The Informational Centrality of Banks

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) The Informational Centrality of Banks Nathan Foley-Fisher, Gary Gorton, St´ephane Verani 2024-006 Please cite this paper as: Foley-Fisher, Nathan, Gary Gorton, and St´ephane Verani (2024). “The Informational Centrality of Banks,” Finance and Economics Discussion Series 2024-006. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.006. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

The Informational Centrality of Banks∗ Nathan Foley-Fisher1, Gary Gorton2, and Stéphane Verani1 1Federal Reserve Board 2Yale School of Management and NBER First version: January 2021; this version: December 2023 Abstract The equity and debt prices of large nonbank firms contain information about the future state of the banking system. In this sense, banks are informationally central. The amount of this information varies over time and over equity and debt. During a financial crisis banks are, by definition of a crisis, at risk of failure. Debt prices became about 50 percent more informative than equity prices about the future state of the banking system during the financial crisis of 2007-2009. This was partly due to investors’ fears that banks might not be able to refinance the firms’ debt. JEL Codes: D82, E44, G14 Keywords: price informativeness, asset pricing, banking system, financial crises ∗Forprovidingvaluablecomments,wewouldliketothank,withoutimplication,Elena Afanasyeva, Celso Brunetti, Eduardo Davila, Borghan Narajabad, Anna Orlik, Georgio Ottonello, Dino Palazzo, Michael Palumbo, Cecilia Parlatore, Coco Ramirez, Skander Van Den Heuvel, participants in the NBER Summer Institute Conference on Capital Markets and the Economy 2023, the Federal Reserve Board Macro-Finance Workshop 2023, and at the Federal Reserve Bank of Philadelphia. The views in this paper are solely the authors’ and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System.

1 Introduction Banks sit at the center of the savings-investment process. But what does it mean that banks are at the “center” of the savings-investment process? To address this question, we ask whether the equity and debt prices of large nonbank firms contain information about the future state of the banking system (“the state of the banking system”). We look at normal times and during the financial crisis of 2007-2009. We find that the equity and debt prices of large nonbank firms do indeed embed information about the state of the banking system. The amount of information embedded in prices varies over time, and over equity and debt.1 A large literature studies financial crises as information events in which short-term debt transits from being information-insensitive to information sensitive – a crisis. For a review of this literature see Dang, Gorton and Holmström(2020). Afinancialcrisisisasystemicevent, thesolvencyofthe entire banking system is threatened. Ben Bernanke made this point in his testimony before the Financial Crisis Inquiry Commission (2012). He said thatduringSeptemberandOctoberof2008“...outofthe13mostimportant financial institutions in the United States, 12 were at risk of failure within 1There is a large literature establishing that firm outcomes depend on conditions in the banking sector, including bank financing constraints, competition, and profitability (Paravisini, 2008; Claessens and Laeven, 2005; Chava and Purnanandam, 2011). At the micro level, individual banks that tighten their loan supply have real effects on firm investment and employment decisions (Bassett, Chosak, Driscoll and Zakrajsek, 2014; Chodorow-Reich, 2014; Castro, Glancy, Ionescu and Marchal, 2022). An earlier strand of the same literature used aggregate bank data, including Owens and Schreft (1991) and Lown and Morgan (2002, 2006). 2

a period of a week or two” (p. 354). We show that during a financial crisis, there is a kind of information regime switch for corporate assets as well. We find that both equity and debt prices are always informative about the banking system. But during the financial crisis of 2007-2009 debt prices were about 50 percent more informative than equity prices. We show that this difference was in part due to investors’ fears that banks might not be able to refinance the firms’ debt. We proceed by estimating price informativeness corresponding to the relative precision of the signal about future states contained in asset prices. In practice, a specific combination of R2 statistics from linear regressions of changes in asset prices on changes in states exactly identifies relative price informativeness (Davila and Parlatore, 2022). In our setting we study whether a single nonbank firm’s asset prices are informative about two unknown states: the firm state—measured as the firm’s future earnings as in Davila and Parlatore (2022)—and the state of the banking system (defined below). Our calculations are analogous to an external observer updating her prior about the state of the banking system after observing changes in a nonbank firm’s asset prices. We show that the observer can identify the information content about the state of the banking system. Under stylized conditions, the observer in our setting is a Bayesian learner applying a Kalman filter to extract information about two unknown linear combinations of the firm and bank states—that is, both the states and 3

their linear combination are unknown. Information about the state of the banking system can be analyzed by comparing two Kalman gains: The first obtained by imposing a constraint on the unknown linear combination of the firm and bank states, and the second obtained from the unconstrained Kalman filter. When an asset price is uninformative about the banking system, the signal-to-noise ratios in the constrained and unconstrained Kalman filters are the same. Our measures of the information content of a firm’s equity and debt prices about the state of the banking system also provide a measure of the relative information content. In other words, whether equity prices are more informative than debt prices about the state of the banking system. We contrast the relative information content during normal times and during the financial crisis 2007-2009, when the entire financial system was on the brink of collapse. We find that debt and equity prices are equally informative in normal times, while debt was more informative than equity during the financial crisis. This suggests that debt holders believed that there was a nontrivial chance that they would suffer losses and so they produced information about the state of the banking system. This information becomes impounded into prices. We use subsampling to conduct inference on our statistical measures of price informativeness. Our measures are at the firm level using rolling windowsof16quartersandaveragedacrossfirmsforeachperiodoftimetto 4

obtain an aggregate time series. Subsampling the distribution of a sample mean at time t based on n firms requires calculating sample means for all the (cid:0)n(cid:1) combinations of firm subsamples of size c at time t, where c < n. c Subsampling yields a consistent estimate of the sampling distribution of the original sample mean under extremely weak assumptions (see Politis, Romano and Wolf (1999)).2 We then investigate why the debt of individual large nonbank firms contains more information about the state of the banking system than the firms’ own equity prices during the financial crisis. We analyze the relative information content about the bank state in equity and debt prices in a panel data setting. We calculate the fraction of each firm’s total debt maturing over the next twelve months as a measure of refinancing risk. We show that during the financial crisis, a firm’s debt prices contain relatively more information about the state of the banking system than its equity prices when that firm’s refinancing risk is higher. Benmelech, Frydman and Papanikolaou (2018) found that during the Great Depression, when public debt markets disappeared, firms with maturing debt at a location where local banks failed reduced their employment by 11 percent to 17 percent. Our findings suggest that investors feared firms might also struggle to 2Subsampling is conceptually different from bootstrapping. Subsampling, by definition, draws samples from the true data generating process, whereas bootstrapping recomputes a statistic over artificial samples that are created from what the researcher assumes is the true data generating process. As we do not know a priori whether asset prices contain information about the state of the banking system, we do not know the ‘true’ model. 5

refinance their debt during the 2007-2009 financial crisis. Note that we are not addressing a question of market efficiency, which would test for information that the researcher a priori believed should be in asset prices. Rather, we are testing whether information about the state of the banking system is reflected in nonbank firms’ asset prices. We also study how price informativeness varies over time, in normal and crisis periods, and across equity and debt. Note also that we are not trying to forecast future bank capital ratios, which would be an atheoretic exercise in time-serieseconometrics. Instead, wearetestingwhetherinvestorsproduce and incorporate information about future bank capital ratios into nonbank firms’ asset prices. Related literature includes Ottonello and Song (2022). These authors showthatchangesinthenetworthofintermediarieshaverealconsequences for nonbank firms. They look at high-frequency changes in the market value of intermediaries in a narrow window around intermediaries’ earnings announcements. They estimate that news of a one percent decline in the net worth of intermediaries results in a 0.2—0.4 percent decline in the market value of nonbank firms. Our approach is different. We show that changes in the future state of the banking system affect the equity and debt prices of nonbank firms and we propose one mechanism through which that happens: refinancing risk. Intermediary asset pricing is another framework that places financial 6

intermediaries centrally in the economy (He and Krishnamurthy, 2013). But in this case the question is whether intermediaries are the marginal asset pricers, such that a representation of their current state can be taken as the stochastic discount factor. In empirical studies of intermediary asset pricing, researchers define the state of the banking system as the leverage of banks. Adrian, Etula and Muir (2014) and He, Kelly and Manela (2017) define leverage differently. We follow He et al. (2017) who define leverage as the equity capital ratio of primary dealers, that is counterparties of the Federal Reserve Bank of New York. These primary dealer banks would be the underwriters for firms’ debt, making them the salient set of financial intermediaries with respect to our sample of large nonbank firms.3 The paper proceeds as follows. Section 2 presents a small model to explain and motivate the empirical analysis. Section 3 describes how our approach can be interpreted as a Kalman filter. Section 4 provides an overview of our data. We report our aggregate-level analysis in Section 5 and our firm-level analysis in Section 6. Section 7 concludes. 2 Model In this section we provide a small parsimonious model to motivate the subsequent empirical work. The equity and debt market protocols in the 3The primary dealers that interact with the Federal Reserve Bank of New York are some of the largest banks in the world. See https://www.newyorkfed.org/markets/ primarydealers for the list. 7

model follow Chousakos, Gorton and Ordoñez (2023). The model is a tractableversionofGrossmanandStiglitz(1980)inwhichthesupplyofthe assetsonthemarketisexogenouslygiven. Theintuitionforthisassumption is that liquidity traders are asymmetric: There can be an urgency to sell, but not the same urgency to buy. 2.1 Setting There are four dates: 0, 1, 2, and 3. There is a mass one of a continuum of ex-ante identical firms (and similarly for other agent types). At t = 0 a representative firm is already financed by debt, D , and equity, E . So, 0 0 total assets are: A = D +E . The existing debt matures at t = 2 and, 0 0 0 at that time, the firm would like to issue new debt, maturing at t = 3. To attempt to refinance its debt, the firm approaches a bank at t = 2. At t = 3 the debt (if the old debt has been refinanced) is repaid and the value of the equity is paid out as a liquidating dividend. At the start of t = 1, a public signal, S, about the future (i.e., t = 2) state of the banking system is delivered. As detailed below, agents can learn the implications of this state variable at a cost.4 Then, at t = 1, an equity market and a debt market simultaneously open. In the t = 1 asset markets, there are liquidity traders who must sell their holdings of thefirm’sequityanddebt. Biddersinthesemarketsareriskneutral. There 4That is, the signal S is fully informative about the future state of the banking system, but not about its implications for the value of the firm. 8

are twice as many bidders for the firm’s debt and twice as many bidders for the firm’s equity.5 Two bidders are randomly matched to each seller. They submit sealed bids and the assets in each market go to the highest bidders. At t = 2, the firm announces its earnings and the firm goes to the bank for the underwriting of new debt. The timeline is as follows: 0 1 2 3 Firm has debt and Public info S Existing debt Final payoffs on equity arrives. Bidders matures. Bank debt and equity decide to produce opens to refinance info or not. Equity debt, if it can. and debt markets Firm announces open. Trade occurs earnings. simultaneously. At t = 2 banks open and based on the realization of the state, Ω, the firm is affected. For example, the firm may not be able to issue new debt to replace the debt maturing at t = 2 if, say, there is a financial crisis. This is refinancing risk. The firm’s interaction with the bank is summarized by the value of the firm at t = 3, which depends on the state of the banking system at t = 2. The realized state of the banking system at t = 2 is one of two possible realizations: C and N, i.e., Crisis and Normal, with probabilities γ and C 5There are four unit intervals of bidders, two for debt and two for equity. For simplicity, the bidders in the equity and debt markets are distinct. As the two markets open simultaneously, prices are formed at the same time and so information from one marketdoesnotinformtradersintheothermarket. Cross-marketinformationexchange could be added at the cost of more complexity. 9

γ , summing to one, so Ω ∈ {C, N}. The value of the firm at t = 3 N depends on the realized state Ω, i.e., V(Ω). But, in addition, for each state the value of the firm can be V (Ω,x) or V (Ω,x), Ω ∈ {C, N}, where H L V (Ω,x) > V (Ω,x) for each Ω, and V (N,x) > V (C,x). The variable x H L L H refers to the firm’s final cash flows at t = 3 from the firm’s project. These cash flows could also be a function of Ω and, in any case, are random, though to simplify notation that will be suppressed. Uninformed agents in t = 2 know the state Ω, but do not know whether the final firm value in t = 3 will be V (Ω) (a “low-type” firm,) or V (Ω) L H (a “high-type” firm), and implicitly they do not know x. They do know the probability the firm is H in each state, γ (Ω), or L, γ (Ω), in state Ω, H L where γ (Ω)+γ (Ω) = 1. H L Table 1 describes how the state of the banking system realized at t = 2 will affect the value of the firm’s t = 3 liabilities. Firms may not be able to refinance their debt at t = 2, so some of the debt in the table may be zero. Agents cannot sell assets short.6 2.2 Asset markets and information production At t = 1, after the fully-informative signal has been delivered and before markets open, debt and equity bidders choose whether to produce private information about the future value of the firm. Conditional on the realized 6D is the expected value taken over all the uncertainty in the table. 0 10

Table 1: Asset values in period t = 2 as a function of the state of the banking system (Ω). State Value of Debt Value of Equity j = H or L j = H or L Ω = C D (C,j) = min{F,V (C)} E (C,j) = max{V (C)−F,0} 2 j 2 j Ω = N D (N,j) = min{F,V (N)} E (N,j) = max{V (N)−F,0} 2 j 2 j state of the banking system in t = 2, the private information is about whether the firm value at t = 3 will be V (Ω) or V (Ω). Based on that, H L informed agents know the corresponding asset value as shown in the table above. The cost of information production is κ and κ in the equity and E D debt markets respectively. So, in t = 1, an informed agent knows D (Ω,j) or E (Ω,j), while an 2 2 uninformed agent only knows the expected value of debt and equity in each state, γ (Ω)D (Ω,H) + γ (Ω)D (Ω,L) ≡ ∆ (Ω) and similarly for H 2 L 2 D equity γ (Ω)E (Ω,H)+γ (Ω)E (Ω,L) ≡ ∆ (Ω).7 An uninformed bidder H 2 L 2 E will always bid p where i is debt or equity (we shortly solve for p ). At i i price p the asset is either overvalued or undervalued. If the firm state is L, i the uninformed bidder overvalues the asset and wins the bid because an informed trader will never buy an overvalued asset. If the state is H, the uninformedbidderundervaluestheasset, buttheinformedbidderbidsp +(cid:15) i 7To be sure, in t = 1, all agents know the state Ω because the signal S is fully informative. Forthisreason,thevaluesandpricesofequityanddebtareinterchangeably dependent on Ω or S. 11

and gets the undervalued asset. When an uninformed bidder faces another uninformed bidder, he buys with probability 1. When the uninformed bidder faces an informed bidder, 2 he never buys an undervalued asset in equilibrium because the informed bidder will bid p +(cid:15) for an undervalued asset. i Let y be the fraction of uninformed bidders and assume for simplicity that an informed bidder knows whether the other bidder is informed or not. An informed bidder always bids the value of the asset when facing another informed bidder (who he knows is informed). Then the asset is allocated to one of the informed bidders with probability 1. Only in this case is the 2 true value of the asset revealed. An informed bidder knows whether the firm is worth V (Ω) or V (Ω) H L and knows the associated asset value D (Ω,j) or E (Ω,j). The value of 2 2 the equity from an informed bidder’s point of view is: (cid:18) (cid:19) 1−y ΠI = y + (E(Ω,H)−p ) E 2 E and similarly for debt: (cid:18) (cid:19) 1−y ΠI = y + (D(Ω,H)−p ). D 2 D From an uninformed bidder’s point of view, the values of equity and 12

debt are, respectively: (cid:104)y (cid:105) (cid:104)(cid:16) y(cid:17) (cid:105) ΠU = (E −p ) + 1−y + (E −p ) , E 2 L E 2 H E (1) (cid:104)y (cid:105) (cid:104)(cid:16) y(cid:17) (cid:105) ΠU = (D −p ) + 1−y + (D −p ) . D 2 L D 2 H D So, informationaboutassetiisproducedif: ΠI−ΠU ≥ κ . Competition i i i among the uninformed ensures that ΠU = 0. So, the price the uninformed i bid, p , makes them indifferent between buying an undervalued asset and i buying an overvalued asset. The probability that the uninformed can buy an H-type firm out of all the available firms is (for each type of asset) is: 1γ (1−y) ω(p |Ω) = 2 H i 1γ (1−y)+(1−γ ) (cid:0) 1− y(cid:1) 2 H H 2 where i = E or D has been suppressed and where the dependence of γ on the signal S has also been suppressed. In the expression above, the uninformed can buy an H-type firm with probability 1 only if facing 2 another uninformed trader. The uninformed can also buy an L-type firm with complimentary probability, 1− 1. 2 So, the fraction of each asset type i that has its true value revealed is y∗, where y∗ is the equilibrium value of y for i, which solves the following i i i 13

equation for each fully-informative signal S: ω∗(1−ω∗)(i −i ) = κ where i = E, D. (2) H L i Proposition 2.1 The price the uninformed bid in equilibrium is: p∗ = ω∗i +(1−ω∗)i , for i = E,D (3) i i H i L and where dependence on S has been suppressed. Proof Competition across uninformed bidders make them bid so their gains are zero, otherwise there are incentives to marginally increase the bid p∗ and discretely raise the probability of buying the average quality i firm. The equilibrium price, p∗, balances the gains of buying a good firm i and the losses of buying a bad one. So, the equilibrium is a pair {y∗, p∗} i i suchthatthemarginaltraderisjustindifferentbetweenbecominginformed or not, as per equation (3). There is a pricing equation for the debt and for the equity in each state of the banking system—equation (2) is obtained by substituting equation (3) into equation (1). 2.3 Price informativeness The fraction y∗ determines the amount of information in the economy i for each asset i and for each signal (suppressed). In other words, in 14

our parsimonious model, there is a clear mapping from the fundamental refinancing risk to information contained in prices.8 For example, in the debt market: ω∗(1−ω∗)(D −D ) = κ (4) H L D for each signal, Normal (N) and Crisis (C), and where κ is the cost of D producing information in the debt market. Proposition 2.2 (1) For each signal (N and C) and for fixed γ if H (D − D ) is small–i.e., strictly less than (cid:15)–then y∗ is low (little H L D information is produced) and conversely if (D −D ) is large (more H L information is produced). (2) For fixed (D −D ), if γ rises, then y∗ rises and conversely if γ H L H D H falls. (3) (1) and (2) also characterize equity, E, with κ being the cost of E producing information in equity markets. Proof 1. Rewrite (3) as ω∗ −(ω∗)2(D −D ) = k . As ω∗ is between H L D zero and one, the squared term is small. It is apparent from the equation defining ω that if (D − D ) is relatively small, then to H L satisfy (3), y must go down. And conversely if (D −D ) is relatively H L large, y must go up. 8Foreachsignalandeachsecurity, y∗(S), i=D orE andS ∈{N,C}, isdetermined i as the solution to equation (2), with ω given by the equation just above equation (2). 15

2. The derivative of ω∗ with respect to γ is positive, so for fixed (D − H H D ), if γ goes up, y∗ must go up to reduce ω∗. L H D 3. The same logic characterizes equity. Proposition 2.3 Define a Crisis as a situation where (E −E ) is small H L (and both levels are low) and γ is low. And for debt, (D − D ) is H H L high, i.e., there is greater uncertainty about the future debt value for fixed γ . Then less information is produced in the equity market (compared to H Normal times) and more information is produced in the debt market. Proof See Proposition 2. Proposition 2.3 provides a roadmap for our empirical work to determine the price informativeness of equity and debt prices about the future state of the banking system. Each type of asset price is a linear function of the two random variables representing the econometrician’s hypothesis about the state of the banking system, Ω, and about the final project payoff, x. The state of the banking system and the state of the firm’s cash flows are systematic risks and so enter the asset pricing equations. For our purposes, the key point is that the equity and debt prices are functions of random variables, x and S, denoting the firm and bank states, respectively. We will take a linear approximation of each equity and debt 16

priceequationtogetassetpricingequationsforourempiricalspecifications: p ≈ a+bx +cx +dS +eS +(cid:15) E 1 2 1 2 E p ≈ a(cid:48) +b(cid:48)x +c(cid:48)x +d(cid:48)S +e(cid:48)S +(cid:15) D 1 2 1 2 D where the numerical subscripts indicate contemporaneous and future periods of time. These equations are what the econometrician sees. The econometrician does not observe the details of how prices are formed i.e., the interactions of the informed and the uninformed traders. The prices contain information (for the econometrician) about the future value of the firm and the future state of the banking system. 3 Measuring banks’ information centrality Our analysis builds on the insights of Davila and Parlatore (2022), who showed that the coefficient estimates and R2 statistics of two linear regressions of firm equity prices on a measure of firm fundamentals are sufficient to identify equity price informativeness about the evolution of the firm state. Their measure of relative price informativeness is the reduction in uncertainty about future firm states, relative to the remaining residual uncertainty about future firm states, after conditioning on realized firm states and equity prices. Relative price informativeness takes values 17

between 0 and 1, rendering it easy to interpret and compare across assets and time.9 Like the theoretical model presented in section 2, the empirical measure used in this section is based on the notion of informativeness in Grossman and Stiglitz (1980). A difference relative to Davila and Parlatore (2022) is that we consider a framework in which firm security prices are informative about a linear combination of the idiosyncratic firm state and the state of the banking sector. Another difference is that we focus on the price informativeness of both firm equity and debt. One key insight of Davila and Parlatore (2022) is that, under stylized assumptions, relative price informativeness corresponds to the Kalman gain of a Kalman filter applied to a linearized system with one noisy signal (price changes) about one unknown state (future earnings). In this section, we show how similar assumptions yield a measure of information about the state of the banking system from firm security prices using the difference between Kalman gains obtained from unconstrained and constrained Kalman filters. We only assume linear laws of motion with Gaussian noise to cast our problem as a Kalman filter and build intuition— we do not need to make these assumptions to obtain the measure of relative price informativeness. 9Other measures of price informativeness, such as forecasting price efficiency (Bond, Edmans and Goldstein, 2012), are valid only after making assumptions about learning process(es)andtheshapesofunderlyingdistributions. Inaddition,thoseothermeasures of price informativeness typically depend on the volatility of states. The relative price informativenessmeasuresuffersnoneoftheseshortcomings. Nevertheless,undercertain conditions, there is a one-to-one mapping from relative price informativeness to those other notions of informativeness (Davila and Parlatore, 2022). 18

Start from the linearized debt and equity pricing equations derived in Section 2 and assume the following laws of motion for the state of the banking sector, S , and for the individual firm state, x : t t ∆x = µ +ρ∆x +u (5) t+1 ∆x t t ∆S = µ +ρ ∆S +w , (6) t+1 ∆S S t t where u ∼ N(0,σ2) and w ∼ N(0,σ2). t u t w Let the index i ∈ {Debt,Equity} denote the asset type. We can then express the log-change in the equilibrium price of asset i as: ∆pi = φ ¯i +φi∆x +φi∆x +φ ¯i∆S +φ ¯i∆S +φi∆ni (7) t 0 t 1 t+1 0 t 1 t+1 n t where the error term ∆ni ∼ N(µ ,σ2 ). t ∆ni ∆ni After substituting the two laws of motion into the equilibrium asset price equations and rearranging, we obtain an expression for the linear combination of innovations in terms of ∆pi and the two state variables: t φ ¯i 1 (cid:16) u + 1w = × ∆pi −φ ¯i +φiµ +φ ¯iµ +φiµ − t φi t φi t 1 ∆x 1 ∆S n ∆ni 1 1 (cid:17) (φi +ρφi)∆x −(φ ¯i +ρ φ ¯i)∆S −φi(cid:15)∆ni 0 1 t 0 S 1 t n t Note that the ratio of parameters φ¯i 1 is unknown. Therefore, u + φi t 1 φ¯i 1w for i ∈ {Debt,Equity} are two unknown linear combinations of two φi t 1 19

unknown states, which we refer to as the combined states. We can express the signal extraction problem in state space form by defining πi as the noisy signal about the combined state, such that: t (cid:18) φ ¯i (cid:19) φi πi = u + 1w + n(∆ni −µ ), i ∈ {Debt,Equity}, t t φi t φi t ∆ni 1 1 which is a linear combination of Gaussian innovations. Without information about the true value of the linear combination of parameters, an investor uses price changes to learn about the combined state. We assume that the innovations to the two states, u and w , are orthogonal, t t which is a testable assumption in our empirical application. We can use the state space form and standard Kalman filter arguments to measure how an external Bayesian observer learns about the unknown combinedstatesfromchangesinassetprices. OurKalmanfilteringproblem has two noisy signals (equity and debt price changes) about two combined states(u +φ¯i 1w withi ∈ {Debt,Equity}). TheKalmangainoftheKalman t φi t 1 filteristheoptimalweightgiventothechangesinassetpricemeasurements and the current-state estimate. Therefore, the Kalman gain is a 2 × 1 matrix where each element measures the informativeness of the i-th asset price change about the i-th combined state.10 For example, when the first 10In more technical terms, the i-th element of the Kalman gain matrix measures the precisionofthei-thassetpricesignalaboutthei-thunknowncombinedstateinnovations relative to the precision of the prior and the signal precision of an external Bayesian observer who only learns about the i-th combined state from a firm i’s asset price changes. 20

elementoftheKalmangainmatrixcorrespondingtothedebtpriceequation is close to 1, the external observer puts a relatively high weight on the information contained in the change in debt prices to revise his or her estimate of the debt-specific combined state u + φ¯D 1 ebt w . t φDebt t 1 From the standard Kalman filter equations, it follows that the i-th element of the 2×1 Kalman gain matrix for this filtering problem is given by: σ2 + (cid:16) φ¯i 1 (cid:17)2 σ2 K = CpHT(H CpHT +C )−1 = u φi 1 w , i i i i i i i,o σ2 + (cid:16) φ¯i 1 (cid:17)2 σ2 + (cid:16) φi n (cid:17)2 σ2 u φi w φi ∆ni 1 1 (cid:16) (cid:17)2 where H = 1 is the measurement sub-matrix, C = φi n σ2 is the i i,o φi ∆ni 1 (cid:16) (cid:17) measurement covariance sub-matrix, and Cp = V u + φ¯i 1w = σ2 + i t φi t u 1 (cid:16) φ¯i 1 (cid:17)2 σ2 is the covariance sub-matrix for the predicted states at time t. φi w 1 Note that there is only one Kalman gain element per asset type because the Kalman filter is underdetermined: Each asset price change is a noisy signal about its respective unknown combined state: u + φ¯i 1w with i ∈ t φi t 1 {Debt,Equity}. We now show how to infer whether asset price changes contain information about the state of the banking sector. We continue to use the Kalman filter and its properties for illustrative purposes only. Suppose the external Bayesian observer knows (or thinks) that the asset price changes do not contain information about the state of the banking sector. If this 21

assumption is true, then a more efficient signal extraction can be obtained by imposing the following linear constraint in the original Kalman filter:   u (cid:18) (cid:19) R t   t  = r t , where R t = 0 1 and r t = 0. (8)   w t The constraint can be added to the original Kalman filter by augmenting the vector measurement equations with one additional observation about the state vector (Doran, 1992). The state space representation of the constrained model can be written as:       πDebt 1 φ¯D 1 ebt   φD n ebt (∆nDebt −µDebt)  t   φDebt   φDebt t ∆n     1  u  1   πEquity   =  1 φEq¯uity 1     t +  φE n quity (∆nEquity −µEquity)   (9)  t   φEquity   φEquity t ∆n     1  w  1      t   r 0 1 0 t Imposing constraint (8) in the Kalman filter is equivalent to imposing theparametricconstraintφ ¯i = 0foralli ∈ {Debt,Equity}inthestructural 1 equation (7). The only change relative to the unconstrained case is that Cp = σ2 so that the i-th element of the 2×1 Kalman gain matrix for the i u constrained problem is given by σ2 K ˆ = u , i ∈ {Debt,Equity}. i (cid:16) (cid:17)2 σ2 + φi n σ2 u φi ∆ni 1 Each element of the constrained 2×1 Kalman gain matrix is less than 22

or equal to its counterpart in the unconstrained 2×1 Kalman gain matrix because each asset price signal is less informative about the state when the external observer erroneously assumes that the bank state is constant. To see this, define ∆K as the i-th element of a new 2 × 1 matrix obtained i by subtracting the 2 × 1 constrained Kalman gain matrix from the 2 × 1 unconstrained Kalman gain matrix: ∆K = K −K ˆ = σ u 2 + (cid:16) φ φ ¯i 1 i 1 (cid:17)2 σ w 2 − σ u 2 , i i i σ2 + (cid:16) φ¯i 1 (cid:17)2 σ2 + (cid:16) φi n (cid:17)2 σ2 σ2 + (cid:16) φi n (cid:17)2 σ2 u φi w φi ∆ni u φi ∆ni 1 1 1 with i ∈ {Debt,Equity}. It is clear that ∆K > 0 when φ ¯i (cid:54)= 0, which i 1 occurs when asset prices are informative about the state of the banking sector. The Kalman filtering problems described above provide intuition for the identification of relative price informativeness about the state of the banking sector. We now describe the empirical measures that we calculate in general terms and link them to the Kalman filtering problems. 3.1 Identification We will run two pairs of regressions for each asset type. For ease of exposition, we temporarily drop the asset type index i. Consider first the 23

pair of regression equations: ∆p = β ¯ +β ∆x +β ∆x +β ∆S +β ∆S +e (R1) t 0 t 1 t+1 2 t 3 t+1 t ∆p = γ¯ +γ ∆x +γ ∆S +eγ (R2) t 0 t 1 t t with the corresponding R2 statistics given by: var(e ) var(γ ∆x +γ ∆S ) R2 = 1− t and R2 = 0 t 1 t ∆,∆(cid:48) var(∆p ) ∆ var(∆p ) t t Substituting the two laws of motion into the structural equation yields: ¯ ¯ ∆p = φ+φ µ +φ µ +φ µ +(φ +φ ρ)∆x + t 1 ∆x 1 ∆S n ∆n 0 1 t (cid:0) φ ¯ +φ ¯ ρ (cid:1) ∆S +φ u +φ ¯ w +(cid:15)∆n. (10) 0 1 S t 1 t 1 t t Comparing equation (10) to regression equation R2 shows that: γ¯ = φ ¯ +φ µ +φ ¯ µ +φ µ , γ = φ +φ ρ, γ = φ ¯ +φ ρ , and eγ = φ u + 1 ∆x 1 ∆B n ∆n 0 0 1 1 0 1 B t 1 t φ ¯ w +(cid:15)∆n. Likewise, comparing regression equation R1 to the structural 1 t t equation shows that: β ¯ = φ ¯ +φ µ , β = φ , β = φ , β = φ ¯ , β = φ ¯ , n ∆n 0 0 1 1 2 0 3 1 and e = (cid:15)∆n. Exploiting the variance decomposition of equation (10) and t t rearranging: (cid:16) (cid:17)2 R2 −R2 σ2 + φ¯ 1 σ2 ∆,∆(cid:48) ∆ = u φ1 w = K . 1−R ∆ 2 σ2 + (cid:16) φ¯ 1 (cid:17)2 σ2 + (cid:16) φn (cid:17)2 σ2 u φ1 w φ1 n 24

In words, the difference R2 −R2 normalized by 1−R2 is identical ∆,∆(cid:48) ∆ ∆ to the Kalman gain from the unconstrained filtering problem. This combination of R2 statistics identifies relative price informativeness, as in Davila and Parlatore (2022). Intuitively, the numerator is the percentage reduction in uncertainty about future combined state after observing the assetpriceandtherealizedvalueofthecombinedstate. Thedenominatoris the residual uncertainty about the future combined state after conditioning on the realized combined state. Consider now a second pair of regression equations corresponding to the hypothesis under which a nonbank firm’s asset price does not contain information about the state of the banking sector: ∆p = α¯ +α ∆x +α ∆x +e¯ (R3) t 0 t 1 t+1 t ∆p = δ ¯ +δ ∆x +e¯δ (R4) t 0 t t with the corresponding R2 statistics given by var(e¯) var(δ ∆x ) R2 = 1− t and R2 = 0 t . ∆x,∆x(cid:48) var(∆p ) ∆x var(∆p ) t t Using a similar argument, we obtain: R2 −R2 σ2 ∆x,∆x(cid:48) ∆x = u = K ˆ . 1−R2 (cid:16) (cid:17)2 ∆x σ2 + φn σ2 u φ1 ∆n 25

This combination of R2 statistics identifies relative price informativeness under the hypothesis that prices are not informative about the bank state– under this hypothesis, the combined state is simply the firm state. This measure is identical to the Kalman gain from the constrained filtering problem i.e., imposing constraint (8). The Kalman gain in this case is the reduction in uncertainty about the future firm state relative to the remaining residual uncertainty about the future firm state after conditioning on the realized firm state. The difference between the two Kalman gains is a statistic—a combination of R2 statistics from four regressions—that we use to test whetherassetpricesareinformativeaboutthebankingsector. TheKalman gain difference, ∆K, can be written as: R2 −R2 R2 −R2 ∆K = ∆,∆(cid:48) ∆ − ∆x,∆x(cid:48) ∆x . (11) 1−R2 1−R2 ∆ ∆x The statistic ∆K is close in spirit to a Wald statistic in the context of two nested linear regression models. That is, ∆K measures the distance betweenthe(random)valueoftheKalmangainintheunconstrainedmodel and the (random) value of the Kalman gain in the constrained model. Under the null hypothesis that asset prices are not informative about the future state of the banking sector, equivalent to satisfying constraint (8), then ∆K is not statistically different from 0. 26

In the remainder of this section, we show how we estimate ∆K from data and construct asymptotically valid hypothesis tests about banks’ information centrality. 3.2 Estimation We estimate time-specific and firm-security-specific measures of ∆K using rolling windows of data for each firm’s debt and equity prices. We continue to suppress the index i for asset type. Let ∆K be the j-th firm’s estimate j,q of∆K, calculatedasthecombinationofR2 statisticsgiveninequation(11) and obtained by estimating the regression equations (R1)-(R4) on a rolling window q ∈ {1,...,T}, where j ∈ {1,...,J}. In our application, we use a 16-quarter rolling window as our baseline. In any given 16-quarter rolling window, we retain firm-security time series that have at least ten contiguous observations and we discard firm-security time series whose (R1)-(R2) maximum leverage is greater than 95 percent. To control for seasonality and the arrival of firm-equity and firm-bond public signals, we estimate regression equations (R1)-(R4) including firm-security-quarter fixed effects. If we knew the distribution of ∆K , we could test whether φ ¯ (cid:54)= 0 j,q 1 for firm j in quarter q by testing if ∆K is statistically different from 0. j,q However, a firm-level test is not feasible, as we only have one observation per firm in a rolling window. 27

Nevertheless, it is possible to obtain a consistent estimate of the distribution of the sample mean of ∆K across nonbank firms within a j,q quarterq underweakassumptions. Toproceed,wetreatthesetoffirm-level estimates {∆K ,...,∆K } as independently and identically distributed 1,q J,q random variables drawn from a common yet unknown distribution. We can then use this estimate of the distribution to construct asymptotically valid hypothesis tests. 3.3 Statistical inference We obtain an asymptotically valid hypothesis test for the sample mean of ∆K in each quarter q using the subsampling method (Politis et j,q al., 1999). In each quarter q, we treat {∆K ,...,∆K } as a set 1,q J,q of J independently and identically distributed random variables taking values in sample space Ω. This approach is justified as we estimate each ∆K independently. The probability law generating the sample j,q {∆K ,...,∆K } is P, which is unknown. Note that P could depend 1,q J,q on q, which we omit from the notation for simplicity. We wish to estimate the true sampling distribution of the sample mean of ∆K in a given j,q quarter q, denoted by θ ˆ , to make inference about θ(P), which, once J,q again, could depend on q. Denote by J (P) the sampling distribution of the normalized statistic J √ J(θ ˆ − θ(P)) based on a sample of size J from P. The corresponding J,q 28

cumulative distribution function is given by: √ ˆ J (x,P) = Prob { J(θ −θ(P)) ≤ x}. J P J,q The basic idea of subsampling is to approximate the sampling distribution of the mean of ∆K in a quarter q based on the means j,q computed over all the possible smaller firm subsets of size c < J from the same quarter. Politis et al. (1999) shows that subsampling behaves well under extremely weak assumptions because each subset of size c taken without replacement from the original sample of size n is a sample of size c from the true model. The only additional assumption needed to construct asymptotically valid confidence intervals for θ(P) is Assumption 1 below: Assumption 1 There exists a limiting law J(P) such that J (P) J converges weakly to J(P) as J → ∞. The subsampling method consists of approximating the sampling √ distribution of J(θ ˆ − θ(P)) with the empirical distribution generated J,q by its subsample counterpart. Let Y ,...,Y be equal to the N = (cid:0)J(cid:1) 1 NJ J c subset of size c of the quarter q sample {∆K ,...,∆K }. Each subset Y 1,q J,q k depends on c and J, which we omit from the notation for simplicity. Let θ ˆ be the average of ∆K calculated over the subset Y . The J,c,t,k j,q k approximation to J (x,P) is defined by J 29

(cid:88) NJ √ L (x) = N−1 1{ c(θ ˆ −θ ˆ ) ≤ x}. J,c J J,c,q,k J,q k=1 Note that our limiting concept is that the number of firms in a quarter q becomes large. Theorem A.1 in Appendix A from Politis et al. (1999) shows that we can derive asymptotically valid confidence intervals for θ ˆ using L (x) J,q J,c because it is a consistent estimator of J(x,P). Then, we can draw asymptotically valid inference about the true θ(P) by exploiting the usual duality between the construction of confidence intervals for the sample mean θ ˆ and the construction of hypothesis tests about θ ˆ . J,q J,q In our application, we wish to test the null hypothesis that each quarter’s θ(P) is 0. That is, the null hypothesis in each quarter is that the average nonbank firm’s asset prices do not contain information about the future state of the banking sector. This test is equivalent to testing whether constraint (8) in the Kalman filtering problem holds on average. If the value of the estimated θ(P) for quarter q falls outside the quarterly confidence interval, we reject the null hypothesis on that date. 3.4 Measurement Thus far, we have shown how to obtain an asymptotically valid test of banks’ information centrality using nonbank firms’ asset prices. Our 30

statistical test answers the question: Do asset price changes contain information about the future state of the banking sector? Moving beyond this hypothesis test, we turn to the issue of measuring the level of bank information in nonbank firms’ asset prices. We are particularly interested in measuring variation in price informativeness across asset types and over time. In other words, when is debt more informative than equity about the state of the banking system? Measuring the level of bank information content in nonbank firms’ asset prices requires obtaining consistent estimates of both K and K ˆ. (Note again that our hypothesis test only requires a consistent estimate of ∆K.) The ordinary least square estimate of K ˆ is consistent if Cov(u ,w ) = 0 j,q t t holds. Under this additional assumption, the residuals in the constrained pair of regression equations R2 and R4 are orthogonal to the regressors. Although it is not obvious whether this assumption holds a priori, it is straightforward to test it and we discuss the details in Appendix C. In the rest of this section, we assume that Cov(u ,w ) = 0 holds. t t From the above discussion, conducting inference on the amount of information contained in asset prices about the state of the banking system is limited by the presence of the unknown and idiosyncratic linear combination parameters φ¯i 1 with i ∈ {Debt,Equity}. This feature φi 1 means that our estimate of ∆K is a lower bound estimate of the average information about the state of the banking sector. To see this, note that 31

the estimate of ∆K partially identifies the information content about j,q the state of the banking system in a nonbank firm j’s asset price (Manski, 2009; Tamer, 2010). More formally, under our stylized linear Gaussian assumptions: (cid:16) (cid:17)2 0 ≤ ∆K ≤ σ u 2 + φ φ ¯ 1 1 σ w 2 − σ u 2 (cid:16) (cid:17)2 (cid:16) (cid:17)2 (cid:16) (cid:17)2 (cid:16) (cid:17)2 σ2 + φ¯ 1 σ2 + φn σ2 σ2 + φ¯ 1 σ2 + φn σ2 u φ1 w φ1 n u φ1 w φ1 n σ2 = w ≤ 1. (cid:16) (cid:17)2 (cid:16) (cid:17)2 σ2 + φ1 σ2 + φn σ2 w φ¯ 1 u φ¯ 1 n This inequality means that the price informativeness about the state of the bankingsystemisboundedfrombelowbyourestimateof∆K andbounded from above by 1. This condition is intuitive because changes in ∆K conditionalonφ ¯ (cid:54)= 0couldbedrivenbychangesineitherthemeasurement 1 or noise process. Or, put differently, ∆K is the most an external observer can learn about the state of the banking sector by analyzing changes in asset prices. Therefore, the sample mean of ∆K is a conservative estimate. When the sample mean of ∆K is statistically different from 0 and asset prices do contain information about the state of the banking system, its value is a lower bound on the true price informativeness. The sample mean of ∆K measures the fraction of an external observer’s precision about the state of the banking sector that is conveyed, on average, by observing asset prices. For example, an average ∆K of 0.3 means that, on average, at least 32

30 percent of investors’ ex-post precision about the innovation to the state of the banking sector comes from conditioning on non-financial firm asset prices. 4 Data Our empirical analysis uses data on equity prices, debt prices, the state of individual nonbank firms, firm-level debt refinancing, and the state of the banking system. We construct our data closely following Davila and Parlatore (2022). In this section, we describe our entire process in detail. For equity prices, we begin with the merged CRSP-COMPUSTAT database provided by Wharton Research Data Services (WRDS). We use monthly equity prices adjusted for equity splits and deflated using the personal consumption expenditure price index (PCEPI) from FRED.11 We winsorize these prices at the 2.5th and 97.5th percentiles. We calculate the three month change in the log equity prices and then lag the data by three months before merging with COMPUSTAT data to ensure that the data were public during the period of trading. For debtprices, webegin withthe dailyICE-IDCdatabase, whichis the leading provider of evaluated prices for the widest range of corporate fixed income securities. We identify a firm by its equity ticker, which restricts 11We restrict the sample to securities listed on the NYSE, AMEX, or NASDAQ for standard, consolidated, domestic firms reporting the industrial format. We link the datasets using the linktypes ’LU’, ’LC’, or ’LS’ and with issue marker ’P’ or ’C’. 33

us to large issuers of corporate debt. We find 5,956 individual bonds for 792 individual firms. We can match 97 percent of these firms to CRSP by ticker.12 We then calculate a weighted-average bond price for each firm, where the weights are the amounts of each bond outstanding. We deflate these prices using the PCEPI from FRED and winsorize them at the 2.5th and 97.5th percentiles. We calculate the three month change in the log bond prices and then lag the data by three months before merging with COMPUSTAT data to ensure that the data were public during the period of trading. 4.1 State of nonbank firms As our measure of the state of the firm we use quarterly earnings before interest and tax (EBIT) deflated using the PCEPI and winsorized as for equity prices. As earnings can be negative, we calculate the growth rate in earnings as:        x x t− t 1 −1 if x t−1 > 0    ∆x t = xt +1 if x < 0 (12)   |xt−1| t−1       NA if x t−1 = 0 12AlthoughwematchICE-IDCtoCRSPbyticker,ourfirm-levelanalysisisdoneusing firms identified by GVKEY from COMPUSTAT, as described later, to avoid concerns about significant changes in firm structure over time. 34

We merge these data with the lagged quarterly change in equity prices using the CCM data crosswalk provided by WRDS. Our analysis uses firm-level debt refinancing that we construct using Moody’s Credit Watch data, which provides detailed information about individual debt actions including when a bond is called or paid down early. We use the firm organization structure from Moody’s to aggregate information about actions on individual bonds issued by all subsidiaries up to the parent company identified by its ticker.13 For each bond, we construct a daily time series of the amount outstanding, paying close attention to calls and paydowns. We then aggregate for each firm the total amount scheduled to mature in the coming twelve months and divide by the total amount outstanding. We restrict our sample to public nonbank firms that we could match in CRSP and IDC. These firms finance their operations by issuing publicly listed equity and public corporate debt. We match by ticker. We identify nonbank firms as those whose two-digit SIC code is not 60-62. There are roughly 250 firms at the end of our sample. Table 2 shows the distributions of their total asset size ($bn) at the end of each year in our sample. Evidently, these are all large firms. Table 3 shows the count of firms by credit ratings at the end of each year in our sample. 13Moody’s also provides the CIK identifier. The final merge of Moody’s data with CRSP-ICE-IDC is done using the GVKEY identifier, which we obtain using the CIK- GVKEY crosswalk from WRDS. 35

Table 2: Firm total assets ($bn) at end of year. Source: Authors’ calculations based on data from CRSP, COMPUSTAT, and ICE-IDC. Year N Min. 1st Qu. Median Mean 3rd Qu. Max. 2005 75 0.01 15.14 27.72 74.79 53.23 1,024.39 2006 83 0.01 13.56 28.44 52.81 52.55 623.03 2007 86 2.12 17.47 33.76 70.65 55.26 1,191.86 2008 91 2.31 17.18 32.56 57.27 54.02 559.59 2009 93 2.30 17.25 32.15 61.10 59.69 871.95 2010 93 2.60 16.67 38.95 71.71 64.36 826.59 2011 99 2.57 16.82 38.02 72.65 65.21 856.78 2012 115 2.80 16.60 37.10 63.69 67.59 720.08 2013 138 0.41 14.36 33.48 61.56 66.29 683.18 2014 152 1.87 11.27 30.33 56.19 60.37 665.77 2015 189 1.03 7.73 22.72 42.24 49.31 566.55 2016 216 0.78 7.22 18.86 44.69 46.92 627.96 2017 231 0.79 6.62 18.08 46.19 48.79 716.09 2018 246 0.75 6.82 14.50 39.40 42.52 670.15 2019 265 0.74 6.87 15.68 46.86 43.79 786.57 36

Table 3: Moody’s credit ratings for firms at the end of each year. Source: Authors’ calculations based on data from CRSP, COMPUSTAT, and ICE-IDC. Year Aaa Aa A Baa Ba B Caa 2005 8 10 38 16 1 0 0 2006 7 12 41 19 0 0 0 2007 8 13 40 21 0 0 0 2008 7 14 37 20 0 0 0 2009 7 14 37 22 0 1 0 2010 7 15 42 28 0 1 0 2011 7 15 45 26 2 0 0 2012 7 17 50 37 1 0 0 2013 7 16 55 52 3 3 0 2014 7 15 55 63 4 3 0 2015 6 17 60 82 7 7 0 2016 7 19 59 85 16 16 0 2017 8 18 59 97 19 20 0 2018 8 17 56 103 26 28 0 2019 9 17 53 106 30 22 1 4.2 State of the banking system Our measure of the state of the banking system is the equity capital ratio of financial intermediaries from He et al. (2017), henceforth HKM.14 An important qualification is that He et al. (2017) calculate the equity capital ratio using only Primary Dealer counterparties of the New York Federal Reserve, rather than all commercial banks. As such, it represents a specific—albeit central—part of the financial system. The equity capital ratio is a measure of financial system leverage, whose effect on the real 14Downloaded in January 2022 from https://voices.uchicago.edu/zhiguohe/. 37

economy has been studied in an extensive literature.15 We calculate its growth rate ∆b = bt −1. t bt−1 In Appendix F, we study the equity capital ratio of the oil and gas sector as a placebo test for the informational centrality of the banking sector. In the next section, we present our main results about the price informativeness of nonbank firms’ asset prices for the future state of the banking system. 5 Informativeness of debt and equity prices Figure 1 summarizes our main findings. In each quarter q from 1999Q3 to 2020Q4, we estimate ∆K for each firm i ∈ {1,...,n } in our matched q,i q sample using quarterly observations from t = q to t = q +15. We choose a 16-quarter rolling window because it is a good compromise between retaining high-frequency variation in information flow and a large enough sample for ordinary least square estimation—see Appendix G for more details. The solid black line in panels A and B of Figure 1 is the average of ∆K taken across firms in quarter q i.e., ∆K = 1 (cid:80)nq ∆K . The i,q q nq i=1 i,q 99 percent confidence interval (CI) of ∆K obtained with subsampling is q represented by the upper and lower dashed red lines. Panels A and B 15See,forexample,Bernanke,LownandFriedman(1991);HancockandWilcox(1998); Van den Heuvel (2008); Meh and Moran (2010). 38

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plot the banking sector information content in nonbank firms’ debt and equity prices, respectively. Whenever the ∆K CI lies above 0, nonbank q firms’ debt or equity contain statistically significant information about the state of the banking sector, on average. Note that, because the information content is partially identified (section 3.4) our estimates are a lower bound for the information content about the state of the banking system. For example, panel A in Figure 1 shows that during the financial crisis 2007- 2009, atleast30percentofinvestors’ex-postprecisionabouttheinnovation tothestateofthebankingsectorcamefromconditioningonnonbankfirms’ debt prices. Table 4 reports summary statistics for the time series reported in Panels A and B of Figure 1. The upper and lower parts of the table show statistics for debt and equity, respectively. The summary statistics suggest that, in normal times, the information content of debt and equity about the future state of the banking system is about the same. By contrast, during the 2007-09 financial crisis, the information content of debt was about 50 percent higher than that of equity. Panel C of Figure 1 expands the analysis of the information content of debt relative to equity about the state of the banking sector. The black line is ∆Kdebt−∆Kequity with the 99 percent CI obtained by subsampling q q represented by the dashed red lines. Whenever the CI is above zero, debt contains more information than equity on average, and vice versa when 40

Table 4: Summary statistics of the information in asset prices about the future state of the banking system. The table shows summary statistics across nonbank firms and time for debt and equity of ∆K , representing the information content in prices about the future state i,q of the banking system, measured as the equity capital ratio of financial intermediaries (He et al., 2017). The period of the crisis is defined as 2007Q3-2009Q4. Source: Authors’ calculations based on data from CRSP, COMPUSTAT, and ICE-IDC. N Mean St. Dev. Min Pctl(25) Pctl(75) Max Debt full sample 9,612 0.154 0.169 -0.525 0.031 0.237 0.978 crisis sample 873 0.224 0.193 -0.361 0.070 0.356 0.964 no crisis sample 8,739 0.147 0.165 -0.525 0.028 0.225 0.978 Equity full sample 9,612 0.140 0.171 -0.480 0.020 0.214 0.985 crisis sample 873 0.161 0.171 -0.265 0.031 0.244 0.985 no crisis sample 8,739 0.138 0.171 -0.480 0.019 0.208 0.957 the CI is below zero. The main takeaway from the difference between the two measures is that debt and equity contain roughly the same information about the banking sector in normal times. However, during the financial crisis of 2007-2009, nonbank firms’ debt prices contained significantly more information about the state of the banking sector. A first-order explanation for this finding is illustrated in Appendix D. During crisis periods, debt prices may become more sensitive than equity prices to changes in the value of the underlying collateral. That said, this explanation falls short of empirically answering why the future state of the banking sector is a driver of the fundamental value of the firm. 41

6 Why was debt more informative? We investigate thedriversof variation inthe relative informationcontentof debtpricesduringthefinancialcrisis2007-2009usingfirm-levelregressions. Informed by our theoretical model in section 2, our analysis focuses on refinancing risk. For each firm in our matched sample we use data from Moody’s to create a quarterly time series of how much of that firm’s corporate debt will mature in the next 12 months expressed as a fraction of that firm’s total amount of debt outstanding. When calculating the total amount outstanding we are careful to account for debt that is called by the firm using detailed rating changes information. Figure 2 plots the distribution across all firms’ debt refinancing ratio in each quarter. The median fraction of debt maturing within 12 months is essentially zero over the sample period. Most of the variation comes from firms in the upper part of the distribution. There is a gradual widening of the distribution in the year leading up to the financial crisis 2007-2009 followed by a substantial contraction during the crisis. We implement our test with the following regression specification: KG_ddiff =β +β crisis +β Debt_12m_roll i,q 0 1 q 2 i,q +β crisis ×Debt_12m_roll +(cid:15) . 3 q i,q i,q Our dependent variable KG_ddiff = ∆Kdebt −∆Kequity is the informai,q i,q i,q 42

Figure 2: Distribution of the ratio of debt maturing within 12 months to total debt outstanding across nonbank firms. The lines in this figure are percentiles in the distribution over nonbank firms for each quarter. Source: Authors’ calculations based on data from Moody’s. tiveness of firm i’s debt prices about the future state of the banking sector relative to the informativeness of the same firm’s equity prices in quarter q. The binary variable crisis takes the value 1 if the quarter q falls in the q range 2007Q3-2009Q4 and 0 otherwise. The variable Debt_12m_roll is i,q the ratio of the par value of firm i’s corporate debt maturing in the next 12 months to the same firm’s total par value of corporate debt outstanding in quarter q (shown in Figure 2). Table 5 contains summary statistics of the regression variables. Table 6 summarizes the regression results. We report bootstrapped 43

Table 5: Summary statistics of regression variables. This table shows summary statistics for the variables used in the empirical analysis reported in Table 6. Source: Authors’ calculations based on data from CRSP, COMPUSTAT, ICE-IDC, and Moody’s. N Mean St. Dev. Min Max KG_ddiff i,q full sample 9,612 0.014 0.213 -0.979 1.101 crisis sample 873 0.063 0.243 -0.751 0.985 no crisis sample 8,739 0.009 0.209 -0.979 1.101 Debt_12m_roll 6,640 0.052 0.098 0 1 i,q crisis 9,612 0.091 0.287 0 1 q standard errors that are clustered at the firm level. In Column 2, the coefficient on the interaction term between Debt_12m_roll and crisis i,q q suggests that a one standard deviation (10 percent) increase in our debt refinancing measure during the 2007-2009 financial crisis implies that investors’ex-postprecisionabouttheinnovationtothestateofthebanking sector was 2 percentage points greater when conditioning on debt prices relative to equity prices. As a benchmark for economic significance, note that the unconditional increase in the difference in informativeness during the 2007-2009 financial crisis was 5 percentage points (Column 1). Columns 4 and 5 include firm fixed effects to analyze the within-firm effect. We find similar results indicating that the relevant variation is within firms over time, rather than across firms. 44

What about equity? It pays to produce information when the uncertainty about future payments is greatest.16 During the Financial Crisis it may well have been the case that investors believed that the prospects for equity values were dire. In that case, it may have been that it was not profitable to produce information. Equity holders knew the situation was dire without producing information. But debt holders wanted to know how bad the crisis would be for their specific firm as it might affect not just the payout, but also the payout timing or recovery values that are outside the scope of our model, which entails producing information about the future state of the banking sector.17 7 Conclusion Banks are at the center of the savings-investment process. Banks are special. One reason is that they produce short-term debt. But another reason is that firms rely on banks for loans and to underwrite their debt. Firms have relationships with banks. Consequently, firms care about the future state of the banking system. We showed that firms’ debt and equity prices reflect information about the future state of the banking system. Financial crises have been viewed as information events in which 16While this is quite intuitive, it cannot be proven analytically in our model. With the same asset price formation protocol, Chousakos et al. (2023) provide a numerical example showing this. 17This is a topic for further research. 45

Table 6: Debt refinancing determines the amount of information about the future state of the banking system conveyed in debt prices relative to equity prices. The dependent variable is KG_ddiff = ∆Kdebt − ∆Kequity, as defined in the text. The first i,q i,q i,q explanatory variable is crisis , which takes the value 1 if the quarter q falls q in the range 2007Q3-2009Q4 and 0 otherwise. The second explanatory variable (Debt_12m_roll ) is the ratio of the par value of firm i’s i,q corporate debt maturing in the next 12 months to the same firm’s total par value of corporate debt outstanding in quarter q (shown in Figure 2). Table 5 contains summary statistics of the regression variables. We report bootstrapped standard errors clustered at the firm level with 2,999 replications. Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 Dep. var.: KG_ddiff (1) (2) (3) (4) i,q crisis 0.051∗∗∗ 0.035∗ 0.053∗∗∗ 0.032 q (0.019) (0.021) (0.019) (0.022) Debt_12m_roll −0.004 0.006 i,q (0.058) (0.073) Debt_12m_roll ×crisis 0.223∗∗ 0.276∗∗∗ i,q q (0.098) (0.104) Constant 0.013∗ 0.013∗ (0.007) (0.008) Firm FE N N Y Y Observations 6,640 6,640 6,640 6,640 Adjusted R2 0.005 0.006 0.173 0.176 46

information-insensitive short-term bank debt becomes sensitive. We showed that corporate debt also displays an important change in information sensitivity during a financial crisis. Corporate debt becomes 50 percent more informative than equity during the 2007-2009 financial crisis. The reason is partly due to refinancing risk: firms are concerned that they will not be able to borrow to refinance existing debt during the crisis. References Adrian, Tobias, Erkko Etula, and Tyler Muir, “Financial Intermediaries and the Cross-Section of Asset Returns,” Journal of Finance, 2014, LXIX, 2557–2596. Bassett, William F., Mary Beth Chosak, John Driscoll, and Egon Zakrajsek, “Changes in Bank Lending Standards and the Macroeconomy,” Journal of Monetary Economics, 2014, 62 (C), 23–40. Benmelech, Efraim, Carola Frydman, and Dimitris Papanikolaou, “Financial Frictions and Employment during the Great Depression,” Journal of Financial Economics, 2018, 133, 541–563. Bernanke, Ben S, Cara S Lown, and Benjamin M Friedman, “The credit crunch,” Brookings Papers on Economic Activity, 1991, 1991 (2), 205–247. Bond, Philip, Alex Edmans, and Itay Goldstein, “The Real Effects of Financial Markets,” Annual Review of Financial Economics, 2012, 4 (1), 339–360. Castro, Andrew, David Glancy, Felicia Ionescu, and Greg Marchal, “What Happens When Banks Tighten C&I Loan Supply?,” FEDS Notes, 2022. 47

Chava, Sudheer and Amiyatosh Purnanandam, “The Effect of Banking Crisis on Bank-Dependent Borrowers,” Journal of Financial Economics, 2011, 99 (1), 116–135. Chodorow-Reich, Gabriel, “The Employment Effects of Credit Market Disruptions: Firm-level Evidence from the 2008-09 Financial Crisis,” Quarterly Journal of Economics, 2014, 129 (1). Chousakos, Kyriakos, Gary Gorton, and Guillermo Ordoñez, “Information Dynamics and Macro Fluctuations,” American Economic Journal: Macro, forthcoming, 2023. Claessens, Stijn and Luc Laeven, “Financial Dependence, Banking Sector Competition, and Economic Growth,” Journal of the European Economic Association, 2005, 3 (1), 179–207. Dang, Tri Vi, Gary Gorton, and Bengt Holmström, “The Information View of Financial Crises,” Annual Review of Financial Economics, 2020, 12, 39–65. Davila, Eduardo and Cecilia Parlatore, “Identifying Price Informativeness,” Working Paper, Yale University 2022. Doran, Howard E, “Constraining Kalman filter and smoothing estimates to satisfy time-varying restrictions,” The Review of Economics and Statistics, 1992, pp. 568–572. Financial Crisis Inquiry Commission, The Financial Crisis Inquiry Report of the National Commission on the Causes of the Financial and Economic Crisis, U.S. Government Printing Office, 2012. Grossman, Sanford and Joseph Stiglitz, “On the impossibility of Informationally Efficient Markets,” American Economic Review, 1980, 70 (3), 393 – 408. Hancock, Diana and James A Wilcox, “The “credit crunch” and the availability of credit to small business,” Journal of Banking & Finance, 1998, 22 (6), 983–1014. He, Zhiguo and Arvind Krishnamurthy, “Intermediary Asset Pricing,” American Economic Review, 2013, 103 (2), 732–770. 48

, Brian Kelly, and Asaf Manela, “Intermediary Asset Pricing: New Evidence from Many Asset Classes,” 126, 2017, pp. 1–35. Lown, Cara and Donald Morgan, “The Credit Cycle and the Business Cycle: New Findings Using the Loan Officer Opinion Survey,” Journal of Money, Credit and Banking, 02 2006, 38, 1575–1597. Lown, Cara S. and Donald Morgan, “Credit effects in the monetary mechanism,” NY Fed Economic Policy Review, 2002, 8 (May), 217–235. Manski, Charles F, Identification for prediction and decision, Harvard University Press, 2009. Meh, Césaire A. and Kevin Moran, “The role of bank capital in the propagation of shocks,” Journal of Economic Dynamics and Control, 2010, 34 (3), 555–576. Ottonello, Pablo and Wenting Song, “Financial Intermediaries and the Macroeconomy: Evidenc from a High-Frequency Identification,” Working Paper 29638, National Bureau of Economic Research January 2022. Owens, Raymond and Stacey Schreft, “Survey Evidence of Tighter Credit Conditions: What Does It Mean?,” Federal Reserve Bank of Richmond Economic Review, 1991. Paravisini, Daniel, “Local Bank Financial Constraints and Firm Access to External Finance,” The Journal of Finance, 2008, 63 (5), 2161–2193. Politis, Dimitris, Joseph Romano, and Michael Wolf, Subsampling, Springer-Verlag, 1999. Tamer, Elie, “Partial identification in econometrics,” Annu. Rev. Econ., 2010, 2 (1), 167–195. Van den Heuvel, Skander, “The welfare cost of bank capital requirements,” Journal of Monetary Economics, 2008, 55 (2), 298–320. 49

Appendix for online publication A Asymptotically valid hypothesis tests of price informativeness In each rolling window q ∈ {1,...,T}, denote by ∆K the j-th firm’s j,q asset (debt or equity) price informativeness of about the future state of the banking sector in rolling window q, where j ∈ {1,...,m}. As in the main text, we drop the asset type index i for readability. In each q, we treat {∆K ,...,∆K } as a sample of m independently and identically 1,q m,q distributed random variables taking values in the sample space Ω—recall thateach∆K isestimatedindependently. Theprobabilitylawgenerating j,q the sample {∆K ,...,∆K } is P, which is unknown. It is understood 1,q m,q that P could depend on q, which we omit from the notation for simplicity. We wish to estimate the true sampling distribution of the sample mean of ∆K in a given rolling window q, denoted by θ ˆ , to make inference j,q m,q about θ(P), which, once again, could depend on q. Denote by J (P) the sampling distribution of the normalized statistic m √ m(θ ˆ −θ(P)) based on a sample of size m from P. The corresponding m,q cumulative distribution function is given by: √ ˆ J (x,P) = Prob { m(θ −θ(P)) ≤ x}. m P m,q 50

As explained by Politis et al. (1999), the only assumption needed to constructasymptoticallyvalidconfidenceintervalsforθ(P)isAssumption2 below: Assumption 2 There exists a limiting law J(P) such that J (P) m converges weakly to J(P) as m → ∞. Let Y ,...,Y be equal to the N = (cid:0)m(cid:1) subset of size c of the time q 1 Nm m c sample{∆K ,...,∆K }. EachsubsetY dependsoncandm, whichwe 1,q m,q k omit from the notation for simplicity. Let θ ˆ be the average calculated m,c,q,k over the subset Y . The approximation to J (x,P) is defined by k n (cid:88) Nm √ L (x) = N−1 1{ c(θ ˆ −θ ˆ ) ≤ x}. m,c m m,c,q,k m,q k=1 Our limiting concept is that the number of firms in a rolling window q becomes large. The following theorem of subsampling (Theorem 2.2.1 from Politis et al. (1999)) follows: Theorem A.1 Assume Assumption 1 and assume that c/m → 0 and c → ∞ as m → ∞. i. If x is a continuity point of J(·,P), then L (x) → J(x,P) in m,c probability ii. If J(·,P) is continuous, then sup |L (x) − J (x,P)| → 0 in x m,c m probability. 51

iii. Let a (1−α) = inf{x : L (x) ≥ 1−α}. m,c m,c Correspondingly, define a(1−α,P) = inf{x : J(x,P) ≥ 1−α}. If J(·,P) is continuous at a(1−α,P), then √ ˆ Prob { n(θ −θ(P)) ≤ a (1−α)} → 1−α as m → ∞. P m,q m,c Therefore, the asymptotic coverage probability under P of the √ ˆ −1 confidence interval [θ − m a (1−α),∞) is the nominal level m,q m,c 1−α. √ ˆ iv. Assume c(θ −θ(P) → 0 almost surely and, for every d > 0, m,q (cid:88) exp{−d(m/c)} < ∞. m Then, the convergence in i. and ii. holds with probability one. Theorem A.1 shows that we can derive asymptotically valid confidence intervals for the average debt or equity price informativeness of about the future state of the banking sector in rolling window q using L (x) because m,c 52

it is a consistent estimator of J(x,P). By exploiting the usual duality between the construction of confidence interval for the sample mean θ ˆ m,q and the construction of hypothesis test about θ ˆ , subsampling allow us to m,q drawasymptoticallyvalidinferenceaboutthetrueθ(P). Inourapplication, we wish to test the null hypotheses that the daily θ(P) equals zero. That is, under the null, the average nonbank firm’s asset prices do not contain information about the future state of the banking sector. If the value zero is outside the daily confidence interval, we reject null hypotheses on that date. A.1 Choosing an optimal subsample block size Although any subsample block size satisfying Assumption 1 is valid asymptotically, we need to ensure that our choice of subsample block size does not materially affect our finite sample confidence interval estimates. We build on the algorithm proposed by Politis et al. (1999) in section 9.3.3 to find an optimal subsample block size that minimizes the variability of the confidence intervals. Let c be the subsample size for quarter t, which t yields a confidence interval {I ,I }. We construct a discrete grid ct,low ct,high of possible values c ∈ {c small, ..., c large}. For each subsample size c t,s t, t, t,s we consider a perturbation of small integer k around the subsample size 53

and calculate a measure of variation in the confidence interval: VI ≡ std.dev.(cid:0) I ,...,I (cid:1) +std.dev.(cid:0) I ,...,I (cid:1) . ct,s c t,s−k ,low c t,s+k ,low c t,s−k ,high c t,s+k ,high Finally, we choose the value of c that minimizes the confidence interval variability for the greatest number of quarters in our entire sample: T (cid:88) c∗ = argmax I where c∗ = argminVI . c {c∗ t =c} t∗ ct,s ct,s t=0 We use a grid of subsampling block size that is 7, 10, 15, 20, 25, 30 and 35 percent of the number of firms present in a given quarter rounded to the nearest integer above. For example, if there are 100 firms in quarter t, then c ∈ {7,10,15,20,25,30,35}. We use a percentage of firm t and not an absolute number of firms because the number of firms in each rolling window varies. We consider a perturbation k equal to one firm. As the number of possible firm combinations in a 10 or 30 percent-sized subsample is large, we randomly draw 1,000 subsamples from the full set of firm combination in each rolling window. We find that the optimal sub-sample size c is 30 percent of the number of firms in each quarter. Nevertheless, the confidence intervals obtained with the optimal block size are not substantially different from those we obtain with the other block sizes we considered. Our optimal block size algorithm also found that a 30 percent block size was optimal when drawing 2,000 subsamples from 54

the full set of firm combination in each rolling window. Table 7 reports the number of quarters each candidate sub-sampling block size is optimal and c∗ appears in bold font. Table 7: Optimal subsampling block size results Block size # of quarter optimal # of quarter optimal (percent of firms) 1,000 draws 2,000 draws 7 9 12 10 10 6 15 8 7 20 5 3 25 11 8 30 28 30 35 10 15 55

B Bootstrapping and subsampling The subsampling method is not as well known as the bootstrap method in economics and finance, which warrants a cursory comparison—see Politis et al. (1999) for textbook-length treatment. The most relevant bootstrap method for our application is the block bootstrap. In a given rolling window q, the block bootstrap draws entire firm-level time series with replacement to form a bootstrap sample of size n and evaluate the statistic of interest on the set of bootstrap samples to estimate its sampling distribution. A key issue with the block bootstrap is establishing consistency for the distribution of a sample mean. In our application, this requires establishing that the distribution of the average ∆K is locally i,q smooth as a function of the unknown model. Therefore, we would either need to assume such smoothness or verify such smoothness by making assumptions about the (unknown) true model. Neither of these options is desirable in our application. A considerable advantage of subsampling is that we do not need to make such assumptions or carry out this type of verification to draw asymptotically valid inference. All that is required is that our (normalized) statistic has a limit distribution under the true model. 56

C Consistent informativeness estimates We show in Section 3.4 that ordinary least squares delivers a consistent estimate of the price informativeness measure K ˆ under the assumption j,q that Cov(u ,w ) = 0. A straightforward way to investigate the validity t t of this assumption is to regress firm state innovations on the bank state innovationswithandwithoutfirmfixedeffects. Theresultsaresummarized in Table 8. The lack of a statistically significant relationship between the firmstateinnovations(∆x )andthebankstateinnovations(∆S )confirms i,t t that the innovations to the two states are uncorrelated. Table 8: Correlation between firm state and bank state innovations. Column 1reports Driscoll-Kraay standard errors and Column 2 reports firm-level clustered standard errors. Dep. var: ∆x (1) (2) i,t ∆S 0.74 0.64 t (0.52) (0.62) Firm FE N Y Standard errors Driscoll-Kraay Y N Cluster by firm N Y R2 0 0.04 Observations 309,645 309,645 57

D Understanding variation across asset prices The schematic diagram shown in Figure 3 illustrates why debt prices are more sensitive to changes in the value of the underlying collateral than equity prices in crisis times. The diagram plots the prices of equity and debt as a function of the underlying collateral (firm) value. During normal times, equity prices are more sensitive to changes in the value of collateral (blue bell curve). During crisis times, debt prices are more sensitive (red bellcurve). Thisstatedependenceisafirst-orderexplanationfortheresults described in Section 5. Figure 3: Debt prices are more sensitive than equity in crisis times. This schematic diagram illustrates an intuition for the results presented in Section 5. We thank Skander Van den Heuvel for suggesting the figure. Collateral value ecirp tessA Normal times: Equity price Crisis: debt is equity is more more sensitive sensitive Debt price 58

E Notes on merging data List of identifiers by dataset: • IDC identifiers are CUSIP and ticker. • CRSP identifiers are permno and ticker. • COMPUSTAT identifiers are gvkey and ticker. • Moody’s debt data identifiers are OrgID, ticker, CIK, and CUSIP. Summary of merging process: • We use CCM (accessed through WRDS) to match CRSP and COMPUSTAT – This is a well-known and often-used match between permno and gvkey that is date dependent. • We use ticker-date to match IDC-CRSP – There are 767 unique ticker matches. – Of the universe of ticker-dates in IDC, 90 percent are matched in CRSP. • The only identifier from our analysis of CRSP-IDC is gvkey. • We use several cross-walks to merge with Moody’s debt data. 1. WRDS provides a crosswalk from gvkey to CIK. – This is a unique match that yields 35,000 firms. 2. SEC provides a crosswalk from CIK to ticker. – There are many related tickers (preferred equity, reinsurer, units...) 3. We constructed a new merged crosswalk ticker-CIK-gvkey that yields 427 unique matches. 4. We then merged this new cross-walk to include many Moody’s OrgID identifiers that represent subsidiaries of the 427 entities identified by ticker-CIK-gvkey. 59

F Placebo test: The oil and gas sector As a placebo test of the informational centrality of the banking sector, we consider instead the oil and gas sector. We construct the equity capital ratio of firms in US SIC Code 13 that primarily engaged in: (1) producing crude petroleum and natural gas; (2) extracting oil from oil sands and oil shale; (3) producing natural gasoline and cycle condensate; and (4) producing gas and hydrocarbon liquids from coal at the mine site.18 The equity capital ratio is constructed following the same methodology as for the banking sector in He et al. (2017). Using the CRSP-COMPUSTAT merged database, we calculate the ratio: (cid:80) Market Equity i i,t (13) (cid:80) (Market Equity +Book Debt ) i i,t i,t where for each firm i at the end of quarter t, Market Equity is the i,t market capitalization from CRSP, and Book Debt is the sum of long i,t term debt plus debt in current liabilities from COMPUSTAT. Note that this calculation uses the full sample of firms in US SIC Code 13 available in CRSP-COMPUSTAT and is not restricted to the sample of CRSP firms that match with the IDC debt data. The results are shown in Figure 4 for debt (Panel A), equity (Panel B), and the difference between debt and equity (Panel C). In contrast to the 18https://www.naics.com/standard-industrial-code-divisions/?code=13 60

main finding of the paper, shown in Panel C of Figure 1, there is no significant increase in the relative informativeness of debt about the future state of the banking system during the 2007-2009 financial crisis. This distinction validates the central role played by banks. Figure 4 shows an increase in the relative informativeness of equity prices about the oil and gas sector during the 2010-2014 shale gas boom. In addition, both debt and equity are informative about the oil and gas sector. One potential explanation is the important role played by the oil and gas industry in the supply chains of nonbank firms. This hypothesis could be tested using sectoral input-output tables, but we leave the implementation of the test for future research. 61

.rotces sag dna lio eht fo etats erutuf eht tuoba secirp tessa mrfi knabnon egral ni noitamrofnI :4 erugiF ecnereffid eht dna ,)B lenaP( ytiuqe ,)A lenaP( tbed rof K∆ fo naem elpmas eht fo seires emit eht swohs erugfi ehT q,i SU ni smrfi fo oitar latipac ytiuqe eht sa derusaem si rotces sag dna lio eht fo etats ehT .)C lenaP( ytiuqe dna tbed neewteb ’srohtuA :ecruoS .slavretni ecnedfinoc gnilpmasbus tnecrep 99 eht wohs erugfi hcae ni senil dehsad der ehT .31 edoC CIS .CDI-ECI dna ,TATSUPMOC ,PSRC morf atad no desab snoitaluclac 62

G Rolling window length There is a trade-off between the number of individual firm time series that can be used to estimate price informativeness about the banking sector in a given rolling window and how much short-term fluctuations are reflected in our rolling window estimate of price informativeness. We retain firms that have at least 10 contiguous observations in any rolling window as it is the minimum to estimate the four pairs of linear regressions. A longer rolling window length tends to yield a greater number of usable firm time series, but produces a greater time-series averaging of price informativeness across short-term fluctuations. Because the 2007-09 financial crisis lasted approximately two years, we chose a rolling window length of 16 quarters as our benchmark. A shorter rolling window would result in less time-series averaging of the crisis effect and would estimate price informativeness using data on fewer firms. Toillustratethispoint, theresultsusinga12-quarterrollingwindoware shown in Figure 5 for debt (Panel A), equity (Panel B), and the difference between debt and equity (Panel C). Figure 5 shows an greater increase in the relative informativeness of debt prices about the banking sector during the 2007-2009 financial crisis. For completeness, we also report the analog of Table 6 for the 12-quarter window estimates in Table 9. 63

ehT .rotces gniknab eht fo etats erutuf eht tuoba secirp tessa mrfi knabnon egral ni noitamrofnI :5 erugiF neewteb ecnereffid eht dna ,)B lenaP( ytiuqe ,)A lenaP( tbed rof K∆ fo naem elpmas eht fo seires emit eht swohs erugfi q,i ehT .slavretni ecnedfinoc gnilpmasbus tnecrep 99 eht wohs erugfi hcae ni senil dehsad der ehT .)C lenaP( ytiuqe dna tbed morf atad no desab snoitaluclac ’srohtuA :ecruoS .atad fo wodniw gnillor retrauq-21 a no desab era elbat siht ni stluser .CDI-ECI dna ,TATSUPMOC ,PSRC 64

Table 9: Debt refinancing determines the amount of information about the future state of the banking system conveyed in debt prices relative to equity prices. The dependent variable is KG_ddiff = ∆Kdebt − ∆Kequity, as defined in the text. The first i,q i,q i,q explanatory variable is crisis , which takes the value 1 if the quarter q falls q in the range 2007Q3-2009Q4 and 0 otherwise. The second explanatory variable (Debt_12m_roll ) is the ratio of the par value of firm i’s i,q corporate debt maturing in the next 12 months to the same firm’s total par value of corporate debt outstanding in quarter q (shown in Figure 2). Table 5 contains summary statistics of the regression variables. We report bootstrapped standard errors clustered at the firm level with 2,999 replications. The results in this table are based on a 12-quarter rolling window of data. Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 Dep. var.: KG_ddiff (1) (2) (3) (4) i,q crisis 0.139∗∗∗ 0.123∗∗∗ 0.132∗∗∗ 0.110∗∗∗ q (0.024) (0.027) (0.026) (0.027) Debt_12m_roll 0.026 −0.001 i,q (0.062) (0.077) Debt_12m_roll ×crisis 0.214 0.296∗∗ i,q q (0.155) (0.149) Constant 0.006 0.005 (0.008) (0.009) Firm FE N N Y Y Observations 6,381 6,381 6,381 6,381 Adjusted R2 0.019 0.020 0.116 0.118 65