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Decomposing Global Bank Network Connectedness: What is Common, Idiosyncratic and When?
Measuring connectedness is of paramount importance in many aspects of financial risk measurement and management. Particularly, following the global financial crisis in 2008–2009, the heightened focus of governments and financial institutions on the significant concerns surrounding the propagation of macro financial risks and its potential impact on financial stability has become increasingly evident. Connectedness measures, such as return connectedness, default connectedness, and system-wide connectedness, are commonly featured in various facets of risk management, including market risk, credit risk, and systemic risk. Nevertheless, the concept of connectedness remained rather elusive in econometric theory until diebold2014network undertook the task of addressing it comprehensively. Their work provided a rigorous definition by introducing measures of connectedness rooted in (generalized) forecast error variance decomposition (FEVD) from approximating, finite order vector autoregressive (VAR) models.\footnote{To clarify: the term approximating refers to the fact that a model should be chosen for the data, and that is never correct; if a dynamic one is chosen then, like a VAR here, a finite length of its past dynamic (i.e., a lag-length) has to be specified. This in itself is another approximation, as it presumes all the series to have the same dynamic.} To elaborate, their approach involves evaluating the distribution of forecast error variance across different actors, such as banks, firms, markets, countries, etc., attributable to shocks originating elsewhere. In simpler terms: if the future variation of e.g., bank $i$, is mostly due to shocks attributable to bank $j$, then the two banks are connected as $j\to i$, and vice versa. Then, the appeal of such approach lies in its ability to address the question of “to what extent the future variation (at different horizons `$H$') of actor $i$ can be attributed not to internal shocks originating within actor $i$ itself, but rather to external factors associated with actor $j$?". To identify uncorrelated structural shocks from correlated reduced form shocks, diebold2014network chose the generalized variance decomposition (GVD) framework introduced in koop1996impulse, pesaran1998generalized. Differently from the identification schemes that orthogonalize the shocks e.g., through Cholesky factorization, and which are dependent on the variables' ordering, GVD avoids forced orthogonalization of the shocks and -under a normality assumption- properly accounts for historically observed correlations among them, while being order-invariant.\footnote{The principle of GVD and their generalized impulse responses is that of treating each variable as if they were the first in the ordered vector of observables, and account for the correlation among shocks by discounting for historical correlations among them, rather than orthogonalize them. In this sense, it does not matter which variable comes first or later in the vector.} Although it clearly depends on the level of aggregation considered, systems of banks, firms, markets, countries etc. are seldom low-dimensional. The likely high-dimensionality of such systems instead introduces some challenges to tackle in order to estimate the approximating --now high-dimensional-- models, imposed on the --now high-dimensional-- vector of observables. Such challenges have been taken on by demirer2018estimating in the context of global bank network connectedness. Using a sparse VAR model of order $p$, VAR($p$), directly on the observables, they employ $\ell_1 + \ell_2$-norm regularization in the form of an (adaptive) Elastic Net. This approach allows them to jointly perform shrinkage, variable selection, and estimation where their goal is that to estimate the high-dimensional connectedness network linking a publicly traded subset of the world's top 150 banks, covering the period from 2003 to 2013. Once an estimate of the high-dimensional VAR coefficient matrix is obtained, the $H$-step generalized variance decomposition matrix can be easily computed and thus the various connectedness relationships as in diebold2014network. These can be between: each pair of banks (pairwise directional connectedness), each bank with all the others -and vice versa- (\emph{total directional connectedness}), and all banks in a total connectedness sense (\emph{system-wide connectedness}, SWC henceforth). We refer to Section (ref) for the mathematical definitions. As any measure based on a model relies on a set of decisions/assumptions on that very model, the connectedness measure of demirer2018estimating is no exception. As observed in diebold2014network, among other factors, estimating connectedness based on FEVD is affected by the type of approximating model to which data is fed to, and forecast error variance is obtained from. The popularized use of sparse-regularization techniques to account for the large dimensionality of such problems can be tempting, if anything for its simplicity, and -aside of demirer2018estimating- it is often employed in the applied literature on financial connectedness yi2018volatility, liu2022high. In this paper, we argue that such a direct \emph{sparsity} assumption on the VAR coefficient matrix might be a (too) strong statement on the data generating process, and becomes (more) reasonable only \emph{after} controlling for common variation within the observables, i.e., after estimation and thus accountancy of the common factors in the data. In fact, common factors are widely recognized to play a fundamental role with financial data and its modeling.\footnote{As observed in bai2006evaluating, e.g., the arbitrage pricing theory is built upon the existence of a set of common factors underlying all asset returns. In the capital asset pricing theory the market return is the common risk factor that has pervasive effects on all assets. Many other examples could be made.} Failure to account for factors, and the use of direct sparsity assumptions when linkages among units are truly non-sparse might induce an underestimation of the degree of connectedness. However, here we do not depart away from high-dimensional \emph{sparse} VARs, but instead build upon the recent literature bridging factors and sparse models fan2021bridging, barigozzi2023fnets, krampe2021factor, where we assume the series to follow an approximate, static factor model whose idiosyncratic term follows a \emph{sparse} VAR.\footnote{We allow both factors and idiosyncratics to have parametric VAR representations. The term “static" stock2002forecasting, bai2003 is to distinguish it from a model where lagged factors enter directly the factor model decomposition: the so-called 'generalized dynamic' factor model forni2000generalized. The term 'approximate' refer to the fact that the idiosyncratics are allowed to exhibit cross-sectional dependence. We refer to Section (ref) for the details.} We also employ the same strategy in the frequency domain in Section (ref), where we describe the frequency dynamics of the connectedness by considering the spectral representation of variance decompositions based on the frequency responses to shocks. \emph{Why a factor model for computing connectedness?} As mentioned, factor models play a fundamental role in financial data analysis, as documented in a nowadays vast literature. Assuming sparsity directly on the coefficient matrix of the VAR is tantamount to force somewhat weaker predictive linkages to be zeroed-out by the LASSO-type technique employed. While regularization promotes parsimony (interpretability) and contrasts overfitting, in the case of connectedness it risks to underestimate the degree of SWC by tossing away connections. A factor model instead, accounts for a common dynamic among all the volatilities. Once that has been accounted for, the idiosyncratic dynamic of connections left is much more sensible to be sparse. Also, we propose here a \emph{joint} treatment of factors and idiosyncratics (not either of). That means our FEVD expression (see (ref) below) contains both moving average (MA) representations of factors \emph{and} idiosyncratics. As a consequence, we can compute high-dimensional IRFs and therefore the connectedness measures proposed in diebold2014network, now disentangled between common and idiosyncratic shocks. Especially, SWC can be divided into SWC \emph{due to the common shocks} and SWC \emph{due to the idiosyncratic shocks}. This helps in addressing questions such as: “what drives SWC in the banking sector?" and also, “is a shock on a single bank (and likewise a global shock) able to (and to what extent) affect the SWC? and when?". \emph{Why factors & idiosyncratics?} First, explicit modeling of the idiosyncratics allows to capture cross-sectional and time dependence, which remains after the factors' estimation. If instead, what remains is only measurement error, this is unnecessary. But while this scenario might be defensible in macroeconomic contexts, it is really not the case in finance acemoglu2012network. Thinking about stock returns daily range-based volatilities for a publicly traded subset of the world’s top 150 banks, as in demirer2018estimating, it is reasonable to assume a common dynamic among these banks' stock price volatilities, i.e., some sort of “market dynamic". Likewise, it is also sensible that a substantial “individual dynamic" of the single banks themselves, or small subsets of them, would play a role. Second, once the common factors are accounted for, the assumption of \emph{sparsity}—which is often considered unrealistic on its own (e.g., giannone2021economic)—becomes far more plausible when imposed on the idiosyncratic VAR coefficient matrices.\footnote{Note how in the literature there exists many papers billio2012econometric, hecq2023granger taking on the challenge of estimating financial networks (not necessarily \emph{connectedness} networks) via direct regularization of high-dimensional VARs, as also done in demirer2018estimating. As sparsity is a non-testable assumption, assuming it directly on the VAR coefficient matrix can be, at times, hard to justify.} Third, controlling for common factors in a first step tends to reduce collinearity among idiosyncratics, which is well known to render LASSO variable selection arbitrary. Fourth, the factor model gets robustified against misspecifications of the number of factors, since the transferred mistake to the idiosyncratics is at least modeled, instead of ignored. About this last point, this type of modeling also attenuates potential worries about rate-weak factors going undetected. barigozzi2024dynamic highlight that factors might remain undetected when their empirical cross-correlations are small. Regardless of whether they are static or dynamic, such weak factors are not lost but instead remain within the empirical idiosyncratic space. Due to their limited correlations, this omission is generally inconsequential—provided that the idiosyncratic component itself is not disregarded.
To illustrate our approach, we begin by re-examining the estimation of global bank connectedness networks using the dataset from demirer2018estimating. This contains stock price volatilities for a large set of global banks, as well as the bond price volatilities of ten major countries, recorded on daily frequency from 2003 until 2014. Next, we construct a more recent dataset covering 2014–2023, which includes nearly the same bank and bond assets, and replicate our analysis. This updated vintage allows us to analyze how SWC has evolved from 2014 until more recently, but also to investigate similarities and differences in the behavior of connectedness across different global crises (e.g., Covid19). We employ an approximate static factor model with sparse VAR idiosyncratics just like the one considered in krampe2021factor. The common factors and loadings are estimated via principal component analysis (PCA), while the (obtained) idiosyncratics are estimated in a sparse VAR by adaptive LASSO. The compound of the two estimates in moving average representation gives the response of the observables, and the sequence of moving average coefficients at different horizons $H$ gives the impulse response of the observables to either a global or an idiosyncratic shock. Consequently, forecast error variance decompositions can be obtained and likewise a measure of SWC, now declined into common and idiosyncratic shocks. Additionally, adapting the framework of krampe2023structural we are able to compute bootstrap confidence bands for the SWC. This is an important addition, as previously no statistical error bands were given over the estimated connectedness. We also extend the analysis to the spectral domain. This entails estimating the spectral densities of factors and idiosyncratics, the first with traditional nonparametric methods, the second leveraging on the VAR structure and using regularization to estimate its (high-dimensional) residuals' precision matrix. Let us note that one could also directly estimate a sparse moving average representation of the idiosyncratics using a high-dimensional version of the local projection (LP) jorda2005estimation. However, even in the low dimensional case “there is no evidence to suggest that local LPs should replace conventional linear VAR models" kilian2017structural. Our empirical findings demonstrate how in calm times SWC is high, but mostly due to idiosyncratic variation at low frequency. When financial turmoils occur, SWC is even higher, and the common component variation spikes upward, driven by a short-run dynamic response to shocks. This is interesting, as it blends with -and contribute to- the economic literature discourse on systemic risk and stability in financial networks. acemoglu2015systemic have shown how there exists a “double-edge knife" component to connectedness in financial networks. On the one hand, highly interconnected financial networks are “shock-absorbing". On the other hand, once a certain unspecified threshold of connectedness is passed, the robustness turns into a “shock propagating" mechanism. What we find is essentially that networks are shock-absorbing as long as their connectedness is driven by an idiosyncratic dynamic. Networks are not anymore shock-absorbing as soon as connectedness starts to be driven more by a common component dynamic at high frequency.
Particularly in the context of systemic risk, measuring connectedness has been extensively explored in the literature, offering a variety of methods, each of which has its own advantages and limitations. Our focus is on showing that employing a factor model with sparse VARs idiosyncratics allows to answer a richer question within the context of estimation of global bank connectedness networks. Therefore, for comparison purposes we employ the same GVD-based identification as demirer2018estimating.
Let us now mention few important related works. barigozzi2017network look at generalized dynamic factor models to study interdependences in large panels of financial series, specifically S&P100. Connectedness networks for the idiosyncratics are built, based on FEVD, but they focus mainly on the idiosyncratics, after controlling (filtering) for the “market effects", i.e., after accounting for the factors. Similarly, ando2022quantile employ a VAR together with a common factor error structure, fitted by quantile regression. Also in this case, their focus is on the analysis of direct spillovers of credit risk, after controlling for common systematic factors. This means that their vector of forecast errors for the target is conditional on the information set (at time $t-1$) and, crucially, on the common factors. Their FEVD is then a measure of the proportion of the $h$-steps-ahead forecast error variance in the $j$-th observable, accounted for by the $i$-th idiosyncratic innovation. Another interesting related approach is that of barigozzi2021time who introduce a time-varying general dynamic factor model for high-dimensional locally stationary processes. Their focus though is on the factors only. Similar to our empirical findings, using a panel of adjusted intra-day (1999-2015) log ranges for 329 constituents of the S&P500, they show how large increases in connectedness (intended as factors-only connectedness) are associated with the most important turmoils in the stock market (e.g., the great financial crisis of 2007–2009).
The main difference between barigozzi2017network, ando2022quantile, barigozzi2021time treatments and the one we propose is that we consider a joint factor-plus-idiosyncratic treatment of the IRFs in order to compute SWC based on FEVD. To elaborate, although we do not concentrate on the tails as in ando2022quantile, nor on locally stationary processes as in barigozzi2021time, we allow both MA representations of factors and idiosyncratics to enter the expression of the FEVD. Thus, connectedness due to factors, idiosyncratics and the summation of both can be properly disentangled, without limiting it to be computed only from either of these sources. Also, we work out a complete extension to the frequency domain, which allows to disentangle further the financial connectedness into long/medium/short term responses to shocks. This is relevant and, to the best of our knowledge, not considered before within this context.
In this section, we first briefly introduce the connectedness measures established by diebold2014network. Then, we discuss how to adapt this framework when the employed model is an approximate static factor model with sparse VAR idiosyncratics. We show how this modeling approach opens up to more flexibility in interpretation as it disentangles the connectedness due to the common component shocks, to that due to the idiosyncratic shocks. We first present the framework in-population, then briefly discuss its estimation strategy in-sample. As in Section (ref) we are going to use this proposed approach on a couple of global banking datasets, throughout this section, whenever we talk about “observables", we have in mind stock return daily range-based realized volatilities\footnote{Throughout, for brevity, we often omit the “realized" and leave only “volatility"; it should always be intended as “realized volatility".} for a (large) set of world banks (details are given in Section (ref)). In what follows, we employ boldface characters for vectors and capital boldface characters for matrices where e.g., $\bm I_N$ is the identity matrix of order $N$. As the notation used in this section is always defined in-text, we refer the reader to the first paragraph of Section (ref) on Technical Details, where a more detailed description of the notation is provided.
Consider a large, $N$-dimensional covariance-stationary stochastic process with MA representation $\bm x_t=\sum_{i=0}^{\infty}\bm \varPsi_i \bm u_{t-i}$, $\bm u_t\sim (\boldsymbol{0}, \bm \varSigma)$, $\bm \varPsi_0=\bm I_N$. Then, bank $j$'s contribution to bank $i$'s $H$-steps ahead generalized error variance, i.e., pairwise directional connectedness is given, in population, by\footnote{Note how since the GVD of koop1996impulse is employed and therefore the variance shares are not guaranteed to add up to 1, each entry of the generalized variance decomposition matrix gets normalized by the row sum $\sum_{j=1}^N \theta_{ij}^g(H)$. This way, $\sum_{j=1}^N C_{i\leftarrow j}^H=1$ and $\sum_{i,j=1}^N C_{i\leftarrow j}^H=N$.}
and where $\BS e_i (\BS e_j)$ is a selection vector with $i (j)$-th element unity and zeros elsewhere and $\sigma_{jj}=\bm e_j^{\top}\bm \varSigma \bm e_j$. There, $\theta_{ij}^g(H)$ for $i,j=1,\ldots,N$, is the forecast error variance decomposition, i.e., the proportion of the $H$-step ahead forecast error variance of the volatility of stock price of bank $i$, accounted for by the innovations in the volatility of stock price of bank $j$. Similarly, for the total directional connectedness $C^H_{i\leftarrow \operatorname{All}(j)}$, $C^H_{\operatorname{All}(j)\leftarrow i}$ and\footnote{We use the notation $\operatorname{All}(j)=\{i: i\neq j\}$, $\operatorname{All}(i)=\{j: j\neq i\}.$} system-wide connectedness $C^H$:
In this paper, in place of assuming a VAR($p$) approximation for $\bm x_t$ as in demirer2018estimating, we first assume that the $N$ time series can be decomposed into a sum of two uncorrelated components: an $N$-dimensional vector of common components $\bm \chi_t$, and an $N$-dimensional vector of idiosyncratic components $\bm \xi_t$, such that:
As for the first, $\bm \chi_t$, it represents the comovements between the $N$ bank stock price volatilities, and it is assumed to be low-rank, i.e., driven linearly by an $r$-dimensional vector of common factors $\bm f_t$, for $r\ll N$. This means there are $r$ factors, common to all the different banks, driving the change of their stock price volatilities. We call this common behavior “the market". Provided a consistent estimate of $\bm f_t$ is obtained, and likewise one for $\bm \varLambda$, i.e., the $N\times r$ matrix of factor loadings on $\bm x_t$, this entails for the common component an effective dimensionality reduction from $N$ to $r$ series.\footnote{Shall be noted here that a factor model in itself is never a dimensionality reduction technique. From $N$ observables to $2N$ with the decomposition. It is a reduction if one assumes both a low rank for $\bm \chi_t$ and white noise for $\bm \xi_t$. The low rank assumption is mostly sensible, the white noise on $\bm \xi_t$ is often not.} Therefore, the common component gets decomposed as $\bm \chi_t=\bm \varLambda \bm f_t$. As for the idiosyncratic component, $\bm \xi_t$, it represents individual features of the series and/or measurement error. E.g., certain stocks might be more exposed to the behavior of their own reference stock exchange, or to the political situation of their origin country, or to the monetary policy decision of the central bank of their origin country, or to the exchange rate risk, etc. For the purpose of forecasting $\bm x_t$, if $\bm \xi_t$ would truly only be made up of measurement errors, its inclusion in the forecasting equation should not be relevant. However, if $\bm \xi_t$ contains individual features of the series, and these are correlated (e.g., two banks listed on the same stock exchange), accounting for idiosyncratics in the forecasting equation becomes paramount. Instead of assuming a stable VAR($p$) on $\bm x_t$, we assume two stable VARs, namely a VAR($p_f$) for $\bm f_t$ and a VAR($p_{\xi}$) for $\bm \xi_t$, such that
Then, the factor model decomposition in (ref) can be re-written as:
where the second line rewrites the VARs in (ref) for factors and idiosyncratics in their infinite moving average representations, for $\bm \varPsi_f^{(0)}, \bm \varPsi_{\xi}^{(0)}=\bm I_r,\bm I_N$, $\bm \varPsi_f^{(j)}, \bm \varPsi_{\xi}^{(j)}=\boldsymbol{0}$ if $j<0$ and $\bm \eta_t:=(\BS u_t^\top,\BS v_t^\top)^\top\overset{iid}{\sim}(\boldsymbol{0}, \BS \Sigma_{\bm \eta})$ such that $\BS \Sigma_{\eta}$ is an $(r+N)\times (r+N)$ block-diagonal matrix with blocks $\BS\Sigma_u$, $\BS \Sigma_v$, i.e., respectively the covariance matrices of factors and idiosyncratics innovations. Within this framework, an impulse response function (IRF) would measure the time profile of the effect of a market and/or an idiosyncratic shock at a given point in time on the expected future values of (any of) the observables in $\bm x_t$. More formally, IRFs here compare the time profile of the effect of an hypothetical $r$-dimensional market-shock $\bm \delta^m=(\delta_1^m,\ldots,\delta_r^m)^{\top}$ and/or an $N$-dimensional idiosyncratic shock $\bm \delta^{id}=(\delta_1^{id},\ldots,\delta_N^{id})^{\top}$ hitting the global banking system at time $t$ (i.e., $\bm u_t=\bm \delta^m$ and/or $\bm v_t=\bm \delta^{id}$), with a base-line profile at time $t+H$, given (i.e., conditional on) the global banking system behavior's history up to before the shock, i.e., $\bm \varOmega_{t-1}$. Letting $\bm \delta=(\bm \delta^{m \top}, \bm \delta^{id \top})^{\top}$, then the IRF captures the following (expected) difference: $IRF(H,\bm \delta,\bm \varOmega_{t-1})=\mathbb{E}(\bm x_{t+H}|\bm \eta_t=\bm \delta, \bm \varOmega_{t-1})-\mathbb{E}(\bm x_{t+H}|\bm \varOmega_{t-1})$, which translated into (ref) means $IRF(H,\bm \delta,\bm \varOmega_{t-1})=
\bm \delta.$ The usual problem with this formulation is that while it is independent of $\bm \varOmega_{t-1}$, the IRF depends on the composition of the vector $\bm \delta$, i.e., the vector of hypothesised shocks. The \emph{generalized} IRF (GIRF) approach of \citet{koop1996impulse, pesaran1998generalized} adopted in \citet{diebold2014network} and by us as well, is that of avoiding orthogonalization of the shocks in $\bm \eta_t$, but instead using the expression for the IRF directly, shocking only one element (say, the $i$th) at a time, and integrating out the (expected) effects of the other shocks $\mathbb{E}(\bm \eta_t|\eta_{i,t}=\delta_i)$ via an assumed or (historically) observed distribution of the errors. Indeed, by assuming $\bm \eta_t$ to be multivariate Gaussian for instance, then by standard properties\footnote{For any two zero mean Gaussian random variables $Y,X$ with variance $\sigma_y, \sigma_x$ respectively, then $\mathbb{E}(Y|X=x)=\sigma_y \rho(x/\sigma_x)$. The normality assumption is mostly for convenience; as noted in \citet{pesaran1998generalized} one can obtain the conditional expectation $\mathbb{E}(\bm \eta_t|\eta_{i,t}=\delta_i)$ by stochastic simulations or resampling techniques.} one gets $\mathbb{E}(\bm \eta_t|\eta_{i,t}=\delta_i)=\bm \varSigma_{\eta}\bm e_i (\bm e_i^{\top}\bm \varSigma_{\eta}\bm e_i)^{-1}\delta_i$, where again $\BS e_i$ is a selection vector with $i$-th element unity and zeros elsewhere. Then, by setting $\delta_i=(\bm e_i^{\top}\bm \varSigma_{\eta}\bm e_i)^{1/2}$, GIRF and FEVD ($\theta^g_{ij}$) are obtained for $H=1,2,\ldots,$ as
Clearly, both GIRF and the FEVD can be now split into a “due to a market shock" and “due to an idiosyncratic shock". This is obtained simply by, respectively, either {specifying $j=1,\ldots,r$ in (ref), (ref) for “market only", and $j=r+1,\ldots, r+N$ in (ref), (ref), for “idiosyncratics only".} As a consequence, the same connectedness measures as in (ref), (ref) can be obtained, now decomposed into: (pairwise directional, total), system-wide connectedness due to a market shock, $C^H_{Mkt}$, due to an idiosyncratic shock, $C^H_{Ids}$, and due to the summation of both $C^H=C^H_{Mkt}+C^H_{Ids}$.
where $C_{i\leftarrow j}^H$ are the same as defined in (ref), but now containing $\theta_{ij}^g(H)$ as in (ref)
All we presented so far was in-population. In order to obtain an in-sample estimate of (ref), a two step procedure as in krampe2021factor is employed here, that estimates the factor(s) and loadings first, and the sparse VAR over the idiosyncratics after. We leave the technical details/assumptions for Section (ref), but the estimation steps and the intuition of how this work in relation to (ref) is now given.
By inverting the estimated VARs for factors and idiosyncratics as of (I) and (II), in their moving average representations, we then obtain $\hat \bm \varPsi_{f}$, $\hat \bm \varPsi_{\xi}$ as of second line of (ref). Note that the inversion is simply an algebraic nonlinear transformation which can result in a sparse VAR being represented as a nonsparse MA. Finally, the covariance matrix of the error $\bm \varSigma_{\eta}$ is estimated by plugging-in the upper left block of the sample covariance of the residuals from the VAR($p_f$) for the factors, $\hat{\bm \varSigma}_u$, and on the bottom right block the sample covariance of the residuals from the regularized VAR($p_{\xi}$) for the idiosyncratics, $\hat{\bm \varSigma}_v$. Importantly, we show that it is also possible to obtain confidence bands for the connectedness measures. The algorithm for computing the bootstrap confidence bands is given in full in Section (ref). The statistical validity of the employed bootstrap is given by results in krampe2023structural. Later, in our empirical application in Section (ref) we are going to focus especially on $C^H$, the SWC, as it is the most interesting in a systemic-risk perspective. demirer2018estimating found that SWC has grown steadily between $2004$ and $2008$, peaking with the financial crisis, only to then decrease again (although not recovering the initial level) all the way to $2013$. The question that we can answer with our framework is: how much of SWC is due to the banking market (i.e., to the factors) and how much is due/driven to/by the single banks behaviors (i.e., by the idiosyncratics). Furthermore, by means of our spectral analysis in Section (ref), we can also decompose SWC (SWC due to common/idiosyncratics) according to the frequency response to shocks.
Inspired by barunik2018measuring, we can extend the idea of Section (ref) to the frequency domain. This is important in economics. In fact, shocks to the economic activity can affect variables at various frequencies, with various degrees of strength. Therefore, being able to disentangle further the financial connectedness ($C^H, C^{H}_{Mkt}, C^{H}_{Ids}$) into long/medium/short term response to shocks, appears of great practical relevance. The idea is rather simple: to describe the frequency dynamics of the connectedness, one can consider the spectral representation of variance decompositions based on frequency responses to shocks, instead of impulse responses to shocks, as done thus far barunik2018measuring. As our proposed approximating model is an approximate static factor model, we have the following structure as the spectral analogue of (ref)-(ref). The population spectral density matrix, at frequency $\omega$, for the factor process is given by
Likewise, for the idiosyncratic component:
Here, the $\sum_{h=0}^\infty \BS \Psi^{(h)}_{f/\xi} \exp(-\mathrm{i} h \omega)$ are the Fourier transforms of the respective MA($\infty$) coefficients, where $\mathrm{i}=\sqrt{-1}$. Therefore, the spectral density (or “power spectrum"), at frequency $\omega\in[0,2\pi]$, of the process $\{\BS x_t\}$ is given by
The $\BS f_x(\omega)$ describes how the variance of $\bm x_t$ is distributed over the frequency components $\omega$, where we note that $\mathbb{E}(\bm x_t\bm x_{t-h}^{\top})=\frac{1}{2\pi}\bigintssss_{-\pi}^{\pi}\BS f_x(\omega)\exp(\mathrm{i}h\omega) d\omega$. Interestingly, given our factor model decomposition of $\bm x_t$, this variance distribution over the frequencies is disentangled into variance from the common component and variance from the idiosyncratic component. In fact, the generalized causation spectrum over the frequencies $\omega\in(-\pi,\pi)$ can be defined as
for $k=1,\ldots,N, \; j=1,\ldots, r+N$ and where as before $\BS e_j (\BS e_k)$ is a selection vector with $j (k)$-th element unity and zeros elsewhere. Here, $(\texttt{f}(\omega))_{kj}$ is the spectral analogue of (ref), and it measures the portion of the spectrum of the $k$th variable at frequency $\omega$, due to shocks in the $j$th variable. Now, in the same way as barunik2018measuring, in order to obtain a decomposition of variance decompositions to frequencies, it is necessary to weight $(\texttt{f}(\omega))_{k,j}$ by the frequency share of the variance of the $k$th variable. Therefore, the weighting function can be defined as $(\Gamma(\omega))_k=\frac{\left( \bm e_k^{\top} \BS f_x(\omega)\bm e_k\right)}{\frac{1}{2\pi}\bigintsss_{-\pi}^{\pi}\left( \bm e_k^{\top} \BS f_x(\lambda)\bm e_k\right) d\lambda}$, representing the power of the $k$th variable at a given frequency $\omega$, summing through frequencies to a constant value of $2\pi.$ It then finally follows that the spectral representation of the variance decomposition from $k$ to $j$ can be stated as
Let us note that $\lim_{H\to \infty}\theta_{kj}^g(H)$ as in (ref) is a weighted average of the generalized causation spectrum $(\texttt{f}(\omega))_{kj}$ which gives the strength of the relationship at frequency $\omega$, weighting by the power of the series on that frequency. Furthermore, to define connectedness at short/medium/long term frequencies, it is necessary to work with frequency bands. Hence, let us define a frequency band $d=(a,b):a,b\in(-\pi,\pi), a<b$, such that the FEVD on frequency band d ($\text{FEVD}_d$) can be defined as
where $C_{k\leftarrow j}^{H,\omega}$ is its scaled version (to sum up to 1, as before). With all this in place, we can explore the frequency connectedness on a frequency band $d$, both for the factors, the idiosyncratics and the summation of both (system-wide). As in Section (ref) for the GIRF and FEVD in (ref),(ref), also in the case of the $\text{FEVD}_d$ in (ref) the distinction between market and idiosyncratics connectedness is obtained by summing $(\texttt{f}(\omega))_{kj}$ over $j=1,\ldots,r$ for “market only", and $j=r+1,\ldots, r+N$ for “idiosyncratics only".
Table (ref) below summarizes these measures:
where $\sum \theta_d, \sum \theta(\infty)$ stands for the sum of all elements of $\theta_d, \theta(\infty)$, respectively.
In order to obtain an in-sample estimate of (ref), both spectral densities for the factors, $\bm f_f(\omega)$, and for the idiosyncratics, $\bm f_{\xi}(\omega)$, need to be estimated.
As $r$ is of fixed dimension, the spectral density $\BS f_f(\omega)$ (or its inverse) can be estimated by classical methods such as non-parametric lag-window estimators wu2018asymptotic. For the idiosyncratics, we can explicitly use the VAR structure to estimate their spectral density. We note though how the natural estimator is the inverse spectral density and an additional assumption of column-wise sparsity of the VAR companion-form matrix is required (see Section (ref), Assumption (ref)). This is an additional requirement when estimating spectral densities, in fact in Section (ref), as in krampe2021factor, we only required row-wise sparsity (see Section (ref), Assumption (ref)). Additionally, a parametric estimation of the spectral density matrix of a VAR process requires an estimate of the covariance, or precision matrix, of the residual process $\{\BS v_t\}$, i.e., $\BS \Sigma_v$ or $\BS \Sigma_v^{-1}$ in (ref). The residuals can be consistently estimated by $\hat {\BS v}_t=\hat {\BS \xi}_t-\sum_{j=1}^{p_{\xi}} \hat {\BS B}^{(j)} \hat {\BS \xi}_{t-j}, t=p_{\xi}+1,\dots,T$. Then, based on these, procedures such as the graphical LASSO of friedman2008sparse, (A)CLIME of cai2011constrained or fused LASSO of dallakyan2023fused can be used in order to obtain a regularized estimator of $\bm \varSigma_v$ or $\bm \varSigma_v^{-1}$ (respectively, we will employ the notation $\hat{\bm \varSigma}_v^{(re)}$, $\hat{\bm \varSigma}_v^{(re)-1}$, where “re" is a shorthand for “regularized"). With this, we get the following estimator for $\BS f_\xi(\omega)^{-1}$:
where $\hat{\BS B}^{(thr,h)}=\operatorname{THR}_{\lambda_\xi}(\hat{\BS B}^{(h)})$ and $\operatorname{THR}_{\lambda_\xi}(\cdot)$ is a thresholding function with threshold parameter $\lambda_\xi$, fulfilling the conditions $(i)$ to $(iii)$ in Section 2 in cai2011adaptive. For instance, such a thresholding function can be the adaptive LASSO thresholding function given by $\operatorname{THR}_{\lambda_\xi}^{al}(z)=z(1-|{\lambda}/z|^\nu)_+$ with $\nu\geq1$. Soft thresholding ($\nu=1$) and hard thresholding ($\nu=\infty$) are boundary cases of this function. These thresholding functions act by thresholding every element of the matrix $\hat{\BS B}^{(h)}$ resulting in a row- and column-wise consistent estimation of the VAR slope matrices. In Section (ref), Lemma (ref), we present the error bounds for $\|\hat{\BS f}_\xi(\omega)^{-1}-{\BS f}_\xi(\omega)^{-1}\|_\infty$ and $\|\hat{\BS f}_\xi(\omega)^{-1}-{\BS f}_\xi(\omega)^{-1}\|_2$. Inversion of (ref) yields the estimator $\hat{\BS f}_\xi(\omega)$. Finally, replacing in $\bm f_x(\omega)=\BS \Lambda \bm f_{f}(\omega)\BS \Lambda^{\top}+\bm f_{\xi}$ both estimated spectral densities discussed above leads to our final estimator of the spectral density matrix $\hat{\bm f}_x(\omega)$. Its error bounds $\|\BS f_x(\omega)-\hat {\BS f}_x(\omega)\|_l$ for $l=1,2,\infty$ are presented in Section (ref), Theorem (ref).
We make use of two datasets comprising a large number of global bank assets.
Dataset (ii) necessarily has some differences with respect to (i). In fact, it comprises 83 stock price volatilities (instead of 96). The remaining ten series are the bond price volatilities of the same ten major world countries as in (i). The reason for the lack of 13 banks in (ii) with respect to (i) is that certain banks considered before are either not traded anymore in the new sample or they have too many missing values. We provide a complete list in Table (ref). Stock prices are from Datastream and Bond prices are from Bloomberg. To compute daily range-based realized volatilities\footnote{This type of volatility is the same computed in demirer2018estimating and is almost as efficient as realized volatility based on high-frequency intra-day data given it is robust to certain forms of micro structure noise, see alizadeh2002range.} we use the formula below, namely
where $H_{i,t}, L_{i,t}, O_{i,t}, C_{i,t}$ are the logs of daily high, low, opening and closing prices for bank stock $i$ on day $t$. We are going to focus on system-wide connectedness $C^H$, {for $H=10$}, and compute the part of it due to the (banking) market: $C^H_{Mkt}$, and the part due to the idiosyncratic shocks $C^H_{Ids}$, such that $C^H=C^H_{Mkt}+C^H_{Ids}$. Following demirer2018estimating, we employ a rolling window of 150 days and the reporting time point corresponds to the final day of the window. We estimate the factors and loadings via PCA, selecting the number of factors and lag-length of the VAR using the extended BIC information criteria of krampe2021factor (see also Section (ref)). This gives us for both Dataset (i) and Dataset (ii) only one common factor ($r=1)$ and $p_f=2$. The idiosyncratics are estimated via adaptive LASSO where initial weights are preliminary plain LASSO weights and the lag-length is estimated to be $p_{\xi}=4$.\footnote{In Section (ref) we show that $r,p_f,p_{\xi}$ can be jointly obtained via minimization of a single information criterion as in krampe2021factor.} The LASSO tuning parameter $\lambda$ is selected via standard BIC (see hecq2023granger for an overview of data-driven techniques to select the tuning parameter). Together with the SWC measure, we also report $95\%$ Bootstrap confidence bands. The full details of their computation are given in Section (ref).
Starting from $C^H$ in Figure (ref), we find an entirely similar shape, and roughly the same magnitude, as the SWC computed in demirer2018estimating (see especially their Figure 9). With the Federal Reserve decision to tighten monetary policy in May-June 2006, we observe an upward trending behavior of the SWC, culminating in the Lehman bankrupcy in 2008. Indeed, Lehman Brothers filed for bankruptcy on September 15, 2008. At that time, our estimated SWC is found on the rise, where on September 16th reaches a level of 74.7%, only to continue towards its highest peak reached on November 25th, at a staggering 89.7% SWC. It will take the whole year of 2009 for the SWC to reassess at a pre-crisis level of roughly 70% SWC. Two other notable SWC jumps correspond to May 2010, due to delays in Greece's rescue package, and another in August 2011, as sovereign debt and banking sector concerns spread to Spain and Italy. While the magnitude of SWC is in the same ballpark as demirer2018estimating, it is though slightly (roughly 5%) higher, and this is especially visible in calmer times.\footnote{Take the beginning of the sample for instance, our estimated SWC starts at a level of 64.8%, while demirer2018estimating estimates it just under the 60% threshold.} One likely reason of this is that the VAR Elastic Net of demirer2018estimating directly “sparsifies" the number of banks in every estimate of the VAR equations, while we only shrink part of the connections among different cross-sections. This entails that if truly factors are playing a role, then such direct sparsity is also implicitly shrinking the loadings. In our case, the factor model does not impose direct sparsity on the linkages of $\bm x_t$, nor on the loadings, but retains one strong factor representing the common behavior of all banks. Instead, we just sparsify the idiosyncratics' dynamic which, as discussed, is a more reasonable consideration. In other words, the assumption of sparsity of demirer2018estimating can potentially lead to underestimation of the degree of connectedness if truly the data linkages are many, and if the factors are strong. Furthermore, the fact that SWC is generally 'quite high' in our results, resonates well with the findings in a.o., allen2000financial, acemoglu2015systemic. Namely, that when the magnitude of the shocks is below a certain threshold, “a more diversified pattern of inter bank liabilities leads to a less fragile financial system". The other way around, if the shocks' magnitude surpasses a certain threshold “highly diversified lending patterns facilitate financial contagion and create a more fragile system" acemoglu2015systemic. These last considerations provide an interesting connection with the spectral analysis. In fact, further decomposing SWC into frequencies, thus considering our spectral-SWC $C_{d}^{H,\omega}$ for $\omega=\{\text{Monthly, Quarterly, Yearly}\}$, it so appears that a medium frequency response to shocks, $\omega=\{\text{Quarterly}\}$, seems to dominate the calmer times, while the more rapid monthly frequency, $\omega=\{\text{Monthly}\}$, leads and overcomes during the crises. The spectral SWC at monthly frequency indeed reaches its highest peak on October 28th 2008 (slightly leading the main peak of the SWC), where it reaches a value of 0.465, i.e., 52.3% of the SWC at the same date (0.889). The remaining 48% is made up of the other frequencies, respectively quarterly (0.271) and yearly (0.162). Equally interesting is to observe by what types of shocks (market/idiosyncratics), when, and at which frequency, the SWC is driven, i.e., looking at $C^H_{Mkt}, C^H_{Ids}$, and the respective spectral versions $C^{H,\omega}_{Mkt,d}, C^{H,\omega}_{Ids,d}$. Interestingly, we observe how the connectedness is mostly driven by idiosyncratic variation. In fact, $C^H_{Ids}$ averages at 0.6, meaning on average drives 80% or more of the SWC, while only 20% or less is left to the common component $C^H_{Mkt}$. However, $C^H_{Mkt}$ does jump upward during crises, or more generally financially turbulent times. We can observe how during the 2008 crisis, on October 28th, the $C^H_{Mkt}$ reaches its maximum peak at 0.36, i.e., roughly 40.5% of the whole SWC at the same date (0.889). While it doubles if compared to pre crisis levels, it is still not driving the majority of the SWC, which is indeed driven 59.5% by $C^H_{Ids}$. In terms of frequency response to shocks, we observe how $C^H_{Mkt}$ is driven for the major part by short frequencies, i.e., by $C^{H,\omega}_{Mkt,d}$, for $\omega=\{\text{Monthly}\}$, whilst the longer frequencies are almost irrelevant (especially, the yearly one). Interestingly, the opposite shall be said about the frequencies response decomposition for $C^H_{Ids}$. Indeed, we find that $C^H_{Ids}$ is predominantly driven by the longest frequency $C^{H,\omega}_{Ids,d}$, $\omega=\{\text{Yearly}\}$, which accounts alone for about 50% of $C^H_{Ids}$, whilst the quarterly and yearly frequencies accounts for less than 30% and 20%, respectively. Overall, we see that, given only one estimated common factor, SWC is driven predominantly, and in the long-run, by idiosyncratic connectedness. Crisis times instead see a surge in common component connectedness, driven purely by short-run dynamics.
Now we discuss the connectedness results in Figure (ref) on our more recent (2014-2023) dataset, containing almost the same variables (some are discontinued, see Table (ref) for a list) as in the previous analysis. The level of SWC (0.69) picks up from where was left after January 2014 in the previous analysis (0.67). $C^H$ exhibits less of a clear trending behavior in this new sample period but more of a level-shift/oscillatory one, around 75%. This level is roughly 10% more than the pre-2008 crisis, as it seems that the lesson of 2008 resulted in a more cohesive inter-bank network. A remarkable shift of the SWC is observed already between 2015 and 2017, where the mean wanders around 80% with heights of 84%. Several events can be associated with this raise of the SWC: from the European banking sector problems (a.o., Greece's debt crisis, italian banks' loans crisis), to low interest rates and negative rates in Europe, Brexit, the Chinese economic slowdown and stock market crash of 2015, just to name a few. The main, hard-to-miss event of relevance within the sample is of course the Covid19 outbreak and the consequent global crisis starting in 2019. In fact, we observe how in correspondence of the Covid19 outbreak, the SWC exhibits a vertical increase of more than 20%, from 72% (February 2020) to over 95% (March 2020). This unprecedented SWC level is maintained roughly until October 2020, after which an equally vertical drop is observed back to a level of roughly 75% or less. While a relapse that touches 80% can be observed in March 2022, in the remaining part of our sample SWC never reaches similar heights as March-October 2020. By observing the decomposition of SWC into frequencies, an entirely analogous picture as for the previous analysis presents itself. The medium frequency response to shocks, $\omega=\{\text{Quarterly}\}$, dominates the calmer times, even during 2015-2017, while the more rapid monthly frequency, $\omega=\{\text{Monthly}\}$, has a vertical increase of almost 50%, from 0.17 on 26th October 2020, to 0.60 in May and even 0.61 in July of the same year, only to drop back to a level of 24% and less from October 2020 onwards, with a short relapse at around 37% in March 2022. Disentangling again SWC into $C^H_{Mkt}$ and $C^H_{Ind}$, we again see how SWC is mostly driven by idiosyncratic variation. In this second sample, $C^H_{Ids}$ averages at roughly 0.65 (5% more compared to the previous sample) corresponding to more than 85% of the whole SWC, leaving only the remaining 15% to $C^H_{Mkt}$. The latter is again predominantly driven by the short frequency response to shocks ($C^{H,\omega}_{Mkt,d}$, for $\omega=\{\text{Monthly}\}$) and spikes upward during crisis. Especially, in the case of the abnormal circumstances of Covid19 we can observe almost an overlap between $C^H_{Mkt}$ and $C^{H,\omega}_{Mkt,d}$, for $\omega=\{\text{Monthly}\}$. Vice-versa, $C^H_{Ids}$ is again mostly driven by the longest frequency (yearly), but one difference can be observed during the Covid19 outbreak. While for the 2008 crisis the idiosyncratic connectedness did not drop dramatically, it is most definitely the case for the Covid19. While the confidence bands are wide, in May 20th, 2020, $C^H_{Ids}$ is estimated at 0.31 (only 32% of the SWC) while $C^H_{Mkt}$ is at 0.64 (67% of SWC), thus practically inverting the levels of one-another.
In the previous analyses we observed two, very different global crises: the 2008 subprime crisis and the 2019 Covid19 pandemic. The former is a financial collapse rooted in the financial sector and its dynamics; the latter is a global health crisis that had unprecedented effects (for the recent history, at least) on the economy and the financial stability of countries and institutions. Both of these, in their own ways, are responsible for an increased level of global economic uncertainty, and in some cases of proper panic-spreading in the financial markets (aside of elsewhere). We have seen how, in line with demirer2018estimating, global crises correspond to an increase in the overall SWC. However, we have also been able to uncover how, when times are calm, SWC is predominantly driven by an idiosyncratic dynamic. When global crises hit instead, a sharp increase in the connectedness due to the common component (market dynamic) is observed, in line with e.g., barigozzi2021time, and this is almost entirely driven by a short dynamic response to shocks. While both crises are in their own way somewhat unprecedented events, the financial crisis in 2008 has undoubtedly quite a different shape with respect to the Covid19 pandemic. Banks seem to have perceived the 2008 collapse quite some years in advance, as visible from the building-up pattern of $C^H$ from late 2004 onwards, all the way to 2008 barrell2008evolution. It also seems that such a crisis has had a more “sticky" effect for the connectedness, which ever since has maintained a slightly higher level of interconnections than before. The Covid19 crisis instead, has created an unprecedented vertical increase in SWC and SWC due to the Market, and correspondingly the most vertical drop in $C^H_{Ids}$. The global panic generated by such a crisis has very rapidly shoot up $C^H$. This finding is in line with e.g., bouri2021return, who also find Covid19 has altered the network of (in their case return) connectedness by generating sudden increases in the system-wide connectedness. The reached peak of $C^H$ is then maintained roughly for the whole duration of the uncertainty created by the pandemic and the connectedness then recovers --almost at the same vertical pace-- the pre-crisis level.
A few words on the notation we employ. For any vector $\boldsymbol{x}\in \mathbb{R}^n$, ${\left\lVert\boldsymbol{x}\right\rVert}_p = \left(\sum_{i=1}^n |x_i|^p \right)^{1/p}$ denotes the $\ell_p$-norm and $\bm e_j$ denotes a unit vector of appropriate dimension with the one in the $j$th position. For a $r\times s$ matrix $\BS A=(a_{i,j})_{i=1,\ldots,r, j=1,\ldots,s}$, $\|\BS A\|_1=\max_{1\leq j\leq s}\sum_{i=1}^r|a_{i,j}|=\max_j \| \BS A \bm e_j\|_1$, $\|\bm A\|_\infty=\max_{1\leq i\leq r}\sum_{j=1}^s|a_{i,j}|=\max_{i} \| \bm e_i^\top \BS A\|_1$ and $\|\BS A\|_{\max}=\max_{i,j}\allowbreak |\bm e_i^\top \BS A \bm e_j|$. $\bm A^i$ denotes the $i$th matrix power of $\bm A$ and $\bm A^{(i)}$ refers to the $i$th element of a sequence of matrices. We denote the largest/smallest absolute eigenvalue of a square matrix $\BS A$ by $\sigma_{\max/\min}(\BS A)$ and $\|\BS A \|_2^2=\sigma_{\max}(\BS A \BS A^\top)$. $\|\BS x\|_0$ denotes the number of non-zero elements of $\BS x$. $\operatorname*{plim}$ denotes convergence in probability.
In estimating the dynamic factor model with sparse VAR idiosyncratic components we closely follow the work of krampe2021factor. We report here a summary of the main assumptions and, importantly, the estimation algorithm. We refer to the said paper for details.
We work with factors $\{\bm f_t\}$ and idiosyncratics $\{\bm \xi_t\}$ being second order, uncorrelated stationary processes, both with bounded $\ell_2$ innovation covariances, and with the idiosyncratics autocovariance matrix bounded in $\ell_2$ norm for increasing $N$. Eight finite moments are assumed on the innovation process $\{(\bm u_t^{\top},\bm v_t^{\top})^{\top}, t\in\mathbb{Z}\}$ and weak factors are ruled out, so each of the factors provides a non-negligible contribution to the variance of each component of $\{\bm x_t\}.$ For the idiosyncratics VAR($p_{\xi}$) coefficient matrix, approximate {row-wise} sparsity is assumed and it is allowed to grow with the sample size. The following Assumption (ref)-Assumption (ref) formalize this, where we use the notation $M_1,\ldots,M_8$ to refer to some positive constants.
For the estimation algorithm it is convenient to stack $\boldsymbol{x}_t,$ $t=1,\dots,T$ row-wise in order to obtain $\boldsymbol{X}=\boldsymbol{\chi}+\boldsymbol{\Xi}$ as a $T\times N$ matrix form of the factors & idiosyncratics decomposition $\boldsymbol{x}_t=\bm \chi_t+\bm \xi_t$. The two step estimation procedure then proceeds as follows:
Let $\hat{\bm \varSigma}_v^{(re)}$ be a regularized version of the sample covariance matrix $\hat{\bm \varSigma}_v$, i.e., using regularization such as thresholding bickel2008covariance, CLIME cai2011constrained, LASSO Cholesky as in Margaritella17122024 or graphical LASSO meinshausen2006high,friedman2008sparse. We use here the graphical LASSO which puts sparsity constraints on $\bm \varSigma_v^{-1}$. The same estimator will be also used in Section (ref) to estimate the idiosyncratic spectral density matrix.
Let $K$ be a kernel estimator of the factors' sample periodogram, fulfilling the following two regularity assumptions\footnote{Let us note how absolute summability of the factors autocovariances is directly implied by Assumption (ref)} (same as Assumption 1, 2 in wu2018asymptotic):
These requirements are quite general, as they hold for most of the commonly used kernels (e.g., the Barlett kernel). Then, a spectral density estimator for the factors is given by
where $\hat { \BS\Gamma}_f(h)=T^{-1}\sum_{t} \hat{\BS f}_{t+h} \hat{ \BS f}_t^{\top}$ is the sample autocovariance function. Consistency of $\hat {\BS f}_f(\omega)$ follows, as formalized in the following Lemma (ref) below.
A word on the obtained estimation rate in (ref). In order to get consistency, it is unsurprisingly needed for both $N$ and $T$ to grow. Additionally, if $N = T^a$ and $0 \leq a \leq \zeta - 4$, we have $g(N, T, \zeta) \leq \frac{1}{\sqrt{NT}} + \frac{1}{T}$ which means $g(N, T, \zeta)$ could be dropped in $O_P$-notation. Likewise, $\frac{{\log(N)}}{{T}}$ and $\frac{\sqrt{\log(N)}}{\sqrt{NT}}$ can also be dropped if $N$ grows only polynomial with respect to $T$, which is a mild requirement. Hence, it remains $\frac{k_\xi}{N}$. First, note how the term $k_{\xi}$ from Assumption (ref) quantifies the linear dependence of the idiosyncratic component. We follow krampe2021factor who notes that $\sqrt{N}$ is an upper bound for the growth rate of $k_{\xi}$, and therefore a rate smaller than $\sqrt{N}$ is most sensible and in line with the factor models literature. Letting $N = T^a$ as above, we can then consider $k_{\xi}=O_P(T^{a/2-\varepsilon})$ for some small $\epsilon>0$. Substituting and simplifying one obtains $\frac{k_\xi \sqrt{T}}{N \sqrt{B_T}}=O_P( T^{1/2-a/2-b_1-\varepsilon})$. Thus, if $1/2\leq/2+b_1+\varepsilon$ the estimation of the factors does not lead to a slower rate for the spectral density estimator.
Proof of Lemma (ref), given in the Appendix, hinges on the fact that the difference between the estimated spectral density $\boldsymbol{\hat{f}}_f(\omega)$, and the rotated version of the true one $\BS H_{NT}\BS f_f(\omega)\BS H_{NT}^\top$, can be bounded above by the difference between the former and an infeasible version of the former, plus the difference between the infeasible version and the true one. The infeasible version here contains in (ref) $\tilde{\BS \Gamma}_{f} (h)=T^{-1}\sum_{t} \BS f_{t+h} \BS f_t^{\top}$, in place of $\hat {\BS \Gamma}_f(h)$, and results from wu2018asymptotic and krampe2021factor can then be straightforwadly applied to yield the consistency. Though it depends on the choice of the kernel, one would want to have an as small as possible bandwidth $B_T$, so as to approximate a parametric rate for ${\left\lVert\boldsymbol{\tilde f}_f(\omega)-\BS f_f(\omega)\right\rVert}_{\max}$, while having an as smooth as possible spectra, i.e., large $q$ where $\lim_{x\to 0} \frac{1-K(x)}{|x|^q}<\infty$ wu2018asymptotic.
Now onto the idiosyncratics. As mentioned, we use here the VAR structure of the idiosyncratic component to estimate its spectral density matrix. In Section (ref), the VAR parameters of the idiosyncratic component can be estimated row-wise consistently (see Assumption (ref)), i.e., consistency of $\BS B$ for the matrix norm $\|\cdot \|_\infty$. However, the estimation of the spectral density requires additional column-wise consistency, that is consistency of $\BS B^{(j)},j=1,\dots,p_{\xi},$ with respect to $\|\cdot\|_1$. Such a column-wise consistency requires additional sparsity assumptions, see also krampe2020statistical for a discussion. Furthermore, a parametric estimation of the spectral density matrix of a VAR process requires an estimate of the covariance or precision matrix of the residual process $\{\BS v_t\}$. Assumption (ref) below formalizes the additional sparsity assumption and the requirements on the residuals covariance matrix.
Now, we can give the error bound for the estimator of the spectral density of the whole process $\bm f_{x}(\omega)$. We only present here explicitly the rate for a simplified case. In the general case, an explicit rate can be obtained by inserting the results of Lemma (ref) and Theorem 1 of krampe2021factor. Since it leads to a lengthy and not insightful expression, we omit it here. The rate is dominated by the estimation error of the sparse VAR and it is similar to the one in Theorem 1 of krampe2021factor. However, the rate is more affected by the sparsity parameter in the sense that its maximum growth rate is less for the spectral density than it is for prediction. Maximum growth rate refers here to the maximal rate of sparsity for which consistency can be achieved.
We decompose the high-dimensional global bank network connectedness index into connectedness due to market shocks, idiosyncratic shocks, and shocks at high, medium, and low frequencies. Instead of regularizing the high-dimensional vector of banks with sparsity-inducing estimators, we use recent literature linking factor models with sparse ones. We estimate a static, approximate factor model with sparse VAR idiosyncratic components, enabling decomposition of connectedness into these parts and providing bootstrap confidence bands. We also analyze the spectral counterpart to disentangle frequency responses to shocks. Our findings show that idiosyncratic variation largely drives the highly interconnected network of bank stock price volatilities, especially during non-turbulent periods and in the long run. However, during major crises like the 2008 financial crisis and Covid-19, bank stock volatilities become more interconnected, with connections driven by short-run market dynamics.