DOCSLIB.ORG

An Experimental Approach to Causal Identification of Spillover Effects

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Experimental Approach to Causal Identification of Spillover Effects An Experimental Approach to Causal Identification of Spillover Effects under General Interference Alexander Coppock∗ and Neelanjan Sircar† Columbia University July 18, 2013 Abstract This paper extends the Rubin Causal Model to a general interference framework. In par- ticular, we develop a framework to define estimands in a general way using "reference assign- ments." This framework allows for the flexible stipulation of the non-interference assumptions required for estimation. We apply this approach to two common estimands, the average di- rect treatment effect and the average indirect exposure effect, recasting the framework in the language of network analysis. We use the network context to demonstrate the relationship between the potential outcomes and edges in a network, as well as to define intuitive non- interference assumptions. Finally, we derive unbiased and consistent estimators of average causal effects under interference and demonstrate their properties with simulated data. ∗[email protected] †[email protected] We would like to thank Peter Aronow, Albert Fang, Donald Green, Dominik Hangartner, Macartan Humphreys, Ryan Moore and participants at European Political Science Association Annual Meeting 2013, the 6th Annual Meeting of the Political Networks Section of the American Political Science Association, and the Columbia University Methods Workshop for useful comments and advice. All errors are our own. 1 1 Introduction Estimation of causal effects has typically relied on a strong assumption of non-interference be- tween experimental units. Many treatments, such as information, monitoring, or transfers, vi- olate this assumption in ways that are of great social scientific relevance, i.e., average spillover effects themselves are estimands of interest. We address the special estimation challenges asso- ciated with interference by extending the Rubin Causal Model to a general interference frame- work. In this paper, we derive a principled approach to the analysis of experiments under interference. We define estimands of interest with respect to "reference assignments," which are treatment assignment vectors that guarantee that we will observe the outcome of interest. The benefit of this approach is that estimands can be decoupled from assumptions about non-interference. Once an experiment has been conducted, we can flexibly state assumptions of non-interference, which allow for unbiased and consistent estimation of relevant estimands. We apply this approach to two common estimands, the average direct treatment effect and the average indirect exposure effect. In doing so, we recast a typical experiment involving spillovers in the language of network analysis. This allows us to comment on the relationship between the edges of a network and the potential outcomes for indirect exposure to a treatment. Furthermore, the network exposition allows us to frame plausible non-interference conditions in an intuitive way. Section 2 discusses relevant literature and a motivation for our approach. Section 3 derives our general framework using set-theoretic and probability measure-theoretic principles. Section 4 explores techniques to define non-interference conditions, as well as the bias and efficiency of estimation strategies, by applying the framework to a network context. Section 5 concludes the paper. 2 Related Literature and Motivation The assumption of "no interference between units" (Cox, 1958) is commonly made in experi- ments and is one component of the popularly assumed stable unit treatment value assumption, or SUTVA (Rubin, 1980). Non-interference, however, may not always hold between experimental units, as with treatments that are likely to spillover such as persuasion, monitoring, or educa- tion. Spillovers, or indirect exposure effects, bias naïve treatment effect estimates and may be of substantive relevance to researchers in their own right. In the past decade, researchers have begun seriously investigating indirect exposure effects through experimental methods. Estimands of interest from such studies include, for instance, the effects of vaccination on infection rates in surrounding areas, the effects of get-out-the-vote campaigns on neighbors’ voting rates, and the effects of educational interventions for students on their classmates. Such studies have often been conducted through multilevel experimentation or double randomization (e.g., Duflo and Saez (2003), Giné and Mansuri (2011), Sinclair, McConnell and Green (2012)). 2 In multilevel experiments, clusters are selected randomly from the universe of clusters of units in the population, e.g. neighborhoods or counties. For a random subset of selected clusters, a fixed percentage of units are assigned to treatment, and in the other clusters no units are assigned to treatment. Crucially, non-interference is assumed to hold across clusters but not within clusters. This yields three types of experimental groups, consisting of: (a) those units that are directly treated; (b) those units that are not directly treated but are located in clusters where units were treated (thus experiencing spillovers); and (c) those units that are not directly treated and are in clusters where no other unit was treated (serving as a control group). The average indirect effect is estimated as the average outcome in group (b) minus the average outcome in group (c). The statistical foundations of estimation via multilevel experimentation are discussed in detail in Hudgens and Halloran (2008). However, multilevel experiments may not always be feasible. The interference structure may not resemble clusters, as is often the case in the study of peer effects and spatial spillovers. In such cases, it is useful to stipulate a network of nodes and edges that describes the paths along which spillovers can occur. Experiments conducted in this more general network setting include Angelucci, de Giorgi, Rangel and Rasul (2010), Chen, Humphreys and Modi (2010), and Ichino and Schuendeln (2012). The framework we develop in this paper is a generalization of the classic Rubin Causal Model (Rubin (1974)) and its extension to multilevel experimentation (Hudgens and Halloran, 2008). In particular, we follow the insights of Rubin (1990) and Sobel (2006) and explicitly define potential outcomes as a function of the vector of treatment assignments resulting from the randomization. We use this idea to define "reference assignments," which are assignment vectors that guaran- tee that we an observe the potential outcome of interest. For example, we are guaranteed to see the control potential outcome for each unit under the assignment vector where no unit re- ceives treatment. Following Sobel (2006), we then define the estimand of interest by dividing differences in the appropriate potential outcomes by the number of reference assignments. The benefit of this approach is that it defines estimands of interest independently of non-interference assumptions. The stable unit treatment value assumption entails one of a set of many possible non-interference assumptions that may be appropriate for a given network structure. A common non-interference assumption is that units’ potential outcomes do not vary with the treatment assignments of those units that they are not connected to over some network. A more complex version of this assumption is often implicit: units’ potential outcomes may vary with the treatment assignments of direct neighbors but they do not vary with the assignments of indirect neighbors. For instance, we might allow a voter’s turnout decision to be affected when a friend is treated with a get-out- the-vote message, but assume that the voter’s decision is not affected when a friend of a friend is treated. Defining a non-interference assumption entails classifying the set of assignment vectors that would yield the same potential outcomes as a reference assignment. We may then use the known randomization distribution from the experiment and a stipulated non-interference assumption as to construct the probability of assignment to particular treatment conditions. We use inverse probability weighting to construct unbiased or consistent estimates of our estimands of interest. The inverse probability weighting approach, with application, is discussed in Chen, Humphreys 3 and Modi (2010). We refer readers to Aronow and Samii (2013b), who discuss the statistical and mathematical underpinnings of the inverse probability approach, including variance estimation, using “exposure mappings." 3 General Framework In this section, we develop a principled approach to constructing estimands of interest under interference. In particular, following Rubin (1990) and Sobel (2006), we define general poten- tial outcomes as a function of each possible vector of assignments to treatment statuses. Our estimands of interest are defined with respect to reference assignments over which we are guar- anteed to observe the relevant potential outcomes. After a set of estimands has been defined, we can flexibly stipulate non-interference assumptions and corresponding estimators. Many of the studies discussed above are interested in two estimands in particular: the direct and indirect effects of treatment. The average effect of direct exposure to a treatment is the average of all unit treatment effects in the absence of any interference. The average effect of (first-order) indirect exposure is the average of all spillover treatment effects, which are defined with respect to individual edges with receiver i and sender j. These quantities can be estimated
Load more

Recommended publications
  • Detecting Spillover Effects: Design and Analysis of Multilevel Experiments
    This is a preprint of an article published in [Sinclair, Betsy, Margaret McConnell, and Donald P. Green. 2012. Detecting Spillover Effects: Design and Analysis of Multi-level Experiments. American Journal of Political Science 56: 1055-1069]. Pagination of this preprint may differ slightly from the published version. Detecting Spillover Effects: Design and Analysis of Multilevel Experiments Betsy Sinclair University of Chicago Margaret McConnell Harvard University Donald P. Green Columbia University Interpersonal communication presents a methodological challenge and a research opportunity for researchers involved infield experiments. The challenge is that communication among subjects blurs the line between treatment and control conditions. When treatment effects are transmitted from subject to subject, the stable unit treatment value assumption (SUTVA) is violated, and comparison of treatment and control outcomes may provide a biased assessment of the treatment’s causal influence. Social scientists are increasingly interested in the substantive phenomena that lead to SUTVA violations, such as communication in advance of an election. Experimental designs that gauge SUTVA violations provide useful insights into the extent and influence of interpersonal communication. This article illustrates the value of one such design, a multilevel experiment in which treatments are randomly assigned to individuals and varying proportions of their neighbors. After describing the theoretical and statistical underpinnings of this design, we apply it to a large-scale voter-mobilization experiment conducted in Chicago during a special election in 2009 using social-pressure mailings that highlight individual electoral participation. We find some evidence of within-household spillovers but no evidence of spillovers across households. We conclude by discussing how multilevel designs might be employed in other substantive domains, such as the study of deterrence and policy diffusion.
    [Show full text]
  • Robust Score and Portmanteau Tests of Volatility Spillover Mike Aguilar, Jonathan B
    Journal of Econometrics 184 (2015) 37–61 Contents lists available at ScienceDirect Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom Robust score and portmanteau tests of volatility spillover Mike Aguilar, Jonathan B. Hill ∗ Department of Economics, University of North Carolina at Chapel Hill, United States article info a b s t r a c t Article history: This paper presents a variety of tests of volatility spillover that are robust to heavy tails generated by Received 27 March 2012 large errors or GARCH-type feedback. The tests are couched in a general conditional heteroskedasticity Received in revised form framework with idiosyncratic shocks that are only required to have a finite variance if they are 1 September 2014 independent. We negligibly trim test equations, or components of the equations, and construct heavy Accepted 1 September 2014 tail robust score and portmanteau statistics. Trimming is either simple based on an indicator function, or Available online 16 September 2014 smoothed. In particular, we develop the tail-trimmed sample correlation coefficient for robust inference, and prove that its Gaussian limit under the null hypothesis of no spillover has the same standardization JEL classification: C13 irrespective of tail thickness. Further, if spillover occurs within a specified horizon, our test statistics C20 obtain power of one asymptotically. We discuss the choice of trimming portion, including a smoothed C22 p-value over a window of extreme observations. A Monte Carlo study shows our tests provide significant improvements over extant GARCH-based tests of spillover, and we apply the tests to financial returns data. Keywords: Finally, based on ideas in Patton (2011) we construct a heavy tail robust forecast improvement statistic, Volatility spillover which allows us to demonstrate that our spillover test can be used as a model specification pre-test to Heavy tails Tail trimming improve volatility forecasting.
    [Show full text]
  • Place Based Interventions at Scale: the Direct and Spillover Effects of Policing and City Services on Crime
    WORKING PAPER · NO. 2019-40 Place Based Interventions at Scale: The Direct and Spillover Effects of Policing and City Services on Crime Christopher Blattman, Donald Green, Daniel Ortega, Santiago Tobón JANUARY 2019 1126 E. 59th St, Chicago, IL 60637 Main: 773.702.5599 bfi.uchicago.edu Place-based interventions at scale: The direct and spillover effects of policing and city services on crime∗ Christopher Blattman Donald Green Daniel Ortega Santiago Tobón† January 15, 2019 Abstract In 2016 the city of Bogotá doubled police patrols and intensified city services on high-crime streets. They did so based on a policy and criminological consensus that such place-based programs not only decrease crime, but also have positive spillovers to nearby streets. To test this, we worked with Bogotá to experiment on an unprecedented scale. They randomly assigned 1,919 streets to either 8 months of doubled police patrols, greater municipal services, both, or neither. Such scale brings econometric challenges. Spatial spillovers in dense networks introduce bias and complicate variance estimation through “fuzzy clustering.” But a design-based approach and randomization inference produce valid hypothesis tests in such settings. In contrast to the consensus, we find intensifying state presence in Bogotá had modest but imprecise direct effects and that such crime displaced nearby, especially property crimes. Confidence intervals suggest we can rule out total reductions in crime of more than 2–3% from the two policies. More promising, however, is suggestive evidence that more state presence led to an 5% fall in homicides and rape citywide. One interpretation is that state presence may more easily deter crimes of passion than calculation, and place-based interventions could be targeted against these incredibly costly and violent crimes.
    [Show full text]
  • The Spillover Effects of Top Income Inequality
    The Spillover Effects of Top Income Inequality∗ Jeffrey Clemensy Joshua D. Gottliebz David H´emousx Morten Olsen{ November 2016 - Preliminary Abstract Since the 1980s top income inequality within occupations as diverse as bankers, managers, doctors, lawyers and scientists has increased considerably. Such a broad pattern has led the literature to search for a common explanation. In this paper, however, we argue that increases in income inequality originating within a few oc- cupations can\spill over"into others creating broader changes in income inequality. In particular, we study an assignment model where generalists with heterogeneous income buy the services of doctors with heterogeneous ability. In equilibrium the highest earning generalists match with the highest quality doctors and increases in income inequality among the generalists feed directly into the income inequal- ity of doctors. We use data from the Decennial Census as well as the American Community Survey from 1980 to 2014 to test our theory. Specifically, we identify occupations for which our consumption-driven theory predicts spill-overs and oc- cupations for which it does not and show that patterns align with the predictions of our model. In particular, using a Bartik-style instrument, we show that an increase in general income inequality causes higher income inequality for doctors, dentists and real estate agents; and in fact accounts for most of the increase of inequality in these occupations. ∗Morten Olsen gratefully acknowledges the financial support of the European Commission under the Marie Curie Research Fellowship program (Grant Agreement PCIG11-GA-2012-321693) and the Spanish Ministry of Economy and Competitiveness (Project ref: ECO2012-38134).
    [Show full text]
  • Dependence-Robust Inference Using Resampled Statistics
    Dependence-Robust Inference Using Resampled Statistics∗ Michael P. Leung† August 26, 2021 Abstract. We develop inference procedures robust to general forms of weak dependence. The procedures utilize test statistics constructed by resam- pling in a manner that does not depend on the unknown correlation structure of the data. We prove that the statistics are asymptotically normal under the weak requirement that the target parameter can be consistently estimated at the parametric rate. This holds for regular estimators under many well-known forms of weak dependence and justifies the claim of dependence-robustness. We consider applications to settings with unknown or complicated forms of de- pendence, with various forms of network dependence as leading examples. We develop tests for both moment equalities and inequalities. JEL Codes: C12, C31 Keywords: resampling, dependent data, social networks, clustered standard errors arXiv:2002.02097v4 [econ.EM] 25 Aug 2021 ∗I thank the editor and referees for comments that improved the quality of the paper. I also thank Eric Auerbach, Vittorio Bassi, Jinyong Hahn, Roger Moon, Hashem Pesaran, Kevin Song, and seminar audiences at UC Davis, USC INET, the 2018 Econometric Society Winter Meetings, the 2018 Econometrics Summer Masterclass and Workshop at Warwick, and the NetSci2018 Satellite on Causal Inference and Design of Experiments. This research is supported by NSF Grant SES-1755100. †Department of Economics, University of Southern California. E-mail: [email protected]. 1 Michael P. Leung 1 Introduction This paper builds on randomized subsampling tests due to Song (2016) and proposes inference procedures for settings in which the dependence structure of the data is complex or unknown.
    [Show full text]
  • A Data-Driven Method for Reconstructing a Distribution from a Truncated Sample with an Application to Inferring Car-Sharing Demand
    A data-driven method for reconstructing a distribution from a truncated sample with an application to inferring car-sharing demand Evan Fields1, Carolina Osorio1,2, and Tianli Zhou2 1Massachusetts Institute of Technology Operations Research Center 2Massachusetts Institute of Technology Civil and Environmental Engineering August 8, 2018 Abstract This paper proposes a method to recover an unknown probability distribution given a censored or truncated sample from that distribution. The proposed method is a novel and conceptually simple detruncation method based on sampling the observed data accord- ing to weights learned by solving a simulation-based optimization problem; this method is especially appropriate in cases where little analytic information is available but the truncation process can be simulated. The proposed method is compared to the ubiqui- tous maximum likelihood (MLE) method in a variety of synthetic validation experiments where it is found that the proposed method performs slightly worse than perfectly specified MLE and competitively with slight misspecified MLE . The practical application of this method is then demonstrated via a pair of case studies in which the proposed detruncation method is used alongside a car-sharing service simulator to estimate demand for round trip car-sharing services in the Boston and New York metropolitan areas. 1 Introduction Car-sharing is a transportation service in which travelers use cars that they do not own; the vehicles are thus shared between users of the car-sharing service. In this work, we focus on round-trip station-based car-sharing, which is one of the most common types of car-sharing worldwide (Barth and Shaheen 2002).
    [Show full text]
  • Spillover Effects in Experimental Data
    Spillover Effects in Experimental Data Peter M. Aronow, Yale Dean Eckles, MIT Cyrus Samii, NYU Stephanie Zonszein, NYU Abstract We present current methods for estimating treatment effects and spillover effects under “interference”, a term which covers a broad class of situations in which a unit’s outcome depends not only on treatments received by that unit, but also on treatments received by other units. To the extent that units react to each other, interact, or otherwise transmit effects of treatments, valid inference requires that we account for such interference, which is a departure from the traditional assumption that units’ outcomes are affected only by their own treatment assignment. Interference and associated spillovers may be a nuisance or they may be of substantive interest to the researcher. In this chapter, we focus on interference in the context of randomized experiments. We review methods for when interference happens in a general network setting. We then consider the special case where interference is contained within a hierarchical structure. Finally, we discuss the relationship between interference and contagion. We use the interference R package and simulated data to illustrate key points. We consider efficient designs that allow for estimation of the treatment and spillover effects and discuss recent empirical studies that try to capture such effects. arXiv:2001.05444v1 [stat.AP] 15 Jan 2020 1 1 Introduction We present current methods for identifying causal effects under “interference”, a term which covers a very broad class of situations in which a unit’s outcomes depend not only on treatments received by that unit, but also on treatments received by other units (Cox, 1958).
    [Show full text]
  • Treatment and Spillover Effects Under Network Interference
    Treatment and Spillover Effects under Network Interference Michael P. Leung˚ July 3, 2016 Abstract. We study identification and estimation of treatment and spillover effects when interference is mediated by a network, focusing on the case of ran- domized assignment. We show that identification of spillover effects is possible with only a single, possibly sampled, network under common shape restrictions on treatment response. Estimation in the single-network setting is nonstan- dard due to correlation between outcomes induced by network spillovers and correlated effects. Nonetheless, under restrictions on moments of the network, nonparametric estimators for treatment and spillover effects are consistent and asymptotically normal, and we can construct consistent variance estimators. We also derive analogous results for GMM estimators, including robust stan- dard errors. JEL Codes: C13, C21 Keywords: social networks, treatment effects, spillovers, SUTVA, Stein’s method ˚USC, Department of Economics and MIT, Institute for Data, Systems, and Society. E-mail: [email protected]. 1 Michael P. Leung 1 Introduction A vast literature in econometrics and statistics studies identification and estimation of treatment effects. In this literature, the stable-unit treatment value assumption (SUTVA) is a fundamental assumption, which states that the ego’s treatment response is invariant to the treatment assignment of any alter. In other words, there is no “interference” or are no “spillovers” in treatment assignment. However, there are many important contexts in which this assumption fails to hold, and the measurement of spillover effects is often of inherent interest. This paper studies identification and estimation of treatment and spillover effects in the absence of SUTVA. In our model, treatment response may depend on the treat- ment assignment of others in a network.
    [Show full text]
  • Identification and Estimation of Spillover Effects in Randomized
    Identification and Estimation of Spillover Effects in Randomized Experiments∗ Gonzalo Vazquez-Bare† June 17, 2021 Abstract I study identification, estimation and inference for spillover effects in experiments where units' outcomes may depend on the treatment assignments of other units within a group. I show that the commonly-used reduced-form linear-in-means (RF-LIM) regression identifies a weighted sum of spillover effects with some negative weights, and that the difference in means between treated and controls identifies a combination of direct and spillover effects entering with different signs. I propose nonparametric estimators for average direct and spillover effects that overcome these issues and are consistent and asymptotically normal under a precise relationship between the number of parameters of interest, the total sample size and the treatment assignment mechanism. These findings are illustrated using data from a conditional cash transfer program and with simulations. The empirical results reveal the potential pitfalls of failing to flexibly account for spillover effects in policy evaluation: the estimated difference in means and the RF-LIM coefficients are all close to zero and statistically insignificant, whereas the nonparametric estimators I propose reveal large, nonlinear and significant spillover effects. (JEL C10, C13, C14, C90) arXiv:1711.02745v7 [econ.EM] 15 Jun 2021 ∗I am deeply grateful to Matias Cattaneo for continued advice and support. I am indebted to Lutz Kilian, Mel Stephens and Roc´ıoTitiunik for thoughtful feedback and discussions that greatly improved the paper. I thank Cl´ement de Chaisemartin, Catalina Franco, Amelia Hawkins, Nicol´asIdrobo, Xinwei Ma, Nicolas Morales, Kenichi Nagasawa, Olga Namen and Doug Steigerwald for valuable discussions and suggestions.
    [Show full text]
  • Identification of Spillover Effects Using Panel Data
    Identification of Spillover Effects using Panel Data Christiern Rose∗ University of Queensland [email protected] Latest version: March 2018. First version: June 2013 Abstract In this paper I use panel data to identify and estimate spillover effects when the underlying network is sparse and unobserved. The outcome of each entity depends on its own covariates, the outcomes of others (endogenous effects), the covariates of others (contextual effects) and an entity-specific fixed effect. Sparsity restrictions are exclusion restrictions of unknown location, and lead to point identification if the network is suitably connected. Prior knowledge that a particular covariate does not generate contextual effects may also lead to point identification. I use the results of Gautier and Tsybakov(2014) for estimation, model selection and inference under sparsity. The results of a Monte Carlo experiment demonstrate applicability in realistic settings. I apply the approach to study R&D spillovers in an oligopoly model, finding that spillovers are sent predominantly by large firms and that R&D stocks are below the social optimum. My results do not depend on pre-specified competition and technology networks. JEL codes: C31, L14, L24 Key words: Networks, spillovers, panel data, peer effects, R&D spillovers, high-dimensional econometrics 1. Introduction Spillover effects occur in many applied settings, including technology adoption, treatment effects, risky behaviors, peer effects in education, economic growth, labor market performance and almost all strategic contexts. There are two types of spillover: endogenous effects, through which the outcomes of interest are simultaneously determined, and contextual effects, through which the outcomes depend on the covariates of others (Manski, 1993).
    [Show full text]
  • Spillover Effects in the Presence of Unobserved Networks
    Spillover Effects in the Presence of Unobserved Networks Naoki Egami Department of Political Science, Columbia University, New York, NY 10027, USA. Email: [email protected], URL: https://naokiegami.com https://doi.org/10.1017/pan.2020.28 . Abstract When experimental subjects can interact with each other, the outcome of one individual may be affected by the treatment status of others. In many social science experiments, such spillover effects may occur through multiple networks, for example, through both online and offline face-to-face networks in a Twit- ter experiment. Thus, to understand how people use different networks, it is essential to estimate the spillover effect in each specific network separately. However, the unbiased estimation of these network- specific spillover effects requires an oen-violated assumption that researchers observe all relevant networks. We show that, unlike conventional omitted variable bias, bias due to unobserved networks remains even when treatment assignment is randomized and when unobserved networks and a network of interest are https://www.cambridge.org/core/terms independently generated. We then develop parametric and nonparametric sensitivity analysis methods, with which researchers can assess the potential influence of unobserved networks on causal findings. We illustrate the proposed methods with a simulation study based on a real-world Twitter network and an empirical application based on a network field experiment in China. Keywords: causal inference, interference, potential outcomes, SUTVA, network experiments 1 Introduction Existing methodologies for causal inference oen assume the absence of spillover effects, that is, people are affected only by treatments directly assigned to them and not by those assigned to oth- ers (Cox 1958; Rubin 1980).
    [Show full text]
  • Handling Spillover Effects in Empirical Research: an Application Using
    Handling Spillover Effects in Empirical Research: An Application using Credit Supply Shocks ∗ Tobias Berg† Daniel Streitz‡§ This version: August 28, 2019 Abstract Despite their importance, the discussion of spillover effects in empirical research misses the rigor dedicated to endogeneity concerns. We show that i) even with random treatment, spillovers lead to an intricate bias in estimating treatment effects, ii) there is a trade-off between endogeneity and spillover concerns, iii) the practice of using individual level regressions to identify direct effects and aggregate level regressions to learn about spillover effects can lead to misleading conclusions. We develop a simple guidance for empirical researchers, apply it to a credit supply shock, and highlight differences in the results compared to current empirical practice. Keywords: Spillovers, Credit Supply, Direct vs. Indirect Effects, Aggregate Effects JEL: C13, C21, G21, G32, R11, R23 ∗We would like to thank Bo Bian (discussant), Benjamin Born, Falko Fecht, Emilia Garcia-Appendini, Thomas Geelen, Paul Goldsmith-Pinkham (discussant), Rainer Haselmann, Katharina Hombach, Kilian Huber (discussant), Michael Koetter, Thorsten Martin, John Mondragon (discussant), Karsten M¨uller, William Mullins, Steven Ongena, Clemens Otto, Lasse Petersen, Markus Reisinger, Zacharias Sautner, Larissa Sch¨afer,Oliver Schenker, Sascha Steffen, conference participants at the 2019 SFS Cavalcade in Pittsburgh, the 2019 FIRS Conference in Savannah, the 8th MoFiR Workshop on Banking, the 2019 Conference on Regulating Financial Markets in Frankfurt, as well as seminar participants at Copenhagen Business School, Frankfurt School of Finance and Management, Humboldt University of Berlin, IWH Halle, Maastricht University, and the University of Zurich for helpful comments and suggestions. Rafael Zincke provided excellent research assistance.
    [Show full text]