables’s j variance is due to shocks in other variable j = k in the economy is given by C j ←• ( u, d ) = 100 × N X k =1 k = j h e θ ( u, d ) i j,k , N X j,k =1 h e θ ( u, ∞ ) i j,k , (9)defining the so-called from connectedness. Note that this quantity can be precisely in-terpreted as from-degrees (often called out-degrees in the network literature) associatedwith the nodes of the weighted directed network represented by the variance decompo-sitions matrix generalized to a time-varying frequency specific quantity. Likewise, thecontribution of variable j to variances in other variables is computed as C j →• ( u, d ) = 100 × N X k =1 k = j h e θ ( u, d ) i k,j , N X j,j =1 h e θ ( u, ∞ ) i k,j (10)and is the so-called to connectedness. Again, this can be precisely interpreted asto-degrees (often called in-degrees in the network literature) associated with the nodes ofthe weighted directed network represented by the variance decompositions matrix. Thesetwo measures show how other assets contribute to the risk of asset j , and how asset j contributes to the riskiness of others, respectively, in a time varying fashion at a chosenfrequency band. Proposition 2 (Reconstruction of Dynamic Network Connectedness) . Denote by d s aninterval on the real line from the set of intervals D that form a partition of the interval ( − π, π ) , such that ∩ d s ∈ D d s = ∅ , and ∪ d s ∈ D d s = ( − π, π ) . We then have that ( u, ∞ ) = X d s ∈ D C ( u, d s ) C j ←• ( u, ∞ ) = X d s ∈ D C j ←• ( u, d s ) C j →• ( u, ∞ ) = X d s ∈ D C j →• ( u, d s ) (11) where C ( u, ∞ ) are local network connectedness measures aggregated over frequencies with H → ∞ .Proof. See Appendix A.In light of the above, all local frequency connectedness measures C ( u, d ) for u = t/T approximate well the time-varying frequency connectedness of the process X t,T , that is C t,T ( d ).Finally, we note that our measures can be have a direct causal interpretation. Ram-bachan and Shephard (2019) provide a discussion about causal interpretation of impulseresponse analysis in the time series literature. In particular, they argue that if an observ-able time series is shown to be a potential outcome time series, then generalized impulseresponse functions have a direct causal interpretation. Potential outcome series describeat time t the output for a particular path of treatments.In the context of our study, paths of treatments are shocks. The assumptions re-quired for a potential outcome series are natural and intuitive for a typical economicand/or financial time series: i) they depend only on past and current shocks; ii) seriesare outcomes of shocks; and iii) assignment of shocks depend only on past outcomes andshocks. The dynamic adjacency matrix we introduce in Proposition 1 is a transformationof generalized impulse response functions. Therefore, the dynamic adjacency matrix andall measures that stem from manipulations of its elements possess a causal interpretation;thus establishing the notion of causal dynamic network measures. In light of the assumptions that underpin our measures, we conjecture that the economy(or market) follows a stable time-varying parameter heteroskedastic VAR (TVP-VAR)model as in (1). To obtain the time-varying coefficient estimates at a fixed time point10 = t /T , Φ ( u ) , ..., Φ p ( u ), and the time-varying covariance matrices, Σ ( u ), we estimatethe model using Quasi-Bayesian Local-Liklihood (QBLL) methods (Petrova, 2019).Specifically, this approach uses a kernel weighting function that provides larger weightsto observations that surround the period whose coefficient and covariance matrices are ofinterest. Using conjugate priors, the (quasi) posterior distribution of the parameters ofthe model are analytical. This alleviates the need to use a Markov Chain Monte Carlo(MCMC) simulation algorithm and permits the use of parallel computing. Note also thatin using (quasi) Bayesian estimation methods, we obtain a distribution of parametersthat we use to construct network measures that provide confidence bands. Details ofthe model and estimation algorithm are in Appendix B. We provide a computationallyefficient package
DynamicNets.jl in JULIA and
DynamicNets in MATLAB that allowsone to obtain our measures on data the researcher desires. We provide some details on estimation here. First, the variance decompositions ofthe forecast errors from the VMA( ∞ ) representation require a truncation of the infinitehorizon with a H horizon approximation. As H → ∞ the error disappears (L¨utkepohl,2005). We note here that H serves as an approximating factor and has no interpretationin the time-domain. We obtain horizon specific measures using Fourier transforms and setour truncation horizon H =100; results are qualitatively similar for H ∈ { , , } .Second in computing our measures, we diagonalize the covariance matrix because ourobjective is to focus on the causal affects of network connections. The Ψ ( u, h ) matrixembeds the causal nature of network linkages, and the covariance matrix Σ ( u ) containscontemporaneous covariances within the off-diagonal elements. In diagonalizing the co-variance matrix we remove the contemporaneous effects and focus solely on causation. To illustrate our methodology, we focus on characterizing network connections amongilliquidity proxies of financial assets. However, it is worth noting that our dynamicnetwork measures can be applied to a wide range of economic and financial data. An Unlike traditional TVP VARs time-variation evolves in a non-parametric manner thus making noassumption on the laws of motion within the model. Typically, the model of Primiceri (2005), and manyextensions, assume parameters evolve as random walks or autoregressive processes. The packages are available at https://github.com/barunik/DynamicNets.jl and https://github.com/ellington/DynamicNets N =496 stocks, we compute their daily returns R t = P Di =1 ( p t,i − p t,i − )and trading volume VOL t = P Di =1 vol t,i on day t where D denotes the total number of12ntraday observations. The subscript i denote intraday observations which we observe at5 minute intervals; and p t,i , vol t,i are the intraday price and trading volume of the assetrespectively. We then convert trading volume into $ trading volume by multiplying bythe closing price of the stock on day t . We then compute each stock’s Amihud (2002)ratio as ILLIQ t = | R t | DVOL t (12)where | R t | is the stock’s day t absolute daily return, and DVOL t is the stock’s dollartrading volume on day t .The Amihud (2002) ratio captures the price impact dimension of illiquidity and mea-sures the elasticity of the stock price with respect to a $1, in our case, change in tradingvolume. This measure has strong theoretical links with the price impact coefficient inKyle (1985). This tracks the sensitivity of asset prices to the order flow; empirically,Goyenko et al. (2009) show that the Amihud (2002) ratio is a good proxy for price im-pact. Furthermore, these price impact ratios link well with the economic mechanism ofliquidity shocks in Kiyotaki and Moore (2019).A feature of a stock’s illiquidity is its positive correlation with volatility. This raisesthe question whether proxies for illiquidity capture illiquidity or effects attributable tovolatility. Also, volatility may contain information regarding future prices and the stateof the economy that bears no link with illiquidity (Bansal et al., 2014). Following Chenet al. (2018), we orthogonalize each stock’s ILLIQ t measure by taking the residuals ofa regression of ILLIQ t on the corresponding stock’s contemporaneous realized volatilitymeasure. Formally for each stock, their volatility-free measure of illiquidity is:ILLIQ ⊥ RV t = ILLIQ t − a − b × RV t (13)where a and b are regression coefficients, and RV t is the stock’s realized volatility measurethat we construct as RV t = qP Di =1 ( p t,i − p t,i − ) with D denoting the number of intradayobservations, and p t,i is the intraday stock price.We estimate the TVP VAR as in (1) on N =496 stock’s volatility-free illiquidity mea-sures with p =2 lags on our T =3275 days of data. We estimate dynamic horizon specificnetwork measures on a 48-core server and for every time period we generate 500 simu-lations of the quasi-posterior distribution; this results in a total estimation time of 10days.We compute dynamic network measures at short-, medium-, and long-term horizons.Short-term captures connections at horizons 1-day to 1-week; medium-term captures13onnections at horizons 1-week to 1-month; and long-term captures at horizons greaterthan 1-month. Figure 3 shows total dynamic network connectedness, we define C dt,T , d = { S , M , L } over the short-term, S; medium-term, M; and long-term, L as an approximationfor the theoretical quantity C ( u, d ) such that C dt,T ≈ C ( u, d ). C S t,T C M t,T C L t,T Figure 3: Illiquidity Dynamic Network Connectedness from July 8, 2005 to August31, 2018 C dt,T , d = { S , M , L } This figure plots the quasi posterior median and one standard-deviation percentiles of to-tal illiquidity network connectedness specific to the short-term, medium-term and long-term C dt,T , d = { S , M , L } for S&P500 stock returns from July 8, 2005 to August 31, 2018. We defineshort-term, S, as connections made over the 1 day to 1 week horizon; we characterize connec-tions over the medium-term as 1 week to 1-month and long-term as horizons greater than 1month. We can see that short-term illiquidity network connectedness is far stronger thanmedium-term and long-term network connectedness. We expect this to occur since ourmeasures of illiquidity are orthogonal to realized volatility that stem from an OLS regres-sion. Furthermore, observe that peaks in short-term and long-term illiquidity networkconnectedness appear in conjunction with key stock market events. In particular, connect-edness rises: i) during the 2007–2009 recession; ii) late 2010 to mid 2011 that correspondswith the European sovereign debt crisis and bear market in 2011; and iii) the 2015-201614tock market sell-off where our measures peak in August 2015.On the whole, our measures provide a useful characterization of illiquidity conditionsinto their respective horizon specific components. We interpret our measures of illiquiditynetwork connectedness as indexes tracking systemic risk that formulates as a result ofilliquidity shock propagation throughout the market. The next section provides twoapplications that demonstrates the use of our measures in the context of return dynamicsand real activity on the basis of theoretical arguments and empirical evidence we outlineearlier.
In this section, we conduct two exercises that demonstrate the usefulness of our dynamicnetwork measures that from on illiquidity proxies of S&P500 constituents. First, we usetotal network connectedness and explore the in-sample dynamics with market returns.We then examine the contemporaneous links of total network connectedness with realactivity and financial uncertainty indicators.
Here we explore the relationship between market returns on total illiquidity connect-edness. In doing so, we construct a market capitalisation weighted index using all ourS&P500 constituents on a daily basis and use the first order log-difference (multipliedby 100) as the market return, R t,T . Within this application our goal is to explore thedynamic relationship between market returns and illiquidity connectedness. This is be-cause studies examining the link between illiquidity and returns suggest that there is anegative link between illiquidity and returns (Chen et al., 2018). Economically this makessense, as liquidity declines prices fall which causes returns to deplete. In the context ofour exercise, as total illiquidity connectedness rises, illiquidity shocks propagate acrossassets. In turn, liquidity dries up causing prices decline and returns to deteriorate. How-ever, literature also documents: i) time-varying liquidity risk premia; and ii) businesscycle contingent predictions of market returns (see e.g. P´astor and Stambaugh, 2003;Watanabe and Watanabe, 2007).In order to account for the above, we estimate univariate specifications of the time-varying parameter methods in Petrova (2019) of which we provide details in the AppendixC. We control for three popular specifications to obtain loadings; namely the market risk15remium; the Fama and French (1993) three factors; and the Fama and French (2015)five factors. All factor data are from Kenneth French’s Data Library. In each regression,we include only one of our network connectedness measures which by definition have highcorrelations. Formally R t,T = α ( t/T ) + β ∆ C d ( t/T )∆ C dt − ,T + β > ( t/T ) X t − ,T + (cid:15) t,T (cid:15) t,T (cid:118) N (cid:16) , σ t,T (cid:17) (14) X t − ,T = MKT t − ,T MKT t − ,T , SMB t − ,T , HML t − ,T MKT t − ,T , SMB t − ,T , HML t − ,T , CMA t − ,T , RMW t − ,T (15)where α ( t/T ) is the time-varying intercept; β ∆ C d ( t/T ) is the sensitivity of our S&P500index return to lagged changes in horizon specific or aggregate network connectednesswith d = { S , M , L , ∞} here ∞ refers to aggregate network connectedness that sums con-nections over horizons; β > ( t, T ) is a vector of coefficients holding loadings to other laggedfactors we use as controls; and X t − ,T is a matrix containing lagged factors. MKT t − ,T is the lagged market risk premium; SMB t − ,T , HML t − ,T , CMA t − ,T , and RMW t − ,T arethe small-minus-big (Fama and French, 1993), high-minus-low (Fama and French, 1993),conservative-minus-aggressive (Fama and French, 2015), and robust-minus-weak (Famaand French, 2015) factors respectively.Figure 4 reports the beta coefficients associated to changes in aggregate networkilliquidity connectedness. In general, it is clear that there is substantial time-variation inthe response of market returns to changes in aggregate illiquidity network connectedness.Specifically, we can see that lagged changes result in lower returns from 2007 to 2009and again in late-2010 and throughout 2011. Returns are most sensitive to changes inaggregate illiquidity network connectedness during 2008 with an economically meaningful(posterior median) estimate hovering around -1.45 in mid-to-late 2008. We also see thatduring late 2010 and throughout the 2011, the aggregate illiquidity network risk betabecomes significant, both statistically and economically, with a posterior median estimatein the range of -0.4 to -0.5. This period coincides with the European Sovereign debt crisis,the 2010 flash crash, and the S&P500’s bear market during spring/summer of 2011.We now turn to the links between horizon specific network connectedness and themarket return. In Figure 5 we plot the betas associated to changes in: i) short-term net-work connectedness; ii) medium term network connectedness; and iii) long-term network16onnectedness. The columns in this figure distinguish between additional controls in eachregression. The leftmost columns control for the market risk premium and the middleand rightmost columns control for the Fama and French (1993) 3-factors and Fama andFrench (2015) 5-factors respectively. − β ∆ C ∞ ( t / T ) Controlled for: CAPM 2005 2010 2016 − Fama French 3-factorsMedian 2005 2010 2016 − Fama French 5-factors
Figure 4: Market return betas with respect to aggregate dynamic network con-nectedness, β ∆ C ∞ ( t/T )This figure plots the quasi posterior median and one standard-deviation percentiles of betacoefficients with respect to changes in aggregate dynamic network connectedness, β ∆ C ∞ ( t/T ),for S&P500 stock returns from July 12, 2005 to August 31, 2018. The leftmost columns arefrom regressions controlling for the market risk premium, CAPM; the middle columns are fromregressions controlling for the Fama and French (1993) 3-factors; and the rightmost columnsare from regressions controlling for the Fama and French (2015) 5-factors. Two main factors emerge from Figure 5. First, the sensitivity of S&P500 returnsto changes in horizon specific dynamic network connectedness are robust to controls weaccount for. In particular, we can see that the time profile of these betas are similaracross all specifications. Second, there are substantial differences in the response ofmarket returns to changes in horizon specific dynamic network connectedness. Prior toand during the 2008 recession, S&P500 returns are significantly and positively associatedto changes in short-term illiquidity network connectedness. Turning our attention tomedium-term and long-term illiquidity network connections during the same period, theimpact is negative. The same behaviour occurs in late 2010 and throughout 2011 thatcorresponds with the European sovereign debt crisis and the short-lived bear market in2011.In general, the mechanism between aggregate illiquidity network connectedness and17
005 2010 2016 − − . . β ∆ C S ( t / T ) Controlled for: CAPM 2005 2010 2016 − − . . Fama French 3-factors 2005 2010 2016 − − . . Fama French 5-factors − . − . . . β ∆ C M ( t / T ) − . − . . . − . − . . . − . − . . . β ∆ C L ( t / T ) − . − . . . Median 2005 2010 2016 − . − . . . Figure 5: Market return betas with respect to horizon specific dynamic networkconnectedness, β ∆ C d ( t/T ) , d = { S , M , L } This figure plots the quasi posterior median and one standard-deviation percentiles of betacoefficients with respect to changes in horizon specific dynamic network connectedness, β ∆ C d ( t/T ) , d = { S , M , L } , for S&P500 stock returns from July 12, 2005 to August 31, 2018.The leftmost columns are from regressions controlling for the market risk premium, CAPM; themiddle columns are from regressions controlling for the Fama and French (1993) 3-factors; andthe rightmost columns are from regressions controlling for the Fama and French (2015) 5-factors.The top row reports sensitivities of the S&P500 returns to changes in short-term network con-nectedness, β ∆ C S ( t/T ); while the middle and bottom rows show S&P500 return sensitivities tochanges in medium-term and long-term network connectedness, β ∆ C M ( t/T ) , β ∆ C L ( t/T ) respec-tively. Here we examine the information content of illiquidity network connectedness for real ac-tivity and financial uncertainty. The financial sector helps the economy function throughintermediation. As we outline earlier, a growing literature exists quantifying macro-financial linkages; particularly through the lens of illiquidity (see e.g. Ellington, 2018).Furthermore, history serves as a reminder that systemic shocks to the financial systemcan cause deep and long recessions (Hubrich and Tetlow, 2015).We select the main three monthly indicators that measure economic activity andfinancial uncertainty, these are: the Aruoba-Diebold-Scotti (ADS) Business ConditionIndex (Aruoba et al., 2009); the Kansas City Financial Stress Index (KCFSI) (see Hakkioet al., 2009); and the Chicago Board of Exchange (CBOE) VIX. We collect them accordingto their availability at a monthly frequency . To match the monthly frequency of our The Aruoba-Diebold-Scotti (ADS) Business Condition Index tracks real business conditions at a highfrequency and it is based on economic indicators. It is collected from: . The Kansas City Financial Stress Index (KCFSI) is ameasure of stress in the U.S. financial system based on eleven financial market variables (see Hakkio et al.,2009). It is collected from .VIX C dt , d = { S , M , L } .Figure 6 reports our real activity and financial uncertainty indicators, along with hori-zon specific illiquidity network connectedness measures C dt , d = { S , M , L } on a monthlybasis from July 2005 to December 2017. We normalize all variables to have zero meanand unit variance so they are interpretable. The leftmost plot shows the ADS businessconditions index. It is clear that our measures of network connectedness have a strongnegative link with real activity. Specifically we can see that trough during the 2008recession corresponds with peaks in our C dt , d = { S , M , L } measures. Turning to themiddle and rightmost plots, there is a clear positive relationship between our measuresof illiquidity connectedness and financial uncertainty with peaks and troughs matchingthroughout these monthly time-series.For our purposes we want to explore the contemporaneous relationships between ourrespective measures of illiquidity network connectedness and real activity (financial un-certainty). We therefore focus on first differences of the series we plot in Figure 6. Intu-itively, we expect that changes in our illiquidity network connectedness measures have anegative (positive) contemporaneous relationship with changes in real activity (financialuncertainty).We relate dynamic illiquidity network connectedness using the following OLS regres-sion ∆ X t = α + γ ∆ C d ∆ C dt + γ ILLIQ ∆ILLIQ
MKT t (cid:15) t , (16)where X t ∈ { ADS , KCSFI , VIX } is one of the macroeconomic and economic uncertaintyindicators and (cid:15) t (cid:118) N (0 , σ ). Naturally, we also control for market illiquidity, ILLIQ MKT t ,that we compute as the Amihud (2002) ratio for individual stocks listed on the NYSE,AMEX and NASDAQ on a daily basis subject to filtering conditions standard in theliterature. To compute a market measure, we take the cross sectional average each dayand then take the time-series average of daily observations in each month. One canloosely interpret the γ coefficients in Equation 16 as capturing the contemporaneouscorrelations between real activity (financial uncertainty) and our measures of illiquidityconnectedness, after controlling for market illiquidity.Table 1 reports various specifications of Equation 16 to quantify the contemporanouslinks between changes in our illiquidity network connectedness measures and real activ- is the Chicago Board of Exchange implied volatility index collected from Bloomberg. We drop the double time index here since we aggregate the daily measures to monthly observationsand estimate linear OLS regressions in this section.
007 2012 2017 − −
202 ADS index C S t C M t C L t − C S t C M t C L t − C S t C M t C L t Figure 6: Monthly real activity indicators and financial uncertainty from July 2005to December 2017
This figure plots real activity and financial uncertainty indicators, along with the quasi posteriormedian dynamic network connectedness measures at short-term, medium-term and long-termhorizons. The leftmost plot shows the Aruoba et al. (2009) (ADS) business conditions index;the middle plot shows the Kansas City Financial Stress Index (KCFSI); and the rightmostplot shows the VIX index. On each plot, we report the horizon specific illiquidity networkconnectedness measures C dt , d = { S , M , L } . All variables are standardized to have mean 0 andstandard deviation 1 to make the series interpretable. C dt , d ∈ { S , M , L } . Three main fac-tors emerge from these results. First, looking at regressions using ADS (KCFSI/VIX),we can see that all γ ∆ C d , d ∈ { S , M , L } are negative (positive) which is consistent withour expectation. Second, short-term illiquidity connections have a statistically significantcontemporaneous link with ADS and KCFSI. Adding to this, the contemporaneous rela-tionship between changes to long-term illiquidity network connectedness and changes toeach respective measure of real activity or financial uncertainty are statistically signifi-cant. Third, controlling for changes in market illiquidity does not subsume the significanceof our network connectedness measures.Overall, this exercise shows that horizon specific connections among stock illiquiditiescontain information directly related to real activity and financial uncertainty; particularlyover the long-term. Economically speaking, one can interpret our measures as indicesfor systemic risk that capture horizon specific illiquidity shock propagation. From atheoretical perspective, our results here suggest: i) that our measures better capture theshock propagation mechanism of liquidity shocks in Kiyotaki and Moore (2019); and ii)that the information content of our measures is inherently different to raw measures ofstock market illiquidity. 22 able 1: Information content of illiquidity network connectedness for real activity and financial uncertainty Notes: This table reports regression results linking horizon specific illiquidity network connectedness to alternative indicators foreconomic activity and financial uncertainty. ADS is the Aruoba et al. (2009) business conditions index. KCFSI is the KansasCity Financial Stress Index (Hakkio et al., 2009) and VIX is the CBOE implied volatility index. α is the regression constant; γ ∆ C d , d ∈ { S , M , L } are the respective betas associated to short, medium, and long term changes in illiquidity network connectedness;and γ ILLIQ is the coefficient associated to changes in market illiquidity. *, **, and *** denote statistical significance at the 10%, 5%and 1% levels respectively. ¯ R is the regression adjusted R-squared. ADS KCFSI VIX α − − − − γ ∆ C S − ∗ ∗∗∗ γ ∆ C M − γ ∆ C L − ∗∗∗ ∗∗∗ ∗∗∗ (0.029) (0.036) (0.405) γ ILLIQ − − − − ∗ R − − Conclusion
In this paper, we characterize measures of dynamic network connections for large financialand economic systems. Specifically, we assume the market or economy is generated by aTVP VAR model and rely on a locally stationary approximation of the data generatingprocess. Our measures rely on causal linkages by viewing the local forecast error variancedecomposition matrices at each point in time as adjacency matrices that completelycharacterize the dynamic network. We decompose the aggregate dynamic network intocomponents that capture horizon specific connections among systems of financial andeconomic variables. This allows the researcher or practitioner to study the impact of shockpropagation that causes network structures over horizons of interest. In our empiricalapplication we show that our methods are readily available to use in problems usingbig data. In doing so we redefine the meaning of “big” in big data, in the context ofTVP VAR models, and track dynamic connections among illiquidity ratios of all S&P500constituents.We then go on to show the usefulness of our dynamic network measures by: i) explor-ing the dynamic relationship between horizon specific illiquidity network connectednessand the market return and ii) the contemporaneous relation of horizon specific networkconnections and real activity (financial uncertainty). The former uncovers a strong linkbetween illiquidity connections and the market return that intensifies during periods offinancial turbulence. We link this to the self reinforcing mechanism between marketand funding liquidity in Brunnermeier and Pedersen (2009) and argue that our measuresmay be useful for portfolio formation and rebalancing. The latter shows a strong nega-tive (positive) link with real activity (financial stress indicators). We link this with thetheoretical mechanism in Kiyotaki and Moore (2019) and conjecture that our illiquiditynetwork connectedness measures are indices that track horizon specific systemic shockpropagation.
Acknowledgements
We thank Oliver Linton, Wolfgang H¨ardle, Catherine Forbes, Luk´aˇs V´acha, RylandThomas, Chris Florackis, Costas Milas, Alex Kostakis and Charlie Cai for invaluablediscussions and comments. We are grateful to Luboˇs Hanus for help in furnishingand converting estimation codes. We acknowledge insightful comments from numer-ous seminar presentations, such as: the 2019 and 2020 Society for Economic Measure-ment Conferences; the Danish National Bank; the 2019 STAT of ML conference; the13th International Conference on Computational and Financial Econometrics; and manymore. Jozef Barun´ık gratefully acknowledges support from the Czech Science Founda-24ion under the 19-28231X (EXPRO) project. For estimation of dynamic horizon specificnetworks, we provide packages
DynamicNets.jl in JULIA and
DynamicNets in MAT-LAB . The packages are available at https://github.com/barunik/DynamicNets.jl and https://github.com/ellington/DynamicNets . Disclosure Statement:
Jozef Barun´ık and Michael Ellington have nothing to disclose.
References
Acemoglu, D., V. M. Carvalho, A. Ozdaglar, and A. Tahbaz-Salehi (2012). The networkorigins of aggregate fluctuations.
Econometrica 80 (5), 1977–2016.Acharya, V. V. and L. H. Pedersen (2005). Asset Pricing with Liquidity Risk.
Journalof Financial Economics 77 (2), 375–410.Amihud, Y. (2002). Illiquidity and stock returns: cross-section and time-series effects.
Journal of Financial Markets 5 (1), 31–56.Aruoba, S. B., F. X. Diebold, and C. Scotti (2009). Real-time measurement of businessconditions.
Journal of Business & Economic Statistics 27 (4), 417–427.Bansal, R., D. Kiku, I. Shaliastovich, and A. Yaron (2014). Volatility, the Macroeconomy,and Asset Prices.
Journal of Finance 69 (6), 2471–2511.Barun´ık, J. and T. Kˇrehl´ık (2018). Measuring the frequency dynamics of financial con-nectedness and systemic risk.
Journal of Financial Econometrics 16 (2), 271–296.Bekaert, G., C. R. Harvey, and C. Lundblad (2007). Liquidity and Expected Returns:Lessons from Emerging Markets.
Review of Financial Studies 20 (6), 1783–1831.Brunnermeier, M. K. and L. H. Pedersen (2009). Market Liquidity and Funding Liquidity.
Review of Financial Studies 22 (6), 2201–2238.Chen, Y., G. W. Eaton, and B. S. Paye (2018). Micro (Structure) before Macro? ThePredictive Power of Aggregate Illiquidity for Stock Returns and Economic Activity.
Journal of Financial Economics 130 (1), 48–73.Chordia, T., R. Roll, and A. Subrahmanyam (2000). Commonality in Liquidity.
Journalof Financial Economics 56 (1), 3–28. 25ahlhaus, R. (1996). On the kullback-leibler information divergence of locally stationaryprocesses.
Stochastic processes and their applications 62 (1), 139–168.Dahlhaus, R., W. Polonik, et al. (2009). Empirical spectral processes for locally stationarytime series.
Bernoulli 15 (1), 1–39.Demirer, M., F. X. Diebold, L. Liu, and K. Yilmaz (2018). Estimating global banknetwork connectedness.
Journal of Applied Econometrics 33 (1), 1–15.Diebold, F. X. and K. Yilmaz (2014). On the network topology of variance decomposi-tions: Measuring the connectedness of financial firms.
Journal of Econometrics 182 (1),119–134.Ellington, M. (2018). Financial Market Illiquidity Shocks and Macroeconomic Dynamics:Evidence from the UK.
Journal of Banking & Finance 89 , 225–236.Elliott, M., B. Golub, and M. O. Jackson (2014). Financial networks and contagion.
American Economic Review 104 (10), 3115–53.Fama, E. F. and K. R. French (1993). Common risk factors in the returns on stocks andbonds.
Journal of Financial Economics 33 (1), 3–56.Fama, E. F. and K. R. French (2015). A five-factor asset pricing model.
Journal ofFinancial Economics 116 (1), 1–22.Florackis, C., G. Giorgioni, A. Kostakis, and C. Milas (2014). On Stock Market Illiquidityand Real-time GDP Growth.
Journal of International Money and Finance 44 , 210–229.Garvey, M. D., S. Carnovale, and S. Yeniyurt (2015). An analytical framework for sup-ply network risk propagation: A Bayesian network approach.
European Journal ofOperational Research 243 (2), 618–627.Geraci, M. V. and J.-Y. Gnabo (2018). Measuring interconnectedness between financialinstitutions with Bayesian time-varying vector autoregressions.
Journal of Financialand Quantitative Analysis 53 (3), 1371–1390.Glasserman, P. and H. P. Young (2016). Contagion in financial networks.
Journal ofEconomic Literature 54 (3), 779–831.Goyenko, R. Y., C. W. Holden, and C. A. Trzcinka (2009). Do Liquidity MeasuresMeasure Liquidity?
Journal of Financial Economics 92 (2), 153–181.26akkio, C. S., W. R. Keeton, et al. (2009). Financial stress: what is it, how can it bemeasured, and why does it matter?
Economic Review 94 (2), 5–50.Herskovic, B., B. T. Kelly, H. N. Lustig, and S. Van Nieuwerburgh (2020). Firm volatilityin granular networks.
Journal of Political Economy (Forthcoming).Hubrich, K. and R. J. Tetlow (2015). Financial Stress and Economic Dynamics: TheTransmission of Crises.
Journal of Monetary Economics 70 , 100–115.Kadiyala, K. R. and S. Karlsson (1997). Numerical methods for estimation and inferencein Bayesian VAR-models.
Journal of Applied Econometrics 12 (2), 99–132.Kiyotaki, N. and J. Moore (2019). Liquidity, business cycles, and monetary policy.
Journalof Political Economy 127 (6), 2926–2966.Kyle, A. S. (1985). Continuous Auctions and Insider Trading.
Econometrica , 1315–1335.L¨utkepohl, H. (2005).
New introduction to multiple time series analysis . Springer Science& Business Media.Næs, R., J. A. Skjeltorp, and B. A. Ødegaard (2011). Stock Market Liquidity and theBusiness Cycle.
Journal of Finance 66 (1), 139–176.P´astor, L. and R. F. Stambaugh (2003). Liquidity risk and expected stock returns.
Journal of Political economy 111 (3), 642–685.Pesaran, H. H. and Y. Shin (1998). Generalized impulse response analysis in linearmultivariate models.
Economics letters 58 (1), 17–29.Petrova, K. (2019). A quasi-Bayesian local likelihood approach to time varying parameterVAR models.
Journal of Econometrics .Primiceri, G. E. (2005). Time varying structural vector autoregressions and monetarypolicy.
Review of Economic Studies 72 (3), 821–852.Rambachan, A. and N. Shephard (2019). Econometric analysis of potential outcomestime series: instruments, shocks, linearity and the causal response function. arXivpreprint arXiv:1903.01637 .Richmond, R. J. (2019). Trade network centrality and currency risk premia.
Journal ofFinance 74 (3), 1315–1361. 27oueff, F. and A. Sanchez-Perez (2016). Prediction of weakly locally stationary processesby auto-regression. arXiv preprint arXiv:1602.01942 .Watanabe, A. and M. Watanabe (2007). Time-varying Liquidity Risk and the CrossSection of Stock Returns.
Review of Financial Studies 21 (6), 2449–2486.28 echnical AppendixA Proofs: New measures of horizon specific networkrisk for large dynamic networks
Proposition 1.
Let us have the VMA( ∞ ) representation of the locally stationary TVPVAR model (Dahlhaus et al., 2009; Roueff and Sanchez-Perez, 2016) X t,T = ∞ X h = −∞ Ψ t,T ( h ) (cid:15) t − h (A.1) Ψ t,T ( h ) ≈ Ψ ( t/T, h ) is a stochastic process satisfying sup ‘ || Ψ t − Ψ ‘ || = O p ( h/t ) for1 ≤ h ≤ t as t → ∞ , hence in a neighborhood of a fixed time point u = t/T the process X t,T can be approximated by a stationary process f X t ( u ) f X t ( u ) = ∞ X h = −∞ Ψ ( u, h ) (cid:15) t − h (A.2)with (cid:15) being iid process with E [ (cid:15) t ] = 0, E [ (cid:15) s (cid:15) t ] = 0 for all s = t , and the local covariancematrix of the errors Σ ( u ). Under suitable regularity conditions | X t,T − f X t ( u ) | = O p ( | t/T − u | +1 /T ).Since the errors are assumed to be serially uncorrelated, the total local covariancematrix of the forecast error conditional on the information at time t − Ω ( u, H ) = H X h =0 Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) . (A.3)Next, we consider the local covariance matrix of the forecast error conditional on knowl-edge of today’s shock and future expected shocks to k -th variable. Starting from theconditional forecasting error, ξ k ( u, H ) = H X h =0 Ψ ( u, h ) h (cid:15) t + H − h − E ( (cid:15) t + H − h | (cid:15) k,t + H − h ) i , (A.4)29ssuming normal distribution of (cid:15) t ∼ N (0 , Σ ), we obtain E ( (cid:15) t + H − h | (cid:15) k,t + H − h ) = σ − kk h Σ ( u ) i · k (cid:15) k,t + H − h (A.5)and substituting (A.5) to (A.4), we obtain ξ k ( u, H ) = H X h =0 Ψ ( u, h ) h (cid:15) t + H − h − σ − kk h Σ ( u ) i · k (cid:15) k,t + H − h i . (A.6)Finally, the local forecast error covariance matrix is Ω k ( u, H ) = H X h =0 Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) − σ − kk H X h =0 Ψ ( u, h ) h Σ ( u ) i · k h Σ ( u ) i >· k Ψ > ( u, h ) . (A.7)Then h ∆ ( u, H ) i ( j ) k = h Ω ( u, H ) − Ω k ( u, H ) i j,j = σ − kk H X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! (A.8)is the unscaled local H -step ahead forecast error variance of the j -th component withrespect to the innovation in the k -th component. Scaling the equation with H -step aheadforecast error variance with respect to the j th variable yields the desired time varyinggeneralized forecast error variance decompositions (TVP GFEVD) h θ ( u, H ) i j,k = σ − kk H X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! H X h =0 h Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) i j,j (A.9)Next, we derive the frequency representation of the quantity in (A.9) using the factthat unique time varying spectral density of X t,T at frequency ω which is locally thesame as the spectral density of f X t ( u ) at u = t/T can be defined as a Fourier transformof VMA( ∞ ) filtered series over frequencies ω ∈ ( − π, π ) as S X ( u, ω ) = ∞ X h = −∞ E h X t + h ( u ) X > t ( u ) i e − iωh = n Ψ ( u ) e − iω o Σ ( u ) n Ψ ( u ) e + iω o > , (A.10) Note to notation: [ A ] j,k denotes the j th row and k th column of matrix A denoted in bold. [ A ] j, · denotes the full j th row; this is similar for the columns. A P A , where A is a matrix that denotes thesum of all elements of the matrix A . Ψ ( u ) e − iω = P h e − iωh Ψ ( u, h )which can be obtained as a Fourier transform of the coefficients with i = √− H → ∞ , we have time varying generalized forecast error variance decompo-sitions h θ ( u, ∞ ) i j,k = σ − kk ∞ X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! ∞ X h =0 h Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) i j,j = AB . (A.11)Starting with frequency domain counterpart of the nominator A , we will use thestandard integral 12 π Z π − π e iω ( r − v ) dω = r = v r = v. (A.12)Using the fact that P ∞ h =0 φ ( h ) ψ ( h ) = π R π − π P ∞ v =0 P ∞ r =0 φ ( r ) ψ ( v ) e iω ( r − v ) dω , we canrewrite (A.11) as σ − kk ∞ X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! = σ − kk ∞ X h =0 n X z =1 h Ψ ( u, h ) i j,z h Σ ( u ) i z,k ! = σ − kk π Z π − π ∞ X r =0 ∞ X v =0 n X x =1 h Ψ ( u, r ) i j,x h Σ ( u ) i x,k ! n X y =1 h Ψ ( u, v ) i j,y h Σ ( u ) i y,k e iω ( r − v ) dω = σ − kk π Z π − π ∞ X r =0 ∞ X v =0 n X x =1 h Ψ ( u, r ) e iωr i j,x h Σ ( u ) i x,k ! n X y =1 h Ψ ( u, v ) e − iωv i j,y h Σ ( u ) i y,k dω = σ − kk π Z π − π ∞ X r =0 n X x =1 h Ψ ( u, r ) e iωr i j,x h Σ ( u ) i x,k ! ∞ X v =0 n X y =1 h Ψ ( u, v ) e − iωv i j,y h Σ ( u ) i y,k dω = σ − kk π Z π − π n X x =1 h Ψ ( u ) e iω i j,x h Σ ( u ) i x,k ! n X y =1 h Ψ ( u ) e − iω i j,y h Σ ( u ) i y,k dω = σ − kk π Z π − π (cid:18)h Ψ ( u ) e − iω Σ ( u ) i j,k (cid:19) (cid:18)(cid:16) Ψ ( u ) e iω Σ ( u ) i j,k (cid:19) dω = σ − kk π Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Ψ ( u ) e − iω Σ ( u ) (cid:21) j,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω (A.13)31ence we have established that A = σ − kk ∞ X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! = σ − kk π Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Ψ ( u ) e − iω Σ ( u ) (cid:21) j,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω (A.14)from (A.11), we use the local spectral representation of the VMA coefficients in the secondstep. The rest is a manipulation with the last step invoking the definition of modulussquared of a complex number to be defined as | z | = zz ∗ . Note that we can use thissimplification without loss of generality, because the V M A ( ∞ ) representation that isdescribed by the coefficients Ψ ( u, h ) has a spectrum that is always symmetric.Next, we concentrate on B from (A.11). Using similar steps and the positive semidef-initeness of the matrix Σ ( u ) that ascertains that there exists P ( u ) such that Σ ( u ) = P ( u ) P > ( u ) . ∞ X h =0 h Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) i = ∞ X h =0 h Ψ ( u, h ) P ( u ) ih Ψ ( u, h ) P ( u ) i > = 12 π Z π − π ∞ X r =0 ∞ X v =0 h Ψ ( u, r ) e iωr P ( u ) ih Ψ ( u, v ) e − iωv P ( u ) i > dω = 12 π Z π − π ∞ X r =0 h Ψ ( u, r ) e iωr P ( u ) i ∞ X v =0 h Ψ ( u, v ) e − iωv P ( u ) i > dω = 12 π Z π − π h Ψ ( u ) e iω P ( u ) ih Ψ ( u ) e − iω P ( u ) i > dω = 12 π Z π − π "n Ψ ( u ) e iω o Σ ( u ) n Ψ ( u ) e − iω o > dω (A.15)That establishes the fact that B = ∞ X h =0 h Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) i j,j = 12 π Z π − π "n Ψ ( u ) e iω o Σ ( u ) n Ψ ( u ) e − iω o > j,j dω (A.16)32rom (A.11), and we have shown that h θ ( u, ∞ ) i j,k = σ − kk ∞ X h =0 h Ψ ( u, h ) Σ ( u ) i j,k ! ∞ X h =0 h Ψ ( u, h ) Σ ( u ) Ψ > ( u, h ) i j,j = σ − kk Z π − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Ψ ( u ) e − iω Σ ( u ) (cid:21) j,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω Z π − π "n Ψ ( u ) e iω o Σ ( u ) n Ψ ( u ) e − iω o > j,j dω (A.17)Finally, focusing on a frequency band d = ( a, b ) : a, b ∈ ( − π, π ) , a < b , we have h θ ( u, d ) i j,k = σ − kk Z ba (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) Ψ ( u ) e − iω Σ ( u ) (cid:21) j,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dω Z π − π "n Ψ ( u ) e iω o Σ ( u ) n Ψ ( u ) e − iω o > j,j dω (A.18)This completes the proof. Proposition 2.
Using the Remark 1 and appropriate substitutions, it immediately followsthat X d s ∈ D C ( u, d s ) = X d z ∈ D N X j,k =1 j = k h e θ ( u, d s ) i j,k , N X j,k =1 h e θ ( u, ∞ ) i j,k = X d z ∈ D N X j,k =1 j = k h e θ ( u, d s ) i j,k , N X j,k =1 h e θ ( u, ∞ ) i j,k = N X j,k =1 j = k h e θ ( u, ∞ ) i j,k , N X j,k =1 h e θ ( u, ∞ ) i j,k = C ( u, ∞ ) (A.19)Similarly, quantities C j ←• ( u, ∞ ) and C j →• ( u, ∞ ) will sum over frequency bands. Thiscompletes the proof. 33 Estimation of the time-varying parameter VARmodel
To estimate our high dimensional systems, we follow the Quasi-Bayesian Local-Liklihood(QBLL) approach of Petrova (2019). let X t be an N × p lags: X t,T = Φ ( t/T ) X t − ,T + . . . + Φ p ( t/T ) X t − p,T + (cid:15) t,T , (B.1)where (cid:15) t,T = Σ − / ( t/T ) η t,T with η t,T ∼ N ID (0 , I M ) and Φ ( t/T ) = ( Φ ( t/T ) , . . . , Φ p ( t/T )) > are the time varying autoregressive coefficients. Note that all roots of the locally station-ary VAR polynomial, lie outside the unit circle, and Σ − t is a positive definite time-varyingcovariance matrix. Stacking the time-varying intercepts and autoregressive matrices inthe vector φ ( t/T ) with ¯ X t,T = ( I N ⊗ x t,T ) , x t,T = (cid:16) , x t − ,T , . . . , x t − p,T (cid:17) and ⊗ denotesthe Kronecker product, the model can be written as: X t,T = ¯ X t,T φ ( t/T ) + Σ − ( t/T ) η t,T (B.2)We obtain the time-varying parameters of the model by employing Quasi-Bayesian LocalLikelihood (QBLL) methods. Estimation of (B.1) requires re-weighting the likelihoodfunction. Essentially, the weighting function gives higher proportions to observationssurrounding the time period whose parameter values are of interest. The local likelihoodfunction at discrete time period s , where we drop the double time index for notationalconvenience, is given by:L s (cid:16) X s | φ s , Σ s , ¯ X s (cid:17) ∝| Σ s | trace( D s ) / exp (cid:26) −
12 ( X s − ¯ X s φ s ) ( Σ s ⊗ D s ) ( X s − ¯ X s φ s ) (cid:27) (B.3)34he D s is a diagonal matrix whose elements hold the weights: D s = diag( % s , . . . , % sT ) (B.4) % st = ζ T,s w st / T X t =1 w st (B.5) w st = (1 / √ π ) exp(( − / k − t ) /H ) ) , for s, t ∈ { , . . . , T } (B.6) ζ T s = T X t =1 w st ! − (B.7)where % st is a normalised kernel function. w st uses a Normal kernel weighting function. ζ T s gives the rate of convergence and behaves like the bandwidth parameter H in (B.6),and it is the kernel function that provides greater weight to observations surrounding theparameter estimates at time s relative to more distant observations.Using a Normal-Wishart prior distribution for φ s | Σ s for s ∈ { , . . . , T } : φ s | Σ s (cid:118) N (cid:16) φ s , ( Σ s ⊗ Ξ s ) − (cid:17) (B.8) Σ s (cid:118) W ( α s , Γ s ) (B.9)where φ s is a vector of prior means, Ξ s is a positive definite matrix, α s is a scaleparameter of the Wishart distribution ( W ), and Γ s is a positive definite matrix.The prior and weighted likelihood function implies a Normal-Wishart quasi posteriordistribution for φ s | Σ s for s = { , . . . , T } . Formally let A = (¯ x , . . . , ¯ x T ) and Y =( x , . . . , x T ) then: φ s | Σ s , A , Y (cid:118) N (cid:18) ˜ φ s , (cid:16) Σ s ⊗ ˜Ξ s (cid:17) − (cid:19) (B.10) Σ s (cid:118) W (cid:16) ˜ α s , ˜Γ − s (cid:17) (B.11)with quasi posterior parameters˜ φ s = (cid:16) I N ⊗ ˜Ξ − s (cid:17) h ( I N ⊗ A D s A ) ˆ φ s + ( I N ⊗ Ξ s ) φ s i (B.12) ˜Ξ s = ˜Ξ s + A D s A (B.13)˜ α s = α s + T X t =1 % st (B.14) ˜Γ s = Γ s + Y D s Y + Φ s Γ s Φ s − ˜Φ s ˜Γ s ˜Φ s (B.15)35here ˆ φ s = ( I N ⊗ A D s A ) − ( I N ⊗ A D s ) y is the local likelihood estimator for φ s . Thematrices Φ s , ˜Φ s are conformable matrices from the vector of prior means, φ s , and adraw from the quasi posterior distribution, ˜ φ s , respectively.The motivation for employing these methods are threefold. First, we are able toestimate large systems that conventional Bayesian estimation methods do not permit.This is typically because the state-space representation of an N -dimensional TVP VAR( p ) requires an additional N (3 / N ( p + 1 / p =2 and a Minnesota Normal-Wishart prior with ashrinkage value ϕ = 0 .
05 and centre the coefficient on the first lag of each variable to 0.1 ineach respective equation. The prior for the Wishart parameters are set following Kadiyalaand Karlsson (1997). For each point in time, we run 500 simulations of the model togenerate the (quasi) posterior distribution of parameter estimates. Note we experimentwith various lag lengths, p = { , , , } ; shrinkage values, ϕ = { . , . , . } ; andvalues to centre the coefficient on the first lag of each variable, { , . , . , . } . Networkmeasures from these experiments are qualitatively similar. Notably, adding lags to theVAR and increasing the persistence in the prior value of the first lagged dependent variablein each equation increases computation time. C Univariate time-varying parameter regressions
We now consider a linear regression with time-varying parameters that has a Normal-Gamma quasi-posterior distribution. Specifically, this is a univariate version of Petrova(2019). Let y t,T = β ( t/T ) + x ,t,T β ( t/T ) + · · · + x l,t,T β l ( t/T ) + (cid:15) t,T , (cid:15) t,T (cid:118) N (cid:16) , σ t,T (cid:17) (C.1) y t,T = x t,T β ( t/T ) + (cid:15) t,T (C.2)where x t,T ≡ (1 , x ,t,T , . . . , x l,t,T ) and β ( t/T ) ≡ ( β ( t/T ) , β ( t/T ) , . . . , β l ( t/T )) > .Now let λ t,T ≡ σ − t,T . The weighted local likelihood function of the sample Y ≡ y , . . . , y T ), using X ≡ (cid:16) x > , . . . , x > T (cid:17) > as a T × l matrix, at each discrete time point s ,where we drop the double time index for notational convenience, is given by L s ( Y | β s , λ s , X ) = (2 π ) − tr( D s ) / λ tr( D s ) / s exp ( − λ s Y − X β s ) > D s ( Y − X β s ) ) (C.3)with D s = diag ( ϑ s, , . . . , ϑ s,T ) (C.4) ϑ s,t = ζ T,s w s,t / T X t =1 w s,t (C.5) w s,t = (cid:16) / √ π (cid:17) exp (cid:16) ( − / k − t ) /H ) (cid:17) , ∀ s, t ∈ { , . . . , T } (C.6) ζ T,s = T X t =1 w s,t ! − (C.7)Now assuming β s , λ s have a Normal-Gamma prior distribution ∀ s ∈ { , . . . , T } β s | λ s (cid:118) N (cid:16) β ,s , ( λ s κ ,s ) − (cid:17) (C.8) λ s (cid:118) G ( α ,s , γ ,s ) (C.9)We can combine L s with the above priors such that β s , λ s have Normal-Gammaquasi-posterior distribution ∀ k ∈ { , . . . , T } β s | λ s (cid:118) N (cid:16) ¯ β s , ( λ s ¯ κ s ) − (cid:17) (C.10) λ s (cid:118) G ( ¯ α s , ¯ γ s ) (C.11)with (quasi) posterior parameters:¯ β s = ¯ κ − s (cid:16) X > D s X b β s + κ ,s β ,s (cid:17) , b β s = (cid:16) X > D s X (cid:17) − X > D s y (C.12)¯ κ s = κ ,s + X > D s X (C.13)¯ α s = α ,s + T X t =1 ϑ s (C.14)¯ γ s = γ ,s + 12 (cid:16) Y > D s Y − ¯ β > s ¯ κ s ¯ β s + β > ,s κ ,s β ,s (cid:17) (C.15)37 he Algorithm