Sophie Béreau

Nicolas Debarsy

Cyrille Dossougoin

Jean-Yves Gnabo

Contagion in the Banking Industry: a Robust-to-Endogeneity Analysis

Keywords: banking, common asset exposures, contagion, endogeneity, Katz centrality, market-price channel, information channel, spatial econometrics, spillover

What drives financial contagion? The empirical literature aimed at modeling financial risk spillovers in crisis periods and documenting the role of contagion channels is subject to an endogeneity issue, as the channel itself can respond to a change in the level of risk. We tackle this issue by using a novel spatial econometric estimation procedure based on a control function approach and offer "robust-toendogeneity " evidence on the role of indirect financial contagion channels in the banking industry. Our estimations, based on on 28 large US banks during the financial crisis (2007Q3-2013Q2), confirm that several channels are endogeneous. Accounting for endogeneity is proved to be important for recovering reliable estimates of transmission mechanisms. Banks similarity in fundamentals, similarity in investment strategy as well as common exposure appear as significant drivers of contagion. Based on relevant transmission's channels, we build a simple systemic risk indicator named "Interaction Based Centrality". We show that it may help forecast vulnerable institutions in times of crisis and could thereby be used for monitoring purposes by regulatory authorities.

Introduction

Contagion cascades in the banking industry build on the premise that banks' losses in a crisis period can be transmitted from one institution to another through their interconnectedness, meaning that a small segment of the market can destabilize the whole industry. In the case of the global banking crisis that broke out in 2007 multiple channels of contagion were potentially at play, including cross-lending in interbank markets, fire-sale externalities and characteristics-based similarities (information channels) to cite a few (see [START_REF] Clerc | Indirect contagion: the policy problem[END_REF]. From that period, growing consensus on the importance of financial sector interconnectedness has spread rapidly in both the research and policy worlds. In the latter, policymakers have pressed forcefully for the introduction of insights from financial interconnectedness and network theory on stress testing [START_REF] Espinosa-Vega | Cross-Border Financial Surveillance; A Network Perspective[END_REF] and identification of Globally Systemically Important Banks (GSIBs). On the other side, the research community has been implicitly entrusted to guide policy decisions by documenting theoretically and empirically the contagion channels as well as the associated transmission mechanisms.

What drives financial contagion? We argue in this paper that the empirical literature aimed at documenting the channels of contagion is subject to an endogeneity issue, which means that careful treatment of the data is required to retrieve reliable evidence.

The problem stems from the response of the transmission channel itself to the distress of a financial institution or a country, depending on the context. Ignoring this form of endogeneity yields inconsistent and biased parameters estimates, as acknowledged in recent studies on spillover effects (see [START_REF] Kelejian | Estimation of spatial models with endogenous weighting matrices, and an application to a demand model for cigarettes[END_REF][START_REF] Qu | Estimating a spatial autoregressive model with an endogenous spatial weight matrix[END_REF].

Relying on a "robust-to-endogeneity" framework recently developed by [START_REF] Shi | A spatial panel data model with time varying endogenous weights matrices and common factors[END_REF], this paper provides new evidence about the drivers of financial contagion in the banking industry, which was at core of the US financial crisis in the late 2000s. In a nutshell, our results -based on a sample of 28 large US institutions over the 2007Q3 -2013Q2 crisis period -unambiguously confirm the endogeneity of several transmission channels discussed in the literature, show the importance of controlling for it in the estimation procedure and confirm the role of common portfolio exposure, along with both investment-based similarity and fundamental-based similarity, as channels of contagion.

While the theoretical or simulation-based literature is vast, econometric studies on interconnectedness and spillover effects in the banking industry are scarcer (see, for an overview of contagion in financial networks, [START_REF] Glasserman | Contagion in financial networks[END_REF]. 1 Among the causes explaining this asymmetry of treatment stand two historical caveats. First, the applied literature has suffered from limited access to data on banks' physical interconnectedness. Second, estimating a contagion model requires an econometric approach that accounts for cross-sectional dependence in the data. In recent years, however, remarkable progress has been made on both fronts. Data have been made increasingly available to researchers, even though relevant information such as cross-lending is still difficult to obtain by most researchers outside central banks.2 Meanwhile, new classes of "system-wide" econometric models have emerged in applied works on macroeconomic or financial contagion, accommodating cross-sectional correlation in the data. Among these models, some remain "agnostic" on the nature of interconnections, such as large VAR models [START_REF] Diebold | Measuring financial asset return and volatility spillovers, with application to global equity markets[END_REF]. Others pertaining to Global Vector Auto-Regression (GVAR) or spatial econometrics models explicitly specify in their reduced form equation the links governing the spillover effects process (see, among others [START_REF] Favero | Modelling and forecasting government bond spreads in the euro area: A GVAR model[END_REF]for GVAR and Blasques et al., 2016[START_REF] Debarsy | Measuring sovereign risk spillovers and assessing the role of transmission channels: A spatial econometrics approach[END_REF] for spatial econometric models). Spatial econometric models in particular display several attractive features in relation to model contagion. This class of models builds on the premise that a specific individual (here a bank characterized by its stock returns' volatility) is affected by her neighborhood. The closer the neighbors, the stronger their influence. Whether other entities populating the system are distant or close is governed by an interaction scheme (matrix) set by the researcher. Modelling propagation of financial turbulence, [START_REF] Jing | The propagation of financial turbulence: interdependence, spillovers, and direct and indirect effects[END_REF] use, for instance, trade and capital flows as interaction matrices. Importantly, the statistical inference regarding the relevance of each interaction scheme can be performed. Compared to "agnostic" models such as large VAR models that do not impose economically-based interaction schemes, these models enable isolating the contribution of specific channels of spillover (i.e. specific matrices). They also allow comparison of their respective importance (see [START_REF] Debarsy | Measuring sovereign risk spillovers and assessing the role of transmission channels: A spatial econometrics approach[END_REF][START_REF] Elhorst | Spillovers in space and time: where spatial econometrics and Global VAR models meet[END_REF] for further discussions on these two classes of models), making them of utmost interest for policymakers.

When it comes to estimating these models, however, one central assumption was often overlooked. It pertains to the exogeneity of the interaction matrix. 3 We argue that this assumption might be violated in a number of cases, be it in the banking industry or, in many other instances in economics and finance more generally. In our context, for instance, the exogeneity assumption implies that interaction matrices (i.e. transmission channels) such as those constructed from cross-lending or characteristics-based similarities (e.g. similarity in size or portfolio composition) do not respond to a change in the financial conditions of an institution. In practice, though, it is likely that the distress of a bank, for instance, will soon affect its lending relationship or the composition of its portfolio, changing its interaction scheme with the rest of the industry. Wang et al. (2019) acknowledge the problem in a contribution, closely related to ours, documenting the role of the information contagion channel with a spatial econometric model.

Quoting their study, they emphasize that " [t]he data-generating process assumes that information contagion happens for given perceptions of the bank similarities. However, if the underlying similarities can change as a result of information spillovers, then simultaneity will lead to biased estimates of the network effects." Eventually, they try to mitigate the problem by playing with the data frequency. We add to their contribution by addressing the problem in full, proposing a "robust" approach to contagion. Robust, in our context, means that we model contagion effects in the banking industry while relaxing the exogeneity assumption on the channel of transmission. Specifically, we proceed in two steps. First, we test for the presence of endogeneity. Second, we interpret the results for the most suited model -that is, with a correction for the presence of endogeneity should the matrix be endogenous or without otherwise.

To do so, we rely on the spatial panel data model with time-varying interaction matrices recently developed by [START_REF] Shi | A spatial panel data model with time varying endogenous weights matrices and common factors[END_REF]. This paper develops a control function approach that accommodates endogenous interaction matrices. It also displays two additional interesting features for modelling the contagion phenomenon. First, it does not require identifying external instruments, which is often a difficult task and a subject of debate. Second, it enables including in a single model both spillover effects and " systematic " effects through the inclusion of common factors. The effects of common risk factors can thus be easily separated from purely cascading mechanisms when the risk is transmitted from one institution to another. To our knowledge, we are the first to use this flexible econometric framework in the field of finance. 4 In line with recent contributions focusing on risk transmission in the banking industry (see, for instance, [START_REF] Balla | Tail dependence and indicators of systemic risk for large US depositories[END_REF][START_REF] Korobilis | Measuring Dynamic Connectedness with Large Bayesian VAR Models[END_REF][START_REF] Patro | A simple indicator of systemic risk[END_REF], we apply our model to a set of 28 large US depository institutions (hereafter, banks), over the period 2007Q3 -2013Q2. The risk of each institution is measured by its stock market volatility, as often 4 Balla et al. (2014) display several similarities with our analysis as they assess the link between interconnectedness and the systemic risk of US banks. A notable difference lies in the role of the transmission channel. [START_REF] Balla | Tail dependence and indicators of systemic risk for large US depositories[END_REF] remain agnostic about the transmission channel, while our econometric set-up allows identification of the channels through which the transmission operates.

in the literature (see, for instance, [START_REF] Korobilis | Measuring Dynamic Connectedness with Large Bayesian VAR Models[END_REF].5 Most bank-level characteristics are extracted from Bloomberg and the Center for research in Security Prices (CRSP). Full information on portfolio holdings is retrieved for each bank at a quarterly frequency from Form 13F for to the Securities and Exchange Commission (SEC).

Our baseline model includes standard bank-specific characteristics along with common factors. The model with spillover effects adds to the baseline equation an interaction component, which is constructed as the weighted sum of the risks of all surrounding institutions. The weights are given by a specific interaction matrix in each model. We concentrate on indirect financial contagion channels. Unlike with direct financial contagion, negative externalities occur without contractual obligation [START_REF] Kiyotaki | Balance-sheet contagion[END_REF] such as cross-lending. Typical causes for indirect financial contagion are common asset exposure and fire sales of assets -market price channel -or when disclosure of financial trouble by one bank leads market participants to make inferences about problems of other banks viewed as similar -information channel -spreading the stress from one institution to others. As expressed in the opening quote from [START_REF] Clerc | Indirect contagion: the policy problem[END_REF], this source of contagion is often cited as critical in explaining the financial crisis that followed the collapse of the US housing market. Documenting empirically the role of the different sub-channels that could have been at play is therefore of utmost importance.

Within this framework, we consider in total 15 alternative interaction schemes belonging to three broad categories: (i) common asset exposures (market price channel), (ii) fundamental-based similarity (information channel), (iii) investment-based similarity, which includes both sector-based similarity and diversification-based similarity (information channel), and cover as a result a wide array of potential contagion channels.

The contribution of each channel is tested separately. We then illustrate how the estimation's results from the spatial model can be used for financial stability monitoring.

First, we compute a simple indicator of interconnectedness, named "Interaction-Based Centrality "(IBC) which is based on the Katz centrality measure. It enables to identify financial institutions which are the most exposed to the risk of the system. Next, we test whether such indicator has significant forecasting power for some of the matrices to spot vulnerable institutions in time of crisis.

Overall, several important outcomes emerge from our analysis. First, our estimation procedure supports the presence of endogeneity in several transmission channels. This finding confirms the need to apply robust-to-endogeneity methods to analyze financial contagion. Looking at the magnitude of the problem created by the endogeneity of the interaction matrix, we show that applying the control approach procedure proposed by [START_REF] Shi | A spatial panel data model with time varying endogenous weights matrices and common factors[END_REF], when needed, may change the value of the coefficients attached to the spillover term by a factor up to four. Second, our framework identifies several active channels of indirect financial contagion in the banking industry. Common exposure emerges as a driver of financial contagion, whereby two banks that have invested in the same assets are more exposed to contagion. Likewise, the similarity in sectoral investments appears as a significant channel of risk spillovers. We also find significant spillovers when we consider similarities in fundamentals such as dividend yield and market value to total asset ratio. Third, our estimation procedure can be used to built indicators as illustrated by our IBC indicator that forecast vulnerable institutions.

The remainder of the paper is as follows. Section 2 introduces the data along with the empirical estimation strategy. Section 3 discusses the empirical results. Section 4 concludes.

Data and methodology

We study risk spillovers in 28 US large banks. 6 We consider the period of 2007Q3-2013Q2 as the crisis period for our baseline model and test other alternatives for the robustness check. 7 Table 1 summarizes the full list of banks considered.

6 Our sample of institutions is comparable in size and bank's profiles to other studies on interconnectedness and spillover effects in the banking industry. [START_REF] Balla | Tail dependence and indicators of systemic risk for large US depositories[END_REF] investigate the extreme loss tail dependence between the stock returns of 27 large US depository institutions. [START_REF] Korobilis | Measuring Dynamic Connectedness with Large Bayesian VAR Models[END_REF] analyze banking interconnectedness by estimating a large Bayesian time-varying parameter vector autoregressive (TVP-VAR) model on 35 US and European financial institutions. [START_REF] Patro | A simple indicator of systemic risk[END_REF] explore the link between the 22 largest bank holding companies and investment banks.

7 Selecting the sample period to characterize the chain reaction or domino effects among banks is not trivial. On the one hand, the model needs enough chronological data to identify regular patterns across time periods. On the other hand, we must be as close as possible to the crisis period to estimate the effects when these mechanisms operate more forcefully. For these reasons, we consider 2007Q3, marked by the problems of two Bear Stearns' hedge funds, as a starting date. This choice is consistent with a large part of the literature (see for instance [START_REF] Balla | Tail dependence and indicators of systemic risk for large US depositories[END_REF][START_REF] Acharya | Measuring systemic risk[END_REF][START_REF] Billio | Econometric measures of connectedness and systemic risk in the finance and insurance sectors[END_REF]. For an end date, we select the statement made by Ben Bernanke before the Congress on May 22, 2013, announcing that the Fed would likely start "tapering" the pace of its bond purchases. The end date also corresponds with the period when the S&P500 resumed to its level of 2007Q3. Finally, we have also tested the robustness of our conclusions to slight alterations of the definition of the end of crisis period. This section presents our main variable of interest and the methodology to assess the role of alternative transmission channels.

Risk measure

Our dependent variable is the level of risk of a banking institution. We take as a risk measure stock market volatility which has been widely used in the literature to assess the risk of either a stock, portfolio or company. For instance, [START_REF] Korobilis | Measuring Dynamic Connectedness with Large Bayesian VAR Models[END_REF], in a study closely related to ours, reconstruct the bank contagion network by modelling stock-market volatility with the Time Varying VAR model (see also [START_REF] Diebold | Financial and Macroeconomic Connectedness: A Network Approach to Measurement and Monitoring[END_REF][START_REF] Sarin | Understanding bank risk through market measures[END_REF].

For each institution, we compute within-quarter historical volatility (standard deviation) from the daily returns collected on the CRSP database.

Baseline econometric model

To assess the role of alternative channels in the transmission of risk across banks, we posit the following econometric model:

y t = X yt β y + Γ y f yt + λW t y t + V t , (1)
where for a given time t, y t is the n × 1 vector of risk measures for all the considered banks. X yt is a n×K matrix of banks' specific characteristics with the associated K ×1

vector of parameters β y . 8 In this study, we consider three types of bank characteristics: business model, liquidity measure and credit risk variables (see Tables 2 and3 respectively for a definition of the variables and descriptive statistics). The spatial lag W t y t captures interactions between banks' volatility and its intensity is quantified through the cross-sectional de-8 In spatial econometrics models, variables form triangular arrays (see [START_REF] Kelejian | A generalized spatial two stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances[END_REF]. All variables should therefore be indexed by n, the number of banks considered. However, we omit this notation for the sake of readability.

9 We delay further details about the specification of the interaction matrix to section 2.3. It is important to account for these effects in the econometric model so as to clean the cross-sectional dependence parameter from their influence. Our specification thus allows for a flexible yet parsimonious way to model common shocks and account for their heterogeneous effect on banks. Following [START_REF] Bai | Panel data models with interactive fixed effects[END_REF], we treat Γ y and f yt as fixed effects parameters to be estimated. Second, the interaction matrix, W t , is time-varying and possibly endogenous. A common and key feature of econometric models in applied research on economic spillovers [START_REF] Dell'erba | Spatial spillovers in emerging market spreads[END_REF][START_REF] Blasques | Spillover dynamics for systemic risk measurement using spatial financial time series models[END_REF][START_REF] Debarsy | Measuring sovereign risk spillovers and assessing the role of transmission channels: A spatial econometrics approach[END_REF] and financial spillovers [START_REF] Eder | Cds spreads and contagion amongst systemically important financial institutions -a spatial econometric approach[END_REF][START_REF] Tonzer | Cross-border interbank networks, banking risk and contagion[END_REF]Wang et al., 2019) is to assume the interaction scheme between individuals to be time-invariant or at least exogenous. These assumptions are the centerpiece of classical spatial econometrics models for which several estimation methods have been proposed.10 However in some circumstances this assumption can be too strong, especially when the interaction structure is constructed from economic or financial variables (see for example [START_REF] Eder | Cds spreads and contagion amongst systemically important financial institutions -a spatial econometric approach[END_REF][START_REF] Eder | Cds spreads and contagion amongst systemically important financial institutions -a spatial econometric approach[END_REF][START_REF] Caporin | Proximity-structured multivariate volatility models[END_REF][START_REF] Caporin | Proximity-structured multivariate volatility models[END_REF].11 A novel generation of models have been very recently introduced [START_REF] Kelejian | Estimation of spatial models with endogenous weighting matrices, and an application to a demand model for cigarettes[END_REF][START_REF] Qu | Estimating a spatial autoregressive model with an endogenous spatial weight matrix[END_REF][START_REF] Qu | Instrumental variable estimation of a spatial dynamic panel model with endogenous spatial weights when t is small[END_REF][START_REF] Han | Bayesian analysis of spatial panel autoregressive models with time-varying endogenous spatial weight matrices, common factors, and random coefficients[END_REF][START_REF] Qu | QML estimation of spatial dynamic panel data models with endogenous time varying spatial weights matrices[END_REF][START_REF] Shi | A spatial panel data model with time varying endogenous weights matrices and common factors[END_REF] accommodating a time-varying and endogenous interaction scheme. As further discussed when presenting the estimation procedure developed by [START_REF] Shi | A spatial panel data model with time varying endogenous weights matrices and common factors[END_REF] we build on these developments to document the role of indirect contagion in the banking industry. Finally, V t is a n × 1 vector of independent and normally distributed error terms with mean 0 and variance

σ 2 v I n .

Characterization of indirect financial contagion channels

A key feature of the spatial econometrics model is the specification of the interaction matrix. The matrix models the channel through which the transmission operates. We rely on the literature to select the set of channels to be tested. More specifically, we focus on indirect contagion. Within this framework, we consider the two main families of channels: (i) market-price channels and (ii) information channels. According to the theoretical literature [START_REF] Duarte | Fire-sale spillovers and systemic risk[END_REF][START_REF] Greenwood | Vulnerable banks[END_REF][START_REF] Cont | Fire sales, indirect contagion and systemic stress testing[END_REF], the market-price channel is a key contributor to financial contagion. The transmission mechanism operates through common asset exposure and fire sales of assets. If a large bank experiences distress and is forced to liquidate its assets with a haircut, the asset price will spiral down, forcing banks holding similar assets in their portfolio to get rid of them under fire-sale conditions. Such a response contributes to reinforcing as well as diffusing the initial shock. This mechanism is studied in [START_REF] Greenwood | Vulnerable banks[END_REF], who propose a theoretical model to analyze the effect of fire sales when a bank has a regulatory leverage constraint. The theoretical foundation of spillovers through information channels can be traced back to the seminal work of [START_REF] Diamond | Bank runs, deposit insurance, and liquidity[END_REF]. The transmission mechanism at work in this case suggests that investors use signals or information about one bank in distress to infer the likelihood of distress in another bank that shares similar characteristics. By adjusting their positions accordingly, investors contribute to spreading of the distress. In their paper, [START_REF] Clerc | Indirect contagion: the policy problem[END_REF] stress the critical role of this channel, arguing that runs are rarely driven by pure hysteria but tend to be related to fundamental changes in information, such as bad news about the health of a bank.

An interesting source of data is the quarterly US data on institutional portfolio holdings as reported through the Form 13F to the Securities and Exchange Commission (SEC).12 According to rule 13(f)-1 of the Equipped with these data, we can construct our connectivity matrices. For the information channel, the underlying mechanism builds on the existence of imperfect information in financial markets. It further supposes that market participants can use the characteristics attached to one financial institution for extracting a signal on those judged as similar. The similarity of two institutions as perceived by the market is the cornerstone of the mechanisms. We consider two criteria here: (i) similarity of investments and (ii) similarity of fundamentals. The components of our similarity-based interaction matrices take the following generic form:

w s ij,t = 1 1 + |z s i,t -z s j,t | , (2)
where z s i,t depends on specific characteristics. For the similarity of investment, we VtPi,t , where P i,t = (P i,1,t , ,P i,S,t ) is a vector of investment shares (that sum to 1) at sector level, Σ t is a vector of volatility measures and V t the associated covariance matrice. Σ t and V t are computed over a rolling window of four quarters using the daily returns on the S&P500 sector indices.

√ P i,t
Bloomberg consider the level of industry concentration (i.e. sector-based similarity matrices), which has been shown to be a critical feature of investment strategy by financial institutions (see [START_REF] Kacperczyk | On the industry concentration of actively managed equity mutual funds[END_REF]. Hence, z s i,t is the ratio between the total amount of assets held by bank i in sector s and its balance sheet total asset at time t, s = 1, . . . , S, where S is the number of sectors (in our case S=10; see Table 5 for the detailed list). 13 Another measure of investment similarity, Portfolio Diversification index -feature the level of diversification of the portfolio (see Table 4 for further details).

For fundamental-based measures of similarity, we consider three standard characteristics of banks: the dividend yield ratio, the debt to total assets ratio and the market capitalization to total assets ratio.

We go on to construct a measure of common exposure based on portfolio holdings. As further explained in Section 2.4, we need to impose some restrictions on the measure to be used in our econometric setting. In particular, the matrix should be constructed as a

13 Assets are categorized following the Global Industry Classification Standard (GICS). for each institution. It gives z ca i and z ca j for banks i and j respectively. Equipped with these bank-specific variables, we use the cross-product in the spirit of [START_REF] Blocher | Network externalities in mutual funds[END_REF] to create our a measure of portfolio proximity. Specifically, we obtain the connectivity at time t between banks i and j by computing the product of the share held in this set of assets by the 2 banks: w ca ij,t = z ca i,t z ca j,t . A high level of the measure means that the two corresponding banks are concentrated in these similar assets. Arguably, a more accurate measure would have been to compute the overlapping for two banks with respect to the whole set of assets included in their portfolio instead of restricting it to the share in the assets held by all banks in our sample. However, as further explained in the next section, our empirical framework enables to deal with endogeneity at the price of a few restrictions for the construction of the interaction matrix. Our way to construct the w ca ij,t , while imperfect, has the advantage to provide insights regarding common exposure effects within this framework.

The resulting w s ij,t defines the constituents of our different transmission channels, W t .

In total, we count 15 matrices. We finally standardize these interaction matrices by the maximum eigenvalue to ensure comparability of the cross-sectional dependence parameter (λ) across specifications.

Econometric approach to the endogeneity of transmission channels

Traditional estimation methods in spatial econometrics rely on the assumption that the interaction matrix is exogenous. When the interaction structure is constructed from economic or financial variables and hence contains economic interpretation (see, for example, the interaction structures in [START_REF] Eder | Cds spreads and contagion amongst systemically important financial institutions -a spatial econometric approach[END_REF][START_REF] Eder | Cds spreads and contagion amongst systemically important financial institutions -a spatial econometric approach[END_REF][START_REF] Caporin | Proximity-structured multivariate volatility models[END_REF][START_REF] Caporin | Proximity-structured multivariate volatility models[END_REF], this exogeneity assumption might be too strong. Ignoring this endogeneity yields inconsistent and biased parameters estimates. We tackle this issue by harnessing the emerging literature on endogenous interaction matrices in spatial econometrics (see [START_REF] Kelejian | Estimation of spatial models with endogenous weighting matrices, and an application to a demand model for cigarettes[END_REF][START_REF] Qu | Estimating a spatial autoregressive model with an endogenous spatial weight matrix[END_REF][START_REF] Han | Bayesian analysis of spatial panel autoregressive models with time-varying endogenous spatial weight matrices, common factors, and random coefficients[END_REF][START_REF] Qu | Instrumental variable estimation of a spatial dynamic panel model with endogenous spatial weights when t is small[END_REF][START_REF] Qu | QML estimation of spatial dynamic panel data models with endogenous time varying spatial weights matrices[END_REF][START_REF] Shi | A spatial panel data model with time varying endogenous weights matrices and common factors[END_REF] and specifically exploit the approach recently developed by [START_REF] Shi | A spatial panel data model with time varying endogenous weights matrices and common factors[END_REF]. In that framework, the elements of the interaction matrix W t are constructed from a variable Z t . 14

14 For instance w ij,t = 1 (zit-zjt) . In their paper, Shi and Lee (2018) allow w ij,t to be constructed from multivariate vector of p variables Z t . However, in this contribution, we consider an univariate Z t .

20

The variable Z t is modeled as follows:

Z t = X zt β z + Γ z f zt + t , (3)
where X tz are k z × 1 regressors with corresponding coefficient vector β zl , f zt which consists of R z × 1 time factors with loading Γ z and t , an idiosyncratic error. Finally, Assumption 2 from Shi and Lee (2018), reproduced below, models the source of endogeneity of the interaction matrix.

The error terms (v it , it ) are independently and identically distributed across i and over t, and have a joint distribution

(v it , it ) ∼ (0, Σ v ), where Σ v =    σ 2 v σ v σ v σ 2    is positive definite, σ 2 v is the variance of v it , σ v is the co-
variance between v it and it and σ 2 is the variance of it . Furthermore

sup n,T sup i,T E |v it | 4+δ and sup n,T sup i,T E | it | 4+δ exist for δ > 0. Denote E (v it | it ) = it δ and define ξ it = v it -it δ, assuming that E (ξ 2 it | it ) = E (ξ 2 it ) = σ 2 ξ , E (ξ 3 it | it ) = E (ξ 3 it ) and E (ξ 4 it | it ) = E (ξ 4 it ).
When σ v = 0, the error terms v it and it are correlated and the interaction matrix W t becomes endogenous. Under this assumption, V t = ξ t + t δ. Substituting this expression in equation ( 1) yields:

y t = λW t y t + X yt β y + Γ y f yt + t δ + ξ t , (4)
where the (n × 1) vector of error terms ξ t are i.i.d across i and t. We also have

E (ξ t | t ) = 0, V ar(ξ t | t ) = σ 2
ξ I n and ξ t uncorrelated with the regressors.

Identifying appropriate instrumental variables, X zt , in empirical applications is often a difficult task. For the problem at hand, the economic theory provides us with limited guidance. Due to the highly nonlinear relation between W t and the exogenous X yt variables, we do not necessarily need to rely on external exogenous variables to instrument Z t .15 Thus, X zt and X yt can share common components (see Shi and Lee, 2018, p.3).

Further, as often done in empirical studies in a similar situation, we use the first order lag of the endogenous variable, Z t-1 as instrument (see [START_REF] Reed | On the practice of lagging variables to avoid simultaneity[END_REF].

Shi and Lee ( 2018) develop and study the asymptotic properties of the quasimaximum likelihood estimator of model ( 4). Denoting θ = (β z , β y , λ, σ ξ , σ 2 , δ) and treating the unobserved heterogeneity components

(Γ y , Γ z , F y , F z ), with F y = (f y1 , • • • , f yT )
and

F z = (f z1 , • • • , f zT )
, as fixed-effects parameters to be estimated, the sample average log-likelihood function of the model is:

1 nT log L nT (θ, Γ y , F T y ) = -log (2π) - 1 2 log σ 2 ξ - 1 2 log |σ 2 | + 1 nT T t=1 log |S t (λ)| - 1 2nT σ 2 ξ σ 2 T t=1 e t e t - 1 2σ 2 ξ nT T t=1 [S t (λ)y t -X t β y -(δ ⊗ I n ) e t -Γ y f yt ] × [S t (λ)y t -X t β y -(δ ⊗ I n ) e t -Γ y f yt ] ,
where S t = (I n -λW t ) and e t = Z t -X zt β z -Γ z f zt .

In the estimation strategy they propose, time factors as well as loading parameters are concentrated out of the above log-likelihood function so that only the parameter vector θ needs to be estimated. Further, even though we can recover the value of the time factors and loading parameters, they are not uniquely defined (but leaving unaffected the value of the other parameters). 16When dealing with unobserved common factors, an important question is the number of factors to consider in the empirical estimation. Indeed, we need to make sure to include enough factors to avoid an omitted variable bias. In this contribution, we rely on the Ahn and Alex (2013) growth ratio, as proposed by [START_REF] Shi | A spatial panel data model with time varying endogenous weights matrices and common factors[END_REF], to find the optimal number of time factors to consider.

An Interaction-Based Centrality indicator

We propose to derive a metric from our spatial econometrics framework assessing how much a bank is exposed to the rest of the system. We call this metrics " Interaction-Based Centrality" (IBC). It consists of the Katz centrality measure [START_REF] Katz | A new status index derived from sociometric analysis[END_REF] where the attenuation factor is estimated from the data. The original Katz centrality measure for observation i, denoted c i , is given by expression (6),

IBC i,t = (I n -λW t ) -1 i. ι n (5) c i = κ n j =i w ij c j + ϕ (6)
where n is the number of observations (banks here), κ a parameter whose value is less than the largest eigenvalue of the adjacency matrix W and ϕ a scalar parameter. This equation can be rewritten into matrix form as follows:

c = δ(I -κW ) -1 ι (7)
with c, the n-dimensional column vector of Katz centralities and ι an n-dimensional vector of one. Each element of c indicates how important a bank is in the sense that it is linked to other important banks. The advantage with our spatial econometrics modelling is that, instead of having to set κ, the so-called "attenuation parameter" (the equivalent for ϕ in our spatial setting) to a predefined value as done usually in the network literature, we can let the data speak and provide rigorous statistical inference about it. In other words, the value of κ is estimated rather than chosen on an ad-hoc basis. However, for convenience, ϕ is set equal to 1 (see Newman, 2010, p.173).

We then illustrate how our indicator can be used for monitoring purposes by assessing whether it helps forecast vulnerable institutions. To do so, we estimate in the spirit of [START_REF] Balla | Tail dependence and indicators of systemic risk for large US depositories[END_REF]; [START_REF] Billio | Econometric measures of connectedness and systemic risk in the finance and insurance sectors[END_REF] and [START_REF] Sedunov | What is the systemic risk exposure of financial institutions[END_REF] several cross-sectional regressions where a standard measure of vulnerability (cumulated daily log-returns over one quarter) is explained by the β and our IBC index computed on alternative interaction matrices (transmission channels) among those emerging as relevant in the first-stage regressions. To have a fair assessment of how much our indicator forecast future financial developments of banks, we need to be careful when selecting the estimation period for both constructing the IBC index and the cumulated returns. First, the parameters of the IBC index should be re-estimated over a period prior to the one of the dependent variable. Otherwise, our procedure would be prone to the so-called look ahead bias.

Second, the estimation period for computing IBC should be long enough to recover estimates of reasonable precision. Against this background, we select the 2007Q3-2013Q1

to re-estimate the spatial model and compute our IBC indexes. Our dependent variable is then computed over the next quarter (2013Q2).

Estimation results

The baseline results are provided in Tables 6 and9. which report estimates for timevarying interaction matrices. Each column represents a specific channel of transmission.

Tables 6 to 8 are based on the Shi and Lee (2018) approach and include the control function parameter, δ, to test and control for the presence of endogeneity in the interaction matrix. By contrast, in Tables 7 and9, we impose the exogeneity of the transmission channel.

Before going further, it is worth clarifying which results should be considered in the case of conflicting outcomes, and when. Recall that the Shi and Lee (2018) approach provides unbiased and consistent estimators regardless of the nature of W t (exogenous or endogenous). In the absence of endogeneity, the estimator is, however, inefficient.

In large samples, estimating the model either with or without controlling for the endogeneity should not matter much if W t is exogenous. In small samples, conversely, marked differences could emerge between the two approaches. In this case, estimating the model under the correct assumption for the interaction matrix should be preferred.

We thus propose to proceed in two steps. First, we estimate the model allowing for -90362.748 -79159.774 -81460.460 -78683.762 -69220.544 -73454.804 -87428.413 -73742.035 -70333.785 -84981.096 -82653.537 Number of factors in y -629.134 -629.685 -624.253 -636.928 -636.200 -637.042 -619.555 -636.513 -636.890 -636.974 -637.312 Number of factors in y T-stats between brackets an endogenous transmission channel (i.e. by including the correction term, δ). After testing for the presence of endogeneity, we are left with two options. If δ is statistically significant, we interpret all the remaining coefficients (Tables 6 and8). If δ is not significant, we re-estimate the model supposing an exogenous (time-varying) matrix that is without correction (Tables 7 and9). We go on to compare, in Tables 10 and11, our baseline results with the case where the channels are treated as exogenous and time-invariant. In that setting, we use the data in 2007Q3 (initial period) to construct the interaction matrices.

1 1 1 1 1 1 1 1
Our controls in Tables 6 to 9 all exhibit the expected signs. In particular, the loan-todeposit ratio, the level of leverage, as well as the loan loss provisions to interest income ratio, are all positive and significant, with the exception of two models over 30.17

We now turn to our main parameter of interest at this stage -the one controlling for the presence of endogeneity in the model, namely δ. We start our analysis with the sector-based similarity matrices. Table 6 shows that δ is statistically significant for 6 sectors over 10. Consumer Staples, Financial, Industrial and Utilities are the sectors for which W t can be assumed as exogenous. Further, the exogeneity of the channel constructed from Portfolio Diversification index is also rejected. Consistently with the above discussion, we proceed by considering the most suited approach for each chan-nel. Table 6 shows that for the 7 endogenous matrices the cross-sectional dependence parameter λ is statistically significant for three of them: Portfolio Diversification, Consumer Discretionary and Materials. We cannot reject the null of no contagion effects for the others. Turning to the exogenous matrices, two models out of four, Consumer

Staples and Financial, display significant cross-sectional dependence. This first set of results confirms the existence of positive and significant externalities stemming from sector-based similarity, which indicates that a stress in a bank tends to spread out to other institutions sharing similar sectoral investment strategies. Looking more carefully at the different sectors, we observe that the results are quite stable over the 4 significant sector-based transmission channels (Consumer Discretionary, Financial, Consumer Staples, Materials) with coefficient estimates for λ ranging from 0.189 (Consumer Discretionary) to 0.136 (Financial). However, the interaction intensity is much higher for the Portfolio Diversification channels, with λ = 0.728.

To assess the influence of endogeneity, we examine the consequences of wrongly assuming W t as exogenous. To do so, we compare the significance along with size of the coefficients with the two approaches for the seven endogenous matrices (Tables 6 and7).

Both approaches reach similar conclusions in four instances. For the Information Technology sector, we cannot find cross-sectional correlation. For three matrices -Portfolio Diversification, Consumer Discretionary and Materials -contagion effects are detected with and without exogeneity (even though results under endogeneity are relatively less striking). In three cases ( Energy, Health Care, Telecommunication Services), we observe conflicting results; that is we reach a wrong conclusion by not accounting for the presence of endogeneity. In each case, we can reject the null of no contagion under exogeneity of the transmission channel, but not with the robust approach. Turning to the size of the coefficients, we consider the three endogenous matrices for which the cross- We now consider the models with transmission channels related to common exposure and fundamental-based similarity matrices (Tables 8 and9).

Here, we cannot reject the null hypothesis of exogeneity in any case. The crosssectional dependence parameter appears always highly significant (except for the channel constructed from debt over asset ratios). This suggests that those characteristics -Dividend yield (DIV _Y LD) and Market value to total asset ratio (M KT _ASS) -are relevant in explaining risk spillovers across banks, although with different intensities.

Finally, the last column of Table 8 reports the results for the market price channel.

The testing procedure does not allow the rejection of the null of exogeneity for this interaction matrix. Here also, λ is strongly significant. The sign is positive, meaning that two banking institutions investing in the same assets are prone to contagious effects. T-stats between brackets.

The bottom panels of Tables 6 and8 As a matter of comparison, we report in Tables 10 and11 the estimation results when transmission channels are treated as time-invariant (and exogenous). We observe several differences with time-varying matrices. Notably, the Common Exposure, Portfolio Diversification and Dividend Yields transmission channels are no longer identified as transmitting risk among banking institutions. In contrast, the channel constructed from the ratio of debt over assets becomes statistically significant. Further, transmission channels based on similarity of investment in the Health Care, Industry and Telecommunication Services sectors become significant when assuming constant and exogenous connectivity matrices. These discrepancies illustrate that relying on timeinvariant transmission channels may lead to inaccurate conclusions.

To close the discussion on the estimation part, we check the robustness of our results

to the sample period. Tables 12 to 15 display the main coefficients of interest for our two main cases: time-varying matrices with and without exogeneity of the transmission channel. In each case, we slightly alter the reference period. Overall, the results are robust. We note, however, a slight loss of significance for the cross-sectional coefficient -636.833 -634.973 -634.323 -637.042 -636.946 -636.938 -617.019 -636.440 -636.894 -636.875 -635.018 Number of factors in y T-stats between brackets.

of sector-based matrices as the time span to estimate the model shrinks.

Equipped with estimated values for the strength of the various channels, we illustrate how the above results can be used for monitoring purposes by constructing an Interaction-Based Centrality index, derived from our spatial model. This index measures the overall exposure of a bank vis-à-vis the whole system, according to equation ( 5). For this exercise, we first estimate our model on the period 2007Q3 to 2013Q1, then construct our IBC index and finally regress the vulnerability of a bank as measured by its cumulated log-returns over 2013Q2 on our IBC index and the β. Table 16 shows that among the eight selected matrices, three -Consumer Staple, Consumer Discretionary and Materials -displays individually forecasting power to predict banks' risk even in the presence of their market β. More precisely, a negative statistical relationship is observed between vulnerability and exposure (measured by the IBC index).18

Conclusion

What drives financial contagion? We empirically document in this paper the role of indirect contagion channels during the financial crisis of the late 2000s. Indirect contagion has been argued to be a "key ingredient in 2007", to quote [START_REF] Clerc | Indirect contagion: the policy problem[END_REF]. But its driving factors can be diverse. The transmission of risk from one institution to another in a crisis period can be due to common exposure (i.e. market price channel), for example, or to similarity in size, diversification, industrial concentration strategy (i.e.

  Figure 1 displays the evolution of the median value of banks' historical stock market volatility over the period 2005Q1-2013Q4. The shaded area represents the inter-quartile range. Unsurprisingly, it can be seen that the banks' risk increases sharply in 2007 at the beginning of the crisis, reaches a peak in early 2009 and slowly decreases until it reaches the pre-crisis period level in 2013Q2.

Figure 1 :

 1 Figure 1: Quarterly stock market historical volatility based on daily data (shaded area represents the interquartile range).

(

  2015); Cohen-Cole et al. (2011); Liedorp et al. (2010), among others.However, we depart from these studies in two main directions. First, we include in model (1) unobserved common factors (labeled "interactive effects" in the literature) Γ y f yt , where f yt is a R y × 1 vector of unobserved factors and Γ y is a n × R y matrix of factor loadings. The interactive effects capture the fact that bank risk can be influenced by common global factors that can have different impacts on individual banks.

  sectional coefficient is significant: Portfolio Diversification, Consumer Discretionary, and Materials. While the estimated values of λ for the first channel are quite similar (0.728 and 0.746), the situation is quite different for the last two. The estimated values are equal to 0.19 (Consumer Discretionary) and 0.18 (Materials) under endogeneity of W t , compared to 0.86 and 0.85 without. These figures show that failing to control for endogeneity may lead, in our context, to inflate the magnitude of contagion effects up to a factor of four.

  finally report the relevant number of considered common factors in the main equation (number of factors in y) and in the control function equation (number of factors in Z). The results show that considering one unobserved factor is sufficient in the main equation, while a second factor is sometimes needed for the control function equation (when Z t is built on Health Care, Dividend Yield and Common Exposure).

  Estimations for fundamental-based and common exposure matrices using the subsample Q3 2007-Q2 2013 (T=24) using constant (2007Q3) and exogenous matrices Parameters DEB_ASS DIV_YLD MKT_ASS COM_EXPO Estimation results of the main SAR model (eq.

DEB_ASS

  for alternative time periods, fundamental-based and common exposure matrices. Results accounting for the possible endogeneity of matrices

Table 1 :

 1 Banks considered

	Label	Firm	Label	Firm
	BAC	Bank of America	UMBF UMB Financial
	JPM	JPMorgan Chase	HBHC Hancock Holding Co
	PNC	PNC financial Services TRMK Trustmark Corp
	STI	Suntrust Banks	FMBI	First Midwest Bancorp
	BBT	BB&T Corporation	PRK	Park National Corp
	RF	Regions Financial	FCF	First Commonwealth Bank
	CMA	Comerica	NBTB NBT Bancorp Inc
	HBAN Huntington Bank NA	WSBC Wesbanco Inc
	SNV	Synovus Financial	SRCE 1st Source Bank
	ASB	Associated Banc-Corp TMP	Tompkins Financial Corp
	CYN	City National Bank	WASH Washington Trust Bank
	CBSH Commerce Bank NA	UVSP Univest Bank
	CFR	Cullen Frost Bankers	SYBT Stock Yards Bank
	BXS	Bancorpsouth Inc	PGC	Peapack Gladstone Financial Corp

Table 2 :

 2 Definition of variables

	Variable	Definition	Source
	Dependent variable
	Volatility	Quarterly stock market historical volatility based on daily data	CRSP
	Control variables
	Loan_Dep Ratio of customer loans to customer deposits Bloomberg
	Leverage	Financial Leverage	Bloomberg
	Loss_Iinc Loan Loss Provisions to Interest Income	CRSP

W t is an n × n matrix modeling time-varying interdependences between banks and represents a specific transmission channel. 9

Table 3 :

 3 Descriptive statistics

		Nobs Mean STD	Min	P25	Median P75	Max
	Volat	672 2.894 1.963 0.689 1.570	2.299 3.427 15.630
	Loan_Dep	672 85.516 14.206 45.001 77.432 87.776 94.577 126.931
	Leverage	672 10.459 1.893 6.976 9.141 10.187 11.338 18.085
	Loss_Iinc	672 18.268 21.544 -6.042 5.587 11.418 21.631 171.491

pendence parameter λ that needs to be estimated. When this estimated parameter is statistically different from 0, one faces evidence of spillover effects through the transmission channel characterized by W t . This way of testing for contagion effects in finance is derived from the wide class of spatial econometric models (see

[START_REF] Anselin | Spatial dependence in linear regression models with an application to spatial econometrics[END_REF][START_REF] Lesage | Introduction to Spatial Econometrics[END_REF]

and was applied in early studies such as

[START_REF] Blocher | Network externalities in mutual funds[END_REF]

; Tonzer

  Security Exchange Act of 1934, any institutional investment manager who exercises investment discretion over 100 million USD or more in securities of a class specified by the SEC has to report its holdings through Form 13F. This form provides information about the issuer of the asset, and the type and value of the asset held, among others. Holdings are reported by a wide range of institutional investors including banks, insurance companies, investment advisors, mutual and pension funds, etc. and data are available since 1980. It is worth noting that only long positions are included in the reports and the assets are mostly equities.These portfolio holdings data have been used in several studies: for instance, Gompers

and Metrick (2001);

[START_REF] Cai | Institutional trading and stock returns[END_REF]

;

[START_REF] Phalippou | Can risk-based theories explain the value premium[END_REF]

;

[START_REF] Koijen | An equilibrium model of institutional demand and asset prices[END_REF]

;

[START_REF] Guo | The topology of overlapping portfolio networks[END_REF]

. Our data are retrieved primarily from Bloomberg and CRSP and we use the official Form 13F filed by institutions and available on the SEC's website to control for missing values and outliers.

Table 4 :

 4 Construction of the interaction matrices

	Category	Variable	Definition	Source
	Common exposure (market price channel)	Common exposure	Share of the set of assets held in all portfolios in the portfolio of each bank	Bloomberg
		Div_Yld	Dividend yield	Bloomberg
	Fundamental-based similarity (informational channel)	Debt_ass Mkt_Ass.	Ratio of total debt to total assets (in percentage) Bloomberg Total current market value of all of a company's outstanding shares stated in USD over total asset Bloomberg
			(in percentage)
		Portfolio holdings by sectors	Value (in USD) of holdings by sector over total asset	Bloomberg
			Index of portfolio diversification computed at sec-
			tor level and accounting for market volatility. For
	Investment-based similarity (informational		a given portfolio i at time t, it is computed as div_idx i,t = P i,t Σt
	channel)	Port. Div.

Table 5 :

 5 List of sectors in the portfolio holding data

	Label Sector
	CD	Consumer Discretionary
	EN	Energy
	HC	Health care
	IT	Information Technology
	MA	Materials
	TS	Telecommunication Services
	CS	Consumer Staples
	FI	Financial
	IN	Industrial
	UT	Utilities

combination of bank i and bank j characteristics. For this purpose, we first identify the set of assets held in all portfolios. Next, we compute the share invested in this category

Table 6 :

 6 Estimations for investment-based similarity matrices (sector-based and diversification-based matrices) using the subsampleQ3 2007-Q2 2013 (T=24)

	UT		0.008	(4.107)	0.069	(5.423)	0.010	(5.573)	0.118	(1.377)	-0.268	(-0.989)
	IN		0.009	(4.581)	0.075	(5.629)	0.010	(5.429)	0.046	(0.541)	-0.221	(-1.801)
	FI		0.008	(4.646)	0.065	(4.835)	0.010	(5.402)	0.144	(1.894)	-0.066	(-0.981)
	EN HC IT TS CS	Estimation results of the main SAR model (eq. 4)	0.008 0.008 0.009 0.008 0.008	(4.349) (4.222) (4.551) (3.910) (4.302)	0.076 0.081 0.079 0.078 0.067	(5.587) (5.176) (5.593) (5.416) (5.428)	0.010 0.010 0.010 0.010 0.010	(5.420) (5.375) (5.408) (5.353) (5.650)	0.092 0.088 0.045 0.094 0.161	(1.124) (0.928) (0.501) (0.918) (2.424)	-0.246 -0.403 -0.295 -1.646 -0.097	(-2.421) (-3.516) (-2.631) (-3.499) (-0.690)
	MA		0.007	(3.777)	0.069	(4.892)	0.010	(5.474)	0.182	(1.966)	-0.600	(-2.255)
	CD		0.007	(3.910)	0.071	(4.935)	0.010	(5.445)	0.189	(2.029)	-0.431	(-2.787)
	Port. Div		0.003	(1.700)	-0.000	(-0.041)	0.009	(5.510)	0.728	(16.051)	0.721	(6.415)
	Parameters		Loan_Dep		Lev		Loss_Iinc		WY		δ	Log-lik

Table 7 :

 7 Estimations for investment-based similarity matrices (sector-based and diversification-based matrices) using the subsample Q3 2007-Q2

		UT		0.008	(4.100)	0.069	(5.421)	0.010	(5.608)	0.125	(1.500)
		IN		0.009	(4.655)	0.073	(5.908)	0.010	(5.529)	0.061	(0.828)
		FI		0.008	(4.747)	0.066	(5.179)	0.010	(5.396)	0.136	(1.970)
	2013 (T=24) using time-varying but exogenous matrices	Port. Div CD MA EN HC IT TS CS Parameters	Estimation results of the main SAR model (eq. 4)	0.007 0.008 0.004 0.008 0.002 0.005 0.004 0.008 Loan_Dep	(4.086) (4.501) (2.858) (4.333) (1.510) (3.149) (2.719) (4.340)	0.065 0.071 -0.005 0.067 0.013 0.009 0.006 0.071 Lev	(5.334) (5.785) (-0.446) (5.413) (1.021) (0.783) (0.511) (5.858)	0.010 0.010 0.004 0.010 0.009 0.003 0.005 0.010 Loss_Iinc	(5.459) (5.585) (2.733) (5.660) (5.090) (1.925) (3.018) (5.568)	0.201 0.100 0.877 0.161 0.746 0.867 0.858 0.130 WY	(2.980) (1.337) (39.652) (2.425) (15.567) (33.102) (34.878) (1.929)	Log-lik

Table 8 :

 8 Estimations for fundamental-based and common exposure matrices using the

	subsample Q3 2007-Q2 2013 (T=24)	Parameters DEB_ASS DIV_YLD MKT_ASS COM_EXPO	Estimation results of the main SAR model (eq. 4)	Loan_Dep 0.009 0.009 0.007 0.009	(5.122) (5.289) (3.506) (5.153)	Lev 0.074 0.051 0.067 0.059	(5.892) (3.845) (4.963) (4.760)	Loss_Iinc 0.010 0.010 0.010 0.009	(5.420) (5.676) (5.572) (4.999)	WY 0.006 0.227 0.220 0.278	(0.106) (3.871) (3.360) (3.880)	δ 0.003 -0.007 -0.005 -0.384	(0.171) (-0.299) (-0.341) (-0.357)	Log-lik -71870.177 -67753.082 -60850.679 -52200.508	Number of factors in y 1 1 1 1	n 28 28 28 28	T 24 24 24 24	Estimation results of the reduced form model for W (eq. 3)	Loan_Dep 0.002 0.005 -0.022 0.000	(0.862) (3.683) (-3.430) (0.205)	Lev 0.000 0.003 -0.180 0.001	(0.002) (0.301) (-4.539) (1.610)	Loss_Iinc -0.002 -0.008 -0.008 -0.000	(-0.814) (-6.130) (-2.418) § §(-2.373)	Z t-1 0.967 0.873 0.668 0.716	(100.402) (75.706) (34.976) (67.111)	Number of factors in Z 1 2 1 2

Table 10 :

 10 Estimations for investment-based similarity matrices (sector-based and diversification-based matrices) using the subsample Q3 2007-Q2

		IN UT		0.008 0.010	(4.458) (4.492)	0.069 0.082	(5.626) (5.855)	0.010 0.010	(5.557) (5.447)	0.121 -0.087	(1.910) (-0.757)
		FI		0.008	(4.261)	0.067	(5.302)	0.010	(5.517)	0.137	(2.041)
	2013 (T=24) using constant (2007Q3) and exogenous matrices	Port. Div. CD MA EN HC IT TS CS Parameters	Estimation results of the main SAR model (eq. 4)	0.008 0.009 0.003 0.008 0.008 0.007 0.004 0.008 Loan_Dep	(4.065) (4.638) (2.413) (4.278) (3.850) (3.774) (2.658) (4.214)	0.065 0.071 -0.005 0.060 0.072 0.052 -0.005 0.073 Lev	(5.222) (5.538) (-0.444) (4.629) (4.977) (4.127) (-0.413) (6.007)	0.010 0.010 0.006 0.010 0.010 0.010 0.006 0.010 Loss_Iinc	(5.612) (5.559) (3.819) (5.683) (5.526) (5.528) (3.443) (5.567)	0.172 0.063 0.841 0.194 0.061 0.306 0.816 0.088 WY	(2.402) (0.859) (37.050) (2.841) (0.534) (4.593) (31.783) (1.318)	Log-lik

Table 12 :

 12 Analysis of Robustness for alternative time periods, investment-based similarity matrices (sectorbased and diversification-based matrices). Results accounting for the possible endogeneity of matrices

	Port. Div CD MA EN HC IT TS CS FI IN UT	0.728 0.189 0.182 0.092 0.088 0.045 0.094 0.161 0.144 0.046 0.118	(16.051) (2.029) (1.966) (1.124) (0.928) (0.501) (0.918) (2.424) (1.894) (0.541) (1.377)	0.721 -0.431 -0.600 -0.246 -0.403 -0.295 -1.646 -0.097 -0.066 -0.221 -0.26	(6.415) (-2.787) (-2.255) (-2.421) (-3.516) (-2.631) (-3.499) (-0.690) (-0.981) (-1.801) (-0.989)	0.728 0.174 0.164 0.088 0.095 0.042 0.065 0.162 0.138 0.046 0.01	(15.714) (1.816) (1.684) (1.036) (1.065) (0.453) (0.602) (2.369) (1.767) (0.512) (0.087)	0.705 -0.483 -0.701 -0.263 -0.590 -0.307 -2.014 -0.105 -0.067 -0.239 -0.564	(6.166) (-2.950) (-2.516) (-2.437) (-3.417) (-2.641) (-3.726) (-0.658) (-0.946) (-1.853) (-3.194)	0.712 0.161 0.158 0.082 0.090 0.013 0.068 0.156 0.127 0.050 0.013	(14.476) (1.641) (1.593) (0.948) (1.005) (0.126) (0.632) (2.248) (1.576) (0.558) (0.119)	0.707 -0.491 -0.702 -0.261 -0.596 -0.223 -1.941 -0.138 -0.060 -0.223 -0.517	(5.987) (-2.934) (-2.512) (-2.424) (-3.303) (-2.912) (-3.497) (-0.821) (-0.829) (-1.680) (-2.984)	0.706 0.173 0.183 0.084 0.092 0.070 0.099 0.031 0.117 0.065 0.087	(13.596) (1.828) (1.888) (0.950) (1.034) (0.823) (0.945) (0.320) (1.446) (0.654) (0.877)	0.647 -0.570 -0.737 -0.263 -0.630 -0.309 -2.015 -0.677 -0.044 -0.141 -0.941	(5.099) (-2.773) (-2.450) (-2.518) (-3.247) (-2.275) (-3.499) (-3.766) (-0.538) (-2.003) (-2.056)	0.688 0.143 0.145 0.062 0.081 0.045 0.069 0.006 0.094 0.054 0.026	(11.902) (1.412) (1.399) (0.661) (0.884) (0.490) (0.616) (0.054) (1.098) (0.574) (0.236)	0.674 -0.612 -0.751 -0.261 -0.616 -0.295 -1.980 -0.677 -0.026 -0.215 -0.695	(5.027) (-2.672) (-2.365) (-2.509) (-2.911) (-2.121) (-3.142) (-3.631) (-0.309) (-1.510) (-2.243)
		WY	δ	WY	δ	WY	δ	WY	δ	WY	δ
			Q3 2007-Q2 2013			Q3 2007-Q1 2013			Q3 2007-Q4 2012			Q3 2007-Q2 2012			Q3 2007-Q4 2011

T-stats between brackets.

Table 13 :

 13 Analysis of Robustness for alternative time periods, investment-based similarity matrices (sectorbased and diversification-based matrices). Results assuming time-varying but exogenous matrices

	Port. Div CD MA EN HC IT TS CS FI IN UT	0.746 0.867 0.858 0.130 0.201 0.100 0.877 0.161 0.136 0.061 0.125	(15.567) (33.102) (34.878) (1.929) (2.980) (1.337) (39.652) (2.425) (1.970) (0.828) (1.500)	0.740 0.867 0.857 0.128 0.199 0.100 0.877 0.162 0.130 0.062 0.115	(14.908) (32.295) (33.969) (1.862) (2.870) (1.311) (38.678) (2.372) (1.824) (0.814) (1.345)	0.710 0.241 0.850 0.125 0.192 0.100 0.872 0.159 0.115 0.062 0.111	(12.995) (3.042) (31.965) (1.807) (2.735) (1.295) (36.562) (2.314) (1.578) (0.809) (1.271)	0.656 0.845 0.253 0.136 0.194 0.116 0.865 0.175 0.101 0.084 0.13	(10.387) (27.052) (3.018) (2.039) (2.811) (1.543) (32.858) (2.583) (1.366) (1.133) (1.495)	0.623 0.215 0.214 0.123 0.172 0.093 0.862 0.155 0.072 0.064 0.089	(8.685) (2.608) (2.365) (1.764) (2.362) (1.184) (30.336) (2.165) (0.918) (0.835) (0.95)
		Q3 2007-Q2 2013 WY	Q3 2007-Q1 2013 WY	Q3 2007-Q4 2012 WY	Q3 2007-Q2 2012 WY	Q3 2007-Q4 2011 WY

T-stats between brackets.

Table 14 :

 14 Analysis of Robustness

Table 15 :

 15 Analysis of Robustness for alternative time periods, fundamental-based and common exposure matrices. Results assuming time-varying but exogenous matrices

	DEB_ASS DIV_YLD MKT_ASS COM_EXPO	0.004 0.227 0.221 0.159	(0.067) (3.912) (3.385) (2.860)	-0.002 0.230 0.211 0.153	(-0.045) (3.875) (3.355) (2.685)	-0.003 0.228 0.207 0.144	(-0.050) (3.761) (3.304) (2.483)	0.008 0.244 0.201 0.137	(0.146) (3.917) (3.402) (2.336)	0.003 0.248 0.216 0.162	(0.056) (3.757) (3.482) (2.680)
		Q3 2007-Q2 2013 WY	Q3 2007-Q1 2013 WY	Q3 2007-Q4 2012 WY	Q3 2007-Q2 2012 WY	Q3 2007-Q4 2011 WY

T-stats between brackets.

Table 16 :

 16 Estimations of the impact of the IBC index on bank vulnerabilityRobust T-stats are reported between brackets. The model includes the market β as a control variable.

	COM_EXPO	-0.221	-0.488	28	0.016
	MA CD CS	-1.071 -1.085 -0.669	(-1.795) (-2.077) -1.643	28 28 28	0.120 0.153 0.104
	Port. Div MKT_ASS DIV_YLD	0.034 0.125 0.197	(0.042) (0.468) (0.536)	28 28 28	0.007 0.016 0.018
	FI	IBC -0.481	(-0.897)	n 28	R 2 0.038

Notable empirical contributions aiming to estimate financial interconnectedness include, for instance,[START_REF] Billio | Econometric measures of connectedness and systemic risk in the finance and insurance sectors[END_REF],[START_REF] Blasques | Spillover dynamics for systemic risk measurement using spatial financial time series models[END_REF][START_REF] Demirer | Estimating global bank network connectedness[END_REF] and[START_REF] Betz | Systemic risk spillovers in the European banking and sovereign network[END_REF]) and, more recently, Wang et al. (2019).

Data on cross-lending for instance are made available on a discretionary basis to central banks (see regulation introduced by the ECB, 2012, and set of data available here) .

Some studies try to mitigate the problem by taking lagged values of the interaction matrix. However, this approach is subject to the same well-known criticisms as traditional endogenous regressors. An alternative consists in using a fixed interaction matrix that disregards their actual dynamics through time.

Volatility is standard as a measure of risk in finance. In the banking literature, alternatives such as the z-score are also often used. However, the z-score is only available at a low frequency, typically yearly, while asset returns volatility is commonly computed at higher frequencies such as monthly, weekly or daily. Volatility appears therefore better suited to analyze fast-changing financial phenomena such as contagion.

[START_REF] Ord | Estimation methods for models of spatial interaction[END_REF];[START_REF] Lee | Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models[END_REF] developed the quasi-maximum likelihood estimator whileKelejian and Prucha (1998, 1999);[START_REF] Lee | Gmm and 2sls estimation of mixed regressive, spatial autoregressive models[END_REF];[START_REF] Lee | Efficient Gmm Estimation Of High Order Spatial Autoregressive Models With Autoregressive Disturbances[END_REF] and[START_REF] Lin | GMM estimation of spatial autoregressive models with unknown heteroskedasticity[END_REF] derived the two stage least square and generalized method of moments estimators. Finally,[START_REF] Lesage | Bayesian estimation of spatial autoregressive models[END_REF] and[START_REF] Lesage | Introduction to Spatial Econometrics[END_REF] developed the Markov Chain Monte Carlo methods.

This exogeneity limit was acknowledged in[START_REF] Tonzer | Cross-border interbank networks, banking risk and contagion[END_REF] and[START_REF] Blasques | Spillover dynamics for systemic risk measurement using spatial financial time series models[END_REF].

See details on http://www.sec.gov/answers/form13f.htm. Information valid on January 5, 2022

There are two sources of nonlinearity in the relation between W t and X yt : 1) W t is a nonlinear function of Z t and 2), the reduced form of model (1) shows a nonlinear relation between W t and X yt .

By imposing stricter restrictions on the factors and the matrix of loading parameters, one could uniquely identify them (see among others,[START_REF] Bai | Panel data models with interactive fixed effects[END_REF]. However,[START_REF] Shi | A spatial panel data model with time varying endogenous weights matrices and common factors[END_REF] do not wish to impose these constraints, leading to non-uniquely identified time factors and loading parameters.

To rigorously interpret the impact of a change in an exogenous determinant on the risk variable, we should compute the reduced form of the model, then calculate matrices of partial derivatives of the dependent variable wrt. the relevant determinant and finally compute direct, indirect and total effects, as shown in[START_REF] Lesage | Introduction to Spatial Econometrics[END_REF]. However, this is not the main objective of this contribution. We can nevertheless say that the direct impact of an increase in whatever measure of indebtedness of the bank will be an increase in its risk, measured by the volatility.

Standard errors in Table11do not account for the fact that IBC has been estimated in a first step. However, due to the small sample size, we lose the statistical significance when the generated regressor correction is applied(Murphy and Topel,

1985).

information channels), etc. Providing sound empirical evidence about the contribution of each specific channel is not trivial, however. As discussed in the paper (see also Wang et al., 2019), standard econometric approaches may be ill-suited to do so because of the endogenous response of certain transmission channels to the risk of a financial institution. This paper addresses this endogeneity issue in order to document the role of 15 indirect channels. To do so, we use a newly-developed spatial econometrics model "robust" to endogeneity in the time-varying interaction matrix (i.e. transmission channel).

Within this framework, we are in addition able to decompose bank risk into systematic risk and spillover risk components while controlling for common factors. Overall, our results confirm that endogeneity characterizes part of our data and can affect the statistical significance along with the magnitude of the results compared to a situation in which the interaction matrix is wrongly assumed to be exogenous. Endogeneity in the transmission channel is therefore a critical feature that needs to be properly taken into account in empirical studies on contagion. Our findings also reveal positive and significant spillover effects through market price (i.e. common asset exposures) and information channels. Two banks that have invested in the same assets, for instance, are more exposed to contagion. Likewise, the similarity in sectoral investments strategy appears as a significant channel of risk spillovers. We also find significant spillovers when we consider similarities in fundamentals such as dividend yield or similarity in diversification strategy. Building on the estimations of the spatial model, we are able to construct an "Interaction-Based Centrality" index, which can be used by regulatory authorities and policymakers to track vulnerable institutions. Wang, D., van Lelyveld, I., Philippon, T., and Richardson, M. (2019). Do information contagion and business model similarities explain bank credit risk commonalities.

Technical Report 94, ESRB Working Paper Series.